1. INTRODUCTION

The three-dimensional (3D) structure of materials is often related to their properties and functionalities. The understanding of such relations at the sub-µm length scale is crucial for many applications, e.g., for controlling the tribological properties of micro-electro-mechanical systems [1] or for studying the electrochemical processes of corrosion and percolation that limit the lifetime of rechargeable batteries [2,3]. Short-wavelength probes are needed to achieve such high resolution. Among them, extreme-ultraviolet (EUV) and x-ray photons can be tuned to specific absorption edges, providing chemical selectivity and sensitivity to the local environment [4–6]. These photons have been utilized to obtain sub-µm resolution images of various subjects, including non-periodic magnetic domains [7,8], artificial nanostructures [9,10], and biological specimens [11,12]. Furthermore, the high brightness of EUV/x-ray free-electron laser (FEL) sources has enabled the 2D single-shot approach [13,14], which is emerging as a crucial tool for probing samples unable to withstand high radiation doses [15–17], and for studying irreversible dynamical processes [18,19]. The conventional way of obtaining 3D structural information relies on the acquisition of a series of views by varying the sample tilt relative to the illumination direction. In this manner, it is possible to obtain a high-resolution and high-contrast x-ray tomographic reconstruction of the investigated object. This powerful approach is to date used as a standard in many contexts, ranging from material science research [20,21] to routine clinical diagnostics [11,22]. Tomographic approaches based on a serial replacement of nearly identical targets [23,24] and complex post-processing based on the classification of the object’s views [25] permit FEL-based 3D imaging at the expense of production of a large number of indistinguishable samples, and time-consuming calculations. Alternatively, Geilhufe et al. [26] proposed to combine single-frame information of Fourier-transform x-ray holography with numerical calculations of wavefront propagation to adjust the focus at different depths. Although this approach provides valuable 3D images, a careful balance between the scattering efficiency of the observed reference and the object is necessary to retrieve high-contrast morphological information. Up until now, the combination of single-shot and 3D imaging has not been reported, although potentially viable at FELs, as demonstrated in the present paper.

The strategy employed here to achieve 3D information in a single-shot fashion relies on the simultaneous determination of multiple 2D views of the same object. This is accomplished by illuminating the sample with time-coincident beams originating from different viewing angles [27,28]. A scheme for the simultaneous collection of x-ray images was theoretically proposed by Schmidt et al. [29], while the first experimental implementation was reported by Duarte et al. [30] using an EUV table-top source. In the latter work, a 3D visualization algorithm calculates disparity maps from two distinct EUV coherent diffraction imaging (CDI) views, producing a 3D point cloud reconstruction of the object. To obtain accurate disparity maps, this approach requires small angles between the two views, which limits its applicability to complex objects with obscured areas. Moreover, the reconstruction algorithm relies on geometrical projections, lacking the ability to convey any physical properties of the sample being investigated.

In this work, we build upon the research conducted by Duarte et al. [30] and present an extension that showcases single-shot 3D imaging at FEL sources. Specifically, we implemented a split-recombination setup to illuminate the sample from different angles and developed a reconstruction algorithm that incorporates physical information about the sample. This enables a larger separation angle between the two views, resulting in a more accurate determination of the 3D structure. The robustness of the algorithm allows us to reconstruct different 3D structures even when the object is exposed to a single FEL shot. Furthermore, the extension to the time domain can be straightforwardly envisioned with the introduction of an optical pulse for pump–probe experiments. The extension to higher photon energies is also feasible by relying on multilayer coatings [31] or transmissive x-ray diffraction optics [32,33], which permits the handling of multiple replicas of FEL pulses simultaneously impinging on the sample from more than two different view angles [34]. This can greatly improve the capabilities of the proposed approach.

