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Extreme-ultraviolet spatiotemporal vortices via high harmonic generation

Nature Photonics (2025)Cite this article

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Abstract

Spatiotemporal optical vortices (STOVs) are space–time structured light pulses with a unique topology that couples spatial and temporal domains and carry transverse orbital angular momentum (OAM). Up to now, their generation has been limited to the visible and infrared regions of the spectrum. During the last decade, it was shown that through the process of high-order harmonic generation, it is possible to upconvert spatial optical vortices that carry longitudinal OAM from the near-infrared into the extreme-ultraviolet (EUV), thereby producing vortices with distinct femtosecond and attosecond structure. In this work, we demonstrate theoretically and experimentally the generation of EUV spatiotemporal and spatiospectral vortices using near-infrared STOV driving laser pulses. We use analytical expressions for focused STOVs to perform macroscopic calculations of high-order harmonic generation that are directly compared to the experimental results. As STOV beams are not eigenmodes of propagation, we characterize the highly charged EUV STOVs in both the near and far fields to show that they represent conjugated spatiotemporal and spatiospectral vortex pairs. Our work provides high-frequency light beams topologically coupled at the nanometre/attosecond scales domains with transverse OAM that could be suitable to explore electronic dynamics in magnetic materials, chiral media and nanostructures.

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Fig. 1: High-order spatiotemporal harmonic vortex generation setup.
Fig. 2: Simulation results for the generation of EUV high-harmonic STOVs and SSOVs.
Fig. 3: Influence of the inhomogeneity parameter in the generation of EUV STOVs given the same transverse OAM l0 = 1.
Fig. 4: Experimental generation of EUV high-harmonic STOVs.

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Data availability

The simulation dataset that supports the findings of this study is available online via figshare at https://doi.org/10.6084/m9.figshare.28714805 (ref. 67). The experimental data are available upon reasonable request.

Code availability

The simulation and data visualization codes to generate the plots and data analysis within this paper are available from the corresponding authors on reasonable request.

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Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 851201). C.H.-G. and L.P. acknowledge funding from Ministerio de Ciencia e Innovación (grant no. PID2022-142340NB-I00) and Junta de Castilla y León and Fondo Europeo de Desarrollo Regional (FEDER), under grant no. SA108P24. Financial support of the Department of Education of the Junta de Castilla y León and FEDER Funds is gratefully acknowledged (Escalera de Excelencia grant no. CLU-2023-1-02). C.-T.L. acknowledges support from AFOSR YIP (grant no. FA9550-23-1-0234) and partial support by DOE BER (grant no. DE-SC0023314). M.A.P. acknowledges funding from Ministerio de Ciencia e Innovación (grant no. PID2021-122711NB-C21). M.M.M. and H.C.K. gratefully acknowledge support from the Department of Energy Basic Energy Sciences grant no. DE-FG02-99ER14982 for the experiments, which were done at JILA, and the US National Science Foundation through JILA Physics Frontiers Center grant no. PHY-2317149 for the experimental facilities. R.M.-H., L.P. and C.H.-G. thankfully acknowledge RES resources provided by BSC in MareNostrum 5, and CESGA in Finisterrae3 to grant nos. FI-2024-2-0010 and FI-2024-3-0035.

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Authors and Affiliations

  1. Grupo de Investigación en Aplicaciones del Láser y Fotónica, Departamento de Física Aplicada, Universidad de Salamanca, Salamanca, Spain

    Rodrigo Martín-Hernández, Luis Plaja & Carlos Hernández-García

  2. Unidad de Excelencia en Luz y Materia Estructuradas (LUMES), Universidad de Salamanca, Salamanca, Spain

    Rodrigo Martín-Hernández, Luis Plaja & Carlos Hernández-García

  3. JILA and Department of Physics, University of Colorado and NIST, Boulder, CO, USA

    Guan Gui, Henry C. Kapteyn, Margaret M. Murnane & Chen-Ting Liao

  4. Department of Physics, Indiana University, Bloomington, IN, USA

    Chen-Ting Liao

  5. Complex Systems Group, ETSIME, Universidad Politécnica de Madrid, Madrid, Spain

    Miguel A. Porras

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  1. Rodrigo Martín-Hernández
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  2. Guan Gui
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  3. Luis Plaja
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  4. Henry C. Kapteyn
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  8. Carlos Hernández-García
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Contributions

C.H.-G., M.A.P., C.-T.L. and M.M.M. conceived the project. R.M.-H, M.A.P. and C.H.-G. performed the theoretical derivations and simulations and analysed the resulting data. G.G. conducted the experiment and analysed the experimental data. C.-T.L., M.M.M. and H.C.K. supervised the experiment. C.H.-G., M.A.P. and L.P. supervised the theoretical simulations. All authors wrote and prepared the manuscript.

Corresponding authors

Correspondence to Rodrigo Martín-Hernández or Chen-Ting Liao.

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Competing interests

H.C.K. has a co-affiliation as CTO of KMLabs Inc. M.M.M. and H.C.K. have a financial interest in KMLabs. The other authors declare no competing interests.

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Nature Photonics thanks Qiwen Zhan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information sections S1–S3 and Figs. 1–3.

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Martín-Hernández, R., Gui, G., Plaja, L. et al. Extreme-ultraviolet spatiotemporal vortices via high harmonic generation. Nat. Photon. (2025). https://doi.org/10.1038/s41566-025-01699-w

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