Attosecond inner-shell lasing at ångström wavelengths

Thomas M Linker^{1

2}, Aliaksei Halavanau³, Thomas Kroll⁴, Andrei Benediktovitch⁵, Yu Zhang⁶, Yurina Michine⁷, Stasis Chuchurka⁵, Zain Abhari⁸, Daniele Ronchetti^{5

9

10}, Thomas Fransson^{6

11}, Clemens Weninger^{12

13}, Franklin D Fuller¹², Andy Aquila¹², Roberto Alonso-Mori¹², Sébastien Boutet¹², Marc W Guetg³, Agostino Marinelli^{6

3}, Alberto A Lutman³, Makina Yabashi^{14

15}, Ichiro Inoue^{14

16}, Taito Osaka¹⁴, Jumpei Yamada^{14

17}, Yuichi Inubushi^{14

15}, Gota Yamaguchi¹⁴, Toru Hara¹⁴, Ganguli Babu¹⁸, Devashish Salpekar¹⁸, Farheen N Sayed¹⁸, Pulickel M Ajayan¹⁸, Jan Kern¹⁹, Junko Yano¹⁹, Vittal K Yachandra¹⁹, Matthias F Kling^{6

12

20}, Claudio Pellegrini³, Hitoki Yoneda⁷, Nina Rohringer^{5

9}, Uwe Bergmann²¹

Affiliations

¹ Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA. tlinker@slac.stanford.edu.
² Department of Physics, University of Wisconsin-Madison, Madison, WI, USA. tlinker@slac.stanford.edu.
³ Accelerator Research Division, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁴ Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁵ Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron, Hamburg, Germany.
⁶ Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁷ Institute for Laser Science, The University of Electro-Communications, Tokyo, Japan.
⁸ Department of Physics, University of Wisconsin-Madison, Madison, WI, USA.
⁹ Department of Physics, Universität Hamburg, Hamburg, Germany.
¹⁰ Max Planck School of Photonics, Friedrich-Schiller University of Jena, Jena, Germany.
¹¹ Department of Theoretical Chemistry and Biology, KTH Royal Institute of Technology, Stockholm, Sweden.
¹² Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
¹³ MAX IV Laboratory, Lund University, Lund, Sweden.
¹⁴ RIKEN SPring-8 Center, Sayo-cho, Japan.
¹⁵ Japan Synchrotron Radiation Research Institute, Sayo-cho, Japan.
¹⁶ Institute for Experimental Physics/CFEL, University of Hamburg, Hamburg, Germany.
¹⁷ Graduate School of Engineering, Osaka University, Osaka, Japan.
¹⁸ Department of Materials Science and Nano Engineering, Rice University, Houston, TX, USA.
¹⁹ Molecular Biophysics and Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.
²⁰ Department of Applied Physics, Stanford University, Stanford, CA, USA.
²¹ Department of Physics, University of Wisconsin-Madison, Madison, WI, USA. ubergmann@wisc.edu.

PMID: 40500439
PMCID: PMC12314423
DOI: 10.1038/s41586-025-09105-9

Attosecond inner-shell lasing at ångström wavelengths

Thomas M Linker et al. Nature. 2025 Jun.

. 2025 Jun;642(8069):934-940.

doi: 10.1038/s41586-025-09105-9. Epub 2025 Jun 11.

