Longitudinal Programming of Modal and Topological States in Free-Space Optical Fields.

Spatially structed optical fields provide a powerful tool for controlling and conducting light matter interactions. Despite the vast number of ways to configure this light, the modal identity of a propagating beam is generally fixed.

We establish a framework in which the propagation direction functions as a longitudinal modal register, enabling discrete transverse mode identities and topological states to be deterministically assigned to preselected axial positions along a single beam. The physical mechanism is a one-to-one geometrical correspondence between radial spatial frequency and axial reconstruction distance inherent to the conical angular spectrum of Bessel beams. By exploiting this radial–axial spectral mapping, the radial spectrum of a Bessel-generating axicon can be transformed into a programmable longitudinal state memory. Using a single static, phase-only SLM, we demonstrate passive free-space beams whose transverse modal identity and topological charge evolve in discrete steps along the propagation direction

Specifically, we realize longitudinal sequences that include cross-basis transitions between Cartesian (HG), cylindrical (LG), caustic (Airy-like), and propagation-invariant (Bessel-like) modes, deterministic hopping and sign reversal of orbital angular momentum, and discrete ring-lattice fields with axially varying site number. All transitions are encoded directly into the radial spectrum of a single axicon phase pattern and unfold autonomously during free-space propagation without dynamic modulation or cascaded optical elements. The ability to program discrete modal identities, topological charge, and lattice geometry along the propagation direction opens new opportunities in volumetric optical manipulation, axial multiplexed communications, three-dimensional photonic fabrication, and free-space preparation of spatially structured quantum states.