Inverse design of Pancharatnam-Berry phase optical elements for all-optical multiple-order spatial differentiation

•The multiple-order differentiation is realized by the PBOE.•The fabricated PBOEs have high diffractive efficiencies and transmittance.•The experimental results conform to theoretical predictions and simulation well.•The method is straightforward and universal and will have various applications. All-optical spatial differentiation has garnered considerable interest within the domain of optical computation for its utility in various applications including edge detection and enhancing contrast in microscopy. Predominantly, research efforts have been concentrated on first- and second-order differentiations, whereas the exploration of higher-order derivatives has been comparatively scarce. Here, we introduce a novel approach for designing Pancharatnam-Berry phase optical elements (PBOEs) to achieve multiple-order spatial differentiation. The PBOEs are fabricated with liquid–crystal (LC) materials and demonstrate exceptional performance, including diffractive efficiencies exceeding 99.55 % and transmittance above 98.32 %. The experimental results corroborate with theoretical predictions and numerical simulations. The proposed method offers a straightforward and universally applicable design process for achieving high-performance multiple-order differentiation, laying the groundwork for such advanced differentiation in practical scenarios involving image processing, enhanced contrast microscopy, and light field modulation.