Controllable auto-focusing characteristics of circular Airy-prime beam with grafted zero-order and fractional-order equivalent vortex phases.

Fractional optical vortex beam possesses a complex phase structure, which provides additional degrees of freedom for orbital angular momentum modulation. In this paper, a grafted zero-order and fractional-order circular Airy-prime beam with controllable auto-focusing characteristics is proposed. The grafted topological charges (TCs) can be flexibly selected in fractional forms, and the grafting angle α can be independently regulated. The focusing performance of the beam can be manipulated by α and the grafted TCs (l1 and l2). Furthermore, the focal intensity of the beam can be enhanced by introducing a thin convex lens at the initial plane. The focal positions and the number of focal spots can be precisely controlled by adjusting the lens focal length f. The work provides a feasible reference for the design of high-performance auto-focusing vortex beams and the precise manipulation of complex light fields.