Multi-slice ptychography (MSP) is a fast three-dimensional ptychography technology developed on the basis of conventional ptychography. With this method, three-dimensional imaging can be achieved without rotating the sample. The prototype multi-slice algorithm can only reconstruct three-dimensional samples with a limited number of slices, which greatly limits the depth range and resolution of sample imaging. Here we reported a virtual depth-scan scheme of MSP in which a thick sample is scanned virtually in the depth direction across its whole thickness range within the reconstruction process, thereby eliminating the restriction on slice number and potentially improving the depth resolution of MSP. This new approach also improves the flexibility of multi-slice ptychography. Both the simulation and experimental results validate the feasibility of our new approach.
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With the development of X-ray source and detector technology, novel X-ray microscopic methods are emerging continuously, and the spatial resolution is improved accordingly. With the rapid progress of third-generation and fourth-generation synchrotron radiation sources, X-ray coherent diffraction imaging (XCDI) has become increasingly popular in many fields because of its ultrahigh resolution. XCDI is a lensless imaging technique that can reconstruct object information from its far-field diffraction pattern and achieve a diffraction-limited resolution. Its resolution is mainly limited by the maximal diffraction angle and X-ray wavelength [1–5]. Ptychography is an important scanning variant of CDI and inherits all its merits [6–19]. In this method, a sample (object) is illuminated by a spatially confined beam (called probe) to produce a far-field diffraction pattern, and the object is moved step by step (or continuously) with a small step size (a fraction of the probe diameter) to obtain a series of diffractive patterns. Then, the dataset is reconstructed iteratively to generate the probe and object images simultaneously. This method has various variants and strong robustness because of the redundant data resulting from the large overlap between adjacent probed areas. The redundant data contain much information more than the 2D object and probe images, such as the partial coherence of the probe, the correct sample positions of ptychography scan, and some information about the third dimension of the object, which results in the emergence of a multi-slice ptychography (MSP) algorithm [14,20–23]. It is certain that part depth information is encoded in each single diffraction pattern . However, the overlapping scan encodes the depth difference information between different slices by different moving rates of the projected (or diffracted) signals across the detector plane. Moreover, the correlation of neighbored diffraction patterns in ptychography makes it easier to extract depth information .
2D ptychography assumes that the thickness of sample is negligible, thereby it cannot handle multiple scattering effects within thick samples . However, the multi-slice algorithm provides a good way to resolve the problem, and it can extend the traditional 2D ptychography to 3D imaging without changing the original experimental setup . In this algorithm, a thick sample is divided into several virtual slices perpendicular to the optical axis and each slice is treated as a thin sample with negligible thickness. The wavefront from the upstream slice propagates freely to the next slice and interacts with it (multiplies with it), which is repeated until the last slice is reached. Then, the exit wave from the last slice propagates freely to the detector plane in the far field. After updating the wave function on the CCD plane, the wavefront is transmitted back through each slice sequentially and updated together with each slice, with the updating equations being the same as that of the 2D ptychography (Fig. 1). In conventional 3D imaging, such as ptychographic X-ray CT (PXCT), the sample thickness is largely limited by the depth of focus . For another 3D imaging method, confocal 3D microscopy , it is essential to use a focusing element of higher performance to obtain a smaller depth of focus (DOF) for better resolution in depth. However, in MSP, the depth resolution is basically determined by the X-ray wavelength and the maximal scattering angle, not strictly limited by DOF. The maximal scattering angle is often closely related to the numerical aperture (NA) or the DOF of the divergent probe. Therefore, a high-quality focusing lens (zone-plate) with a small DOF is also necessary for multi-slice 3D ptychography in the experiment.
In recent years, the multi-slice algorithm has been demonstrated and applied in visible light and X-ray regimes. Maiden et al. demonstrated multi-slice imaging of Spirogyra algae with visible light [14,20]. Akihiro Suzuki applied this algorithm in the imaging of a two-layered object with a 105 µm gap using hard X-rays . This method was also used with the focusing optics of Laue lenses in hard the X-ray regime . Furthermore, this algorithm was combined with tomography to achieve a much higher depth resolution [28–30] but using a much smaller dataset (much smaller number of rotation angles) compared to the conventional CT.
