Revealing hidden periodicity in momentum-encoded metasurfaces

Structured planar materials known as metasurfaces enable versatile manipulation of electromagnetic waves at subwavelength scales. Especially, metasurfaces can serve as spatial sampling lattices that encode prescribed in-plane momentum profiles, as commonly employed in phase-gradient and Pancharatnam-Berry metasurfaces. However, such momentum-encoded metasurfaces typically lose global periodicity, complicating rigorous analyses. Here, we introduce a universal geometric framework that interprets momentum-encoded metasurfaces as directional Moiré lattices formed by imposing one-dimensional periodic perturbations onto two-dimensional Bravais lattices. We derive explicit commensurability conditions that reveal hidden periodicity in otherwise aperiodic structures and enable their representation as compact periodic superlattices. This supercell representation allows the discretization of radiation channels into defined diffraction orders, enabling clear, intuitive performance characterization. We validate the framework by applying it to local metasurfaces, obtaining diffraction results consistent with two-dimensional sampling theory, and to non-local metasurfaces with spin-dependent Rashba momentum, precisely identifying their superlattice conditions and their valley-addressable character. We further experimentally demonstrate the predicted valley addressability using fabricated momentum-encoded photonic crystal slabs. Our method thus provides an efficient and generalizable basis for analyzing complex metasurface architectures and exploring diverse lattice phenomena in wave physics.