Compact Source of Focused X-ray Radiation

Free-electron interactions with van der Waals heterostructures: a source of focused X-ray radiation
Compact Source of Focused X-ray Radiation

The present coherent X-ray sources require facilities with large footprints such as kilometer-long free-electron lasers, strongly limiting the widespread application of X-rays from fundamental science to industry and medicine. A series of works1–5 in recent years demonstrate that parametric X-ray radiation (PXR), excited by free electrons traversing three-dimensional (3D) crystals, is a promising mechanism for compact coherent X-ray sources, to serve applications that do not need the high photon fluxes of large-footprint coherent X-ray sources.   

By playing a key role in the coherent excitation of PXR, the crystal structure provides a means of fine-tuning the properties of free-electron radiation. A well-known analogue in the microwave to the ultraviolet range is Smith-Purcell radiation, where one-dimensional gratings  extract coherent radiation from free electrons. In the visible optical  regime, it is possible to design and fabricate artificial gratings with subwavelength precision to shape the outgoing Smith-Purcell radiation67. However, this approach is not feasible for X-rays due to their ultrashort wavelengths necessitating atomic-scale grating structures. The advances of van der Waals (vdW) materials, particularly vdW heterostructures8, make it possible to engineer the crystal structure at the atomic scale, opening the avenue of shaping PXR via sophisiticated material engineering techniques.

Our paper in Light: Science & Applications9 presents a theoretical scheme for a compact, free-electron-driven X-ray source that leverages the coherence of PXR and the precision of atomic scale engineering in vdW heterostructures to realize focused X-ray beams. Besides the inherent wavelength tunability of free electron-based radiation sources, our proposed X-ray source is unique in allowing the X-rays to be directly shaped and focused from the source, circumventing the need for bulk and lossy X-ray optics. This innovative development paves the way to more versatile and efficient ways of generating X-rays for applications from industrial inspection to medical imaging.

Parametric X-ray radiation

The mechanism of PXR can be explained using both classical and quantum frameworks. In the classical framework, a swift electron is considered an evanescent source of supercontinuum light. Under the nonrecoil approximation, the electron’s velocity remains constant as it moves across each layer. Meanwhile, the response of bound electrons to the external field can be modeled by susceptibility or dipole moments, which are adopted to the study PXR in periodic4 and aperiodic crystals5, respectively.

The quantum framework treats PXR as spontaneous emission, including first-order processes between a free-electron and a photonic quasiparticle in crystals10. A vital insight from the quantum framework is that the final electron and emitted photon are entangled, meaning the coherence of the emitted photon is derived from the electron11–13. While this feature is not critical for collimated light sources, it plays a significant role in limiting the generation of focused X-ray radiation, which requires a high degree of spatial coherence. 

Illustration of parametric X-ray radiation. In the classical framework, the electron moves uniformly across the crystal, exciting polarization currents of bound electrons. These polarization currents, in turn, give rise to X-ray radiation. In the quantum framework, PXR is represented as spontaneous emission of a photonic quasiparticle in crystal by a free electron.

van der Waals heterostructure: a platform for engineering crystal structure

Photon emission is closely linked to the crystal structure.  This is revealed in our experiments, where we show that adjusting the interlayer spacing can be used to tune the emission direction of collimated beams1. Complex radiation patterns can be created by engineering the crystal structures. Similar ideas have been implemented in Smith-Purcell radiation, where unconventional light sources with designed spectral and angular shapes have been achieved by superlattice and chirped gratings6,7. Unlike the hundreds of nanometers or larger cells used in Smith-Purcell radiation, the crystal structures are of ~1 nm in scale or smaller. vdW heterostructures bring the necessary tools to engineer crystal structures at the atomic scales. The weak vdW force allows for stacking highly disparate atomic layers, creating a wide range of heterostructures without the constraints of lattice matching and processing compatibility typically encountered in traditional materials. Consequently, in principle, one can create vdW heterostructures with designed interlayer spacing.

In addition to stacking, the interlayer spacing can be adjusted reversibly through methods such as intercalation, pressure, temperature, and optical excitation. Take intercalation as an example, the addition of foreign species, such as polyethylene oxide (PEO), to MoS2 during the exfoliation/restacking steps can expand the interlayer spacing. The extent of this expansion is controlled by the type of species and their infiltration densities, offering a versatile approach to fine-tune the crystal structure.