There are several limitations to the multi-slice ptychography, including the small number of slices that can be reconstructed, the high overlap degree or redundancy of the dataset required [31,32]. If there are many slices, reconstruction will be difficult or even impossible. In the frequently-used 3PIE multislice algorithm , because the update process of the object and the probe starts from the slice close to the detector, the reconstruction quality of upstream slices depends on the reconstruction accuracy of downstream slices, and the convergence speed of the upstream slices will be slower or even stagnated.
Therefore, the prototypical multi-slice method can only work for a limited slice number, usually less than ten in X-ray regime. The optical sectioning technique employing a way of progressively inserting slices during reconstruction can effectively increase the slice number for visible light MSP . However, the effectiveness of this technique is greatly reduced for X-ray MSP, due to the much lower scattering intensity and much fewer diffraction signal amount of X-ray than that of visible light. Here we developed a new approach, the virtual depth-scan multi-slice ptychography, to solve this problem and achieve the full 3D reconstruction of thick samples, with an effectively improved resolution in the depth direction.
2. Reconstruction algorithm
In 2D ptychography, the object and probe interact by a simple multiplication, forming an exit surface wave that propagates to the detector in the far field. A series of diffraction patterns Ij measured by a CCD detector with a distance d from the sample can be expressed as:
However, the premise of 2D ptychography is that the specimen can be considered as a 2D thin sample. When the specimen thickness is non-negligible or its 3D structure needs to be characterized, this algorithm is no longer valid. The multi-slice ptychography is a recently developed method for modeling the interaction between the probe and the thick object [14,21], and the first proposed MSP algorithm is called 3PIE . In this model, the sample is split into several slices perpendicular to the optical axis, each of which can be viewed as an approximation of a 2D thin sample. The propagation of illumination between adjacent slices is considered to be free-space propagation and is modeled by the angular spectrum method . The performance of the algorithm depends to a large extent on the accuracy of this propagation model and the redundancy of the dataset. The diffraction patterns measured by the detector can be expressed as follows :
Here, Δzn = zn+1-zn denotes the distance between the nth and (n+1)th slices, and TΔzn is the angular spectrum propagator over the distance Δzn, which can be described as
The resolution of MSP is limited only by the wavelength and the NA of the detector (the effective NA is related to the maximum diffraction angle measurable). The 2D and depth resolutions (in the x and z directions) can be expressed, respectively, as 36]. When the signal-to-noise ratio of the high frequency signal is higher, we can theoretically reconstruct a 3D sample image with a better depth resolution.
For this algorithmic scheme, if a 10 µm thick specimen is split into three slices that are spaced at 3.3 µm distances, then the depth resolution can only be given as approximately 3.3 µm, and the specific resolution cannot be discussed at all. If the photon energy of the X-ray is 600 eV, the pixel size of a 2048×2048 CCD is 13.5 µm×13.5 µm, and the distance between the detector and the sample is set to 70 mm, which ensures that the measured diffractions are in the Fresnel diffraction regime, then the depth resolution can be calculated as 109 nm according to Eq. (5). This value is much smaller than the depth resolution determined by the slice spacing, as the algorithm usually performs.
We proposed a new scheme of MSP based on 3PIE algorithm to overcome one of the above-mentioned shortcomings of MSP, that is, the limited number of slices, and improve the depth resolution to be close to the theoretical limit of Eq. (5). This new scheme is called virtual depth-scan multi-slice ptychography (VDSP). It will be a promising complementary technique to multi-slice ptychography, hopeful to extend the capability and application of MSP.
