Missing harmonic dynamics in Generalized Snell’s Law: revealing full-channel characteristics of gradient metasurfaces

Conceptual illustration of Spatial Harmonic-expanded Generalized Snell's Law (SH-GSL). a, The schematic of the metasurface for abnormal-harmonic reflection. b, The schematic of the metasurface for multi-beam splitting. c, The schematic of the metasurface for perfect multi-channel retroreflection. d, The illustration of the proposed SH-GSL for harmonics manipulation. The fundamental harmonic can be manipulated by gradient metasurfaces based on the generalized Snell’s law (d left), while high-order harmonics need to be analyzed by the Floquet period theory (d left). By combining the phase gradient (GSL) and the period property (Floquet period theory), the abnormal reflection of harmonics can be manipulated according to a deterministic Floquet-engineered momentum compensation mechanism of spatial harmonics (d middle). Furthermore, by utilizing multi-harmonics simultaneously, the multi-channel retroreflection based on multiple harmonics can be achieved (d right). The 0th harmonic is controlled by the generalized Snell’s law in the red box, the −1st harmonic is denoted as the mirror mode in the yellow box, and the other order harmonics can be manipulated by the Floquet period theory in the blue box. @ Yueyi Zhang et al.

Since the Generalized Snell's Law (GSL) was proposed, planar metasurfaces have achieved remarkable progress in optical and electromagnetic wavefront manipulation by leveraging phase gradients. The Generalized Snell’s Law primarily focuses on the influence of phase gradients on the fundamental wave components while neglecting higher-order spatial harmonics generated by inter-element coupling and periodicity, often limiting metasurfaces to "single-channel" devices and constraining their applications in high efficiency, multi-angle, and multi-channel scenarios. Therefore, there is an urgent need to establish a deterministic theory that systematically analyzes the relationship among phase gradients, supercell periodicity, and Floquet harmonics to fully unlock the potential of metasurfaces in comprehensive wavefront manipulation.

In a new paper published in Light: Science & Applications, a team of scientists, led by Professor Chaohai Du from the Center for Carbon-Based Electronics and the State Key Laboratory of Photonics and Communications, School of Electronics, Peking University, and Professor Hongsheng Chen from Key Lab. of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, College of Information Science and Electronic Engineering, Zhejiang University, and co-workers introduced the Spatial Harmonic-expanded Generalized Snell’s Law (SH-GSL), which is a deterministic theoretical framework that fills a critical gap in gradient-metasurface theory. For the first time, SH-GSL rigorously accounts for the dynamic roles of higher-order spatial harmonics by unifying phase-gradient control with Floquet periodicity. Rather than treating these harmonics as parasitic, the framework promotes them to independent, addressable degrees of freedom via a Floquet-engineered momentum-compensation mechanism. This represents a fundamental paradigm shift: from designs that try to avoid inter-unit coupling to designs that precisely harness and regulate strong nonlocal coupling for new functionality.

SH-GSL is validated by analysis, full-wave simulation, and microwave experiments at 14 GHz, demonstrating unprecedented, harmonic-selective control in three representative devices: Floquet-engineered abnormal single-sided harmonic reflection with angular precision better than 5°, harmonic-selected dual and quad beam-splitting, and a three-channel retroreflector retroreflecting energy into three discrete directions with measured peak efficiency up to 99%. Detailed investigation reveals how harmonic order, nonlocal coupling, and fabrication tolerance govern efficiency and purity, providing actionable design criteria.

By delivering a concise rule that links supercell periodicity, phase gradient, and harmonic excitation, this paradigm shifts traditional gradient metasurface research from "avoiding inter-unit coupling" to "precisely regulating strong inter-unit coupling". The work systematically elucidates the dynamics of high-order spatial harmonics in gradient metasurfaces, shaping the core physics of "full-channel metasurfaces," and provides theoretical and engineering pathways for ultra-dense beamforming, reconfigurable multichannel sensing, and generalized metasurface device design under strong coupling paradigms.

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

Yueyi Zhang, Fengyuan Han, Yibing Xiao, Ziwen Zhang, Jitao Yang, Yulu Lei, Fei Gao, Hongsheng Chen & Chao-Hai Du

https://www.nature.com/articles/s41377-025-02009-3

Changchun Institute of Optics, Fine Mechanics And Physics, CAS