Authors

Affiliations

¹ Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA. tlinker@slac.stanford.edu.
² Department of Physics, University of Wisconsin-Madison, Madison, WI, USA. tlinker@slac.stanford.edu.
³ Accelerator Research Division, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁴ Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁵ Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron, Hamburg, Germany.
⁶ Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
⁷ Institute for Laser Science, The University of Electro-Communications, Tokyo, Japan.
⁸ Department of Physics, University of Wisconsin-Madison, Madison, WI, USA.
⁹ Department of Physics, Universität Hamburg, Hamburg, Germany.
¹⁰ Max Planck School of Photonics, Friedrich-Schiller University of Jena, Jena, Germany.
¹¹ Department of Theoretical Chemistry and Biology, KTH Royal Institute of Technology, Stockholm, Sweden.
¹² Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, CA, USA.
¹³ MAX IV Laboratory, Lund University, Lund, Sweden.
¹⁴ RIKEN SPring-8 Center, Sayo-cho, Japan.
¹⁵ Japan Synchrotron Radiation Research Institute, Sayo-cho, Japan.
¹⁶ Institute for Experimental Physics/CFEL, University of Hamburg, Hamburg, Germany.
¹⁷ Graduate School of Engineering, Osaka University, Osaka, Japan.
¹⁸ Department of Materials Science and Nano Engineering, Rice University, Houston, TX, USA.
¹⁹ Molecular Biophysics and Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.
²⁰ Department of Applied Physics, Stanford University, Stanford, CA, USA.
²¹ Department of Physics, University of Wisconsin-Madison, Madison, WI, USA. ubergmann@wisc.edu.

PMID: 40500439
PMCID: PMC12314423
DOI: 10.1038/s41586-025-09105-9

Abstract

Since the invention of the laser, nonlinear effects such as filamentation¹, Rabi cycling^2,3 and collective emission⁴ have been explored in the optical regime, leading to a wide range of scientific and industrial applications^5-8. X-ray free-electron lasers (XFELs) have extended many optical techniques to X-rays for their advantages of ångström-scale spatial resolution and elemental specificity⁹. An example is XFEL-driven inner-shell Kα₁ (2p_3/2 → 1s_1/2) X-ray lasing in elements ranging from neon to copper, which has been used for nonlinear spectroscopy and development of new X-ray laser sources^10-16. Here we show that strong lasing effects similar to those in the optical regime can occur at 1.5-2.1 Å wavelengths during high-intensity (>10¹⁹ W cm^-2) XFEL-driven Kα₁ lasing of copper and manganese. Depending on the temporal XFEL pump pulse substructure, the resulting X-ray pulses (about 10⁶-10⁸ photons) can exhibit strong spatial inhomogeneities and spectral splitting, inhomogeneities and broadening. Three-dimensional Maxwell-Bloch calculations¹⁷ show that the observed spatial inhomogeneities result from X-ray filamentation and that the broad spectral features are driven by sub-femtosecond Rabi cycling. Our simulations indicate that these X-ray pulses can have pulse lengths of less than 100 attoseconds and coherence properties that provide opportunities for quantum X-ray optics applications.

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Conflict of interest statement

Competing interests: The authors declare no competing interests.

Figures

**Extended Data Fig. 1 |. Filamentation in different samples.**
Examples of filamentation taken at LCLS for different MnCl₂ and Mn metal foils.

**Extended Data Fig. 2 |. Observation of Mollow triplets.**
Examples of Mollow triplets spectra at SACLA taken for Cu 7μm Foils. Further discussion and simulations in the Supplementary Information is provided to describe their formation.

**Extended Data Fig. 3 |. Typical strong lasing spectra at SACLA.**
25 random Strong Lasing Shots for Cu 20 μm Foils which all show broad and inhomogeneous spectra.

**Figure 1.. Experimental setup and concept of stimulated emission.**
(a) The highly directional stimulated emission signal is analyzed with a flat Si(220) crystal with the Bragg angle centered at the Mn or Cu Kα₁ followed by a 2D CCD detector. (b) Stimulated emission process is initiated by a SASE pump pulse ejecting many 1s electrons leading to emission of Kα₁ photons when 2p_3/2 electrons fill the 1s_1/2 core hole. During stimulated emission, Kα₁ photons emitted along the forward direction stimulate emission of more Kα₁ photons resulting in exponential gain. (c) State diagram employed in 3D Maxwell-Bloch theory to simulate stimulated emission process. Atoms are excited from an initial ground state |g⟩ to set of upper levels |u⟩ corresponding to the two possible 1s_1/2 core hole states with differing magnetic quantum (m = ±1/2). The Kα₁ transition is to a set of 4 lower levels |l⟩ corresponding to 4 possible 2p_3/2 core hole final states with differing magnetic quantum numbers (m = ±3/2, ±1/2).