It is found that the result reconstructed by the multi-slice algorithm is closely related to the initial estimate of a defocused probe, and is largely dependent on the defocus length (the distance between the focus and the sample plane) setting of the initial probe. For example, if we use a model probe with a 1 µm diameter downstream of the focus as the initial probe, which corresponds to a defocus length of 14.5 µm at the depth position z along the optical axis, the first slice of the reconstructed sample can be considered as the 2D cross section of the sample at this depth position z. This slice contains the structural information within the adjacent thickness range of the specimen near this z plane. Therefore, if this defocused position or depth position of the cross-section is virtually moved (in the reconstruction algorithm) through the sample in the optical axis direction (i.e. in the depth direction) by sequentially setting a series of defocus lengths of the probe for a series of multi-slice reconstructions, then, slice images at any depth position of the specimen can be obtained, and the restriction on reconstructed slice number can be removed. During this process, it is only necessary to constantly adjust the defocus distance of the initial probe for a series of independent multi-slice reconstructions to obtain a large number of slice images that fill the entire specimen space (Fig. 2). This process is similar to several virtual planes scanning across the entire sample in the depth direction; therefore, this reconstruction approach can be called virtual depth-scan ptychography.
By fully exploiting (scanning) the third dimension information buried in the redundant ptychography dataset, this new scheme of MSP can effectively improve the resolution of 3D reconstruction. Because we still use a small number of slices to reconstruct a thick sample every time a new defocus length value is set, it means that each slice is actually a projection of the sample volume within a certain thickness onto a 2D plane perpendicular to the optical axis. Within this sample volume, the parts that are closer to this 2D plane will gain a relatively clearer projection, which is similar to the focus/defocus phenomenon. To further improve the lateral and depth resolution of our reconstruction approach, we can remove blurry features that are originated far away from the given slice from the image by using an algorithm called focal stack imaging [37,38].
A full flowchart of the virtual depth-scan ptychography algorithm is outlined in Fig. 3. At each depth position, an N-slice ptychography reconstruction is performed (N usually less than 10). The result of individual N-slice reconstructions is completely dependent on the defocus position or depth position of the initial probe, which leads to different few-slice (N-slice) reconstructions in different z positions within the thick object. By picking out the phase image of the middle slice from each few-slice reconstruction result, and combining these picked middle-slice images at different depths, a 3D reconstructed image can be realized. The focal stack method can be used to process all these phase images by thresholding them with a properly selected or calculated threshold, to further improve the reconstruction quality and depth resolution of the final 3D image.
Before applying VDSP to a real-world specimen, we performed a relevant soft X-ray simulation to verify its effectiveness. First, a 3D fiber specimen model with mutually interlaced filaments was constructed (shown in Fig. 4(a)). These filaments were set as cylinders with a diameter ranging from 60 nm to 140 nm and were randomly distributed within a cuboid but perpendicular to the optical axis. The size of the cuboid was 5 µm×5 µm×10 µm, and the sampled 3D matrix had 512×512×512 voxels. The probe was generated by a 200-µm-diameter Fresnel zone plate (FZP) with a centerstop of 50 µm diameter and an outermost zone width of 10 nm. The incident photon energy was chosen as 600 eV to model a soft X-ray experiment with a photon flux of 109 photons/s. The focal length of the FZP was calculated to be 968 µm. We chose an FZP with a small focal length to demonstrate good imaging results, although this is difficult to implement in realistic experiments.
A CCD detector of 512×512 pixels with a pixel size of 54 µm was used to capture diffractive patterns, which results in a 27.65 mm × 27.65 mm detecting area for reciprocal space signals. The detector was placed 70 mm downstream of the sample, meeting the far-field diffraction condition. The theoretical limit depth resolution was calculated as 104.9 nm based on NA of CCD detector. A 1 µm diameter incident probe was modeled by the Fresnel-Kirchhoff diffraction theory [34,35] and used to illuminate the specimen. Diffraction patterns were generated from illuminated positions of a slightly distorted 10×10 raster grid with a random offset amplitude of 75 nm (To meet the overlap requirement of ptychography, the moving step size was set to ∼0.2 µm). To fully exploit the depth information of a ptychographic dataset, the virtual depth scanning step of VDSP was set to be 50 nm, which was smaller than the theoretical resolution of MSP (Eq. (5)). For each depth position, a reconstruction of 3-slice ptychography with a slice spacing of 2 µm was performed for 100 iterations. The total time spent for the data production and reconstruction was about 12 hours using a Python implementation of VDSP accelerated with a GPU.
