## Abstract

Ptychographic X-ray computed tomography (PXCT) is a potential tool for visualizing three-dimensional (3D) structures of large-volume samples at high spatial resolution. Currently, both the requirement of a large number of views and the narrow depth of field limit the range of applications of PXCT. Here, we propose an improved 3D reconstruction algorithm for PXCT that is based on 3D iterative reconstruction and multislice phase retrieval calculation. Computer simulations showed that the proposed algorithm can reduce the number of required views without degrading the spatial resolution. In a synchrotron experiment, ptychographic diffraction data sets of a flat and thick processor specimen were collected under a limited-angle condition, and then high-resolution multislice images of the Cu multilevel interconnects were clearly reconstructed using the proposed algorithm. The proposed algorithm is expected to open up a new frontier of large-volume 3D nanoimaging in various fields.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Demand for the three-dimensional (3D) visualization of structures with a size of 10 *μ*m order at nanometer resolution is increasing in the fields of cell science, materials science, and device engineering [1]. Computed tomography (CT) is one of the technologies capable of non-destructively and three-dimensionally observing the internal features of an object, in which a 3D image is reconstructed from multiple projections measured at various incident angles [2]. In particular, X-ray microscopy is suitable for the CT imaging of a specimen of micrometer size at nanometer resolution owing to its high penetration power and short wavelength [3]. X-ray ptychography [4] is a promising method for the nanoscale imaging of thick objects, which provides projection images with resolution better than 10 nm [5] and a field of view wider than 10 *μ*m [6]. In X-ray ptychography, a localized probe is scanned so that adjacent positions overlap, and then both an illumination wave field and a sample image are reconstructed by phase retrieval calculation. X-ray ptychography has already been extended to the 3D observation method of ptychographic X-ray CT (PXCT) [7]. PXCT has been used to observe various materials [8,9]. There are currently two difficulties in achieving PXCT with single-nanometer resolution over a field of view of 10 *μ*m order. The first is the requirement of a large number of views measured at various incident angles. The second is the limitation of the sample thickness due to the narrow depth of field (DOF).

The Crowther criterion [10], *πD*/*d*, gives the minimum number of views required to reconstruct an object of diameter *D* at a resolution of *d*. For example, when *D* and *d* are 10 *μ*m and 10 nm, respectively, at least 3142 of views are required, which is equivalent to 0.05° angular sampling. Coarser angular sampling results in the appearance of artifacts in reconstructions. Moreover, the limited available range of the tilt angle due to the sample holder, called the “missing-wedge problem”, also causes the appearance of artifacts. To suppress the appearance of artifacts, various tomographic reconstruction techniques, classified into the analytical type such as filtered back projection (FBP) and the iterative type such as algebraic reconstruction technique (ART) [11] and simultaneous iterative reconstruction technique (SIRT) [12], have been proposed. The analytical type is based on the Radon transform and filtering processes in the Fourier space. The iterative type uses multiple repetitions, in which the current solution converges towards a better solution. The use of prior information, such as the initial volume, can reduce various artifacts due to sparse and incomplete data, although the computational load is greater than that for the analytical type [13].

The DOF determines the tolerable sample thickness at the desired spatial resolution. The DOF in X-ray ptychography is given by *D* ≤ 5.2*d*^{2}/*λ*, where *λ* is the X-ray wavelength, in the Ref. [14]. To circumvent the limitation of the sample thickness, a phase retrieval algorithm using a multislice approach has been proposed and demonstrated in the visible light [15,16] and X-ray regimes [17], in which the object is modeled as a sequence of slices separated by distances smaller than the DOF and the free propagation between the slices is calculated. More recently, a precession method [18,19], in which a multislice calculation is performed for a tilt-series data set and with a small tilt used to approximate the linear shift of each layer, and the use of a Laue lens with a narrow DOF as a focusing optics [20] have been reported. These techniques can potentially be used for the 3D imaging of flat specimens such as integrated circuits without laborious sample preparation even though the longitudinal resolution is an order of magnitude lower than the transverse resolution. Recently, the combination of full-rotation tomography and multislice ptychography in the visible-light regime [21] has been reported, in which the projections at neighboring angles are calculated by parallel translations of reconstructed layers at each angle, thus reducing the number of angular samples required for the tomographic reconstruction. The acceptable angular range for the translations depends on both the transverse resolution and the number of slices. In the X-ray regime, it is difficult to apply this algorithm since the acceptable angular range is narrow. Therefore, the development of a reconstruction algorithm for multislice PXCT is a crucial issue.

