Fig. 3: The UMA for complicated beam shaping with space-varying wave properties.

a–c UAM for Airy beam generation. a Schematic illustration of the UMA for generating the Airy beam at the m = −1 harmonic frequency in free space. The Airy beam requires a spatial-varying phase profile as \(\frac{4}{3}{a}^{1/2}{\xi }_{m}{(-x)}^{3/2}\). b The required aperture phase profile of the Airy wavefront and the corresponding normalized time shifting for the meta-atoms at different positions. c The measured E-field intensities in the xz-plane at different harmonic frequencies. The green dashed line represents the theoretical parabolic trajectory of the Airy beam x = az², where the acceleration factor a = 0.003 in this case. Only the target m = −1 harmonic frequency possesses high field intensity, whereas other harmonic frequencies are all highly suppressed in free space. d–h UAM for focused beam generation. d Schematic illustration of the UMA for wave focusing at the m = −1 harmonic frequency. In this case, the local momentum or output angle of the extracted PWs from all meta-atoms are different. e The required normalized local momentum k_x/ξ_m as a function of the meta-atom position for different focal spot positions from F₁ to F₅, where F₁ = (‒60, 0, 60) mm, F₂ = (‒30, 0, 60) mm, F₃ = (0, 0, 60) mm, F₄ = (30, 0, 60) mm, and F₅ = (60, 0, 60) mm. f Measured cross-section intensity distributions at the m = −1 harmonic frequency in different transversal planes for the designed focal point F₃ (0, 0, 60) mm. g The measured E-field intensities in the xz-plane at the m = −1 harmonic frequency for different focus spots. h The corresponding 1-D normalized intensity distributions on the focal plane z = 60 mm. The input frequency and modulation frequency are f₀ = 23.5 GHz and f_M = 1.2 MHz, respectively.