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

doi:10.1038/s41377-025-02009-3

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

Original Articles

- Missing harmonic dynamics in generalized Snell's law: revealing full-channel characteristics of gradient metasurfaces
- Light: Science & Applications Vol. 14, Issue 11, Pages: 3479-3495(2025)
- Affiliations：
  
  1.Center for Carbon-Based Electronics, School of Electronics, Peking University, Beijing 100871, China
  2.State Key Laboratory of Photonics and Communications, School of Electronics, Peking University, Beijing 100871, China
  3.Key Lab. of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China
- Author bio：
  
  Hongsheng Chen (hansomchen@zju.edu.cn)
  Chao-Hai Du (duchaohai@pku.edu.cn)
- Funds：
- DOI：10.1038/s41377-025-02009-3
  CLC：
- Received：08 April 2025，
  
  Revised：2025-07-23，
  
  Accepted：13 August 2025，
  
  Published Online：15 September 2025，
  
  Published：30 November 2025
- Accepted：
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Zhang, Y. Y. et al. Missing harmonic dynamics in generalized Snell's law: revealing full-channel characteristics of gradient metasurfaces. Light: Science & Applications, 14, 3479-3495 (2025).
DOI：

Zhang, Y. Y. et al. Missing harmonic dynamics in generalized Snell's law: revealing full-channel characteristics of gradient metasurfaces. Light: Science & Applications, 14, 3479-3495 (2025). DOI： 10.1038/s41377-025-02009-3.

摘要

Abstract

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.

关键词

Keywords

references

Chen, H. S. et al. Left-handed materials com posed of only S-shaped resonators. Phys. Rev. E 70 , 057605 (2004)..

Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312 , 1780–1782 (2006)..

Chen, H. S. et al. Electromagnetic wave interactions with a metamaterial cloak. Phys. Rev. Lett. 99 , 063903 (2007)..

Zhang, B. L. et al. Rainbow and blueshift effect of a dispersive spherical invisibility cloak impinged on by a nonmonochromatic plane wave. Phys. Rev. Lett. 101 , 063902 (2008)..

Xi, S. et al. A one-directional perfect cloak created with homogeneous material. IEEE Microw. Wirel. Compon. Lett. 19 , 131–133 (2009)..

Xu, S. et al. Experimental demonstration of a free-space cylindrical cloak without superluminal propagation. Phys. Rev. Lett. 109 , 223903 (2012)..

Chen, H. S. et al. Ray-optics cloaking devices for large objects in incoherent natural light. Nat. Commun. 4 , 2652 (2013)..

Silva, A. et al. Performing mathematical operations with metamaterials. Science 343 , 160–163 (2014)..

Liu, L. X. et al. Broadband metasurfaces with simultaneous control of phase and amplitude. Adv. Mater. 26 , 5031–5036 (2014)..

Cai, T. et al. High-efficiency and full-space manipulation of electromagnetic wave fronts with metasurfaces. Phys. Rev. Appl. 8 , 034033 (2017)..

Overvig, A. C. et al. Dielectric metasurfaces for complete and independent control of the optical amplitude and phase. Light Sci. Appl. 8 , 92 (2019)..

Xu, H. X. et al. Chirality-Assisted High-Efficiency Metasurfaces with Independent Control of Phase, Amplitude, and Polarization. Adv. Optical Mater. 7 , 1801479 (2019)..

Khorasaninejad, M. et al. Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging. Science 352 , 1190–1194 (2016)..

Khorasaninejad, M. et al. Polarization-insensitive metalenses at visible wavelengths. Nano Lett. 16 , 7229–7234 (2016 )..

Chen, W. T. et al. A broadband achromatic metalens for focusing and imaging in the visible. Nat. Nanotechnol. 13 , 220–226 (2018)..

Shrestha, S. et al. Broadband achromatic dielectric metalenses. Light Sci. Appl. 7 , 85 (2018)..

Wang, S. M. et al. A broadband achromatic metalens in the visible. Nat. Nanotechnol. 13 , 227–232 (2018)..

Liu, Y. Q. et al. Broadband high-efficiency plasmonic metalens with negative dispersion characteristic. Photonics Res. 12 , 813–820 (2024)..

Wu, P. C. et al. Versatile polarization generation with an aluminum plasmonic metasurface. Nano Lett. 17 , 445–452 (2017)..

Wu, P. C. et al. Visible metasurfaces for on-chip polarimetry. ACS Photonics 5 , 2568–2573 (2018)..

Wu, Z. N., Ra'di, Y. & Grbic, A. Tunable metasurfaces: a polarization rotator design. Phys. Rev. X 9 , 011036 (2019)..