In this paper, we propose an improved reconstruction algorithm for multislice PXCT, in which 3D iterative reconstruction and multislice phase retrieval are alternately repeated. The proposed algorithm not only maintains the transverse resolution, but also provides more reliable 3D reconstruction from incomplete data sets. We investigated the performance of the algorithm under the full-rotation and limited-angle conditions. We also performed ptychographic measurements of an Intel processor under a limited-angle condition and successfully reconstructed 31 slabs with 240 nm thickness.

## 2. Three-dimensional iterative reconstruction algorithm for multislice PXCT

Figure 1 shows the procedure of the proposed algorithm. The algorithm alternately repeats 3D iterative reconstruction (IR) and multislice phase retrieval calculation. The multislice reconstructions are effectively used for 3D reconstruction without calculating a projection. Let us define an *N* × *N* × *N* 3D complex function ${O}_{i}^{(k)}(x,y,z)$, corresponding to the *N* slabs of *N* × *N* 2D transmission functions along to the *z* direction, which is updated by using the *i*th ptychographic data set, where *k* is the iteration number. The reconstruction procedure includes the following seven steps.

- Divide ${O}_{i,\mathrm{\Theta}}^{(k)}(x,y,z)$ into
*L*slices so that each slice is thinner than the DOF. In Fig. 1,*L*is set to 3. - Input the divided slices in the multislice phase retrieval calculation as initial transmission functions ${S}_{i,l}^{(k)}(x,y)$ and update to ${{S}^{\prime}}_{i,l}^{(k)}(x,y)$. In this paper, we use 3D ptychographic iterative engine (3PIE) [15] for the multislice phase retrieval calculation.
- Calculate the 3D backprojection ${M}_{i,\mathrm{\Theta}}^{(k)}(x,y,z)$ by filling the two-dimensional (2D) images between layers. The 2D image of the
*l*th layer is expressed as ${{S}^{\prime}}_{i,l}^{(k)}(x,y)$ divided by ${S}_{i,l}^{(k)}(x,y)$, raised to the power of 1/*N*, where_{l}*N*is the number of voxels along the propagation direction in the_{l}*l*th layer,*i.e*., ${M}_{i,\mathrm{\Theta}}^{(k)}(x,y,z)$ is given by - Steps 1–6 are repeated at various angles until the 3D reconstruction converges.

## 3. Computer simulation

#### 3.1. Full-rotation condition

We investigated the performance of the proposed algorithm by the computer simulation. In this simulation, the optical parameters were selected to simulate our experiments at SPring-8. The incident X-ray beam was two-dimensionally focused to ∼500 nm at an X-ray energy of 6.5 keV. The pixel size of the 2D detector was 200 *μ*m and the camera length was set to 2.193 m. The achievable transverse resolution and DOF were 10 nm and 2.73 *μ*m, respectively. Figures 2(a) and 2(b) show the 3D model of the sample and an *x* − *z* cross section, respectively. Au particles (*ρ*=19.32 g/cm^{3}) were included in the cylindrical styrene-butadiene rubber sample (*ρ*=0.930 g/cm^{3}) and the volume percent of particles was ∼1.4 %. The diameter of the sample was 3.2 *μ*m. The mean size of the Au particles was ∼25 nm and the particle size had a Gaussian distribution with *σ*=0.2. Ptychographic diffraction patterns were calculated at 11 × 11 raster-grid positions with a step size of 300 nm from two layers separated by 1.6 *μ*m. The ptychographic diffraction patterns were calculated at 150 angles between −90° and 90° with 1.2° spacing, which was much smaller than the number of data sets required according to the Crowther criterion of ∼1005. The image reconstruction was performed by the following four methods: by combining FBP and 3PIE, ART and 3PIE, IR and ePIE [22], and IR and 3PIE, *i.e*., the proposed algorithm. In the combination of FBP or ART with 3PIE, projected phase images were first derived by the multislice phase retrieval calculation of a two-layered object with 500 iterations at 150 angles, and then the 3D images were reconstructed from the 150 projections by FBP or ART. In ART, the 3D reconstruction process was performed five times. In the combinations of IR and ePIE, and IR and 3PIE, the number of iterations in the multislice phase retrieval calculation was 100 at each angle, and then the whole iterative reconstruction process was performed five times. Figures 2(c)–2(f) respectively show the *x* − *z* cross-sectional phase images reconstructed by using the combinations of FBP and 3PIE, ART and 3PIE, IR and ePIE, and IR and 3PIE. In Fig. 2 (c), line-shaped artifacts appeared in the styrene-butadiene rubber, and the spherical shape of the Au particles, indicated by red arrows, was deformed. It is well known that sufficiently fine sampling uniformly distributed over 180° is required to produce accurate reconstructions in FBP [2]. On the other hand, iterative reconstruction methods, such as ART and the proposed algorithm, can prevent the artifacts induced by incomplete angular sampling. The iterative reconstruction methods enhance the low-frequency information with good contrast; however they lose the high-frequently information [23]. The phase shifts of the styrene-butadiene rubber were uniform in Figs. 2(d), 2(e), and 2(f). However, the spherical shapes in Fig. 2(d) were slightly deformed due to incomplete angular sampling. The spherical shapes in Fig. 2(e) were blurred owing to the narrow DOF. The proposed algorithm allowed us to more quantitatively reconstruct the phase structures as shown in Fig. 2(f). Figure 2(g) shows the Fourier ring correlation (FRC) curves [24] between the model image and the reconstructions. For the combination of IR and 3PIE, the FRC remains between 0.6 and 1. For the other combinations, the FRC rapidly decreases at a frequency of approximately 20 *μ*m^{−1}, which indicates that the spatial resolution in Figs. 2(c), 2(d), and 2(e) was degraded by insufficiently fine angular sampling and by thickness effects. The proposed algorithm provides a more robust 3D reconstruction of samples thicker than the DOF without degrading the transverse resolution under sparse angular sampling.

