Fig. 1: Working principle of the BVML for multiple-order radial differential operations.

a Schematic of the proposed BVML based on a dielectric metasurface. The metasurface device modulates the incident object light field and outputs a lth-order differential image at the imaging plane according to the order of the Bessel vortex encoded in its profile. b Amplitude and phase distributions of the BVML combining a Bessel vortex and a focusing lens. c–f Simulated CTF curves for the first-, second-, third-, and fourth-order BVMLs and the corresponding fitting curves using the form \(\left|t\left({k}_{r}\right)\right|=a{\left|{k}_{r}\right|}^{l}\). The abbreviation “Sim” has the full name of “Simulation”. g–j Simulated differential processing results of the first-, second-, third-, and fourth-order BVMLs. The input object is part of the 1951 USAF test chart. The enlarged insets highlight the number of edges. Scale bars: 50 μm.