Dark-field phase retrieval under the constraint of the Friedel symmetry in coherent X-ray diffraction imaging

Amane Kobayashi; Yuki Sekiguchi; Yuki Takayama; Tomotaka Oroguchi; Masayoshi Nakasako

doi:10.1364/OE.22.027892

1. Introduction

Coherent X-ray diffraction imaging (CXDI) is a technique for analyzing the structures of non-crystalline particles with sub-micrometer to micrometer sizes at a resolution of a few tens of nanometers [1–3]. In CXDI experiments, an isolated particle is irradiated by a spatially coherent X-ray beam, and the Fraunhofer diffraction pattern is recorded under the oversampling (OS) condition [4]. The electron density map is then, in principle, reconstructed by applying phase-retrieval (PR) algorithms to the oversampled diffraction pattern [5–8]. Several CXDI studies have analyzed structures of non-crystalline particles from material science and biology using third-generation synchrotron radiation sources [9–18]. Penetrating deeply into materials, X-rays with short wavelengths enable us to visualize the internal structures of thick samples at a resolution exceeding optical microscopy’s limit.

In recent years, CXDI experiments using X-ray free-electron laser (XFEL) sources have commenced at the Linac Coherent Light Source [19] and the SPring-8 Angstrom Compact free electron LAser (SACLA) [20]. The XFEL sources provide completely coherent X-ray pulses with a duration of tens of femtoseconds. In CXDI experiments using XFEL (XFEL-CXDI), diffraction patterns are recorded using single shots under the “diffraction-before-destruction” scheme [21]. The potential of the XFEL-CXDI technique is demonstrated with structural analyses of non-crystalline particles, such as viruses [22], soot [23], cellular organelles [17], and bacterial cells [24]. In XFEL-CXDI experiments, a set of thousands of high-quality diffraction patterns collected within a few hours provides the size distribution of sample particles in addition to the structures of individual particles [25, 26].

However, still serious issues occur in CXDI structural analyses. One such issue is PR from diffraction patterns that miss data in small-angle regions. Because the direct beam is intense to cause fatal damage in X-ray area detectors, the beam must be blocked by a beam stopper. In addition, in the small-angle region where diffraction intensities are very strong, detector pixels are saturated and cannot record diffraction data. In XFEL-CXDI experiments, because the XFEL pulse destroys sample particles, the diffraction pattern of a particle can be recorded only once. Consequently, even when saturation occurs in detector pixels, it is impossible to retake the data for these pixels with appropriate attenuation. Diffraction patterns in the small-angle regions contain structural information regarding the overall shape of and total electrons in sample particles. Therefore, the absence of data in the small-angle region makes it difficult to retrieve the electron density maps of sample particles. To overcome this difficulty, the dark-field phase-retrieval (DFPR) method has been introduced for structural analysis in an XFEL-CXDI experiment [27]. The DFPR method estimates the overall shape of a sample particle from a diffraction pattern with data missing in the small-angle region, and this method has been applied to structural analyses of soot particles.

Recently, we have started XFEL-CXDI experiments for non-crystalline particles with sub-micrometer to micrometer dimensions. A huge number of single-shot diffraction patterns are collected using our custom-made diffraction apparatus KOTOBUKI-1 [17] [Fig. 1(a)]. In our experiments, saturation of detector pixels in the small-angle region is frequently observed, particularly in diffraction patterns from aggregates of nano-metal particles and cellular organelles with large scattering cross-sections [Fig. 1(b)]. Because we use an X-ray with a short wavelength (2.25 Å), the Ewald sphere is approximated as a plane up to a 13-nm resolution for 250-nm particles, for example [28]. Therefore, the collected diffraction patterns satisfy Friedel centrosymmetry up to the resolution when X-ray absorption is negligible [Fig. 1(c)].

Fig. 1 (a) Schematic illustration of our XFEL-CXDI experiments using the KOTOBUKI-1 diffractometer and the two MPCCD detectors. The raster scan of each specimen disk using the goniometer stage is controlled by the IDATEN software suite. The white dotted circle on the diffraction pattern in the MPCCD-Octal detector represents a border, inside which the Ewald sphere is approximated as a plane. (b) A typical diffraction pattern from a cluster of gold colloidal particles after merging the diffraction patterns recorded in the two MPCCD detectors using the G-SITENNO data processing suite. Even using the aluminum attenuator with a 50-μm thickness, the diffraction pattern in the small-angle region of S < 5 μm⁻¹ in the MPCCD-Dual detector exceeds the saturation limit of approximately 2500 X-ray photons at 5.5 keV/pixel. In the vertical and horizontal stripes with widths of approximately 5 pixels, diffraction patterns are missed because of gaps between the detector panels or between the non-sensitive frame regions of the detector panels. (c) An illustration of parameters for defining the resolution limit where the Ewald sphere is approximated as a plane.

