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Abstract
We propose a method of single-frame coherent diffraction imaging using a triangular aperture, which can not only reconstruct the projection image of extended objects from a single-frame coherent diffraction pattern, but also improve the image of the wavefield of the probe. In this method, a plane-wave illuminates a triangular aperture. An object is placed immediately after the aperture or in the image plane of the aperture through a lens. A far-field coherent diffraction pattern is collected by a two-dimensional detector. The object image is reconstructed from the single-frame diffraction pattern using a phase retrieval algorithm without support constraints. We simulate feasible experimental setups in the hard X-ray regime and show that this method can be practical use for single-frame coherent diffraction imaging. The present method has the potential exploring dynamic phenomena in materials science and biology with high spatiotemporal resolution using synchrotron radiation/free-electron lasers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Holography [1] and coherent diffraction imaging (CDI) [2,3] are coherent lensless imaging based on the collection of scattered intensity patterns and computerized image reconstruction. In holography, an interference pattern of reference and object waves, called a hologram, is recorded and the object image is reconstructed by applying a single Fourier transform to the hologram pattern. In CDI, an object is irradiated with a coherent beam with a well-defined wavefront, the diffraction pattern is recorded, and the object image is reconstructed from the diffraction pattern using an iterative phase-retrieval calculation. In particular, coherent lensless imaging in the X-ray regime is considered to be promising for visualizing samples at high spatial resolution beyond the performance limits of the lens [4]. One of the important applications of coherent lensless X-ray imaging is to visualize dynamic phenomena with high spatiotemporal resolution.
X-ray holography is suitable for observation at high temporal resolution, which can be classified into two main methods: in-line holography [5] and Fourier transform holography (FTH) [6]. For in-line holography, a reference light source is placed upstream of the sample. On the other hand, FTH has a reference light source placed in the same plane as the sample. The spatial resolution of both methods is basically limited by the size of the reference source. Thus far, biological imaging by in-line holography using a waveguide [7] and magnetic imaging by FTH using a small pinhole [8] have been reported. The use of an extremely small reference source decreases the flux of the reference wave and the signal-to-noise (S/N) ratio of the measured hologram pattern. Therefore, several ways of improving the S/N ratio of the hologram patterns in FTH using multiple pinholes [9], multiple dots [10], binary uniform redundant arrays [11], and sharp edges [12–14] as a reference have been proposed and experimentally demonstrated using X-rays. It has also been reported that the achievable spatial resolution of FTH exceeds the limits of nanofabrication accuracy by using an iterative phase retrieval algorithm with the support constraint [15]. However, these reports are not yet beyond the realm of proofs of principle. This is mainly because the sample and reference are on the same substrate, which makes sample preparation very complex.
X-ray CDI offers significant advantages over X-ray holography, as it allows imaging with dose-limited resolution [16]. Plane-wave CDI is a classical geometry of CDI [4], in which an isolated sample is irradiated by an X-ray plane-wave and the diffraction intensity pattern is collected. The projection image of the sample is reconstructed from the diffraction pattern using a phase retrieval calculation with the real space constraint that restricts the imaged object to a confined region called the “support”, that is, a support constraint. To date, plane-wave CDI based on single-frame data acquisition has been applied to the imaging of an isolated biological specimen using an X-ray free-electron laser [17,18]. One of the remaining challenges for CDI in the X-ray regime is to establish a single-frame CDI method for observing extended objects. Thus far, two approaches have been proposed and demonstrated. One approach is the production of a top-hat beam to use support constraints in the phase retrieval calculation. The X-ray beam is focused and the object is placed downstream of the focus so that it is illuminated by a divergent wave. It is possible to illuminate small isolated areas within a large object [19]. In addition, the use of apodizing slits can suppress the side-lobe intensity of the focused beam [20]. However, it remains difficult to produce the perfect top-hat beam. Another approach is the use of randomized illumination that removes the inherent ambiguities of CDI. A phase modulator was used to reconstruct the image of an extended object with a relaxed support [21]. This approach uses the phase and amplitude of the modulator’s transmission function as a priori information, which is measured with ptychographic CDI prior to single-shot measurement. This method also requires extremely high optical stability and is still in the development stage.
