Two-Color X-Ray Coherent Diffraction Imaging of Helium Nanodroplets

TL;DR

This paper demonstrates a novel two-color X-ray diffraction technique to image helium nanodroplets with femtosecond temporal resolution, effectively separating overlapping images using energy sensitivity and pattern recognition.

Contribution

It introduces a method to disentangle simultaneous two-color X-ray diffraction images of nanoscale particles, validated on helium nanodroplets, enabling ultrafast structural imaging.

Findings

Successful separation of two-color diffraction images using energy-sensitive detection.

Validation of the method through Mie theory fitting of spherical helium nanodroplets.

High agreement between experimental data and theoretical models, especially in outer regions.

Abstract

Two-color X-ray imaging with Free Electron Laser pulses offers a powerful approach for probing ultrafast structural dynamics in nanoscale systems, combining (near-)atomic spatial resolution with femtosecond temporal precision. The first X-ray pulse captures the object's initial state, while the time-delayed second pulse records its subsequent evolution. A key challenge lies in disentangling the two views simultaneously recorded by the same detector. We demonstrate the realization of this approach on structurally varying nanoscale particles using two X-ray pulses of different photon energies, 1 and 1.2~keV. Sub-micrometer helium nanodroplets generated in vacuum are irradiated by the two X-ray pulses separated in time by up to 750 femtoseconds. Taking advantage of the high energy sensitivity of the imaging detector, we separate the overlapping images by analyzing individual pixel counts and applying pattern recognition. The helium nanodroplets' spherical shape allows us to cross-validate this approach by fitting the radial scattering profiles with Mie solutions for a bichromatic field. The excellent agreement between the two methods, particularly in the sparsely illuminated outer regions of the diffraction images where fine structural information is encoded, highlights the quality of this approach and its potential for future advanced X-ray movie techniques.

Figures

Tables

Table 1. TAB. SM1. : Overview of single photon patterns found in the brightest 75 shots. Naming as in Fig.  Influences on the recall .

Shape	Fraction [%]
single	46
horizontal	21
vertical	17
fuL	3
uL	3
L	3
fL	3
quadrant	4

Equations

$$S_{\text{pump}}$$

$$S_{\text{probe}}$$

$$S_{n,m}$$

$$\text{for }n,m\in \mathbb{N}.$$

Full text

Coherent diffraction imaging (CDI) using intense X-ray pulses produced by Free Electron Lasers (FELs) is a powerful technique to determine the size and structure of individual unsupported nanoparticles [1, 2, 3]. Prominent examples include biological samples such as living cells [4], viruses [5] and proteins [6], as well as aerosol nanoparticles [7, 8], faceted metal nanoparticles [9, 10], and superfluid helium nanodroplets (HNDs) [11, 12, 13, 14, 15, 16, 17, 18, 19].

A key advantage of using ultrashort X-ray pulses for CDI is the ability to capture and track ultrafast structural dynamics of single nanoparticles via pump–probe excitation schemes. Various types of photoinduced dynamics of nanoparticles have been studied, including the expansion and fragmentation of rare-gas nanoclusters [20, 21, 22], anisotropic HND surface broadening [23], melting, bubble-like expansion, and explosion of superheated metal nanoparticles [24].

In cases where the initial size and shape of each nanoparticle are precisely characterized, it is sufficient to generate only an image of the evolved nanostructure with the probe pulse. However, most types of nanoparticles exhibit some degree of variability in size, shape, and orientation, such that the nanoparticles’ initial state is not precisely known. As a result, structural models and dynamical simulations are required to interpret the recorded images. This introduces uncertainties and limits the reliability of conclusions. Ideally, at least two distinct images of the initial and final state should be recorded to obtain a genuine X-ray movie of the light-driven dynamics.

This ambition has spurred significant effort to devise and implement two-color X-ray imaging schemes. Advances in FEL technology now enable the generation of intense, independently tunable two-color X-ray pulses [25, 26, 27, 28, 29, 30]. Since the temporal delay between the two consecutive pulses is extremely short, fast switching of detectors to record the two images individually is impossible. Therefore, various methods to disentangle the two images of the same nanoparticle have been explored, including spectral filtering at the detector [31, 32] and spatial separation onto two different detectors [33]. These approaches, however, suffer from substantial photon loss in the separation process, which limits their efficiency.

Here, we introduce a computational image separation method in post-processing, taking advantage of the European XFEL’s (EuXFEL) ability to generate two co-propagating X-ray pulses with different photon energies [29], combined with a specialized pn-junction charge-coupled device (pnCCD) imaging detector available at the Small Quantum Systems scientific instrument. Its key characteristics are high photon-energy sensitivity and low readout noise [34, 35]. Sub-micrometer HNDs, produced by expanding liquid helium into vacuum through a small nozzle [19], are used as an ideal model target system. HNDs exhibit a broad size distribution and some variety of shapes, but predominantly have a perfect spherical shape [12, 36]. Two X-ray pulses, the pump and the probe, of different photon energies illuminate the same individual droplet, separated in time by up to 750 femtoseconds. The two superimposed diffraction patterns recorded within the same frame are separated by analyzing individual pixel counts. The spherical shape of most HNDs allows us to cross-validate the pixel-based color-separation approach by fitting the radial profiles of the scattering intensity with two-color Mie simulations, with excellent agreement between the two methods.

