Focusing characteristics of Bessel–Gaussian vortex beams with heterogeneous phase modulation

Using vector diffraction theory, we analyze how heterogeneous phase modulation affects the focusing behavior and propagation of Bessel–Gaussian beams. We systematically analyze the influence of beam parameters, topological charge L, phase factor n, and focusing angle phase parameter B on the normalized intensity distribution. The results reveal that the topological charge L can precisely control the opening degree of the spiral-shaped curve on the focal plane: as L increases, the ring opening widens, and a second ring emerges. Increasing the phase factor n concentrates beam energy toward the central spot. The focusing angle phase parameter B strongly affects peak intensity locations: larger B shifts energy toward bilateral regions, and further increase displaces the highest-intensity spot outward along the y = x direction. Increasing beam parameters causes focal plane intensity peaks to separate, expanding the overall focal pattern outward. We also examine how negative B values alter peak intensity positions. As B decreases, peak intensity locations rotate clockwise. These findings have potential applications in optical trapping, particle manipulation, and modern medical technologies.