In this paper, we propose a practical method for single-frame CDI of extended objects. In this method, the sample is irradiated with a wavefield through a triangular aperture of low-symmetry shape with sharp line edges, the diffraction pattern is recorded in the far field, and the image is reconstructed using an iterative phase retrieval algorithm without support constraints. This method can be considered a fusion of in-line holography and CDI. Feasible experimental setups in the hard X-ray regime are simulated computationally. We show that the method can not only reconstruct the phase image of an extended object from a single-frame diffraction pattern, but also improve the image of the irradiated probe. Finally, we discuss the potential for imaging dynamic phenomena in materials science and biology using synchrotron radiation/free-electron lasers.
2. Principle of single-frame CDI
Figure 1 shows a schematic view of a single-frame CDI system. The $y$ direction is parallel to the optical axis. An aperture with diameter $w$ in the $x$-$z$ plane on the optical axis is introduced and the beam is cut out of the incident wavefront into an arbitrary shape. The beam propagates a distance $d$ and illuminates a portion of the extended object. Here, the exit wavefield $\psi (\textbf {r})$ of the sample can be represented as
where r denotes the real-space coordinate vector, $P(\textbf {r})$ the complex probe function on the sample, and $T(\textbf {r})$ the complex transmission function of the sample under the projection approximation. The exit wavefield of the sample $\psi (\textbf {r})$ produces a diffraction pattern in the far field, and a two-dimensional detector collects the diffraction intensity pattern downstream of distance $L$ from the sample plane. The diffraction intensity $I(\textbf {q})$ can be expressed as where $\textbf {q}$ denotes the reciprocal-space coordinate vector, and $\widetilde {\Psi }(\textbf {q})$ the wavefront in the detector plane. $\mathcal {F}$ represents the Fourier transform (FT) operator.For the image reconstruction, we employ a phase retrieval algorithm based on the reciprocal-space constraint and subsequent updates in the real space. Fig. 2 shows the flowchart of image reconstruction in single-frame CDI using the phase retrieval algorithm. First, the initial probe function $P_0(\textbf {r})$ is derived by propagating the exit wave of the aperture using the angular spectrum method (ASM) [22]. The initial transmission function $T_0(\textbf {r})$ is set to a uniform matrix. The exit wave function $\psi _j(\textbf {r})$ at the $j$th iteration cycle is the multiplication of $P_j(\textbf {r})$ and $T_j(\textbf {r})$, where $j = 0, 1, 2,\ldots ,$ and produces the diffraction wave $\widetilde {\Psi }_j(\textbf {q}) = \mathcal {F}[\psi _j(\textbf {r})]$ in the reciprocal space by its forward FT. Here, the reciprocal-space constraint is expressed as
The following steps are used to enforce improvements to the probe function, which is closely connected to the three-plane method [24]. First, the updated probe function $P^{\prime }_j(\textbf {r})$ propagates backward to the aperture: $\widetilde {P}^{\prime }_j(\textbf {r})$. Second, the wavefield behind the aperture $\widetilde {P}_j(\textbf {r})$ is updated according to the following equation:
3. Numerical simulation
3.1 Conditions
We performed numerical simulations of the proposed single-frame CDI with optical parameters for X-ray experiments. The incident X-ray energy was $5.0\,\rm {keV}$. The sample was the “Albert Einstein” image (©Yousuf Karsh) with 24-bit grayscale ranging from $-0.54$ to $0$ rad in 512$\times$512 pixels of a 10$\times$10 $\rm {\mu m^{2}}$ area, as shown in Fig. 3(a). A phase shift of $0.54\,\rm {rad}$ corresponds to a phase shift of $5\,\rm {keV}$ X-rays with respect to the $200\,\rm {nm}$ thickness of Tantalum (Ta), which is a typical specification of an X-ray test chart. For the image reconstruction with known probes, $\alpha$ and $\gamma$ were set to $0.8$ and $0.0$, respectively. For the image reconstruction with probe improvement, $\alpha$ and $\gamma$ were set at $0.2$ and $0.8$, respectively. The number of iterations was $10,000$ for all reconstruction calculations. We evaluated the reconstructed exit wave $\psi (\textbf {r})$ in the reciprocal space using the diffraction error $E_\textrm{diff}(j)$ of the $j$th iteration cycle that is given as
Fig. 3. Aperture-shape dependence in single-frame CDI. (a) Original sample image (left) and its magnified image in the center (right), which has a phase distribution of $-0.54$ to $0$ rad in $512\times 512$ pixels. (b) Images of five types of apertures with $12.6\,\rm {\mu m^{2}}$ area in the $10\times 10\,\rm {\mu m^{2}}$ area. (c) Images of the wavefields of the five probes on the sample plane that propagated $0.5\,\rm {mm}$ from the apertures. (d) Diffraction intensity patterns for each aperture with fixed dynamic range from $\rm {10^{0}}$ to $\rm {10^{8}}$ photons/pixel. (e) Reconstructed images for each aperture. The phase is displayed in the range from $-0.54$ to $0$ rad. (c–e) Each image in the same row shows the results for each aperture in the left, respectively. (f) Progress of the diffraction error $E_\textrm{diff}$ for $10,000$ iterations. (g) Fourier ring correlation plot of the reconstructed images for each aperture. The scale bars in (a), (b), (c), and (e) indicate the length of $2\,\rm {\mu m}$, and in (d), the spatial frequency of $10\,\rm {\mu m^{-1}}$.