Six 2D phase images of the reconstructed slices in the simulation are shown in Figs. 5(a–f). They are 4 µm, 5 µm, 7 µm, 8 µm, 9 µm and 10 µm away from the front surface of the sample, respectively. By comparing the two slice images Figs. 5(a–b), it can be observed that Fiber 1 in Fig. 5(a) is more clearly displayed than that in Fig. 5(b), indicating that Fiber 1 is closer to the depth position of slice 5(a) than to that of slice 5(b). On the contrary, Fiber 2 is closer to the depth position of slice 5(b) than to that of slice 5(a).
Figure 4(b) shows a 3D reconstructed image obtained using the VDSP algorithm. It is also animated in a video (Visualization 1). Within this 3D image, the distance between adjacent reconstructed slices was 50 nm, which is equal to the depth scanning step of VDSP. The 3D matrix of this image had 512×512×200 voxels, and was obtained by combining all the second slices extracted from their corresponding 3-slice ptychography reconstructions at their respective depth positions. After interpolation, the voxel size became consistent with the pixel size of the 2D slice images. It can be observed that the reconstructed 3D image is in agreement with the original sample model, and many fairly fine fibers were reconstructed very well. The depth resolution and lateral resolution were calculated as 0.59 µm and 10.85 nm, respectively, by Fourier line correlation (FLC) and Fourier ring correlation (FRC) analyses [39,40], as Figs. 6(b) and 6(d) show.
To verify the experimental applicability and robustness of the VDSP algorithm for real data, a soft X-ray ptychography experiment was carried out at the scanning transmission X-ray microscopy (STXM) endstation of Shanghai Synchrotron Radiation Facility (SSRF). An X-ray beam of 540 eV photon energy was used to illuminate a sample which was a film consisting of cellulose nano-fibrils (CNF), similar to the modeled sample in the above simulation. In the experiment, we used a Fresnel zone plate with a diameter of 240 µm and an outermost zone width of 35 nm to focus the X-ray, and a CCD of 1024×1024 pixels with a pixel size of 27 µm to record the diffraction data. The detector was placed 58.5 mm downstream of the sample. A set of diffraction patterns were collected from a grid of 26×26 scan positions on the sample with a probe of 2-µm nominal diameter and a step of 0.2 µm. The theoretical limit depth resolution was calculated as 81 nm according to the NA of CCD chip.
According to the VDSP algorithm, a total of 300 slice images with a depth scan step of 152 nm were obtained by combining all the third slices extracted from 300 five-slice ptychography reconstructions with 50 iterations for each 5-slice reconstruction at its corresponding depth position. The total time consumed for the reconstruction was about 115 hours using a Python implementation of VDSP accelerated with a GPU. For the 5-slice reconstruction at each depth, the distance between adjacent slices was set to 10 µm. As a result of the weak absorption and noise, the quality of the reconstructed slice amplitudes was rather poor. Therefore, the phase images of the reconstructed slices were chosen to form the final 3D reconstruction. Six representative slice phase images are shown in Fig. 7. We can see that Fiber 1 is more clearly displayed than Fiber 2 in Fig. 7(d), indicating that Fiber 1 is closer to the depth position of 34.29 µm away from the front sample surface than Fiber 2, while Fiber 2 is closer to the depth position of the slice shown in Fig. 7(e). Figure 7(f) shows that both fibers are severely blurred or almost disappear, indicating that these two fibers are far from the slice in Fig. 7(e). These characteristics show the high resolution of the VDSP imaging method in the depth direction.
