Fig. 2: Design of Jones matrix with eight DOFs and measurement results.

a Flow chart of the gradient descent optimization algorithm for design of Jones matrix with eight DOFs. The input variables are the Jones matrix components of the two single layer metasurface (\({J}_{ij}^{1}({x}_{1},{y}_{1}),{J}_{ij}^{2}({x}_{2},{y}_{2})\), ij = 11, 12 (21), 22). The defined loss includes two parts: the mean of the absolute squared differences of EQJM components between the prediction (\({J}_{ij}\)) and target (\({J}_{ij}^{t}\)), and a boundary constraint loss\({L}_{bnd}\). b Loss value as a function of the iteration number in the gradient descent optimization algorithm. c Unit cell of the single layer metasurface to construct Jones matrix with six DOFs. Each unit contains of four dielectric nanopillars, and the two sets of nanopillars AB and A’B’ can individually and independently construct Jones matrix with six DOFs. Under oblique incidence or oblique scattering, detour phases of \({e}^{-i\varDelta \varphi }\) and \({e}^{i\varDelta \varphi }\) are imposed on the two sets of nanopillars AB and A’B’. To compensate the detour phases, the Jones matrixes of the two set of nanopillars are chosen as \({J}_{AB}^{s}={e}^{i\varDelta \varphi }J\) and \({J}_{A{\prime} B{\prime} }^{s}={e}^{-i\varDelta \varphi }J\) to constructively generate the designed Jones matrix J for the unit cell. The period of the squared unit pixel is P = 800 nm. d Schematic of light propagation through the bilayer metasurface with the oblique incidence-oblique detection measurement strategy. For each layer, the scattering light (indicated by wavy line arrows) is deflected by applying a gradient phase distribution \({e}^{i{k}_{x}x}\) on it and the residual zero order maintains its propagation direction as the incident one. With such measurement, the influence of the residual zero orders can be totally eliminated in the detection. RZO: residual zero order. e Scanning electron microscopy (SEM) images of the fabricated metasurface (partial view). The dashed red square indicates one unit pixel with individually designing the two sets of nanopillars. f Experimental results of the measured nanoprinting and holographic images. The incident and analyzed polarizations are indicated at lower left and lower right of each panel. All panels share the same scale bar in the lower left one.