Evaluation of the uncertainty of phase-difference measurements in (quasi-)Fourier transform digital holographic interferometry

Abstract

Estimation of the uncertainty is an essential requisite for high-end measurement systems. In this communication we derive an expression to evaluate the standard uncertainty of the phase-difference measurements resulting from Fourier and quasi-Fourier transform digital holographic interferometry. We apply the law of propagation of uncertainty, as defined in the “Guide to the expression of uncertainty in measurement” (GUM), to the digital reconstruction of two holograms by Fourier transformation and to the subsequent calculation of the phase change between the holographic reconstructions. The resulting expression allows the evaluation of the uncertainty of the phase difference at every pixel in the reconstruction plane in terms of the measured hologram brightness values and their uncertainty at the whole of the pixels of the original digital holograms. This expression is simplified by assuming a linear dependence between the uncertainty and the local value of the original holograms; in that case, the local uncertainty of the phase difference can be evaluated from the local complex values of the reconstructed holograms. We assess the behavior of the method by comparing the predicted standard uncertainty with the sample variance obtained from experiments conducted under repeatability conditions, and found a good correlation between both quantities. This experimental procedure can be also used to calibrate the parameters of the linear function relating the uncertainty with the local value of the digital holograms, for a given set of operational conditions of the acquisition device.