Since the constant phase offset in the object does not affect the diffraction pattern on the detector but may disturb the quantitative analysis of the final image, the phase images need to be normalized based on the phase sampling of the object-free area. After normalizing and registering these slice images, the 300 slices were combined in the depth position sequence and thresholded by focal stack processing to form a 3D structure. After rendering the visualization, the 3D CNF structure with interlaced nanofibers is shown in Fig. 8(a). It is also animated in movies (Visualization 2 and Visualization 3). It can be observed that the sample is approximately 45 µm thick, which is obviously too thick to be measured by TEM. The depth resolution and lateral resolution were calculated as 1.34 µm and 9.48 nm, respectively, by an FLC and FRC analysis (Fig. 8(c, e)). A more accurate thickness determination of the sample can be achieved via the curve of the reconstruction error versus the depth position of the virtual multi-slice object or the defocused probe (Fig. 8(f)). Convergence of ptychography is typically monitored by computing the summed squared error [14,21] between the predicted intensity and measured intensity. When the given slices are all inside the sample, the reconstruction error is minimal and almost constant. As the number of slices inside the sample decreases, there will be fewer slices to model the multiple scattering within the sample, and thus the reconstruction error will gradually increase until all slices are located outside the sample. Then the error reaches maximum and starts to remain constant. The sample thickness can be determined as 42.67 µm by the difference of the depth positions where the error curve turning occurs (at a high level of the reconstruction error, as denoted by the red arrows in Fig. 8(f)) with the slice gaps subtracted.
The lower absorption rate of carbon nanofibers makes absorption-based methods, such as STXM, difficult to utilize. Compared to the tomography method, this new 3D imaging approach does not require rotation of the sample, and the required data amount is quite small, just as small as that of a 2D ptychography. Therefore, the new approach will significantly decrease the measuring time and operational complexity for 3D imaging of thick samples if a high depth resolution is not required.
It can be found from Eq. (5) that the depth resolution of MSP is determined by the maximum scattering angle and X-ray wavelength. Experimentally, a probe with a smaller DOF means a more divergent incident light wave which is more conducive to increase the scattering angle of diffraction signal than low-divergence or parallel light probe, thereby significantly improving 3D imaging quality of MSP  and VDSP. This is the reason why Eq. (5) is very similar to the DOF definition:
A smaller DOF or a larger divergence angle will cause the wavefront to change faster during propagation between slices, or undergo significant propagation effect over a smaller distance, making the wavefield interacting with adjacent slices be more different, which allows the slices with a closer spacing to be more easily disambiguated by the phase retrieval algorithm. Therefore, if one wants to obtain a reconstructed image of MSP with sufficiently high quality, i.e. without much crosstalk between adjacent slices, the slice spacing should be well larger than 1 DOF, usually several times of DOF. This is also a major reason why the traditional MSP is difficult to reconstruct a 3D structure with a large number of slices. It should be pointed out that this does not mean DOF is a rigorous limiting factor to the depth resolution, as the maximal scattering angle detectable in the experiment is also closely related to the scattering properties of the sample. As a consequence, the VDSP technique proposed in this study could potentially go beyond the DOF limitation.
If the imaged sample is very thick, the boundary slice for a certain probe defocus might be very far away from the exiting/incident plane of the object, which may cause a great difference between the free-space propagated wavefront (probe) and the actual wavefront at the boundary slice of the virtual N-slice object. If this difference is too large, it probably results in poor reconstruction quality with many artifacts. Yet these artifacts should mainly exist in the boundary slices, so we can extract only the middle slices of respective five-slice reconstructions for the final 3D image assembly to avoid this problem. In such case, boundary slices are supposed to approximate the diffraction between themselves and the object boundary as transmission functions (one or two slice functions), thus they are not considered as the correct slice images. Therefore, if one boundary slice is not enough to approximate the transmission process of the thick “slice-less” portion, we would use two slices to approximate the process, then the third slice will be adopted for the final assembly, just as what we did for the experiment data processing. We can use more slices in each N-slice run to further decrease the influence of the problem. Another solution to the mismatch of initial probes and actual probes is to compute the defocused probe by doing a multislice propagation using the previously reconstructed upstream slices, though the computation burden can be significantly increased.