Our experimental findings, supported by molecular dynamics simulations, indicate that within the achieved resolution of the experiment, no structural damage occurs in the HNDs up to the maximum pump and probe delay (750 fs) because of their weak absorption of the pump pulse. Thus, HNDs serve as a near-static target, allowing for detailed benchmarking of our imaging method.

Systematic scattering simulations are used to quantify and test the limitations of the pixel-based color-separation method. The validity of the assignment is excellent in image regions where the average area density of photon hits on the detector remains below 0.1 detected photons per pixel. For higher photon densities, the spill-over of charges across pixel borders as well as detector noise impose limitations. However, the high reliability of our color-separation method in the sparsely illuminated regions of the diffraction images, where fine structural information is encoded, is a particular advantage towards developing more sophisticated approaches. Additionally, future experiments will benefit from the weak interaction of HNDs with X-ray pulses when highly absorbing embedded nanostructures and their X-ray or laser-induced dynamics are to be probed [13, 14, 17, 18, 19]. This pixel-based two-color X-ray imaging method therefore has the potential to become a key building block for future advanced X-ray movie techniques based on two-color XFEL pulses.

Results

Two-color diffraction images of pure HNDs were recorded for various experimental parameters, notably different intensity ratios of pump and probe pulse intensities and time delays between the two pulses, as well as varying HND sizes (Fig. Influences on the recall). A typical experimental diffraction image of a pure HND is shown in Fig. 1 (a). The image consists of a superposition of two patterns for the two colors with a nominal intensity ratio of 1 to 4 ($4.2$ to $1.9\times 10^{16}\>$W/cm2 calculated based on beamline diagnostics [37, 38]) and a pump-probe delay of 750 fs. Since the EuXFEL operates in the self-amplified spontaneous emission (SASE) mode [39, 40], the photon energies slightly vary from shot to shot (1 % of mean) around the mean value of $992\text{\,}\mathrm{eV}$ for the pump pulse and $1192\text{\,}\mathrm{eV}$ for the probe pulse. For simplicity, from now on we will call them “red” and “blue” pulses, respectively. To better visualize the image structure in the outer, sparsely illuminated regions of the detector, the radial intensity profile (black line) is added as an inset in the first quadrant of Fig. 1 (a). It is obtained by angular integration around the center of the diffraction pattern. Note that the image and the radial profile in the inset share the same radial scale.

Owing to the near-perfect spherical symmetry of HNDs and the absence of observable structural changes under the present experimental conditions, the radial intensity distributions of the measured diffraction images can be fitted by Mie’s solution of Maxwell’s equations for scattering by a hard sphere [41, 42]. Molecular dynamics simulations, as further detailed in the supplementary materials, corroborate the latter assumption: Due to the low X-ray absorption cross-section of helium at the employed photon energies [43], an initial fraction of only $0.1\%$ of helium atoms is photoionized by the pump pulse, leading to negligible structural dynamics within the first picosecond given the spatial resolution of our experiment. Therefore, we can fit the radial profiles obtained from two-color scattering patterns by the sum of two simulated profiles, assuming a perfect sphere of unknown size and arbitrary contributions of both colors (see Methods). The dashed pink line in the inset in Fig. 1 (a) is the result of the fit for the radial intensity distribution of a bichromatic field scattered off a hard sphere [44, 45] to the radial profile of the experimental pattern shown here. See Fig. Influences on the recall for more examples of such radial profiles and fits.

Fig. 1 (b) is a magnified view of the same graph, including the individual color contributions to the fit as solid red and blue lines. The sum of red and blue components, represented by the dashed pink line, matches the experimental data very well up to a scattering angle $\theta\approx$1.5\text{,}{}^{\circ}$$, where the scattering intensity approaches the noise level. The decaying fringe pattern clearly shows a beat note that results from the superposition of fringes created by the blue and red pulses, where the fringe spacing differs by the ratio of photon energies, i.e., a factor of 1.2. The good agreement between the experimental data and the fit allows us to accurately quantify the HND’s size and the absolute contributions of the two colors to the total diffraction pattern. We also tested the fitting of the time-evolved profiles with smoothed-surface model shapes. However, the results were consistent with hard edges in the evolved droplet within the resolution limit of our experiment.

While the comparison to Mie simulations allows us to separate the two color components of the radial intensity distributions in the given case of spherical targets, it is desirable to develop a generally applicable approach for separating the two-color two-dimensional diffraction images. Therefore, we developed a pixel-based separation method based on the analysis of individual pixel counts and pattern identification. The amount of charge on the detector created by the photoelectric effect scales with the photon energy; the pnCCD records these photon-induced signals in analog-to-digital units (ADU). By decomposing each pixel ADU value into multiples of the corresponding charges, one can assign an illuminated pixel to a unique combination of those photons. This method relies on the detector’s high sensitivity to photon energy and low readout noise. An intrinsic challenge is the spill-over of charges across pixel boundaries, creating clusters of lit pixels. Further details are described in the Methods.

