Propagation of the measurement uncertainty in Fourier transform digital holographic interferometry

Abstract

Abstract. The derivation of expressions to evaluate the local standard uncertainty of the complex amplitude of the numerically reconstructed field as well as of the phase-change measurements resulting from Fourier-and quasi-Fourier transform digital holographic interferometry is presented. Applying the law of propagation of uncertainty, as defined in the "Guide to the expression of uncertainty in measurement," to the digital reconstruction of holograms by Fourier transformation and to the subsequent calculation of the phase change between two such reconstructions results in a set of expressions, which allow the evaluation of the uncertainties of the complex amplitude and of the phase change at every pixel of the reconstruction in terms of the measured values and their standard uncertainty in the pixels of the original digital holograms. These expressions are increasingly simplified by first assuming a linear dependence between the squared uncertainty and the local value of the original holograms, and then considering that the object beam is a speckle pattern. 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 find a good agreement between both quantities.

Propagation of the measurement uncertainty for the numerical reconstruction of holograms in Fresnel approximation

“…These values may be further propagated to measurements of the local amplitude or phase change. 5,6 Our preliminary experiments show that the values of the uncertainty calculated with the proposed expressions closely agree with an independent evaluation by a Monte Carlo method, with the advantage that the application of the former is typically many thousands of times faster than the latter.…”

“…Our first approach to the problem of uncertainty propagation in the reconstruction process was for Fourier transform and quasi-Fourier transform holograms. 4,5 That is the simplest situation because in that case the reconstruction is achieved by evaluating the discrete Fourier transform of the hologram which, as aforementioned, is typically real-valued and the propagation of uncertainty in the fast Fourier transform of real data had been reported in other application contexts. 6 In this communicaton, we move a step forward by deriving a set of expressions to propagate the uncertainty of measurement in the reconstruction of digital holograms in Fresnel approximation implemented with the Fresnel diffraction integral, 7,8 which is the most frequently used variant for digital holographic metrology.…”

“…5 i. e. of the local value of the hologram h m,n ,u 2 (h m,n ) = K h m,n + u .0 − K µ y.dark.0 (22)where K is the overall system gain of the digital camera -expressed in counts per electron (DN/e − )-, σ 2y.dark.0is the value of the variance of its dark noise when the exposure time approaches zero -including the effects of quantization and expressed in DN 2 -and µ y.dark.0 is the expected value of its dark signal when the exposure time approaches zero -expressed in DN. These three parameters, as well as the methods to measure them, are specified in the EMVA 1288 camera characterization standard 15.…”

An optical-digital method of noise suppression in digital holography

“…• zero-order and twin image [32,33]; and • digital camera noise and the characteristics of the signal registration [34][35][36][37][38][39][40][41][42][43][44][45].…”

“…Speckle noise [1,[29][30][31] and unwanted (zero-order and twin image) diffraction orders [32,33] are the main negative factors. Also, recording systems have their own noises and features, such as discretization and pixels fill factors [34,35], temporal [36][37][38][39] and fixed-pattern [39,40] camera noises, quantization noise [41][42][43], dynamic range [39,44], and bit depth [45].…”