Phase-gradient metasurfaces (PGMs) empowered by the parity-dependent diffraction effect provide new degrees of freedom for the arbitrary manipulation of electromagnetic waves. However, the specific roles and mechanisms of the phase gradient ξ and reciprocal-lattice vector $G$ in wave-front reshaping require clarification for further development and applications. In this work, we propose and demonstrate a theoretical framework to elucidate how the interplay between the phase gradient ξ, reciprocal-lattice vector $G$, and unit-cell number $m$ governs the diffraction behavior of PGMs. A general reversal diffraction phenomenon and a diffraction cycle are demonstrated, with the typical parity-dependent diffraction in conventional PGMs with $\xi =G$ as a special case. In addition, we clarify the roles of the reciprocal-lattice vector $G$ and the phase gradient ξ in PGM diffractions, where the reciprocal-lattice vector $G$ determines the number of allowable diffraction orders in far-field diffraction, whereas the phase gradient ξ determines the selection rule for the diffraction order. This study provides not only unique physical insights into the diffraction mechanisms of PGMs but also a complete theory for wave-front manipulation based on PGMs.

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• Metagratings
• Metasurfaces
• Nanophotonics