Fig. 2 (a) and (b) shows the result for the separation of the original diffraction image, Fig. 1 (a), into its red and blue components, respectively. The separated components’ radial intensity profiles are presented in Fig. 2 (c) and (d). Clearly, the number of illuminated pixels is significantly reduced in the color-separated images. ADU values are discarded if they are identified as noise or if their assignment to integer multiples of the red or blue elementary ADU values is ambiguous. See Methods for details. To quantify the level of ambiguity, we introduce a threshold parameter for the confidence in assigning an ADU value to either noise or a particular combination of red and blue photons. In this work, we choose a threshold of $50\,\%$ for the most likely photon combination (further details in Methods). For this value, the majority of identified pixel groups correspond to single-photon events, and $\sim 10$ % of the color-separated photon events are attributed to two- or three-photon hits.

This identification of color contributions by pixel count is most effective in the outer areas of the image, i.e., at scattering angles $\theta\gtrsim$0.8\text{,}{}^{\circ}$$ where lit pixels are sparse. At smaller angles, the probability for larger lit pixel clusters is higher, and pixel assignment becomes ambiguous. Thus, close to the center, intensity profiles generated from the color-separated images drop off, showing a significant signal loss. However, at high scattering angles, the sum of intensities for both red and blue pixel-based separated patterns presented in Fig. 2 (a) and (b) amounts to 87% of the intensity for the original image in Fig. 1 (a). Further examples of similarly successful single-color pattern retrievals are given in Fig. Influences on the recall.

To test the limitations of this method and to provide boundaries within which it is reliable, we systematically generate synthetic diffraction images (see Methods) and color-separate them using the same algorithm used for processing the experimental images. This procedure allows us to benchmark the pixel-based image separation quantitatively, since, unlike real experimental data, the true photon distribution is known. To determine a threshold value for reliable pixel-based color assignment, we compute recall and precision, which are standard performance metrics for classification tasks [46]. In our case, recall quantifies the ratio between the number of correctly assigned photons to the total number of photons. Precision measures the ratio of correctly assigned photons to all assigned photons, i.e., after discarding the ambiguous pixels. Fig. 3 (a) displays the recall as a function of scattering angle, calculated from a set of eleven synthetically generated patterns with radial profiles identical to those of the measured diffraction image in Fig. 1. The steep rise at an average photon density of $0.1$ detected photons per pixel indicates that efficient separation is feasible and reliable up to this threshold.

Discussion

Our results demonstrate the feasibility of separating the initial-state diffraction pattern and the time-evolved diffraction pattern of the same nanoscale target obtained from the superimposed image generated by two X-ray pulses of different photon energies via computational methods. Although two-color Mie fits are only applicable to spherical particles, the high-quality separation achieved with Mie fitting in this work is a notable result on its own. The fits unambiguously yield the accurate droplet size and the absolute intensities of both X-ray colors at the position of the HND. In this work, we use the method for benchmarking the pixel-based color assignment. Still, the valuable information gained from the Mie fits could be included in more sophisticated combined approaches or for studying, e.g., ultrafast changes of optical properties, where the droplet’s refractive index is introduced as a fit parameter.

In contrast, pixel-based color separation is independent of target shape, and can be applied directly either as a stand-alone method or as a key step in the analysis pipeline in future two-color X-ray pump-probe experiments. Under the experimental conditions of this study, the separation works best for scattering angles of $0.8\text{\,}{}^{\circ}$ and higher, or equivalently, for areas of the detector detecting on average 0.1 photons per pixel and less.

A deeper discussion of our tests with simulated diffraction patterns helps to deduce guidelines for generalizing to other cases. The pixel-based assignment is found to quickly lose its applicability in regions with high photon counts, as detected at low scattering angles in our diffraction patterns; the recall, shown in Fig. 3 (a), exhibits a sharp increase at approximately $0.8\text{\,}{}^{\circ}$. The origin of the clear step is the exponentially dropping intensity of small-angle diffraction from a homogeneous particle. This cut-off angle for unambiguous pixel identification depends in practice on photon flux per X-ray pulse, the nanospecimen’s size and shape, and the detector’s distance from the interaction region.

The recall reaches $90\,\%$ in the outer regions, where it is no longer constrained by large contiguous illuminated areas or high photon counts per pixel. This trend is further supported by the local photon density indicated on the top axis in Fig. 3 (a). Specifically, when the photon density falls below one photon per nine pixels – meaning that, on average, no neighboring pixels are illuminated – the recall begins to rise. As detailed further in the SM, this local photon density represents a universal parameter governing the effectiveness of the separation method.

The level of the recall plateau depends on the choice of the certainty threshold. Increasing the required certainty, i.e., enhancing the selectivity to ensure that assigned events are correct, leads to a decrease in recall, as shown in Fig. 3 (b). In this figure, recall is evaluated at scattering angles $>$2\text{,}{}^{\circ}$$ to assess the performance of the separation method irrespective of the issue of overlapping pixel clusters. In this regime, the recall is influenced primarily by the spread of the charge over multiple pixels and the intrinsic width of the resulting electron signal, which in turn determines the spread in ADU values. The total number of photons in Fig. 3 (b) (on the right axis in yellow) behaves similarly to the recall but deviates slightly, because, unlike the recall, it is calculated for the whole image. The precision, i.e., the fraction of correctly assigned photons among all assigned photons, is also calculated over the entire pattern and increases almost linearly from $70\,\%$ to $95\,\%$.

