Ultrafast Electron Beam X-ray Tomography

• Ultrafast electron beam X-ray computed tomography is a technique that obtains 3D volumetric data of dynamic phenomena using MHz XFEL pulses and micrometer resolution.
• It employs diamond-crystal beam-splitting optics with multi-projection imaging and deep-learning reconstruction to enable single-shot, non-scanning tomography.
• Demonstrated with colliding water microdrops, the method achieves sub-microsecond temporal resolution and ~5% mean relative error in volumetric reconstructions.

Ultrafast electron beam X-ray computed tomography refers to the acquisition and reconstruction of three-dimensional (3D) volumetric data of dynamic phenomena at megahertz (MHz) temporal rates and micrometer spatial resolution, using X-ray pulses generated by electron-beam-driven sources such as the European X-ray Free-electron Laser (XFEL). By exploiting the unique MHz pulse structure, multi-projection imaging techniques, diamond-crystal beam-splitting optics, and advanced deep-learning-based reconstruction, this methodology enables single-shot, non-scanning tomography of non-reproducible and stochastic phenomena with a temporal resolution orders of magnitude beyond traditional rotating-sample CT methods (Villanueva-Perez et al., 2023).

1. Experimental Infrastructure and X-ray Source Characteristics

Ultrafast electron beam X-ray computed tomography leverages the European XFEL, a superconducting linear accelerator operating at 17.5 GeV, capable of delivering up to 4.5 MHz X-ray pulses within a 600 µs radio-frequency train, repeated at 10 Hz (up to 2700 pulses per train). Each X-ray pulse, generated via self-amplified spontaneous emission (SASE) due to electron beam microbunching in the undulator, attains durations of only a few tens of femtoseconds and can contain $10^{11}$–$10^{12}$ photons at 10 keV. The pulse-to-pulse separation may reach 222 ns at the maximum rate; in practical proof-of-concept implementations, MHz frame rates (e.g., 1.128 MHz with 886 ns frame intervals) are typically dictated by detector electronics and readout constraints.

A single XFEL pulse is split into multiple spatially distinct beamlets using diamond crystals in Laue geometry. A 100 µm-thick diamond C(111) acts as the first splitter, transmitting 70–80% of the incident beam and deflecting the remainder by $\theta_1 = 35.0^\circ$. The transmitted beam is further split by a diamond C(220) splitter, yielding $\theta_2 = 58.8^\circ$ and a direct beam. Auxiliary diagnostics are performed via a bent-crystal spectrometer (diamond (440), Bragg geometry) on the direct transmission. Thus, each SASE pulse provides four sub-beams: two multi-projection beamlets, the direct beam, and a diagnostic spectrometer channel.

2. Multi-projection Imaging Geometry and Detector Implementation

All beamlets are geometrically aligned to intersect at a common interaction point occupied by the sample, with the two useful multi-projection arms separated by $\Delta\theta = \theta_2-\theta_1 = 23.8^\circ$. The sample coordinate system is defined such that $z$ follows the direct beam, while $x$ and $y$ are horizontal and vertical, respectively. Two indirect, high-speed detectors (Shimadzu HPV-X2, 128 frames/train) capture 2D projections at MHz rates: $p_1(x, y, t)$ for the C(111) arm (pixel size 3.2 µm) and $p_2(x, y, t)$ for the C(220) arm (pixel size 6.4 µm, employing lower numerical aperture optics due to beamlet fluence constraints).

3. Forward Modeling and Discrete Tomographic Operators

The measured X-ray attenuation for each projection follows the Beer–Lambert law, parameterized by the space-time varying absorption coefficient $\mu(x, y, z, t)$:

$$I_i(x, y, t) = I_{0, i}(x, y, t) \cdot \exp\left(-\int_{z_{min}}^{z_{max}} \mu(x', y, z', t) dz'\right)$$

Defining the projection image $p_i(x, y, t) = -\ln[I_i/I_{0, i}]$, this simplifies to:

$p_i(x, y, t) = \int \mu(x', y, z', t) dz'$

along the corresponding ray direction, which may involve a shear transformation for off-axis projections.