**Figure 2.. Filamentation of stimulated Emission.**
(a, b) 2D profiles taken at LCLS for NaMnO₄ displaying hotspots in their spatial direction (ϕ). (c) Real space (xy) simulation of stimulated emission leaving the medium with 2 hot spots along the y axis. (d, e) Simulation of 2D profiles based on (c) with the spatial direction (ϕ) corresponding to either the x axis (d) or the y axis (e). The figures show that hot spots are more dominant in the 2D profile when they are aligned with the spatial direction (ϕ). (f) Temporal profile of pump (dashed) and stimulated emission (red) for simulation shown in (c). (g) Snap shots depicting xy profile of the stimulated emission simulation shown in (c) as it propagates in the gain medium showing the self-focusing and filamentation process. (See Supplementary Information for full video.)

**Figure 3.. Transition to strong lasing and Rabi cycling.**
(a) Measured spectral width versus number of stimulated emission photons indicating transition to strong lasing regime. (b) Experimental 2D profile for NaMnO₄ solution in the strong lasing regime (red point in a) with shoulder peak indicative of Autler-Townes splitting (top) compared to simulation showing same asymmetric splitting (bottom). (c) Phase dynamics of the stimulated emission. Top: 1D simulation showing symmetric splitting with equal intensity in each emission peak. Bottom: 3D simulation showing modulation during the Rabi cycling resulting in asymmetric, red-shifted spectrum. The Rabi ringing is no longer temporally coherent with the parent emission spike indication self-phase modulation of the radiation during 3D propagation. This asymmetry is consistent with the experimental 2D profile (b).

**Figure 4.. Broadening through Rabi Cycling at High Intensities.**
(a) Experimental 2D profile displaying large spatial and spectral inhomogeneities/broadening for MnO film. (b) Simulation of 2D profile showing similar features using a SASE pump pulse and same average Mn density as in the experiment (a). (c) Temporal profile of the SASE pump pulse (dashed) and stimulated emission signal (red) used in the simulation showing 120 attosecond FWHM pulse length. (d) Experimental 2D profile displaying large spatial and spectral inhomogeneities/broadening and the onset of spectral fringes with ~2eV spacing for MnO film. (e) Simulation of 2D profile showing similar features using a SASE pump pulse and same Mn density as in the experiment (d). (f) Temporal profile of the SASE pump pulse (dashed) and stimulated emission signal (red) used in the simulation (e). The resulting signal shows a strong pulse with 100 attosecond FWHM length, and a much smaller second pulse, delayed by ~2 fs, corresponding the fringe spacing. (g) Experimental 2D profile displaying large spatial and spectral inhomogeneities/broadening and the more pronounced spectral fringes with ~4 eV spacing for Cu foil. (h) Simulation of 2D profile showing similar features using a SASE pump pulse and same Cu density as in the experiment (g). (i) Temporal profile of the SASE pump pulse (dashed) and stimulated emission signal (red) used in the simulation (h). The resulting signal shows two strong pulses with 90 and 100 attosecond FWHM lengths. The pulses are delayed by ~1 fs, corresponding to ~ 4eV fringe spacing.

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References

1. Hercher M. Laser-induced damage in transparent media. J. Opt. Soc. Am 54, 563 (1964).
1. Rabi II Space quantization in a gyrating magnetic field. Phys. Rev 51, 652 (1937).
1. Gray HR, Whitley RM & Stroud CRJ Coherent trapping of atomic populations. Opt. Lett 3, 218–220 (1978). - PubMed
1. Bonifacio R. & Lugiato L. Cooperative radiation processes in two-level systems: Superfluorescence. Phys. Rev. A 11, 1507 (1975).
1. Boyd RW, Gaeta AL & Giese E. Nonlinear optics. in Springer Handbook of Atomic, Molecular, and Optical Physics 1097–1110 (Springer, 2008).

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Attosecond inner-shell lasing at ångström wavelengths

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Attosecond inner-shell lasing at ångström wavelengths

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