3.2 Aperture-shape dependence
We prepared five types of aperture: an equilateral triangle with 5.4-$\rm {\mu m}$-long edges, a square with 3.5-$\rm {\mu m}$-long edges, a pentagon with 2.7-$\rm {\mu m}$-long edges, a nonagon with 1.4-$\rm {\mu m}$-long edges, and a circle of 4.0-$\rm {\mu m}$ diameter, that have the same area of 12.6 $\rm {\mu m^{2}}$ as shown in Fig. 3(b). $d$ and $L$ were 0.5 mm and 3 m, respectively. Figure 3(c) shows the images of the wavefields of the five probes. The propagated probes showed typical features of Fresnel diffraction: the phase distribution is smooth, and the intensity distribution is jagged and lazily decays at the beam end. Figure 3(d) shows the calculated diffraction patterns, in which photon shot noises were induced in diffraction intensity, and the dynamic range of the diffraction pattern was fixed at 1 photon/pixel : $\rm {10^{8}}$ photons/pixel. Figure 3(e) shows the reconstructed images for each aperture, in which the probes were assumed to be known. The reconstructed image for the triangular and pentagonal aperture represented a similar contrast with the original sample image in the illuminated area. In contrast, the reconstructed images for the squared and circular apertures showed many artifacts, and thus the feature of the original image cannot be recognized. The reconstructed image of the nonagonal aperture also showed some artifacts, but are fewer than the image of the squared and circular ones. Figures 3(f) and 3(g) show the progress of the diffraction error and the FRC plot of the reconstructed images for each aperture, respectively. The triangular and pentagonal aperture shows the low diffraction error and the large FRC values in the entire spatial frequency range than the other three apertures. When the squared and circular apertures are used, the FRC values in the low spatial frequency range are smaller. The nonagonal aperture shows similar trends of the FRC values with the sqaured and circular ones in the low spatial frequency range, however, it also shows high values like the triangular and pentagonal ones in the high spatial frequency range.
The present result indicates two important findings; One is, the aperture shape with no point-symmetry achieved the object image reconstruction. Even with non-point symmetry, artifacts appear as the image becomes closer to circular. The other is, when the triangular and pentagon aperture is used, the image of an object (exit wavefield) is reconstructed from the modulus of its Fourier transform without using a support constraint. It is considered that the in-line hologram works to reconstruct the image and its convergence is strongly related to the aperture shape. Therefore, we selected a triangle shaped aperture, which is the simplest non-point symmetrical and most different from the circle, as a useful aperture candidate for the single-frame CDI.