#### 3.2. Limited-angle condition

We also performed the computer simulation under a limited-angle condition using a planar model with the same material as that under the full-rotation condition. Under the limited-angle condition, a double-tilt geometry [18] improves the fidelity of the 3D reconstruction by reducing the “missing wedge” to a “missing pyramid” [25]. 121 ptychographic diffraction pattern sets were calculated at 11 × 11 raster-grid positions with a step size of 300 nm with the angles *φ* and *ω* changed from −5° to 5° with 1° steps, where *φ* and *ω* are the angles relative to the *x* and *y* axes, respectively. The longitudinal resolution is given by [26]

*d*

_{x,y}and

*d*are the achievable resolutions along the transverse and propagation directions, respectively, and

_{z}*α*is the maximum tilt angle. In this simulation,

*d*was calculated to be 200 nm; thus, the phase images were obtained from the reconstructed 3D functions by integrating over 16 slabs with 200 nm thickness. Figures 3(a)–3(e) show the 1st and 9th layers of the model and the reconstructed phase images. As shown in Figs. 3(b), 3(c) and 3(d), artifacts due to the missing data appeared in styrene-butadiene. In addition, the structures were compressed in the vertical direction in Fig. 3(b) and halo-shaped artifacts appeared around the Au particles in Figs. 3(c) and 3(d). In contrast, each layer was almost completely reconstructed in Fig. 3(e). This result was also confirmed via the FRC as shown in Fig. 3(f). For the combination of IR and 3PIE, the FRC remains approximately 1 throughout the frequency range except around 0. These results indicate that the proposed algorithm is effective not only for maintaining the transverse resolution but also for suppressing the appearance of artifacts. This algorithm has the potential for the 3D observation of planar specimens such as integrated circuits at an axial resolution of 100 nm order without laborious sample preparation.