2. EXPERIMENT

The experiment was conducted at the DiProI beamline situated at the FERMI FEL facility in Trieste, Italy [35,36]. For this purpose, we utilized a system previously employed in FEL-based transient grating experiments that has been comprehensively described elsewhere [37,38]. This setup permits the generation of two FEL pulses and their recombination at the sample, precisely setting the crossing angle ($\theta = 40^\circ$) and ensuring temporal coincidence. The spot size at the sample for each crossed FEL beam was approximately $60 \times 120\;\unicode{x00B5} {{\rm m}^2}$ full width at half-maximum (FWHM). The FEL wavelength was 20.8 nm, and the pulse duration was 50 fs (FWHM). The typical intensity at the sample was approximately 1.1 µJ per shot, with circular polarization. We investigated various types of 3D structures, including the five-fold helicoidal structure shown in Fig. 1(b). The sample was positioned such that its surface was perpendicular to one of the two FEL beams. Adjacent to each sample were properly oriented holographic pinholes and an extended HERALDO reference [39,40]. CCD detectors were placed on-axis relative to each of the two FEL beams at a distance of 100 mm from the sample. Hereafter we will refer to these two CCDs as CCD 40° and CCD 0°, where the latter is the one placed on-axis with the FEL beam orthogonal to the sample surface. Further details on the setup and the composition of the samples can be found in Supplement 1.

 

Fig. 1. (a), (c) Diffraction patterns from the sample in (b) collected, respectively, at 0° and 40° view angles; the image in (b) was recorded by scanning electron microscopy (SEM). (d), (e) Holographic reconstruction of the sample (inside the red circle) and, at the same time, a glimpse of the shape of the support (red circle plus red rectangle). (f), (g) HERALDO reconstruction used as the first guess in the iterative phase retrieval algorithm, which led to the CDI reconstructed images in (h) and (i).

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3. METHODS AND EXPERIMENTAL RESULTS

Figures 1(a) and 1(c) present examples of diffraction patterns that have undergone post-processing corrections (refer to Section 2 of Supplement 1). These patterns were captured by CCD 0° and CCD 40°, respectively, for the sample illustrated in Fig. 1(b). For each pair of such simultaneously collected diffraction patterns, we extract two holographic projections by exploiting the reference pinholes [Figs. 1(d) and 1(e)] and the HERALDO line [Figs. 1(f) and 1(g)]. These holographic views are further refined using CDI reconstruction algorithms [41,42]. During this process, the holograms serve as input for the phase retrieval code and to define the support, which sets the spatial position of the unknown object during the CDI reconstruction of the diffracted phase. The 2D images obtained via CDI are presented in Figs. 1(h) and 1(i), for normal and 40° tilted views. To determine an absolute value of the sample transmission, for each iteration of the CDI reconstruction, the 2D images are renormalized by the average value of the clear aperture in the sample [brighter regions in Figs. 1(h) and 1(i)], i.e., the area where photons are not absorbed by the sample. The normal incidence view clearly resolves the five-fold geometry of the helicoidal structure. In contrast, the tilted view displays almost fully open structures, where the impinging photons do not interact with the sample (bottom left part of the image), as well as obscured areas (top right part of the image) where the lamellae of the object overlap with the substrate beneath along the viewing direction, due to the significant angle of view.

The 3D reconstruction algorithm uses the information contained in the two 2D CDI views, as illustrated in Fig. 2(a). This information is essentially the EUV transmission ($I/{I_0}$) and phase ($\phi$), as shown in Fig. 2(b) for the sample displayed in Supplement 1, Fig. S1b). In the reconstruction process, the values of $I/{I_0}$ and $\phi$ for the set of voxels along the directions defined by the two views must match CDI data. The final solution is selected based on this matching, as evaluated by comparing both $I/{I_0}$ and $\phi$ of the two CDI views with those computed from the 3D reconstructed sample. This evaluation is performed using a ray tracing approach recently developed for the 3D stereographic reconstruction based on 2D x-ray fluorescence images [43–45] that assumes an approximated knowledge of sample thickness and composition. More information on the 2D and 3D reconstruction algorithms, as well as on their relation, can be found in Supplement 1.

 

Fig. 2. (a) 3D rendering of the ray tracing reconstruction algorithm. (b) $I/{I_0}$ and $\phi$ maps of the two CDI views, as computed from the 3D reconstructed sample.

