Limited scan antenna systems using phased-array feeds (versus direct radiating array apertures) require fewer array elements; or do they?

It is generally accepted that a limited-scan antenna system with a phased-array feed uses significantly fewer elements than a direct-radiating, phased-array aperture (planar array) for similar scan coverage. This paper presents an analysis showing this is not necessarily the case. The author derives the relative number of elements for both cases (phased-array fed lens and direct radiating aperture) using two grating lobe conditions: (1) elimination of grating lobes in "real space"; and (2) elimination of grating lobes within the antenna coverage area. The former grating lobe condition does result in fewer array elements for the phased-array feed as expected. But imposing the latter grating lobe condition, often of interest in GEO satellite communication, yields no benefit in element count between a direct-radiating, phased-array aperture and the phased array fed, dual-lens antenna system used in the analysis.