_{z}## 4. Synchrotron experiment

We performed the ptychographic measurement of an Intel processor fabricated with 14 nm technology under a limited-angle condition at BL29XUL in SPring-8. Figure 4 shows a schematic of the experimental setup. A 6.5 keV monochromatic X-ray beam was two-dimensionally focused to ∼500 nm by Kirkpatrick–Baez (K–B) mirrors. The bottom of the sample was thinned to ∼30 *μ*m by dry etching. Forward diffraction patterns were recorded using a hybrid pixel array detector (EIGER 1M, Dectris) [27]. The main specification of the detector is as follows: the pixel size 75 × 75 *μ*m^{2}, the sensitive area 77.2 × 79.9 mm^{2}, single-photon counting with zero readout noise, maximum count rate of 5 × 10^{8} photons/s/mm^{2}, frame rate capability up to 3 kHz, and a negligible deadtime between frames of 4 *μ*s. The detector was positioned at a distance of 2.219 m from the sample. To measure the bright field and increase the effective dynamic range, an 88-*μ*m-thick Si attenuator of ∼800 × 800 *μ*m^{2} size was placed approximately 830 mm upstream from the detector [28]. The acquisitions were performed at two positions on the detector to measure the diffraction intensities in the high-Q region with a good signal-to-noise ratio and complement the dead pixels between the modules of the detector. The sample was raster-scanned at 9 × 9 points with a step size of 300 nm. Ptychographic diffraction data sets were collected at 21 angles, (*φ*, *ω*)=(0°,0°), (1°,0°), (2°,0°), (3°,0°), (4°,0°), (5°,0°), (−1°,0°), (−2°,0°), (−3°,0°), (−4°,0°), (−5°,0°), (0°,1°), (0°,2°), (0°,3°), (0°,4°), (0°,5°), (0°, −1°), (0°, −2°), (0°, −3°), (0°, −4°), and (0°, −5°). Under these measurement conditions, the achievable transverse resolution, longitudinal resolution, and DOF were estimated to be 12 nm, 238 nm, and 3.92 *μ*m, respectively.

Then, we performed the reconstruction at (*φ*, *ω*)=(0°,0°) by ePIE and 3PIE. In 3PIE, we first set the number of layers to two. The upstream and downstream images were reconstructed while varying the gap between layers with a 1 *μ*m step, and then the gap was set to 6 *μ*m because it provided the clearest reconstructions. Consequently, we estimated the thickness of Cu multilevel interconnects to be 12 *μ*m and finally set the number of layers and the gaps between the layers to 4 and 2 *μ*m, respectively. Figures 5(a) and 5(b) show the projected phase images reconstructed by ePIE and 3PIE, respectively. The wiring in Fig. 5(b) was more clearly reconstructed than that in Fig. 5(a). Figure 5(c) shows the profiles through the red lines indicated in Figs. 5(a) and 5(b) and the estimated transverse resolutions. The profile of 3PIE was steeper and the transverse resolution was estimated to be 12.4 ±1.2 nm, which was better than that of ePIE of 13.9±2.5 nm.

Next, we performed the reconstruction by the combination of IR and 3PIE. The number of iterations in the multislice phase retrieval calculation was 100 and whole iterative reconstruction process was performed three times. The 3D complex function was successfully reconstructed and 31 slabs with 240 nm thickness were calculated, in which the slab thickness was determined based on Equation (5). An animation of the full stack of 31 phase images can be seen in Visualization 1. Figure 6 shows the phase images of the 10th, 16th, 18th, 21st, and 25th slabs shown in Visualization 1. One can observe five types of circuits, whose finest structure was approximately 25 nm, and the through-silicon via structures were also resolved as indicated by the red arrow. The proposed algorithm worked well for the experimental data set under the limited-angle condition.

## 5. Conclusion

We have proposed an improved 3D reconstruction algorithm for PXCT, which is based on a combination of 3D iterative reconstruction and multislice phase retrieval calculation. It was shown by computer simulations that the proposed algorithm can reduce the number of required views to less than 1/5 of the numbers required accordingly to the Crowther criterion without degradation of the resolution. In the near future, 3D imaging of an object over 100 *μ*m in size at single-nanometer resolution is expected to be realized using this algorithm. This algorithm also worked well under a limited-angle condition. This is a major advantage for a wide range of applications of PXCT. It is easy to perform *in situ* measurements using the proposed technique in various environments since cylindrical samples are not required. In addition, radiation damages of samples, which is induced by X-ray ionizing, can be reduced owing to the decreasing of the number of measurements. We believe that this technique will make it easier to three-dimensionally observe whole structures of specimens at nanometer resolution in various fields, such as cell science, materials science, and device engineering.

## Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Nos. JP16J00329, JP16K13725, JP17J016730, and JP18H05253).

## Acknowledgments

We thank T. Ishikawa, M. Takata, K. Yamauchi, and K. Endo for many stimulating discussions, as well as Y. Kohmura for help in the synchrotron experiment. We also thank Toshiba Nanoanalysis Corporation for preparing the planar sample of an Intel processor.

