A Footnote to “On the Relationship Between the Directivity and the Half-Power Beamwidth in Quasi-Symmetric Pencil-Beam Radiation Patterns” [Education Column]

Summary form only given. In an earlier note (Mahoney, 2012), it was shown how the result due to J. D. Kraus for the directivity as a function of the half-power beamwidth could be more or less established by considering only main-beam effects in radiation from a uniform distribution on large apertures, both square and circular. The aim of this short comment is to point out that in the light of this, it is unlikely that the directivity behavior due to Kraus can be realized from the viewpoint of radiation from a large aperture. To appreciate this, it will suffice to return to the previous note to see that the directivity behavior associated with only the main beam in these distributions is of the form D_K = η4πA/λ², where A denotes the area of the aperture. η is in effect a multiplicative efficiency factor that is given by equation and equation in the case of square and circular apertures, respectively. In both cases, the efficiency factors are now greater than unity, and the associated aperture directivities are consequently greater than the maximum achievable for a large aperture, which is 4πA/λ². It is thus questionable that Kraus´s result can in effect be realized in the context of radiation from large apertures. It is also relevant to point out that the associated figure of merit due to Kraus, viz., 32400(4/π) ≈ 41253, was queried earlier by R. S. Elliott in his book, Antenna Theory and Design.