In the discrete representation, the volume $$\mu \in \mathbb{R}^{N_x \times N_y \times N_z \times N_t}$$ and detector images $p \in \mathbb{R}^{2 \cdot N_x \cdot N_y \cdot N_t}$ relate by the forward operator $A$:

$p = A\mu$

with each block $A_1, A_2$ encoding line integrals along angles $\theta_1, \theta_2$. In component form:

$$(A\mu)_{i,x,y,t} = \sum_{k=1}^{N_z} w_{i,x,y,k} \cdot \mu_{x_i, y_i, k, t}$$

where $w_{i,x,y,k}$ are ray-tracing weights, unity for parallel-beam geometry, and $(x_i, y_i)$ map entrance coordinates.

4. Volumetric Reconstruction Algorithms

The central inverse problem is to recover the 4D absorption distribution $\mu$ from simultaneous multi-angle projections:

$\min_{\mu} \|A\mu - p\|_2^2 + \lambda R(\mu)$

with $R(\mu)$ a regularizer favoring physically plausible solutions, such as 3D total variation (TV):

$$R(\mu) = \sum_{x, y, z, t} \sqrt{(\partial_x \mu)^2 + (\partial_y \mu)^2 + (\partial_z \mu)^2}$$

Rather than classical gradient-descent, a neural-implicit reconstruction (ONIX) models $\mu$ as a 4D deep learning function, enforcing consistency with all views. Optimization employs stochastic gradient descent (Adam, $lr = 10^{-4}$) to jointly minimize mean squared error loss $\|A\mu - p\|^2$ and an adversarial loss encoding realistic volume priors. Convergence is typically achieved within ~2 hours on a single NVIDIA A100 GPU, producing temporally resolved volumetric sequences $\mu(x, y, z, t)$ synchronized to pulse bursts.

5. Spatial–Temporal Resolution, Trade-offs, and Quantitative Metrics

The experimentally obtained spatial resolution is set by detector pixel dimensions: 3.2 µm for the C(111) arm, 6.4 µm for C(220), with potential for 1.6 µm by employing higher-NA optics. Temporal intervals of 886 ns (1.128 MHz) were demonstrated, with the XFEL supporting up to 4.514 MHz (222 ns spacing) without configuration changes. Resolution trade-offs arise from finite X-ray pulse energy: increasing the number of projections improves 3D reconstruction fidelity but splits photons among more beamlets, decreasing per-view dose and signal-to-noise ratio (SNR). Detector limitations, such as 10-bit dynamic range, further constrain sensitivity; 16-bit readout and smaller pixels would enhance performance.

Quantitatively, C(111)-arm projections exhibit ~2x higher SNR and contrast than C(220), attributable to higher-order reflection properties and polarization effects. Simulation-based re-projections of reconstructed 3D volumes yield mean relative errors ≤ 5%, reflecting robust fidelity.

6. Empirical Demonstrations and Application Scope

The MHz-XMPI technique was validated by reconstructing 3D movies of colliding water microdrops, each approximately 70–80 µm in diameter, generated by piezo-driven nozzles and intersecting in flight at ~2.4 m/s (Weber number ~7.4, impact parameter ~0.12). Within a 128-pulse burst, 127 usable time points per pulse train were acquired and reconstructed volumetrically, yielding sub-microsecond resolved sequences that detail merging, neck formation, and separation of droplets. Visualization of these highly time-resolved datasets elucidates fast, non-reproducible fluid dynamic processes previously inaccessible to rotating-sample CT.

State-of-the-art synchrotron-based time-resolved X-ray tomography ("tomoscopy") achieves up to ~1 kHz (1 ms per tomogram), requiring rotation and scanning. MHz-XMPI delivers sub-microsecond 3D frames without sample rotation—three orders of magnitude faster—enabling capture of stochastic and irreversible events in situ.

Application domains opened by MHz-XMPI include fluid dynamics (cavitation, multiphase mixing), additive manufacturing, shockwave and bubble collapse in energetic materials, transient foaming and materials processing, and select biomedical settings (ultrafast drug delivery, cell-wall rupture). The capability to probe opaque systems with temporal resolutions previously unattainable positions MHz-XMPI as a methodological advance in four-dimensional imaging (Villanueva-Perez et al., 2023).