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

This communication presents a set of expressions to evaluate the standard uncertainty and covariance of the real an imaginary parts of the complex-valued field resulting from the reconstruction of digital holograms by using the Fresnel approximation. These expressions are derived by applying the law of propagation of uncertainty as defined in the “Guide to the expression of uncertainty in measurement” to the numerical evaluation of the Fresnel integral, understood as a linear function of the values in the digital hologram. The expressions are eventually applied to holograms produced by the interference of speckle patterns with uniform reference beams, and assuming that the square of the standard uncertainty in the digitized hologram depends linearly with its local values, according to the noise model adopted in the EMVA 1288 camera characterization standard. The resulting uncertainties and covariance of the real and imaginary parts of the reconstructed fields can be subsequently propagated to measurements of the phase change between holograms by following the procedure already presented in our previous work on the propagation of the measurement uncertainty in Fourier-transform digital holographic interferometry.