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To explicitly apply our experimental condition to the DFPR method, we propose using the Friedel symmetry of diffraction patterns as an additional constraint for the electron density maps in the iterative PR calculations. In the present study, we conducted a set of model simulations to examine the effectiveness of this constraint. In addition, to evaluate the practical performance of this method in structural analyses of non-crystalline particles, the proposed method was applied to diffraction patterns collected in XFEL-CXDI experiments.

2. Theoretical background

This section first provides a brief overview of the theoretical basis for the DFPR method, which was developed by Martin et al. [27]. We then explain how we incorporate Friedel symmetry into the DFPR method as a constraint for calculating PR. Finally, we describe the computational protocols for PR that were used throughout this study.

2-1. Theory of DFPR

Here, we treat diffraction from a non-crystalline particle with electron density $ρ (r)$ , which is a real number. In the DFPR method, a diffraction pattern is prepared by multiplying a Gaussian mask with an observed diffraction pattern in the high-angle region, where the diffraction intensity collected by a detector is not saturated. The Gaussian mask is defined as

M (S) = \exp [- {(S - α)}^{2} χ^{2} / 2],

where

S

is a scattering vector. The length of

S

is expressed as

| S | = 2 \sin θ / λ

using diffraction angle

2 θ

and the wavelength of X-ray

λ

. As illustrated in Fig. 2(a), vector

α

specifies the peak position, and

χ

represents the spatial extent of the mask. The Gaussian mask reduces the diffraction intensity around the missing area to almost zero.

Fig. 2 (a) Three types of masks used in this study. The single mask in the left panel is used in the DFPR protocol. The Fs-DFPR protocol uses the symmetry mask in the center panel and the annular mask in the right panel. The details of the parameters are described in Section 2-2. (b) The PR protocols for reconstructing the electron density maps examined in the present study.

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From the masked diffraction pattern, an electron density map $ρ_{dark} (r)$ , which is referred to as “dark image”, is reconstructed using the iterative PR method. $ρ_{dark} (r)$ is a complex number because it is the convolution of $ρ (r)$ and the complex point-spread function [the inverse Fourier transform of $M (S)$ ], expressed as

ρ_{dark} (r) = \exp (- \frac{2 π^{2} r^{2}}{χ^{2}}) \exp (- 2 π i r \cdot α) \otimes ρ (r) .

Here, we designate this mask

M (S)

as “single mask”, which is used in the DFPR method. In

ρ_{dark} (r)

, components of

ρ (r)

with high spatial-frequencies are significantly enhanced.

2-2. The DFPR method under the constraint of Friedel symmetry

In the absence of anomalous scatterers, the distribution of diffraction intensity from the sample particle displays Friedel centrosymmetry. Experimentally observed diffraction patterns are the part of three-dimensional diffraction intensities, which intersect the Ewald sphere [Fig. 1(c)]. The centrosymmetry in diffraction patterns is observed up to a resolution of $S_{max}$ , which satisfies the following equation:

S_{sep} = \frac{1}{λ} - {(\frac{1}{λ^{2}} - S_{\max}^{2})}^{\frac{1}{2}} < \frac{1}{d},

where

d

is the dimension of the particle in the direction of the incident X-ray.

S_{sep}

is the distance between point

S_{max}

on the Ewald sphere and the plane normal to the incident X-ray.

We now introduce Friedel symmetry into the DFPR method, and we designate this version of the DFPR method as Fs-DFPR. In the Fs-DFPR method, we use a mask named “symmetry mask” $M_{sym} (S)$ [Fig. 2(a)], which comprises a pair of two symmetry-related single masks, $M (S)$ and $M (- S)$ . $M_{sym} (S)$ is given by

M_{sym} (S) = M (S) + M (- S) = \exp [- {(S - α)}^{2} χ^{2} / 2] + \exp [- {(S + α)}^{2} χ^{2} / 2] .