3.3 Oversampling requirement
We evaluated the $d$-value dependence in the use of a triangular aperture to investigate oversampling requirement. Figure 4(a) shows the reconstructed images when $d$ is 0 mm, 0.1 mm, 0.5 mm, 1.0 mm, and 5.0 mm. As the value of $d$ increases, the quality of the reconstructed image decreases. Here, the sample image is reconstructed in a region that roughly satisfies $|P(\textbf {r})|/|P(\textbf {r})|_\textrm{max}>0.03$, which can be regarded as an effective irradiation region as shown in Fig. 4(b). We introduce the oversampling ratio ($\sigma$), which is defined as
Fig. 4. Oversampling requirement in single-frame CDI using a triangular aperture. (a) $d$-value dependence of reconstructed images. (b) Effective irradiation region of illumination function for each $d$ value where $|P(\textbf {r})|/|P(\textbf {r})|_\textrm{max}>0.03$. The scale bars in (a) and (b) indicate the length of 2 $\rm {\mu m}$.
3.4 Probe improvement
Next, we improved the probe function by using the algorithm described in Fig. 2. The blurred probe functions were prepared as shown in Fig. 5(a), which convoluted the ideal probe with a Gaussian function with a standard deviation of $19.4\,\rm {nm}$ corresponding to 1 pixel in the image, $58.2\,\rm {nm}$ to 3 pixels, $97.0\,\rm {nm}$ to 5 pixels, and $194\,\rm {nm}$ to 10 pixels from upper to bottom, respectively. Figure 5(b) shows the sample image reconstructed from the diffraction pattern using a triangular aperture shown in Fig. 3(c) without improving the blurred probes. The sample image is poorly reconstructed, and the reconstructed image is degraded with respect to the blurring of the probe function. These results indicate that the image reconstruction in this algorithm is very sensitive to the accuracy of the probe function.
Fig. 5. Probe improvement in single-frame CDI. (a) Image of blurred triangle probe function with a Gaussian function with a standard deviation of $19.4\,\rm {nm}$, $58.2\,\rm {nm}$, $97.0\,\rm {nm}$ and $194\,\rm {nm}$ corresponding to 1, 3, 5, and 10 pixels in the image from upper to bottom, respectively. (b) Reconstructed sample image from the diffraction intensity in Fig. 3(c) without improving each probe function of (a), respectively. (c) Image of the improved probe function. (d) Reconstructed sample image from the diffraction intensity in Fig. 3(c) with improving each probe function of (a), respectively. (e) Progress of the diffraction error $E_\textrm{diff}$ for reconstructions with using the ideal probe function and with/without improving the blurred probe function. (f) Fourier ring correlation plot for reconstructions with using the ideal probe function and with/without improving the blurred probe function. (e–f) The case of probe convolved with a Gaussian with a standard deviation of 194 nm is presented. The scale bars in (a), (b), (c), and (d) indicate the length of 2 $\rm {\mu m}$.
Figures 5(c) and 5(d) show the probe and sample images reconstructed from the diffraction pattern using a triangular aperture shown in Fig. 3(c) with improving each blurred probes, respectively. All the images are clearly reconstructed. Here, the update weights $\alpha$ and $\gamma$ in Eq. (4) and (5) were set at $0.2$ and $0.8$, respectively, as addressed in the Section 3.1. We determined these values empirically in the simulation. The high weight for probe update such as $0.8$ helps the probe reconstruct to the optimum solution at a very early stage. It is crucial for the object function reconstruction. Moreover, it shows a tendency to interrupt the probe reconstruct at an early iteration stage when we give too high weight of object update, which becomes a fatal reason for the object reconstruction failure. Figures 5(e) and 5(f) show the progress of the diffraction error and the FRC plot of the reconstructed images with/without improving the probe, respectively. Here, we present the case of the probe blurred with a Gaussian with a standard deviation of 194 nm (10 pixels) for an example. The diffraction error is reduced by improving the probe and the FRC value is close to that of the known probe. It is clear that the present algorithm for probe improvement works well.
In real experiments, a fairly accurate probe function can be derived by ptychographic CDI. The present algorithm could be useful when the flux of incident X-rays fluctuates during single-frame CDI measurements and cannot be monitored accurately.
3.5 Accuracy of triangular aperture
Next, we investigated the fabrication accuracy required for the triangular aperture for single-frame CDI. Here, for simplicity, the probes were simulated as being known. In practice, it is difficult to create a triangular aperture with an ideal shape. For example, when a triangular aperture is formed by processing a metal foil with a focused ion beam (FIB), the corners become rounded and the cross section of the aperture edge becomes smooth.
