Structured Pearcey-prime Gaussian beams and phase-controlled trajectory engineering.

We present a unified framework for generating structured Pearcey-prime Gaussian beams (PePGBs) with controllable spatial profiles and propagation dynamics. By introducing a generalized radial mapping function, diverse beam structures, including circular, elliptical, petal-shaped, oscillator-like, star-shaped, and lattice configurations, are constructed at the input plane through controlled redistribution of transverse spatial frequencies. Phase modulation is further employed to engineer the beam trajectory without altering its initial structure. The results show that the beams propagate along parabolic paths, where the curvature is determined by the parameter q and its sign directly controls the bending direction. Using representative initial beams, we demonstrate that distinct propagation behaviors can be achieved under different phase conditions. The intensity and phase distributions confirm that the beam structure remains robust during propagation, indicating a decoupling between spatial formation and dynamical control. This approach provides a flexible strategy for structured light design and beam steering, with potential applications in optical manipulation and programmable photonic systems.