Zhu, H. et al. Graphene-based metasurfaces for switching polarization states of anomalous reflection and focusing. Opt. Lett. 44 , 5764–5767 (2019)..

Tao, H. et al. Highly flexible wide angle of incidence terahertz metamaterial absorber: design, fabrication, and characterization. Phys. Rev. B 78 , 241103(R) (2008)..

Liu, X. L. et al. Infrared spatial and frequency selective metamaterial with near-unity absorbance. Phys. Rev. Lett. 104 , 207403 (2010)..

Watts, C. M., Liu, X. L. & Padilla, W. J. Metamaterial electromagnetic wave absorbers. Adv. Mater. 24 , OP98–OP120 (2012)..

Landy, N. I. et al. Perfect metamaterial absorber. Phys. Rev. Lett. 100 , 207402 (2008)..

Guo, W. L., Liu, Y. X. & Han, T. C. Ultra-broadband infrared metasurface absorber. Opt. Express 24 , 20586–20592 (2016)..

Qian, C. et al. Deep-learning-enabled self-adaptive microwave cloak without human intervention. Nat. Photonics 14 , 383–390 (2020)..

Fan, Z. X. et al. Holographic multiplexing metasurface with twisted diffractive neural network. Nat. Commun. 15 , 9416 (2024)..

Yu, N. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334 , 333–337 (2011)..

Zhou, L. X., Kahn, J. M. & Pister, K. S. J. Corner-cube retroreflectors based on structure-assisted assembly for free-space optical communication. J. Microelectromech. Syst. 12 , 233–242 (2003)..

Ma, H. F. & Cui, T. J. Three-dimensional broadband and broad-angle transformation-optics lens. Nat. Commun. 1 , 124 (2010)..

Sun, S. L. et al. Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves. Nat. Mater. 11 , 426–431 (2012)..

Koh, J. et al. Fabrication and integration of microprism mirrors for high-speed three-dimensional measurement in inertial microfluidic system. Appl. Phys. Lett. 105 , 114103 (2014)..

Li, H. P. et al. X-Band Phase-gradient Metasurface For High-gain Lens Antenna Application. IEEE Trans. Antennas Propag. 63 , 5144–5149 (2015)..

Arbabi, A. et al. Planar metasurface retroreflector. Nat. Photonics 11 , 415–420 (2017)..

Yin, L. Z. et al. High-Efficiency Directional Excitation Of Spoof Surface Plasmon Under Arbitrary Illumination. IEEE Antennas Wirel. Propag. Lett. 20 , 1215–1219 (2021)..

Pan, W. K. et al. High-efficiency generation of far-field spin-polarized wavefronts via designer surface wave metasurfaces. Nanophotonics 11 , 2025–2036 (2022)..

Pesarakloo, A. & Khalaj-amirhosseini, M. Planar, wide-band omnidirectional retroreflector using metal-only transmitarray structure for TE and TM polarizations. Sci. Rep. 12 , 11279 (2022)..

Tan, Q. et al. Broadband spin-locked metasurface retroreflector. Adv. Sci. 9 , 2201397 (2022)..

Dai, C. H. et al. Multiplexing near- and far-field functionalities with high-efficiency bi- channel metasurfaces. Photonix 5 , 11 (2024)..

Liu, S. et al. Convolution operations on coding metasurface to reach flexible and continuous controls of terahertz beams. Adv. Sci. 3 , 1600156 (2016)..

Cao, W. K. et al. Tunable acoustic metasurface for three-dimensional wave manipulations. Phys. Rev. Appl. 15 , 024026 (2021)..

Liu, G. H. et al. Ultra-wideband low-detectable coding metasurface. Chin. J. Electron. 28 , 1265–1270 (2019)..

Epstein, A. & Rabinovich, O. Unveiling the properties of metagratings via a detailed analytical model for synthesis and analysis. Phys. Rev. Appl. 8 , 054037 (2017)..

Ra'di, Y., Sounas, D. L. & Alù, A. Metagratings: beyond the limits of graded metasurfaces for wave front control. Phys. Rev. Lett. 119 , 067404 (2017)..

Ra'di, Y. & Alù, A. Metagratings for efficient wavefront manipulation. IEEE Photonics J. 14 , 2207513 (2022)..

Epstein, A. & Eleftheriades, G. V. Floquet-Bloch analysis of refracting Huygens metasurfaces. Phys. Rev. B 90 , 235127 (2014)..

Gao, L. -H. et al. Broadband diffusion of terahertz waves by multi-bit coding metasurfaces. Light Sci. Appl. 4 , e324–e324 (2015)..

Wang, Y. X. et al. Perfect control of diffraction patterns with phase-gradient metasurfaces. ACS Appl. Mater. Interfaces 14 , 16856–16865 (2022)..

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