Twisted optical needle array sourced by a circular hyperbolic umbilic beam

In this Letter, we propose an ultralong twisted optical needle array, derived from a circular hyperbolic umbilic beam (CHUB) array. When propagating within a parabolic potential, the CHUBs periodically focus on optical needles. By extending the Heisenberg equation from quantum mechanics to optics, we theoretically demonstrate that introducing a linear chirp phase and offset modulation causes the optical needles to propagate along spiral trajectories. Furthermore, the Heisenberg equation reveals that these modulations on the centroid of a beam propagating in a parabolic potential can be generalized to arbitrary beams, offering broad applicability and insight into propagation dynamics.