Generation of optical vortex arrays using discrete-shaped beams.

Optical vortex arrays (OVAs) inherit the intrinsic properties of individual vortex beams while introducing an additional degree of freedom through spatial arrangement, thereby demonstrating broad applicability in optical communication, microparticle manipulation, and optical machining. However, conventional OVA design typically constrains the shapes of the constituent sub-vortices to parametric equations, which limits the exploitation of shape-related degrees of freedom within OVAs and hinders their adaptability to emerging materials with novel circularly symmetric crystalline structures. In this work, we propose a discrete beam shaping method that integrates discrete path integration with the Fourier shift theorem, enabling flexible customization of both closed and open geometric patterns, including both parametric and non-parametric shapes. Experimentally, we demonstrate polygonal OVAs composed of tailored sub-vortices (e.g., triangle, square, hexagon), achieving full spatial utilization in close-packed OVAs and genuine self-similarity between fractal OVAs and their sub-vortices. Furthermore, we introduce a polar lattice coordinate system to generate radially distributed OVAs featuring open linear segments. This advancement provides a strategy for precise control over multiple degrees of freedom in OVAs, thus paving the way for expanded applications in structured light fields.