Researchers have proposed a new way to generate and control extreme-ultraviolet (EUV) light carrying intricate three-dimensional polarization structures—objects known as spatiotemporal skyrmions and vector hopfions. The theoretical study, led by Songlin Zhuang and Yi Liu of the University of Shanghai for Science and Technology, shows how high-order harmonic generation (HHG) could transfer complex topological patterns from an infrared driving field into much higher-energy EUV radiation. The approach may provide a new route to structured attosecond light for ultrafast imaging, precision measurement, chiral sensing and emerging quantum technologies.
Optical skyrmions are commonly described as light fields whose polarization or phase forms a continuous topological texture. In a conventional spatial skyrmion, this structure is mapped across a plane perpendicular to the direction of propagation. It can be created by coherently combining a vortex beam, which carries orbital angular momentum, with a Gaussian beam, while arranging their spin angular momentum states in opposite directions. The resulting polarization pattern cannot be smoothly transformed into a uniform field without disrupting its topology.
The concept becomes even more unusual when the vortex component is replaced by a spatiotemporal optical vortex, or STOV. Unlike an ordinary vortex, whose phase winds around a spatial axis, an STOV has a phase singularity embedded in the space-time structure of an optical pulse. Its orbital angular momentum is transverse to the usual propagation direction. When this field is combined with a Gaussian component of opposite spin, the resulting topological texture extends into a plane parallel to the optical axis. These structures are known as spatiotemporal skyrmions because their polarization geometry is distributed across both spatial and temporal coordinates.
Generating such patterns directly in the EUV region is difficult. Most conventional optical components strongly absorb EUV radiation, while nanoscale devices capable of shaping these wavelengths are challenging to design and fabricate. The researchers therefore turned to HHG, a nonlinear process in which an intense laser pulse interacts with matter and produces radiation at integer multiples of the driving frequency. HHG can convert relatively low-energy infrared photons into high-energy EUV photons while preserving, and sometimes multiplying, properties such as polarization, phase and angular momentum.
The proposed scheme uses a two-color driving field containing two carefully coordinated optical components. According to simulations based on the strong-field approximation, the topological structure of this combined field can survive the highly nonlinear harmonic-generation process. The calculations indicate that different harmonic orders can carry different spin and transverse orbital angular momentum states, allowing several varieties of EUV spatiotemporal skyrmions to be produced by adjusting the driving-field parameters.
The mechanism depends on the way photons from the two colors are absorbed during HHG. The team developed a qualitative analysis based on photon absorption probability to identify the dominant nonlinear pathways contributing to each harmonic. In this picture, a particular harmonic is not produced randomly from all possible combinations of absorbed photons. Instead, certain channels become dominant, selecting specific angular momentum states. The final EUV mode is determined by the combined requirements of energy conservation, spin angular momentum conservation and transverse orbital angular momentum conservation.
This conservation-based picture explains why the topological texture can be transferred so effectively despite the complexity of HHG. The electric field first drives electrons away from their parent atoms or molecules. The electrons then accelerate in the laser field before returning and recombining, releasing high-energy photons. During this process, the electron trajectory and recombination radiation retain information about the driving field’s spin–orbit structure. When the two colors are properly configured, this information is encoded into the phase and polarization of the emitted harmonics.
The simulations also identify a crucial condition involving the direction of the Stokes vectors of the two-color fields. Stokes vectors provide a compact description of the polarization state of light, including its degree of linear, circular and elliptical polarization. When the relevant Stokes-vector directions are matched, the two components maintain an ordered spin–orbit coupling that supports robust skyrmion generation. If the directions are mismatched, the coupling dynamics become disorganized and the desired harmonic modes can be weakened or destroyed. This finding offers a practical control parameter for designing future experiments.
A second challenge appears when the generated STOV-containing fields propagate away from the HHG region. Because STOV modes are related to tilted Hermite-lobed modes through Fourier transformation, their appearance changes in the far field: the vortex-like component can split into two lobes. Without additional optical processing, this may make the original skyrmionic structure difficult to observe or use outside the focus. The researchers propose a spatiotemporal mode-conversion system designed to reverse this transformation and reconstruct the near-field topology in the far field. Such a system could make EUV skyrmions accessible to downstream experiments without requiring the beam to be refocused at every stage.
The study further extends the two-dimensional concept into three dimensions, predicting the formation of EUV vector hopfions. In a hopfion, field lines are linked in a topology resembling interlocking rings, creating a more complex structure than a skyrmion. Although the work is based on computational modeling rather than an experimental demonstration, it establishes a theoretical framework for generating and manipulating these fields with high-order harmonics. If realized experimentally, EUV spatiotemporal skyrmions, hopfions and their attosecond pulse trains could offer an unusually precise way to probe matter. Their rapidly changing polarization patterns may encode information about time-dependent, polarization-sensitive material responses, opening possibilities in attosecond electron dynamics, chiral recognition, precision metrology and time-varying photonics. Their topological robustness may also prove useful for transmitting structured information in demanding quantum and ultrafast optical environments.
Subject of Research: Computational simulation/modeling
Article Title: Generation and control of extreme-ultraviolet spatiotemporal skyrmions and vector hopfions with high harmonic generation
News Publication Date: 23-Jun-2026
Web References: https://doi.org/10.1186/s43074-026-00260-4
References: PhotoniX, DOI: 10.1186/s43074-026-00260-4
Image Credits: University of Shanghai for Science and Technology / Jiahao Dong
Keywords
Extreme-ultraviolet light, spatiotemporal skyrmions, vector hopfions, high-order harmonic generation, attosecond science, optical vortices, transverse orbital angular momentum, structured light, spin–orbit coupling, PhotoniX
