Missing harmonic dynamics in generalized Snell's law: revealing full-channel characteristics of gradient metasurfaces.

The conventional generalized Snell's law (GSL), derived from classical laws of optical reflection and refraction, governs wavefront manipulation via phase gradients but neglects higher-order spatial harmonics inherently excited by the mutual coupling among meta-atoms on a metasurface. Here, we introduce a spatial harmonic-expanded GSL (SH-GSL) framework by unifying phase-gradient control with Floquet periodicity, establishing spatial harmonics as independent degrees of freedom rather than conventional parasitic disturbances. The SH-GSL framework rigorously identifies the intrinsic harmonic dynamics inherent to metasurfaces, which is a critical feature absent in GSL. Furthermore, this framework further reveals that all gradient-phase metasurfaces inherently function as multichannel platforms due to full spatial harmonics, with this multifunctionality rooted in nonlocal Floquet-Bloch modal interactions. Experimental validation demonstrates: abnormal spatial-harmonic reflection with angular precision ( < 5° deviation), multi-beam splitting (dual/quad configurations) via the relationship between specific harmonics and compensation wave vectors, and a perfect three-channel retroreflector achieving up to 99% efficiency, where parasitic harmonics are confined to near-field plasmonic regimes. This framework establishes a deterministic Floquet-engineered momentum compensation mechanism to simultaneously activate target harmonic channels while confining parasitic harmonics to near-field plasmonic regimes. Experimental validation confirms the framework's accuracy and scalability, bridging momentum-space physics with practical meta-plasmon systems. This work redefines metasurface engineering paradigms, unlocking advancements in ultra-dense beamforming, sensing, and meta-photonics through harmonic-division multiplexing.