The field radiated by a rectangular periodic array of dipoles (the Array Green's function) with weakly tapered excitation is derived and physically interpreted in terms of uniform high-frequency field contributions. The field, usually obtained by direct summation over the contributions from the individual radiators, is here reconstructed into a double spectral integral whose high frequency asymptotic reduction yields a series of propagating and evanescent Floquet Waves (FWs) together with corresponding FW-modulated diffracted fields, wihch arise from FW scattering at the array edges and vertex. Emphasis is given to the analysis and physical interpretation of the vertex diffracted rays. A sample calculation is included to demonstrate the accuracy of the asymptotic algorithm.