Experimental generation of a photonic toroidal vortex

The study addresses whether a three-dimensional photonic toroidal vortex—featuring a closed-loop vortex line and a rotating spatiotemporal spiral phase—can be generated as a solution to Maxwell’s equations using optical conformal mapping. The context builds on advances in spatiotemporal optical vortices and conformal transformation optics, aiming to transform a spatiotemporal vortex tube into a toroidal vortex ring. The purpose is to realize and characterize such a field structure experimentally and to highlight its importance for toroidal electrodynamics, light–matter interaction, and photonic symmetry and topology.

The work draws on optical coordinate transformations and conformal mapping (e.g., optical conformal mapping and conformal transformation optics) to reshape optical fields. It references sorting and manipulation of orbital angular momentum (OAM) states using Cartesian-to-log-polar transformations and spiral transformations. Prior studies on non-radiating anapole and toroidal excitations in matter (metamaterials, nanoparticles, plasmonic systems) are cited to motivate interest in toroidal modes. The literature on spatiotemporal optical vortices (generation, propagation, angular momentum, and characterization) provides the foundation for creating a spatiotemporal vortex tube that can be conformally mapped into a toroidal vortex. Broader context includes vortex rings in diverse physical systems (fluids, magnets) and historical perspectives on vortex phenomena.

A dispersion-managed mode-locked fibre laser at 1,030 nm produces a chirped pulse that is split into a signal and a reference pulse (BS1). The reference is dechirped to a transform-limited pulse (~90 fs) via a grating pair on a precision stage (Zolix MAR20-65, 5 µm step) and used as a probe. The signal passes through a 2D reflective pulse shaper in a 4f configuration (grating, cylindrical lens, and a reflective SLM; Holoeye GAEA-2-NIR-069). A spiral phase is applied on SLM1 to convert the Gaussian pulse into a spatiotemporal optical vortex (STOV) with a spiral phase in the x–t plane. The STOV is elongated along the vortex line (y direction), forming a spatiotemporal vortex tube. The beam is expanded with an afocal cylindrical beam expander. Two additional SLMs (SLM2 and SLM3) implement an afocal conformal mapping system using programmed phase masks corresponding to equation (2) and equation (3) (Cartesian-to-log-polar transformations) with parameters a and b chosen to occupy much of the liquid-crystal area. This maps the elongated STOV tube into a photonic toroidal vortex. For characterization, the toroidal vortex is interfered with the short reference pulse via BS4 and recorded on a CCD. By scanning the temporal delay with a movable mirror, interference is obtained for each temporal slice of the toroidal vortex pulse (~3 ps). Using a known 3D pulse characterization method, amplitude and phase are retrieved from the interference fringes for each slice and stitched to reconstruct the 3D iso-intensity and phase, revealing the ring-shaped vortex core and the rotating spatiotemporal spiral phase. Simulations accompany the experiment, showing the conformal mapping and the formation of a vortex ring with a circulating spatiotemporal phase and local OAM density tangential to the ring.

• Demonstration of a photonic toroidal vortex generated from a spatiotemporal vortex tube via optical conformal mapping using two phase masks (Cartesian-to-log-polar type) implemented on SLMs.
• Experimental reconstruction of the 3D iso-intensity structure shows a ring-shaped vortex line (toroidal core) with semitransparent visualization distinguishing the outer isosurface (purple) and the core (red).
• The spatiotemporal spiral phase circulates around the toroidal core with topological charge l = 1 in the radial–temporal plane; three radial slices confirm identical spiral phase profiles of charge 1.
• Local orbital angular momentum (OAM) density lies tangent to the vortex ring due to dominant phase gradients in the radial–temporal plane.
• The toroidal vortex pulse duration is ~3 ps; the reference pulse is ~90 fs, enabling time-slice interferometric reconstruction.
• Propagation in air (weak positive GVD) over short distances introduces only small distortions, consistent with expectations.
• Simulations corroborate the experimental observations, showing successful mapping from a spatiotemporal vortex tube to a toroidal vortex ring and subsequent collimation by a second phase mask.

The findings confirm that optical conformal mapping can convert a spatiotemporal vortex tube into a toroidal vortex, addressing the central question of realizing a toroidal photonic field with a closed-loop vortex line and rotating spatiotemporal phase. The experimental measurement of phase and intensity validates the topological structure (charge 1) and the OAM distribution predicted by the mapping design. This realization provides an accessible free-space photonic platform for exploring toroidal electrodynamics, symmetry, and topology, and complements prior matter-based toroidal excitations by enabling free-space toroidal modes with controlled phase and intensity distributions. The robustness against small propagation-induced distortions in weakly dispersive media supports potential applications in structured light–matter interactions and advanced beam shaping.

The study experimentally demonstrates a photonic toroidal vortex—an optical field with a ring-shaped vortex core and circulating spatiotemporal phase—as a new solution to Maxwell’s equations enabled by optical conformal mapping. The approach reliably maps an elongated spatiotemporal vortex tube into a toroidal structure and allows full 3D characterization. The results open avenues for studying toroidal electrodynamics, light–matter interactions, and photonic symmetry and topology. The conformal mapping concept is general and can be extended to other spectra and wave systems (electron beams, X-rays, acoustics, hydrodynamics, aerodynamics). Future work may explore different topological charges, propagation dynamics in various dispersive media, robustness over longer distances, and interaction with materials or particles.

• The demonstrated toroidal vortex has topological charge 1; other charges or more complex toroidal structures were not reported.
• Characterization relies on interferometric time-slicing with a short reference pulse; accuracy depends on alignment and fringe retrieval, and stitching of slices may introduce reconstruction artifacts.
• Propagation was discussed for air with weak positive GVD over short distances; long-distance robustness and behavior in different media were not explored.
• Specific efficiencies and loss metrics of the SLM-based conformal mapping system were not detailed.