For the analysis in this work, we choose a certainty threshold of $50\,\%$. This value strikes a balance between recall and precision, and it marks the point at which recall starts to decrease. One might want to decide differently in other experiments, e.g., increase the probability for correct assignment of events by setting a higher threshold. For bright small-angle coherent diffraction patterns, as presented here, the accumulation of signal in contiguous central regions will always pose a challenge for pixel-based separation. In other types of diffraction experiments, e.g., containing Bragg peaks, it might be possible to also assign higher photon counts to pixels, as the areas with high signal are spatially confined and thus not overlapping on the detector for appropriate choices of photon energies.

The inability to separate the two patterns towards the center of the image using the pixel-based separation method results in the loss of information at low momentum transfer, which corresponds to large spatial scales in the real-space object. To disentangle overlapping contributions in this central region, additional prior knowledge about the sample is required, such as in our case, the assumption of spherical shape and negligible change of the HND by the first X-ray pulse, allowing us to fit with Mie scattering solutions. This will certainly pose a challenge when dynamic processes are expected to occur across extended regions of the sample, e.g., in studies of ultrafast changes in the scattering cross section of materials near resonant photon energies [47, 48]. However, for smaller-scale changes in HNDs, such as surface softening or expansion [49, 23], the higher scattering angles provide the most relevant information. In this regime, the pixel-based separation method performs particularly well.

One pathway to improve experimental conditions for pixel-based color separation is to increase the spatial resolution of the detector; either the pixel density could be enhanced or the distance between the interaction region and the detector could be increased (see Fig. Pixel-based Cluster Assignemt). Then, the same diffraction image will spread out over more pixels, enlarging the area covered by sparse events. Additionally, the method could be further improved by optimizing the photon energy selectivity for the two X-ray pulses, ensuring that one-, two-, and potentially three-photon events are well separated across all combinations of two-color photons (see Fig. Influences on the recall for various combinations). In this context, improving the signal-to-noise ratio of the detector will be crucial, too.

In future experiments, the presented method has the potential to become a key building block for the analysis of more sophisticated two-color CDI approaches. Photon energies could be tuned across electronic resonances and inner-shell absorption edges, where ultrafast electron dynamics on the femtosecond timescale have recently been shown to significantly and transiently alter X-ray responses  [48] to resolve such intricate electron dynamics in time and space. Another way to exploit resonances and absorption edges would be to locally induce high degrees of ionization within a composite nanotarget, such as a helium nanodroplet with an embedded dopant nanostructure, to drive structural dynamics in a controlled way. Dynamics could also be induced by an additional intense optical laser pulse. In this way, it would be possible to image melting dynamics and nanoplasma expansion [20, 21, 22, 23, 24] while simultaneously capturing the particle’s initial state with a preceding X-ray pulse that changes the dynamics in a negligible way. Having a reliable and versatile tool at hand to separate the two images without the need for any model assumptions for the initial state paves the way for further approaches to recording genuine ultrafast X-ray movies with unprecedented resolution in time and space.

Methods

Experimental Setup

The experiment was carried out at the SASE3 soft X-ray branch of the EuXFEL [50] using the Nano-sized Quantum System (NQS) endstation at the Small Quantum Systems (SQS) instrument [51]. The EuXFEL was operated at an electron energy of $16.3\text{\,}\mathrm{GeV}$ with a pulse repetition rate of $10\text{\,}\mathrm{Hz}$. Pairs of X-ray pulses with mean photon energies of $992\text{\,}\mathrm{eV}$ and $1192\text{\,}\mathrm{eV}$ were generated by simultaneously operating undulator sections U1 and U2 [29], see the schematic representation of the setup in Fig. 4. A magnetic chicane positioned between undulator sections U1 and U2 enabled control of the time delay between the two X-ray pulses (FWHM durations of 15–20 fs), allowing delays in the range of 50 to 750 fs. The two X-ray beams were focused by a Kirkpatrick–Baez mirror system to two spots with diameters of $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (FWHM), separated by approximately $60\text{\,}\mathrm{mm}$ around the interaction point [29, 52]. This displacement arises from the distinct source points of the two X-ray beams in the U1 and U2 undulator sections. To ensure approximately equal intensity ratios of the two X-ray pulses across the interaction region, the HND jet was aligned to intersect the X-ray beams at right angles midway between the two focus spots. Thus, the effective size of the X-ray spot is estimated to about $14\text{\,}\mathrm{\SIUnitSymbolMicro m}$ FWHM.

Due to slight fluctuations of the HND positions within the interaction volume, the X-ray intensity ratio of the two-color pulses in the diffraction image varied strongly from shot to shot. Many images even show only one color, as determined from the analysis of pixel values. In Fig. 1, for example, the nominal energies of the red and blue pulses were $0.16$ and $0.73\text{\,}\mathrm{mJ}$, respectively, as measured with a gas monitor in the tunnel, taking into account the beamline transmission of $0.6$. This corresponds to a nominal intensity ratio of $1:4.5$. In contrast, analysis of areas in the pixel-based separated diffraction images where recall is high revealed an intensity ratio of $2:3$, indicating that the HND was located closer to the red focus than to the blue focus in this specific event; see Fig. Influences on the recall.

