Design of a linearly chirped circular Airyprime beam with adjustable dual foci

The auto-focusing phenomenon of structured beam has a wide range of applications. However, a single structured beam usually undergoes only one auto-focusing behavior during propagation. If a single structured beam can generate two tunable auto-focusing phenomena, leading to the birth of dual adjustable focal points, it will significantly expand its application scenarios. Therefore, what we believe to be a novel beam model that achieves a distinct dual auto-focusing effect is proposed. Leveraging the enhanced auto-focusing properties of circular Airyprime beams, a single structured beam comprising two phase components encoded with distinct linearly chirped factors <i>v</i>₁ and <i>v</i>₂ is constructed. In this configuration, each focal point can be independently controlled. Independent modulation of <i>v</i>₁ and <i>v</i>₂ enables precise control over the focal positions and focal intensities of the two foci-an advancement beyond conventional single-focus structured beams. With the aid of a control parameter <i>P</i>, two foci with precisely equal intensity can be accurately engineered. By jointly tuning <i>v</i>₁, <i>v</i>₂, and the initial ring radius <i>r</i>₀, the beam supports full control over the three-dimensional (3D) intensity distribution. This enables the formation of uniform 3D optical dark regions bounded by the two focal points, with customizable physical dimensions and equal intensity at both ends. The proposed beam architecture offers significant advantages for advanced optical micro-manipulation. Its ability to localize and tune dual foci within a single structured beam facilitates applications such as dual plane particle trapping, the construction of optical cages, and controlled energy deposition. This model introduces what we believe to be a new degree of freedom for structured beam in space and opens avenues for optical control in micro- and nanoscale environments.