Terahertz (THz) waves, typically spanning the frequency range from 0.1 to 10 THz, possess unique characteristics such as low photon energy, broad bandwidth, and strong penetration through nonpolar materials, enabling important applications in wireless communication, biomedical sensing, and nondestructive evaluation. Efficient manipulation of the amplitude, frequency, and phase of THz radiation is essential for the development of high-performance functional devices. Metasurfaces composed of subwavelength artificial meta-atoms provide a versatile platform for THz wavefront engineering through tailored electromagnetic responses. However, most reported THz metasurfaces operate in the linear regime and rely on propagation or geometric phase, which limits their functionality to the fundamental frequency and prevents the integration of frequency conversion and wavefront control. Second-harmonic generation (SHG) offers an effective approach for extending the accessible THz spectral range. Conventional nonlinear crystals, however, suffer from low nonlinear efficiency and stringent phase-matching requirements in this frequency region, which restrict device miniaturization and integration. Resonant artificial microstructures in semiconductors provide an alternative nonlinear mechanism through strong local electromagnetic field enhancement, where the Lorentz-force-driven nonlinear current gives rise to an effective second-order response. Nevertheless, the phase of nonlinear radiation remains difficult to control independently of the radiation amplitude. Therefore, establishing a deterministic relationship between the resonance phase and the nonlinear radiation phase is critical for nonlinear wavefront engineering. In this work, a GaAs-based metasurface is designed to correlate the resonance phase at the fundamental frequency with the phase of the second-harmonic radiation, enabling phase-engineered nonlinear wavefront construction.

The metasurface consists of gold split-ring resonators patterned on a silicon-doped n-type GaAs epitaxial layer grown on a semi-insulating GaAs substrate. Under normal incidence of an x-polarized THz wave, the resonant current in the metallic ring generates a localized magnetic field perpendicular to the surface and a strongly enhanced electric field in the split gap. The electric field drives free carriers in the n-GaAs layer, while the localized magnetic field exerts a Lorentz force on the moving carriers, resulting in a nonlinear current oscillating at twice the fundamental frequency. Three-dimensional full-wave simulations based on the finite-element method were carried out to optimize the structural parameters and analyze both the linear resonance and nonlinear time-domain response. An anisotropic conductivity tensor was introduced to describe the magneto-electric coupling induced by the Lorentz force. Four meta-atoms with equally spaced resonance phases were designed, and an additional π phase shift was obtained by rotating the structures by 180°, resulting in eight discrete nonlinear phase states covering the full 0-2π range. The samples were fabricated using metal-organic chemical vapor deposition, photolithography, magnetron sputtering, and lift-off processes. The nonlinear response was characterized using a home-built strong-field terahertz time-domain spectroscopy (THz-TDS) system, in which the second-harmonic signal was extracted by spectral filtering and detected via electro-optic sampling.

Frequency-domain simulations show that all metasurface structures exhibit pronounced resonances near 0.8 THz, ensuring that the nonlinear phase comparison is performed under nearly identical linear response conditions. At the resonance frequency, strong local electric and magnetic field enhancements are observed, providing the physical origin for the Lorentz-force-driven nonlinear current. Time-domain simulations and experimental measurements confirm the generation of a cross-polarized second-harmonic signal at 1.6 THz. The four designed meta-atoms provide a phase interval of 0.25π, and the corresponding rotated structures introduce an additional π phase shift, resulting in eight discrete nonlinear phase states with nearly uniform spacing over the entire 0-2π range. Although slight amplitude variations exist among different unit cells, the measured phase evolution follows the designed linear dependence, indicating that the nonlinear phase can be effectively modulated through the resonance phase. Based on the experimentally extracted complex responses of the meta-atoms, an equivalent surface current model was employed to reconstruct the far-field radiation. By arranging the discrete phase elements according to the helical phase distribution, the reconstructed far-field exhibits a doughnut-shaped intensity profile and a continuous azimuthal phase variation of 2πl, which are the characteristic features of vortex beams. These results confirm that the experimentally obtained nonlinear phase library enables nonlinear helical wavefront construction through ordered spatial arrangement of the meta-atoms.

A nonlinear metasurface capable of phase-engineered terahertz second-harmonic radiation has been designed and experimentally demonstrated. By correlating the resonance phase at the fundamental frequency with the phase of nonlinear radiation, eight discrete nonlinear phase states covering the full 0-2π range were achieved through meta-atom design and structural rotation. Based on the experimentally extracted complex responses, nonlinear helical wavefronts were reconstructed through ordered spatial arrangement of the phase elements. The results verify that nonlinear wavefront engineering can be achieved through the cooperative strategy of meta-atom design and ordered arrangement. This approach enables the integration of frequency conversion and wavefront shaping within a planar artificial structure and provides a feasible route for phase engineering in nonlinear THz metasurfaces as well as for integrated structured-light sources in THz communication, imaging, and information processing.