The HND jet was produced by an Even-Lavie-type cryogenic pulsed valve [53] operated using ultrapure helium gas (6.0) at a temperature of $T_{1}=$5 or $T_{2}=$10\text{\,}\mathrm{K} and at $p_{1}=$50 or $p_{2}=$80\text{\,}\mathrm{bar} stagnation pressure in two series of acquisitions. The resulting mean HND radii are estimated as $\overline{R_{1}}=$370\text{,}\mathrm{nm} and $\overline{R_{2}}=$297\text{\,}\mathrm{nm}, see Fig. Influences on the recall for the whole size distributions. Under these conditions, the average rates for hitting an HND and detecting a bright diffraction pattern were $0.3\,\%$ and $3.2\,\%$ per pulse, respectively. The rate of detecting diffraction images of spherical HNDs illuminated by two X-ray pulses with comparable intensities (intensity ratio $\leq 3$) was 0.01 % and 0.26 %, corresponding to a total of $39$ and $93$ bright images for the two respective data sets.

A large-area pnCCD detector (PNSensor GmbH) [35] was used for recording the two-color diffraction images. This detector is capable of counting single photons while offering a high dynamic range and low readout noise. This unique combination of detector properties is crucial for the effectiveness of the pixel-based separation method described in the following section. The pnCCD detector comprises two movable modules that can be separated to create a central gap, allowing the X-ray beam to pass through. The distance between the detector plane and the interaction region is adjustable and was set to 340 mm in the present experiments.

Pixel-based Separation of Images

The pixel-based color-separation method exploits the pnCCD detector’s ability to function as an X-ray imaging spectrometer, offering an energy resolution of $145$ eV (FWHM) at a photon energy of $1.4$ keV at a detector gain of $1/64$  [35]. Before applying the separation method to individual CDI frames, several correction procedures are performed, including background subtraction, masking of defective pixels, noise gating, common mode correction, corrections for variable gain values, and charge transfer efficiency (see SM for a detailed description). These corrections were applied to a set of the 25 brightest two-color diffraction images, and the resulting average distribution of analog-to-digital unit (ADU) values is shown as the dark green line in Fig. 5 (a). As seen, the distribution is broadened due to charge splitting of a single photon event across multiple detector pixels. By applying the split-event correction procedure (detailed below and in  [54]), it becomes possible to distinguish single-, double-, and multiple-photon events, as illustrated by the black line in Fig. 5 (a). Clearly, the histogram of split-event corrected results is dominated by noise for ADU values up to approximately 200. The double-peak structure around 300 ADU corresponds to single-photon events from each color, while the broad maximum between 500 and 770 ADU arises from pixels where two-photon events generate overlapping charge clouds, resulting in ADU values approximately twice those of single-photon events. As the number of photons of each color increases within the same detector area, the ADU peaks in the histogram broaden roughly in proportion to the square root of the total ADU intensity (see Fig. Influences on the recall (b)), which in turn diminishes the ability to separate the two colors accurately. Simultaneously, averaged over the whole diffraction image, events involving two or more photons contributing to a single pixel count are approximately ten times less probable than single-photon events. To validate the color assignment indicated by the black line in Fig. 5 for the two-color mode, reference data were collected for each color individually under comparable experimental conditions, represented by the red and blue lines in the figure.

A close-up view of the ADU distribution up to 500 for the single-shot two-color split-event corrected diffraction image, shown in Fig. 1, reveals a pronounced double-peak structure in the range of 200–400 ADU (black line in Fig. 5 (b)). This structure is due to single-photon events of both the red and the blue colors, reflecting the photon energy distributions of the colors, which are distorted (or superimposed) by the pnCCD detector’s instrument function. The assignment of the peaks in the ADU distributions to each color is confirmed by the comparing to ADU distributions from single-color diffraction images. This step is essential, as it determines the mean ADU values corresponding to single-photon detection for each individual color, which are subsequently used in the pixel-based separation method as reference values.

The pixel-based separation procedure of the measured total diffraction image into two individual components produced by the red and blue X-ray pulses is illustrated in Fig. 6. Panel (a) provides a close-up view of the marked red area in Fig. 1 (a). Illuminated pixels are shown here in a gradient of green, with darker pixels corresponding to higher ADU values, indicating enhanced charge generation within the pixel. As the first step, our routine identifies all connected exposed pixels and assigns them to distinct clusters (see Fig. 6 (b)). Since a photon-generated charge cloud may distribute charge among up to four neighboring pixels, depending on its point of origin (see Fig. Influences on the recall,  [54]), we identify these events in the second step and categorize them according to standard patterns (see light yellow pixels in Fig. 6 (b)). If more than four neighboring pixels are connected, they cannot be assigned to standard patterns and are marked as dark yellow pixels in Fig. 6 (b).

In the next step, the entire cluster is replaced by a pixel whose value corresponds to the sum of ADUs within the cluster. This pixel is positioned at the location closest to the barycenter of its corresponding cluster. Each pixel in the new map is assigned to detector noise (green), a single red or blue photon, or a combination of red and blue photons (see Fig. 6). The color gradient reflects the total accumulated charge, representing the number of photons per color. Due to the overlap of the finite-width one-photon ADU distributions (see Fig. 5 (b)), photon combinations can lead to ambiguous assignments. While separation is still feasible for single or double photons, higher photon counts make unambiguous individual color resolution increasingly difficult.

To reduce the risk of assigning incorrect combinations of red and blue photons, we apply a certainty threshold in the pixel-based separation method. This threshold defines a minimum confidence level required for photon assignment at each pixel. If a pixel’s total ADU value falls within a region of ambiguity, where the likelihood of a correct match is below threshold, the cluster is excluded from further analysis by setting its value to zero, as illustrated by the gray areas in Fig. 6(c).