Our paper

In our recent work, we propose a scheme to create an X-ray source that emits self-focused X-ray radiation by manipulating the interlayer spacing of heterostructures. To generate a diffraction-limited focused beam, a linear chirp of the interlayer spacing is required, with a good degree of approximation. Parameters of the foci, such as beam width, focal depth, numerical aperture, and focal length, depend on the sample thickness and the rate of chirp. Compared to conventional X-ray sources, the generated X-rays are directly shaped at the source, eliminating the need to additional lossy and/or bulk X-ray optics for shaping X-rays.

The focusing performance is resilient to the chirp of the interlayer spacing. As a result, heterostructures that approximately adhere to the chirp requirement while greatly reducing fabrication complexity are an exciting avenue to explore. We theoretically design a 300 nm thick heterostructure by stacking nine kinds of vdW materials. The resulting focused beam has a focal length of ~3 um and a beam width of ~10 nm, which is close to the diffraction limit.

a Illustration of focused X-ray radiation generated by an electron traversing a heterostructure with a linear chirp in the interlayer spacing. b A practical scheme of the heterostructure by stacking vdW materials. The focused X-ray beam achieves a width of ~10 nm at the focal point, approaching the diffraction limit of an optical lens.

Looking ahead

We envision that free-electron-driven PXR in nanophononics offers a promising platform for developing compact, versatile and coherent X-ray sources2. Compared to the widely used X-ray tubes, PXR-based sources provide superior spatial coherence, resulting in high intensity at a narrow direction and narrow bandwidth. In our work, only one dimension of the crystal structure has been modulated, but we anticipate that more intricate X-ray patterns could be produced by manipulating the crystal structure in multiple dimensions. Our quantum mechanical considerations reveal that the radiation is tied to the electron wavefunction due to the entanglement between the final electrons and the emitted photons. Consequently, tailoring the electron wavefunction11 and post-selection of the electrons14 present more degrees of freedom that can be leveraged to shape X-ray emission.


  1. Shentcis, M. et al. Tunable free-electron X-ray radiation from van der Waals materials. Nat. Photonics 14, 686–692 (2020).
  2. Wong, L. J. & Kaminer, I. Prospects in x-ray science emerging from quantum optics and nanomaterials. Appl. Phys. Lett. 119, 130502 (2021).
  3. Huang, S. et al. Quantum recoil in free-electron interactions with atomic lattices. Nat. Photonics 17, 224–230 (2023).
  4. Huang, S. et al. Enhanced Versatility of Table-Top X-Rays from Van der Waals Structures. Adv. Sci. 9, 2105401 (2022).
  5. Shi, X. et al. Free-electron-driven X-ray caustics from strained van der Waals materials. Optica 10, 292–301 (2023).
  6. Remez, R. et al. Spectral and spatial shaping of Smith-Purcell radiation. Phys. Rev. A 96, 61801 (2017).
  7. Karnieli, A. et al. Cylindrical Metalens for Generation and Focusing of Free-Electron Radiation. Nano Lett. 22, 5641–5650 (2022).
  8. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).
  9. Shi, X. et al. Free-electron interactions with van der Waals heterostructures: a source of focused X-ray radiation. Light Sci. Appl. 12, 148 (2023).
  10. Rivera, N. & Kaminer, I. Light–matter interactions with photonic quasiparticles. Nat. Rev. Phys. 2, 538–561 (2020).
  11. Wong, L. J. et al. Control of quantum electrodynamical processes by shaping electron wavepackets. Nat. Commun. 12, 1700 (2021).
  12. Karnieli, A., Rivera, N., Arie, A. & Kaminer, I. The coherence of light is fundamentally tied to the quantum coherence of the emitting particle. Sci. Adv. 7, eabf8096 (2021).
  13. García de Abajo, F. J. & Di Giulio, V. Optical Excitations with Electron Beams: Challenges and Opportunities. ACS Photonics 8, 945–974 (2021).
  14. Dahan, R. et al. Creation of Optical Cat and GKP States Using Shaped Free Electrons. arXiv 2206.08828 (2022).

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