Obtaining the Transfer Function of optical instruments using large calibrated reference objects

It has been suggested recently that the Transfer Function of instruments such as Coherence Scanning Interferometers could be measured via a single measurement of a large spherical artefact [Appl. Opt. 53(8), 1554-1563 (2014)]. In the current paper we present analytical solutions for the Fourier transform of the 'foil' model used in this technique, which thus avoids the artefacts resulting from the numerical approach used earlier. The Fourier transform of a partial spherical shell is found to contain points of zero amplitude for spatial frequencies that lie within the Transfer Function. This implies that the Transfer Function is unmeasurable at these points when a single spherical artefact is used, in situations where the foil model is a valid representation of the physical system. We propose extensions to the method to address this issue.