To compute the certainty threshold, we evaluate the probability that a given ADU value originates from a combination of photons, e.g., two red photons or one red and one blue photon. The combination with the highest probability is then assigned to the pixel. Each photon combination is modeled as a Gaussian distribution the width of which is determined from the averaged histogram for the brightest 25 shots (see Fig. Influences on the recall). The height and exact position of each Gaussian are obtained from fitting the histogram of for each individually image. This procedure accounts for shot-to-shot intensity fluctuations caused by fluctuations of the position of the X-ray beams and the sample jet as well as jitter of the central wavelength inherent to the SASE mode. The final certainty is then calculated as the relative contribution of the Gaussian fit with the highest amplitude (i.e., the assigned photon combination) to the sum of all contributing photon combinations at that ADU value.

Fig. 7 shows the individual contributions of different combinations of photons, along with the resulting certainty level indicated by a black line. For illustrative purposes, the same intensities as in the experimental image are chosen. As an example, we focus on the left wing of the “one red photon peak”, around 250 ADU. As seen, the “one red photon event” dominates the likelihood distribution, contributing nearly unity. Therefore, assigning a cluster with an ADU value of 250 to a single red photon is unambiguous. In contrast, around 320 ADU, where the probabilities for red and blue photons are approximately equal and each contributes less than one, the assignment becomes less certain. Clusters with ADU values in this ambiguous range cannot be reliably attributed to any specific photon color and should be excluded from further analysis when a high certainty threshold is required. At higher ADU values, near the “one blue photon peak” at 349 ADU, the likelihood is again dominated by a single contribution, making the assignment to a single blue photon event reliable.

Synthetic Image Generation

To evaluate the pixel-based separation method and to guide the selection of an appropriate certainty threshold, we generate synthetic image data that closely replicate the experimental results. We assume a static and spherical target and the experimental conditions described above.

The intensities of the red and blue images are determined from the components of the Mie fits to the radial intensity profiles of the experimental images (see next section). For both colors, the corresponding ADU values are fitted by Gaussian distributions using data from the 25 brightest images to infer the mean value and the spread. To incorporate photon statistics, the mean ADU values are used to assign the number of photons per pixel randomly according to a Poisson distribution. To account for detector noise, the ADU values for each photon are then drawn from Gaussian distributions based on their respective spreads. As explained earlier, each photon generates an electron cloud that can spread across up to four neighboring pixels, creating characteristic patterns (see Fig. Influences on the recall). We use the experimentally determined distribution of these pixel patterns to randomly assign a pattern to every photon event (Table Influences on the recall). For each pattern, the total ADU value is distributed evenly across the pattern’s pixels, with the original pixel where the photon event was placed according to Poisson statistics receiving the highest value. The total diffraction image is obtained by summing all single-photon events of both colors and adding dark noise, as observed in the experimental images. In this process, stray light signal seen by the detector, as well as detector artifacts such as hot pixels and broken rows, are neglected because these are typically masked and excluded from analysis of the experimental data.

Fig. 8 presents a direct comparison between an experimental pattern and a corresponding synthetic image. The synthetic image (b) closely reproduces the features of the original experimental image (a). The radial intensity profiles of both images, shown in panel (c), also compare favorably, demonstrating that the synthetic approach provides a reliable approximation of the experimental data.

Separation of Images by Mie Solutions

Owing to the nearly ideal spherical shape of HNDs and the resulting radial symmetry in their diffraction patterns, an alternative method for separating the two color components of the diffraction images can be employed. The scattering of light from a spherical HND can be accurately described by Mie theory [41, 42]. Here, we use Mie solutions to fit the radial intensity distributions of the measured diffraction images. Specifically, we employ the Python library nux-utils which is based on C++ code [45, 44]. The optical properties of liquid helium are used [55]. Since the measured images result from the superpositions of two independent scattering events caused by red and blue X-ray pulses interacting with the same particle, we use three fit parameters: two for the intensities of each pulse and one for the size of the HND. The center of each individual diffraction image depends on the position of the HND relative to the wavefront of the incident X-ray pulse. To account for shot-to-shot jitter, the center of each image is determined by a fitting procedure before further analysis. Since Mie solutions assume perfectly spherical particles, a preselection of radially symmetric diffraction patterns is required. An example of the fit result is shown in Fig. 1 (b), where the black line represents the radial profile of the measured image. The red and blue lines correspond to the two components of the fit model, with their sum shown as a purple line. In the case of scattering from a spherical object under the condition $qR\gg 1$, which holds here with $qR=16$ for $R=300$ nm and $\theta=0.5^{\circ}$, the scattered intensity falls off steeply with the magnitude of the scattering vector, following the relation $I(q)\propto q^{-4}$, where $q=\frac{2\pi}{\lambda}\sin{\theta}$ denotes the magnitude of the scattering vector, $\lambda$ is the photon wavelength, and $\theta$ is the scattering angle between the incoming beam and the detected light on the detector. Therefore, the residual is scaled by $q^{4}$ to equally weight data points across the entire $q$-range considered in the fit. Additionally, the Fourier spectra of both the fit and the data are incorporated into the error function. This enhances the robustness of the fit procedure and improves the accuracy of HND size determination, particularly in the presence of noise.

