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In the present paper, we investigate a system of strongly interacting bosons confined in two-dimensional optical lattice. We propose a combination of the Bogoliubov method with the quantum rotor approach and determine the spatial atom-atom correlations. This allows us to calculate time-of-flight absorption images, which exhibit all the characteristic features present in experimental results, namely transition from superfluid peaks to Mott insulating blob.

Ultra-Cold Bosons in Optical Lattice: Time-of-Flight Imaging of Atom-Atom Correlations