Alternating Projections Method for Physically-Realizable Tensor Impedance Synthesis of Metasurface Antennas

This work addresses the inverse design of metasurface antennas through a physically consistent synthesis of tensorial surface reactance profiles. The method relies on solving the Electric Field Integral Equation (EFIE) via the Method of Moments (MoM), assuming an anti-Hermitian tensor surface impedance to ensure passive and reciprocal implementation. A key challenge in this context is recovering a finite-aperture current distribution that radiates a given far-field pattern, i.e. generates a given spectrum over the visible region, despite the inherent ambiguity due to the unspecified spectral content outside this region. We propose an iterative synthesis strategy inspired by the Gerchberg-Papoulis (GP) alternating projection method to reconstruct the missing spectral content of the current, while enforcing both spatial confinement and physical realizability of the impedance. The resulting synthesis process operates directly in the EFIE framework, enabling consistent updates of the current and the impedance profile at each step. The method is implemented using subdomain Gaussian ring basis functions, which provide localized radial resolution and enable efficient analytical evaluation of the MoM matrices entries. The resulting algorithm is therefore extraordinarily fast. Several examples, including single and multiple source configurations, demonstrate the effectiveness of the proposed strategy in producing complex and highly directive far-field patterns.