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Figure 3 shows an example of 3D reconstruction of the five-fold helicoidal structure shown in Fig. 1(b). These reconstructions were obtained using different datasets acquired by exposing the sample to varying numbers of FEL shots, ranging from single-shot to 750 shots. Additional 3D reconstructed objects, including the sample depicted in Fig. 2(b), are shown in Supplement 1. In all cases, the combination of holographically guided CDI and our 3D ray tracing projection algorithms successfully retrieves a realistic stereographic representation of the sample. As quantitatively discussed in more detail in Section 4, which includes an analysis of the lateral and depth resolution (refer to Fig. 4), the image quality of the object shown in Fig. 3 does not substantially degrade when the number of exposures is reduced. This demonstrates the computational robustness of the stereographic retrieval procedure

 

Fig. 3. 3D reconstructions of the sample shown in Fig. 1(b), as obtained (a) from a single-shot exposure, (b) by averaging two different single-shot exposures, (c) from 25-shot exposures, and (d) by averaging 30 different datasets, each being a 25-shot exposure. Interactive version at [46].

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Fig. 4. (a) Radially averaged PRTF for the reconstructions displayed in Fig. 3. Yellow, violet, blue, and red lines refer to reconstructions based on, respectively, single-shot exposure, two averaged single-shot exposures, single 25-shot exposure, and the average of 30 different 25-shot exposures. (b) Normalized sum of least squares difference as a function of $\Delta z$ for the same reconstructions considered in (a). The vertical line indicates the estimated optimal resolution along the $z$ coordinate (sample depth).

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4. DISCUSSION

To obtain a quantitative assessment of the lateral resolution (${L_{{\rm res}}}$), we computed the phase retrieval transfer function (PRTF) for the 1060 iterations of the CDI code. Figure 4(a) illustrates the comparison of the radial average of the PRTF for the five-fold helicoidal structure presented in Fig. 1(b). The analysis includes reconstructions under different exposure conditions shown in Fig. 3, namely, (i) single-shot, (ii) the average of two images, each corresponding to a single shot, (iii) a 25-shot acquisition, and (iv) the average of 30 images, each corresponding to a 25-shot acquisition. For the latter case, the value of the PRTF radial average is well above the $1/e$ threshold considered by Chapman et al. [14] for the range in the exchanged momentum up to $|q| = 5.4 \; \unicode{x00B5} {{\rm m}^{- 1}}$. The radial average of the PRTF remains above $1/e$ for most of the $|q|$ range, regardless of the number of shots used for reconstruction. This indicates that the actual resolution does not substantially decrease from the multi-shot to single-shot regime. Defining the lateral resolution according to the aforementioned $1/e$ level, ${L_{{\rm res}}}$ ranges from approximately ${90} \pm {10}\;{\rm nm}$ for multi-shot averaged images (i.e., 750 shots in total) to around ${140} \pm {20}\;{\rm nm}$ for the single-shot scenario. These results are consistent with the lateral resolution provided by conventional tomography approaches, which typically range from 38 to 125 nm for similar-sized samples [11,24]. They are also in line with the lateral resolutions proposed by Geilhufe et al. [26] and Duarte et al. [30] at approximately 50 and 127 nm, respectively. Moreover, the achieved resolution surpasses that of single-shot ptychography, which typically resolves in the µm range [27,28].

The resolution (${L_{{\rm depth}}}$) along the $z$ coordinate (sample depth) for the 3D reconstructed sample shown in Fig. 3 was evaluated by varying the sampling step along $z$ within the range of $\Delta z = 10$ to 100 nm, generating an ensemble of associated solutions. These solutions were then assessed using the structural similarity index measure (SSIM) [47], as described in more detail in Section 4 of Supplement 1. The SSIM analysis identified the solutions that exhibited the closest resemblance to the input 2D views, in terms of the least squares difference between the simulated and reconstructed $I/{I_0}$ and $\phi$ maps. We defined the resolution in the $z$ axis as the value of $\Delta z$ at which the least squares differences reach a minimum. This indicates that the two CDI reconstructed views, in both amplitude and phase, do not provide any additional information to the ray tracing stereographic code. Figure 4(b) illustrates the dependence of the least squares difference, obtained from the likeliest reconstructions selected by SSIM analysis, on $\Delta z$ for the four 3D reconstructions presented in Fig. 3, which are associated with different numbers of FEL shots. The SSIM analysis reveals a consistent value of ${L_{{\rm depth}}}\; = 33 \pm 5\;{\rm nm}$, which remains nearly unchanged regardless of the number of shots. This indicates that 3D ray tracing algorithms based on a priori knowledge of material properties such as absorption length and layer thickness, are less affected by the natural degradation of the signal-to-noise ratio caused by a reduced number of exposures. This observation aligns with recent numerical simulations, where the 3D reconstruction process successfully recovered accurate topography even when input thickness maps contained spatially homogeneous white noise [45]. Therefore, our approach delivers both ${L_{{\rm depth}}}$ and ${L_{{\rm res}}}$ values in the range of 10s to 100s nm, which is remarkably better compared to other 3D imaging methods, where ${L_{{\rm depth}}}$ typically falls significantly behind ${L_{{\rm res}}}$ [11,26]. We attribute this improved depth resolution to the careful computational optimization of our 3D retrieving algorithm, which is based on approximate knowledge of the composition and thickness of each individual layer of the sample.

