Critical distance for grating lobe series

Estimating a transition or critical distance above a planar periodic array of point sources radiating into an unbounded medium is considered. It is shown that as an observation point approaches a periodic planar phased array of point sources, the corresponding spectral-domain Green's function, or grating lobe series, converges much more slowly than an equivalent mixed-domain representation exhibiting Gaussian convergence. For a given argument of the Green's function, the critical transition distance above the array for which both representations take the same time to compute can be estimated numerically. Since phased array antenna structures with cavity- or waveguide-type backings often have dimensions that fall well below the critical distance, investigations of such structures would seem to benefit significantly from formulations incorporating such hybrid Green's functions