Tunable rotational transverse multifoci generated by catastrophe fields with discrete phase and coordinate rotation

Autofocusing structured fields have important potential in parallel optical manipulation and configurable focal field. However, it is still challenging to regulate the transverse multifoci array, as superimposed catastrophe fields tend to concentrate energy near the center or mainly exhibit longitudinal autofocusing. Here, we propose a transverse multifocal higher-order catastrophe field by engineering the discrete phase relation among the constituent subfields. This strategy enables coplanar rotational multifocal arrays with tunable radius, focal spacing, focal width, and residual central intensity through the subfield number 𝑁 and structural parameter 𝑐 . We show that the transverse focusing pattern is governed by the combined effects of azimuthal coordinate rotation and discrete phase modulation: removal of the prescribed phase relation leads to enhanced central energy accumulation, whereas the discrete phase sequence promotes the formation of separated peripheral focal regions. The design theory has been well corroborated by experiment using off-axis computer-generated holography. The phase assigned to the 𝑚 th subfield follows 𝜙 𝑚 = 2 ⁢ 𝜋 ⁢ ( 𝑚 − 1 ) / 𝑁 , corresponding to an 𝑁 -level discretization of a first-order azimuthal vortex phase. Phase, transverse energy-flow, and orbital angular momentum (OAM) analyses reveal orientation evolution, circulating transverse energy flow, and a multimodal OAM response associated with this discrete azimuthal phase structure. The established results suggest that the discrete subfield phase engineering can be considered as a versatile strategy for tailoring rotational multifoci catastrophe fields, opening opportunities for structured-light engineering, parallel optical manipulation, and topological light-field design.