5. CONCLUSIONS

We combined a non-collinear EUV split-recombination setup (compact, conceptually simple, and easily reproducible) with a data processing protocol based on both CDI phase retrieval and ray tracing algorithms. This experimental setup allowed us to demonstrate the capability of retrieving information about the 3D structure of an object using a single shot of EUV light at moderate fluence, achieving a resolution of the order of 100 nm. This approach enabled us to perform the experiment well below the typical damage threshold of most materials. The structures employed in this pilot experiment were specifically designed but are representative of potential samples, such as magnetic nanostructures or biological specimens, in terms of size of structural details (ranging from 10s to 100s of nm) and overall dimensions (a few µm). This highlights the potential of FEL stereographic imaging for investigating this class of samples.

The ray tracing approach presented in this study demonstrates promise for reconstructing 3D samples due to its robustness and flexibility, even with the minimum number (two) of viewing angles, and a large angular separation between such views. This system can be directly applied to time-domain studies, expanded to include EUV spectroscopy, and can be further enhanced by incorporating additional views, either through multiple time and space coincident beams or, if a single-shot capability is not required, by rotating the sample.

This approach opens up possibilities for time-domain 3D imaging studies, such as investigating structural changes in chemical reactions or biological processes, phase transitions, or spin texture dynamics. It is particularly valuable in scenarios where high efficiency is crucial, and for non-reproducible dynamics where it is inherently necessary. The introduction of an optical pulse (for pump–probe experiments) in this setup is straightforward since it is routinely employed in EUV transient grating experiments [37,38]. Time-domain stereographic imaging techniques would be essential for studying ultrafast morphological changes in individual 3D objects. Such changes can be challenging to capture using current state-of-the-art 3D imaging methods at FELs, which rely on the continuous replacement of nearly identical samples and intrinsically reconstruct “average” sample features [23,24]. Extending this approach to higher photon energies, such as the water window range (300–500 eV) and even beyond, towards the L-edges of 3D transition metals (600–1000 eV), is technically feasible by exploiting state-of-the-art multilayer coatings [31]. Advances in the fabrication of transmissive x-ray diffraction optics [32,33] allow for envisioning split-recombination systems based on such optics, making this approach compatible even with hard x rays. By utilizing x-ray diffractive optics, it becomes possible to handle multiple replicas of FEL pulses that simultaneously impinge on the sample from more than two different angles [27,28,34]. This multiplex approach can significantly enhance the 3D reconstruction process, which in the current version assumes approximate prior knowledge of the sample composition and the absence of gaps along the photon propagation direction. Moreover, in cases where the sample is purely absorbing and its composition is uncertain, it becomes challenging to determine from a limited number of views whether the light is passing through a shorter path of highly absorbing material or a longer path through a less absorbing one. However, in CDI reconstruction, both phase and amplitude are derived. Hence, the complex scattering quotient, which is the ratio of the reconstructed phase to the natural logarithm of the reconstructed amplitude, can be employed to determine the local material content [48]. This uncertainty in the sample composition could potentially account for the minor discrepancies observed between the real 3D samples used in this study and the reconstructed models since the density and the composition distribution were assumed to be known a priori.