Joint phase control in metasurfaces for optical convolution operations

Combining the propagation and geometric phases in a metasurface facilitates the independent control of multiple parameters of the light field. However, the geometric phase often displays a random distribution, making it difficult to observe directly. We introduce a frequency-dependent phase response: at frequency f₁, there is a superposition of the geometric and propagation phases, whereas at frequency f₂, the propagation phase remains constant, and only the geometric phase is applied. The superposition can be interpreted as a convolution process in far-field Fraunhofer diffraction, enabling convolution metasurface devices to generate complex orbital angular momentum beams array and patterned array. Notably, the geometric phase aligns with the characteristic distribution of orbital angular momentum beams, allowing direct observation of the loaded geometric phase. These findings open what we believe to be new avenues for manipulating and calculating complex vector optical fields, optical information coding, controlling light-matter interactions, and enhancing optical communication.