Fig. 1: Schematic of a dual-layer cascade liquid crystal Pancharatnam-Berry lens.

a Schematic of two cascaded Pancharatnam-Berry (PB) lenses to achieve three-wavelength achromatic function. A cell is formed by two face-to-face “alignment- indium tin oxide (ITO) - substrate” structures, and a liquid crystal (LC) layer is injected into the cell to form a single PB lens. The other PB lens has the same structure, except that the liquid crystal layer is different. Alternating current (AC) voltage is applied to each PB lens to change the focus intensity. b Diagram of the arrangement states of two liquid crystal molecules for pixel-to-pixel alignment. The two liquid crystal molecules have different pointing vectors (red arrows), which lead the different rotation angles in two x-y planes. c The phase profiles of PB lenses with designed wavelengths 396.8 and 1064 nm will form a coupled phase profile for target wavelengths 632.8 nm. Switching the sequence order of coupling between wavelengths 396.8 and 1064 nm will result in a pair of coupled phases of opposite concave-convex characteristics. d Schematic of the polarizations of the foci formed by a pair of opposite chiral plane waves incident on different PB lenses with opposite phase profile.