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

We provide an effective deterministic solution to the canonical and, until now, unsolved problem of the optimal synthesis of 2-D mask-constrained shaped beams. In particular, for 2-D arrays on planar grids and continuous aperture sources with circular support, we identify the minimum source size required to achieve a given mask-constrained performance and, subsequently, synthesize such a source. Conversely, the proposed method can also determine and achieve the optimal performance attainable by a source of fixed size. The approach, based solely on convex programming and polynomial factorization, can accommodate completely arbitrary upper and lower bounds on the radiated power pattern. As such, it constitutes the first method capable of solving this problem without any restriction on the beam shape and footprint while completely avoiding the use of global optimization algorithms. The results from representative test cases demonstrate the ability of the method to find sources with provably minimal size, leading to significant reductions in the number of array elements compared with alternative designs. The effectiveness of the approach is further confirmed by the successful incorporation of full-wave simulated radiating elements, including mutual coupling effects.