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