The major limitation of VDSP is the much lower depth resolution than the lateral resolution. The depth resolution of this technique is usually limited by the DOF of the probe for weak-scattering samples, or by the NA of diffraction signal for strong-scattering samples. To further improve the depth resolution of VDSP to the scale comparable to the lateral resolution, it must be combined with sample rotation or tomography, which will in turn help the ptychographic tomography to reduce the rotation angle number and increase the data acquisition efficiency.
The property of the sample, such as the sparsity or absorption rate, is also a limitation of the technique to some extent, but this limitation can be relaxed in some ways. The VDSP is indeed more applicable to sparse samples or low-absorption samples. If a part of a non-sparse object is too absorptive, it will occlude a cone of light behind it, which can cause artifacts in the reconstructed slice image, and the artifact generation is closely related to the slice spacing . With the slice distance increasing, the quality of the individual slices improves. The dark cone due to the strong absorbent only affects the imaging of the downstream parts close to the absorbent, and the parts that are far from the strong absorbent can be illuminated by the scanning probe in other positions, thus not affected by the strong absorbent. Therefore, we can improve the reconstruction quality for non-sparse or strong absorbent samples by increasing the slice spacing to ensure that all slices can be adequately illuminated at least in part of the scanning positions. Another way to relax the low-absorption limitation is to use a probe of smaller DOF, which will effectively reduce the absorbed photons by a single strong absorbent. Although the sample sparsity is not a necessary condition for VDSP in theory, it is still a relatively weak constraint or limitation to our technique and an object with a higher sparsity or homogeneity can be reconstructed better and easier.
In this study, we proposed a virtual depth-scan ptychography approach for 3D imaging of thick samples based on the multi-slice ptychography method. This new approach could potentially achieve a better resolution in the depth direction than the prototype MSP by eliminating the limitation to the slice number within the sample thickness and fully exploiting the depth information hidden in the redundant diffractions. This is a unique 3D imaging scheme that does not require rotation of the sample, and the required data amount is as small as that of the 2D ptychography, indicating high efficiency and operational simplicity in experiment. We demonstrated this approach using experiment data from a soft X-ray synchrotron radiation beamline, achieving a real 3D structure of a carbon nanofiber sample with a micro-scale resolution in depth and nanoscale resolution in lateral directions. If a Fresnel zone plate with a higher focusing power (a larger NA of the probe) is used, the depth resolution of the approach can be expected to improve further.
VDSP is a promising complementary technique to MSP and ptychographic tomography. It originated from the MSP technique, helping to extend the capability and application of MSP. On the other hand, VDSP method can be combined with the tomography technique, which will greatly improve the depth resolution of VDSP to be comparable to its lateral resolution, and significantly reduce the rotation angle number in ptychographic tomography. The combination way of the two methods should be similar to that of conventional MSP with ptychographic CT as reported in the references [28–30]. The work presented here would provide some guidelines for 3D imaging of thick samples with low absorption contrast that are difficult to image by other methods, such as STXM and Transmission X-ray microscopy.
Appendix - Cellulose nano-fibrils microscopic image by STXM
Here we show the cellulose nano-fibrils images (Fig. 9) obtained using scanning transmission X-ray microscopy (STXM). The two images show sample structures at locations different from that of VDSP images shown in Fig. 7, but we can see that the structure in STXM images is quite similar to that in VDSP images, and they can be thought to be qualitatively equivalent. Therefore, the two STXM images can be used to justify the results of VDSP reconstruction, though their quality is much worse than that of VDSP images.
National Natural Science Foundation of China (11875316, U1832146, U1832154, 51971160); Ministry of Science and Technology of the People's Republic of China (2017YFA0403400, 2016YFA0400902).
Authors thank BL08U1A beamline of Shanghai Synchrotron Radiation Facility and soft X-ray spectro-microscopy (SM) beamline (10ID-1) of Canadian Light Source for providing beamtime. They also thank Jinyou Lin (Shanghai Advanced Research Institute) for providing the specimen used in the experiment.
The authors declare no conflicts of interest.
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