Minimal-Size 2-D Sources for Mask-Constrained Synthesis of Shaped Patterns

We provide an effective deterministic solution to the canonical and up to now unsolved problem of the optimal synthesis of 2-D mask-constrained shaped beams. In particular, for 2-D arrays on planar grids and apertures with circular support, we identify the minimum source size required to fulfill a given maskconstrained performance and, subsequently, synthesize such a source. Conversely, we can also determine and achieve the optimal performance attainable by a source of fixed size. Our approach, which relies just on convex programming and polynomial factorization, is highly general as it accommodates completely arbitrary upper and lower bounds on the radiated power pattern, including arbitrary beam footprints. As such, the proposed technique is the first one capable of solving the problem without any kind of restriction on the shape of the beam footprint or the azimuth cuts of the pattern, all while avoiding global optimization algorithms. The given theory is validated by numerical examples that include full-wave simulated radiating elements. Also, comparisons demonstrate significant reductions in the number of array elements compared to state-ofthe-art methods and the capability of dealing with the design of asymmetric fields for the Earth coverage from satellite.