First, we investigated whether sharp corners are required. Figures 6(a) and 6(b) show the image of the triangular aperture with a corner of $0.5\,\rm {\mu m}$ curvature radius and the image of the probe function in the sample plane, respectively. The curvature radius of $0.5\,\rm {\mu m}$ can be achieved by FIB processing. Figure 6(c) shows the calculated diffraction patterns with noises in the same dynamic range of Fig. 3(d). Figure 6(d) shows the sample image reconstructed using the triangular aperture with the rounded corners, providing a similar contrast to the image in Fig. 3(e). The diffraction error and the FRC plot for the apertures with rounded corners are comparable to those for the apertures with sharp corners, as shown in Figs. 6(e) and 6(f). It can be seen that the sharpness of the corners of the triangular aperture does not affect the reconstruction.
Fig. 6. Image reconstruction in single-frame CDI using a triangular aperture with rounded corners. (a) Image of triangular aperture with a corner of $0.5\,\rm {\mu m}$ curvature radius. (b) Image of the probe function at the sample plane. (c) Diffraction intensity pattern with fixed dynamic range from $10^{0}$ to $10^{8}$ photons/pixel. (d) Reconstructed sample image in the range from $-0.54$ to $0$ rad. (e) Progress of $E_\textrm{diff}$ for a triangular aperture with and without rounded corner. (f) Fourier ring correlation plot for a triangular aperture with and without rounded corner. The scale bars in (a), (b), and (d) indicate the length of $2\,\rm {\mu m}$, and in (c), the spatial frequency of $10\,\rm {\mu m^{-1}}$.
Second, we investigated whether sharp edges are required. Figures 7(a) and 7(b) show the images of the triangular apertures with three types of smoothed edge and their probe function images in the sample plane, respectively. The edges were smoothed by convolution using a Gaussian function with standard deviations of $19.4\,\rm {nm}$ corresponding to 1 pixel in the aperture plane image shown in the upper image, $58.2\,\rm {nm}$ (3 pixels) in the middle image, and $97.0\,\rm {nm}$ (5 pixels) in the bottom image. Figure 7(c) shows the calculated diffraction patterns with noises in the same range with Fig. 3(d). It is clear that the long flares seen in Fig. 3(d) drastically decreased as the amount of smoothing at the aperture edges increases. Figures 7(d)–7(f) show the reconstructed images, the diffraction errors, and the FRC plot for each aperture, respectively. The sharper the edges, the smaller the diffraction errors, and the higher the resolution of the reconstructed image as evident from the FRC plot. The FRC values significantly decreased to 0 at nearly the full width at half maximum (FWHM) of the convolved Gaussian function, with standard deviations of $19.4\,\rm {nm}$ for FWHM of $45.7\,\rm {nm}$, $58.2\,\rm {nm}$ for $137\,\rm {nm}$, and $97.0\,\rm {nm}$ for $228\,\rm {nm}$ in the spatial resolution. These results indicate that there is a strong correlation between the edge sharpness of the aperture and the spatial resolution of the reconstructed images.
Fig. 7. Image reconstructions in single-frame CDI using triangle apertures with smoothed edges. (a) Images of the triangular apertures with three types of smoothed edge: upper, middle, and bottom images show $19.4\,\rm {nm}$ (1 pixel), $58.2\,\rm {nm}$ (3 pixels), and $97.0\,\rm {nm}$ (5 pixels) as standard deviations of Gaussian functions, respectively. All of the corners are rounded with a curvature radius of $0.5\,\rm {\mu m}$. (b) Image of the probe function of each aperture at the sample plane. (c) Diffraction intensity patterns of each aperture with fixed dynamic range from $10^{0}$ to $10^{8}$ photons/pixel. (d) Reconstructed sample images in the range from $-0.54$ to $0$ rad. (e) Progress of $E_\textrm{diff}$ for reconstructions using the triangular apertures of three types of smoothed edge. (f) Fourier ring correlation plot for reconstructions using the triangular apertures of three types of smoothed edge. The scale bars in (a), (b), and (d) indicate the length of $2\,\rm {\mu m}$.
