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Poisson's spot and Gouy phase

Recently there have been experimental results on Poisson spot matter wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for the Poisson's spot with matter waves based on Babinet principle in which we use the results for a free propagation and single slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium ($D_{2}$) molecules.