## References

**1. **G. E. Ice, J. D. Budai, and J. W. L. Pang, “The race to X-ray microbeam and nanobeam science,” Science **334**, 1234–1239 (2011). [CrossRef] [PubMed]

**2. **A. C. Kak and M. Slaney, “*Principles of computerized tomographic imaging*,” (IEEE Press, 1988).

**3. **E. Maire and P. J. Withers, “Quantitative X-ray tomography,” Int. Mater. Rev. **59**, 1–43 (2014). [CrossRef]

**4. **J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-Ray Lensless Imaging of Extended Objects,” Phys. Rev. Lett. **98**, 034801 (2007). [CrossRef] [PubMed]

**5. **D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics **8**, 765–769 (2014). [CrossRef]

**6. **M. Guizar-Sicairos, I. Johnson, A. Diaz, M. Holler, P. Karvinen, H.-C. Stadler, R. Dinapoli, O. Bunk, and A. Menzel, “High-throughput ptychography using Eiger: scanning X-ray nano-imaging of extended regions,” Opt. Express **22**, 14859–14870 (2014). [CrossRef] [PubMed]

**7. **M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature **467**, 436–439 (2010). [CrossRef] [PubMed]

**8. **M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature **543**, 402–406 (2017). [CrossRef] [PubMed]

**9. **J. C. da Silva, P. Trtik, A. Diaz, M. Holler, M. Guizar-Sicairos, J. Raabe, O. Bunk, and A. Menzel, “Mass Density and Water Content of Saturated Never-Dried Calcium Silicate Hydrates,” Langmuir **31**, 3779–3783 (2015). [CrossRef] [PubMed]

**10. **R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A **317**, 319–340 (1970). [CrossRef]

**11. **R. Gordon, R. Bender, and G. T. Herman, “Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. **29**, 471–482 (1970). [CrossRef] [PubMed]

**12. **P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theor. Biol. **36**, 105–117 (1972). [CrossRef] [PubMed]

**13. **M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica **28**, 94–108 (2012). [CrossRef]

**14. **E. H. R. Tsai, I. Usov, A. Diaz, A. Menzel, and M. Guizar-Sicairos, “X-ray ptychography with extended depth of field,” Opt. Express **24**, 29089–29108 (2017). [CrossRef]

**15. **A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A **29**, 1606–1614 (2012). [CrossRef]

**16. **T. M. Godden, R. Suman, M. J. Humphry, J. M. Rodenburg, and A. M. Maiden, “Ptychographic microscope for three-dimensional imaging,” Opt. Express **22**, 12513–12523 (2014). [CrossRef] [PubMed]

**17. **A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, “High-resolution Multislice X-Ray Ptychography of Extended Thick Objects,” Phys. Rev. Lett. **112**, 053903 (2014). [CrossRef] [PubMed]

**18. **K. Shimomura, A. Suzuki, M. Hirose, and Y. Takahashi, “Precession x-ray ptychography with multislice approach,” Phys. Rev. B **91**, 214114 (2015). [CrossRef]

**19. **K. Shimomura, M. Hirose, and Y. Takahashi, “Multislice imaging of integrated circuits by precession X-ray ptychography,” Acta Crystallogr. Sect. A **74**, 66–70 (2018). [CrossRef]

**20. **H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica **5**, 601–607 (2018). [CrossRef]

**21. **P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports **8**, 2049 (2018). [CrossRef]

**22. **A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1262 (2009). [CrossRef] [PubMed]

**23. **Y. Chen, Y. Zhang, K. Zhang, Y. Deng, S. Wang, F. Zhang, and F. Sun, “FIRT: Filtered iterative reconstruction technique with information restoration,” J. Struct. Biol. **195**, 49–61 (2016). [CrossRef] [PubMed]

**24. **M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” Jounal Struct. Biol. **151**, 250–262 (2005). [CrossRef]

**25. **P. Penczek, M. Marko, K. Buttle, and J. Frank, “Double-tilt electron tomography,” Ultramicroscopy **60**, 393–410 (1995). [CrossRef] [PubMed]

**26. **M. Randermacher, “Three-Dimensional Reconstruction of Single Particles From Random and Nonrandom Tilt Series,” Jounal Electron Microsc. Tech. **9**, 359–394 (1988). [CrossRef]

**27. **I. Johnson, A. Bergamaschi, H. Billich, S. Cartier, R. Dinapoli, D. Greiffenberg, M. Guizar-Sicairos, B. Henrich, J. Jungmann, D. Mezza, A. Mozzanica, B. Schmitt, X. Shi, and G. Tinti, “Eiger: a single-photon counting x-ray detector,” J. Instrumentation **9**, C05032 (2014). [CrossRef]

**28. **R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. Sect. A **69**, 490–497 (2013). [CrossRef]