Test for Dynamics by Mie Fits

To cross-check our assumption of static HNDs, we perform additional Mie fits. In these cases, we add an additional degree of freedom that mimics typical dynamics: The first possibility would be a softening of the droplet surface that reflects as a steeper drop of the radial profile toward larger scattering angles in the diffraction image [49]. This effect is modeled by weighting the simulated probe Mie profile with a Gaussian function peaked at zero. The second possibility would be an increase in droplet size. This can be easily accounted for by lifting the restriction that both Mie profiles must be for the same size of the particle. Both cases were tested but no statistically relevant differences were found for any delay ($50$ and $750$ fs). We take this result as a confirmation of the lack of structural dynamics of the HNDs under the present experimental conditions, in agreement with the presented molecular dynamics simulations, see below. An upper limit for changes of HND size is given by the spatial resolution of the detector, $\pi/q=$5\text{,}\mathrm{nm}$$, determined by the detector geometry and photon energy [56]. However, due to the relatively low pulse energies in the two-color mode, the largest scattering angle up to which the signal-to-noise ratio is sufficient is significantly reduced to $\theta=1.5^{\circ}$, resulting in an effective resolution of $20\text{\,}\mathrm{nm}$.

Molecular Dynamics Simulations

To gain insight into the ionization dynamics of HNDs irradiated by X-ray pulses, we performed classical molecular dynamics (MD) simulations. A detailed description of the method can be found in Refs. [57, 58, 59, 60]. Due to computational constraints, the simulated HNDs were significantly smaller than those used in the experiment, ranging from $10^{4}$ He atoms ($R=4.3$ nm) to $8\times 10^{4}$ He atoms ($R=8.6$ nm).

Given the experimental photon fluence of approximately $2\times 10^{17}$ photons/cm2 per pulse and the photoionization cross section of helium at 1 keV of $5.6\times 10^{-22}$ cm2  [43], we estimate that only about 0.01 % of the He atoms in the HNDs are photoionized. At such low degree of photoionization, no measurable structural change of the HNDs is expected within the timescale of the experiment (750 fs). To explore the intensity regime required for inducing detectable structural dynamics, the MD simulations were performed at significantly higher fluences.

As an example, 2 % of the He atoms were photoionized by 1 keV photons, assuming a uniform probability distribution across the HND. Photoionization was modeled as an instantaneous, ad hoc process at $t=0$, and the MD trajectories were propagated up to $t=750$ fs. Fig. 9 (a) shows the radial distributions of the number density of electrons bound to He atoms, $\mathrm{He}^{+}$ ions, and quasi-free electrons, plotted up to a distance of 1.25 times the HND radius ($R=8.6$ nm). The radial density function of the HND at $t=0$ is indicated by the purple curve (Minor fluctuations around the constant value of the bulk density originate from a partition of the HND into spherical shells of finite thickness). The highest $\mathrm{He}^{+}$ ion density (red curve) is observed near the droplet center. It is consistent with previous simulations for Xe clusters [57, 58] which suggest that secondary electron impact ionization, triggered by initial photoionizations, occurs preferentially in the cluster interior. The curves for $\mathrm{e}_{\mathrm{free}}$ (pink) and $\mathrm{He}^{+}$ (red) almost perfectly overlap, indicating net charge neutrality across the interior and in the vicinity of the HND. The droplet periphery consists mainly of neutral He atoms (teal curve). The resulting $\rho_{e}(r)$ curve (orange) features a dent inside the HND, which spreads into the surrounding outer region of the HND with time as the HND ejects He atoms, ions, and electrons. The dynamics of X-ray photoionized HNDs will be further studied by MD simulations in view of future pump-probe imaging experiments.

The radial distribution of scattered photons shown in Fig. 9 (b) is computed from the distribution $\rho_{e}(r)$ by Fourier transformation. As the particle size is smaller than the typical cluster sizes studied in this experiment due to computational limitations, the photon energy is scaled accordingly by a factor $0.05$ so that the diffraction fringes appear at the same range of scattering angles as in the experiment. Most prominently, the maxima of the radial profile at $t=750$ fs are slightly shifted to smaller scattering angles compared to the profile at $t=0$, and their amplitudes drop off faster towards larger angles. This is mainly due to the softening of the surface edge of the HND as the electron density expands into the vacuum.

Acknowledgements

L.H., B.B., A.L., D.R., and M.Mu. thank the Danish Agency for Science and Higher Education for support through the instrument center DanScatt. A.H. gratefully acknowledges financial support from the Basque Government (project IT1584-22), as well as computational resources and manpower support provided by the DIPC computation center. The research leading to these results was supported by the COST Action CA21101 “Confined Molecular Systems: From a New Generation of Materials to the Stars (COSY)” and by SNSF project grant 10003697. F.S. acknowledges funding by the DFG of the RTG DynCAM (RTG2717). S. K. is grateful for the support from the Department of Science and Technology (India-DESY and Core Research Grant) as well as Ministry of Education (Institute of Eminence Scheme) and high-risk-high-reward programme of IIT Madras. T.F. and C.P. acknowledge funding by the DFG via CRC 1477 “Light-Matter Interactions at Interfaces” (project ID: 441234705). The authors acknowledge EuXFEL in Schenefeld, Germany, for provision of X-ray free-electron laser beamtime at SQS SASE3 (proposal 2857) and would like to thank the staff for their assistance.