Third, we investigated the misalignment tolerance of the aperture perpendicular to the X-ray incident beam. Not only the aperture shape itself but also the position can significantly affect the creation of probe function in the experiment. We performed a numerical simulation to give an aperture as a 20 $\rm {\mu m}$–thick Platinum (Pt) plate and tilt it 1 degree against the $z$ direction, in which the inner wall of the aperture becomes a taper along with the aperture’s thickness. The absorption of 5 keV X-rays was calculated to be similar to the edge-smoothed aperture using a Gaussian with a standard deviation of 14 nm. According to the calculation, we expect that the misalignment over $\pm 1$ degree from the perpendicular direction of the incident beam causes a similar resolution degradation with the upper image of Fig. 7(d).
4. Discussion
The proposed single-frame CDI has the following important features.
1. The sample image is reconstructed when the shape of the aperture is not point symmetrical.
2. The resolution of the reconstructed image is improved the sharper the edges of the aperture.
3. No real space constraints, $i.e.$, no support constraints, are used in the image reconstruction.
Features 1 and 2 are similar to those in image reconstruction from the magnitude of the Fourier transform using support constraints in plane-wave CDI [27,28]. In a separate simulation performed, the image was reconstructed even when the aperture was regular pentagon. However, the convergence was reduced compared to the triangular aperture. Since regular polygons approach a circle as the number of sides increases, convergence should be better with a smaller number of sides. Feature 3 means that phase information is recorded as an in-line hologram in the diffraction pattern. Therefore, the present method can be considered a fusion of in-line holography and CDI. In this method, the sample and aperture are separated, and sample preparation is easier than in the FTH method.
One important application of single-frame CDI is in imaging dynamic phenomena in materials science and biology using synchrotron X-ray/X-ray free-electron lasers, where increasing the flux density of the incident coherent X-rays is necessary to improve the spatiotemporal resolution. Here, we propose an experimental arrangement for a single-frame CDI installed with imaging optics. Figure 8(a) shows a schematic view of single-frame CDI with an imaging system as a conceptual diagram of the arrangement. The size of the aperture ($w$) should be equal to the spatial coherence length of the incident X-rays. The imaging device such as Fresnel zone plate (FZP) is used to reduce the size of the aperture ($w'/w<1$), and the sample is placed on the imaging plane. For example, the triangular aperture of $10\,\rm {\mu m}$ per side is reduced to $5.4\,\rm {\mu m}$ per side, which is the same size of the triangular aperture in Fig. 3(b), where the number of lens apertures is $0.0012$ corresponding to the FZP with an outermost line width of 100 nm, and $a/b$ is $1.86$. The diffraction intensity pattern is then acquired under the same conditions as in the previous simulation. Figures 8(b) and 8(c) show the reconstructed image and its FRC plot, respectively. The object image can be seen to be almost completely reconstructed. It is noteworthy that the resolution of the reconstructed image is not limited by the number of lens apertures. Therefore, it can be assumed that only the sharpness of the aperture edge determines the lower limit of resolution, even when a focused beam is used.
Fig. 8. Single-frame CDI installed with imaging optics. (a) Schematic view of single-frame CDI with an imaging system. (b) Reconstructed image using a triangular aperture imaging probe. The scale bar indicates the length of $2\,\rm {\mu m}$. (c) Fourier ring correlation plot for the reconstructed image in (b).
5. Conclusion
We have proposed a method of single-frame CDI using a triangular aperture that can not only reconstruct the projection image of extended objects from the single-frame coherent diffraction pattern, but also improve the image of the wavefield of the probe. In this method, the sample image is reconstructed from a single-frame diffraction intensity pattern without real-space constraints. The reconstruction mechanism is not well understood, but may involve the symmetry of the aperture and the sharpness of its edges. This technique is expected to be applied to various nanoscale observations using synchrotron radiation and free-electron lasers. For example, using a free-electron laser, we can observe localized areas of living cells. In addition, synchrotron radiation allows us to dynamically observe changes in the nanostructure of a material. We believe that in the near future, this method will be experimentally demonstrated, and become applicable in the fields of biology and materials science.
Funding
Japan Society for the Promotion of Science (JP18H05253, JP19H05814, JP20K15375, JP20K20523); Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT).
Acknowledgments
We thank M. Hirose and T. Ishikawa for stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
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