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

Abstract

The science and technology of X-ray optics have come far, enabling the focusing of X-rays for applications in high-resolution X-ray spectroscopy, imaging, and irradiation. In spite of this, many forms of tailoring waves that had substantial impact on applications in the optical regime have remained out of reach in the X-ray regime. This disparity fundamentally arises from the tendency of refractive indices of all materials to approach unity at high frequencies, making X-ray-optical components such as lenses and mirrors much harder to create and often less efficient. Here, we propose a new concept for X-ray focusing based on inducing a curved wavefront into the X-ray generation process, resulting in the intrinsic focusing of X-ray waves. This concept can be seen as effectively integrating the optics to be part of the emission mechanism, thus bypassing the efficiency limits imposed by X-ray optical components, enabling the creation of nanobeams with nanoscale focal spot sizes and micrometer-scale focal lengths. Specifically, we implement this concept by designing aperiodic vdW heterostructures that shape X-rays when driven by free electrons. The parameters of the focused hotspot, such as lateral size and focal depth, are tunable as a function of an interlayer spacing chirp and electron energy. Looking forward, ongoing advances in the creation of many-layer vdW heterostructures open unprecedented horizons of focusing and arbitrary shaping of X-ray nanobeams.

Fig. 1. X-ray beam focusing created by free-electron interaction with a custom-made van der Waals (vdW) heterostructure.

This illustration uses ray optics to compare the monochromatic X-ray emission produced by a free electron passing through either a a crystal with constant interlayer spacings or b a crystal with a chirp in the interlayer spacings. In panel (a), a collimated X-ray beam is generated, while in panel (b), the beam focuses on a point. The illustrations are not to scale

Fig. 2. Focused X-ray beam from a vdW heterostructure with chirped interlayer spacings.

We compare b a focused X-ray beam profile with a a collimated one based on the material TaSe₂. The resulting X-ray beam width is ~10 nm in (b) and ~300 nm in (a). The inset in the top-right corner of each panel shows the interlayer spacing chirp along the z direction. The color scales are the same on both panels, emphasizing the field enhancement at the focused hotspot in (b). The ρ-s frames are rotated clockwise relative to the x-z frame, so that the ρ direction points along the emitted beam axis. The enlarged figures highlight the transverse distribution of the beam profiles. The sample thicknesses (300 nm), photon energy (4 keV), and electron kinetic energy (1 MeV) are the same in both panels

Fig. 3. Width and focal depth of focused X-ray beams.

a Sketch defining the beam width and focal depth. The latter is the distance between the two transverse planes in which the beam width is √2 larger than at the focal spot. b, c Distributions of beam width and focal depth as a function of chirp in the interlayer spacings and photon energies. The horizontal axis shows the maximum difference between the interlayer spacings at the top and bottom layers of the heterostructure. The contours (cyan curves) are labeled by the respective values. The electron kinematic energy is set to 1 MeV and the minimum interlayer spacing is 12.70 Å

Fig. 4. Comparison of the intensity profiles of focused X-ray beams and collimated ones, showing their dependence on the incident e-beam parameters.

a Gaussian e-beam parameters: root-mean-square (rms) divergence angle δθ and spot size δr. b Transverse intensity profiles of the focused and the collimated X-ray beams at the distance ρ = 10 μm from the source. In the left column, the e-beam spot size is δr = 0 and the divergence angles are δθ = 10 mrad and 20 mrad. In the right column, the e-beam divergence angle is δθ = 0 and the spot sizes are δr = 20 and 50 nm

Fig. 5. Examples of focused X-ray beams created in multilayer heterostructures comprising vdW materials of similar crystal structure but different compositions.

a, b Two different X-ray focusing schemes with the material configurations listed in the respective tables. The emitted X-rays self-focus along the respective ρ axes, which are rotated clockwise by angles of a 86.4° and b 86.5° relative to the electron trajectory. The observed focal lengths are a 3 μm and b 1 μm. The sample thickness is a 300 nm and b 100 nm. 1T and 2H denote two phases of vdW materials with hexagonal crystal structure, in which there are either one (1T) or two (2H) layers in each vertical unit cell. Twice the interlayer spacing of crystals in the 1T phase is counted to compare with the interlayer spacings of crystals in the 2H phase. Both panels share the same photon energy (4 keV) and e-beam kinetic energy (1 MeV)

Fig. 6. Limits to the obtainable X-ray numerical aperture (NA) due to wave optics and quantum mechanical effects.

From wave optics: the focal depth and length are inversely proportional to NA² and NA, respectively. Therefore, there is a lower bound for the NA, because the focal depth should be smaller than the focal length. The two vertical yellow lines delineate the condition of the ratio focal depth⁄focal length = 1 for samples of thicknesses T = 1000 and 300 nm. The ratio decreases for larger sample thicknesses and larger chirp in the interlayer spacings. A smaller ratio corresponds to a shorter axial focal region (i.e., a less elongated hotspot). From quantum mechanical considerations: the photon coherence at the focal spot is tied to the electron coherence. The red curves indicate the lower bound for the electron transverse momentum range required for photons to interfere coherently at the focal spot. Generally, a larger electron transverse coherence is needed for achieving a bigger NA. Intriguingly, when the focused X-ray beam is emitted along directions normal to the electron velocity (cyan curve), it requires relatively smaller electron coherence. The calculation is based on the platform of Fig. 2b, with a minimum interlayer spacing of 12.70 Å

Fig. 7. Quantum coherence underlying the formation of focused X-ray beams.

In panel a, we show the parameters k, p, p′, and g are the wave vectors of the emitted photon, the initial and final electron, and the reciprocal lattice vector, respectively. Panel b sketches processes associated with electron-driven radiation from a heterostructure with designed interlayer spacings. Each path of arrows denotes a transition satisfying momentum and energy conservation. The initial electron states, which transition to the final electron states |p′〉 by emitting fixed-energy photons, are distributed along an isoenergetic energy surface (black curve). Different photon coherent states (blue arrows) are entangled to the same electron final state |p′〉. The variation of in-plane reciprocal lattice vector g assists in the connection of quantum coherence between the incident electron and the emitted photon. The right column of (b) represents emission processes corresponding to a zero in-plane reciprocal lattice vector