Author contributions statement

M.Me., Y.O., M.Mu. conceived the experiment. L.H., Y.O., B.B., T.M.B., A.C., S.Du., A.D.F., S.D., S.K., B.K., B.L., T.M., S.M., C.M., T.P., B.S., K.S., F.S., R.M.P.T., P.T., S.U., D.R., M.Mu. conducted the experiment. L.H., Y.O., A.L., A.C., R.H., A.H., M.Mu. analyzed the results. L.H., Y.O., A.L., K.K., M.Me., D.R., M.Mu. prepared the manuscript, with input from all authors.

Additional information

Data recorded for the experiment at the EuXFEL are available at doi:10.22003/XFEL.EU-DATA-002857-00.

The authors declare no competing interests.

Supplemental Material

Detector Corrections

We thoroughly processed the scattering images before applying the pixel-based separation method. How the detector works is described in detail in [35]. Below, the correction steps are listed and briefly described. The first two steps are applied to all data; all further steps are updated during the experiment:

Charge transfer efficiency map: As the charge is read out at one of the two lateral edges, every pixel’s charge needs to be transported to the outside. In doing this, some charge is lost during the transport process. Using the known charge transfer efficiency, the signal of the more central columns is increased accordingly. 2. 2.

Gain map: Every detector row has an individual gain. Before the experiment, we created a homogeneous signal through fluorescence at a signal level where, on average, only single photons are sparsely distributed on the detector. Collecting row-wise all events with only a single pixel activated creates an ADU distribution for every row. The gain map multiplies every row so that the average ADU value of a single photon is the same for all rows. 3. 3.

Background image: Regularly, we took background runs, meaning no scattering target was introduced into the interaction region. We take 20000 images and delete every pixel’s brightest and darkest ten values. Afterwards, the remaining values are averaged. This background image is then subtracted from the data. 4. 4.

Mask: As the detector already has some damaged areas, we mask some parts. We use the mean and standard deviation of every pixel to determine which pixels to mask. Generally, we mask generously by choosing strict values and excluding surrounding pixels, as we want to support pixel-based separation as much as possible. We exclude a pixel if

1. (a)

the mean value is above a threshold (hot pixel). 2. (b)

the standard deviation is too big (flickering pixel) or too small (stuck pixel). 3. (c)

the ratio between mean and standard deviation is too big or too small. 5. 5.

Common-mode correction. During the readout process of a single image, the electronics are subjected to noise. This results in a different baseline for every row of pixels. As this is constant for the entire row, it can be removed afterwards by subtracting the mode of each row, provided the signal does not cover most of the row. 6. 6.

Noise rejection: We create a noise map from the background run. After background subtraction and common-mode correction of each image, we calculate the standard deviation of every pixel. If a pixel’s value in the data is below twice the standard deviation, we mark it as detector noise and remove it. 7. 7.

Detector plate positioning. The detector halves are movable in the detector plane and were moved to reduce stray light on the detector. Using spherical hits as a reference, we add pixels in the center of the image matrix and align the plates laterally by adding blank pixels at the edges. 8. 8.

Center finding. The exact center of the scattering image fluctuates from shot to shot. It depends on the position of the HND in the pulse’s bent wavefront. We fit a center position to every image using the point symmetry of the scattering image.

Pixel-based Cluster Assignemt

The processed raw images are passed through a series of steps to classify the pulse components of different parts of the image. First, the clusters of bright pixels in each image are labeled, noting the total brightness and their position. The relative intensity of the red and blue pulses is estimated. Finally, each cluster is assigned photon components based on its ADU value.

Split events, where photons partly activate more than one neighboring pixel, comprise about a third of all single photon hits in flat illumination. The image detection of the clusters was done in two steps. The one-photon patterns can only create specific 1-4 pixel patterns. These were first labeled by convolving the image with matrices chosen to capture each shape. Afterwards, the remaining clusters were identified using a standard connected component analysis algorithm in SciPy.

Because each diffraction image can result from scattering off droplets anywhere between the two foci of the pulses, the resulting images contain different ratios of red and blue photons. Using the histogram of the cluster ADU values, similar to Fig. 5, the relative sizes of the one-photon and noise peaks are found. For these fits on individual images, the heights and the positions of the peaks are varied. The other parameters determining their shape were assumed to be the same for all images.

We now consider the signal in each cluster as resulting from a single-photon or multi-photon event, with a mean value of a photon $\mu$ and a standard deviation $\sigma$. Under the assumption that the signal from each pump or probe photon is normally distributed ($N$), the resulting distribution $S_{\text{n, m}}$ of a given multiphoton event can be estimated.

[TABLE]

For each cluster with its associated ADU value $x_{i}$, we calculate the $S_{n,m}(x_{i})$ using the image-specific one-peak heights and widths found by fitting Gaussians to the ADU histogram. The highest of these $S_{n,m}(x_{i})$ is assumed to be the origin of the cluster. Summing all these different $S_{n,m}(x_{i})$ gives a measure of the total value attributed to this cluster, from all considered combinations. A confidence value for the found combination is estimated by dividing the highest $S_{n,m}(x_{i})$ by the total. The reconstructed components of all clusters with a confidence above a set threshold are then used to construct a red-only and a blue-only image.