Pencil and shaped beams obtained by planar density tapering

Interest in uniformly illuminated density-tapered planar antenna arrays has recently revived. For arrays consisting of concentric rings of elements, we present an algorithm that calculates ring radii for generation of pencil beams without the need for prior specification of the number of rings. If element phases are allowed to differ, the algorithm also synthesizes density-tapered arrays generating axially symmetric shaped beams.