Dark image

ρ_{dark}^{sym} (r)

is then the convolution of

ρ (r)

and the point-spread function [the inverse Fourier transform of

M_{sym} (S)

], expressed as

ρ_{dark}^{sym} (r) = 2 \exp (- \frac{2 π^{2} r^{2}}{χ^{2}}) \cos (2 π r \cdot α) \otimes ρ (r) .

The centrosymmetry of

M_{sym} (S)

ensures that the point-spread function is a real number, and therefore, dark image

ρ_{dark}^{sym} (r)

becomes a real number. It should be noted that

ρ_{dark}^{sym} (r)

is not positive definite.

A larger mask is likely more advantageous for increasing the quantity of diffraction data available in the PR calculation [27]. Here, we prepare a large mask by arranging several pairs of symmetry masks into an annular shape [Fig. 2(a)]. “Annular mask” $M_{annular} (S)$ is expressed using $M_{sym} (S)$ as

M_{annular} (S) = \int_{| α |} \exp [- {(S - α)}^{2} χ^{2} / 2] d α = \frac{1}{2} \int_{| α |} M_{sym} (S) d α .

The annular mask is defined by only two scalar parameters:

| α |

and

χ

.

Because of Poisson noise in the detection of X-ray photons, experimental diffraction patterns approximately satisfy Friedel symmetry. Thus, we would examine the feasibility of the Fs-DFPR method with respect to the robustness against the increase in detector noise.

2-3. Implementation of Fs-DFPR

In the DFPR method, we apply the following constraint to both the real and imaginary parts of dark image $ρ_{dark} (r)$ in the Hybrid-Input-Output (HIO) iteration [5]:

ρ_{k + 1} (r) = {\begin{cases} ρ'_{k} (r) r \in support \\ ρ_{k} (r) - β ρ'_{k} (r) otherwise \end{cases},

where

ρ_{k} (r)

is a dark image at the beginning of the k^th cycle.

ρ'_{k} (r)

is the inverse Fourier transform of the structure factor with the amplitude of the masked diffraction pattern and phase from

ρ_{k} (r)

.

ρ_{k + 1} (r)

is the dark image for the next cycle. Weight

β

is fixed at 0.9 throughout the HIO iteration. In contrast, in the Fs-DFPR method, because a dark image must be reconstructed as a real number [Eq. (5)], the imaginary part of dark image

ρ_{dark}^{sym} (r)

is constrained to be zero, and the original constraint in Eq. (7) is applied only to the real part of

ρ_{dark}^{sym} (r)

.

To process masked diffraction pattern under the constraint of Friedel symmetry in the Fs-DFPR calculation, we modified our custom-made PR software ZOCHO [26], which uses the HIO algorithm in combination with the Shrink-Wrap (SW) algorithm [29]. In the modified version of the PR software, a two-step procedure is applied to the diffraction pattern with data missing in the small-angle region [Fig. 2(b)]. In the first step, we reconstruct a dark image from a random density map by applying the DFPR or the Fs-DFPR calculation to the diffraction pattern, which has been multiplied by a mask function. In the second step, using the dark image as the initial density map, the electron density map of the particle is retrieved from the observed diffraction pattern. Hereafter, we will refer to the second step as bright-field phase retrieval (BFPR).

3. Setup of model simulation

We first conducted a series of model simulations to examine whether the Fs-DFPR method is feasible for reconstructing correct electron density maps from diffraction patterns with data missing in small-angle regions. As the model structure for the simulations, we used the crystal structure of protein disulfide isomerase (PDI, Protein Data Bank accession code 2B5E) [30] [Fig. 3(a)]. According to the procedure and conditions reported previously [31], a diffraction pattern with dimensions of 1024 × 1024 pixels was calculated up to a resolution of 2.1 Å at the edge. The OS ratio of the calculated pattern was 64 [Fig. 3(b)]. The wavelength was assumed to be 0.03 Å to satisfy Eq. (3) at the corner of the diffraction pattern.

Fig. 3 (a) Crystal structure model of PDI illustrated with the space-filling model. When calculating a diffraction pattern, an X-ray beam is assumed to be incident to the model along the direction of the z-axis. (b) A calculated diffraction pattern for the incident X-ray intensity of 9.39 × 10²⁷ photons/μm²/pulse. (c) The green line profile of the diffraction intensity along the black line in panel (b) demonstrates the effect of Poisson noise at the noise level of NL1. The other two line profiles are from diffraction patterns calculated for incident X-ray intensities of 9.39 × 10²⁶ photons/μm²/pulse (red) and 9.39 × 10²⁵ photons/μm²/pulse (blue). The line profiles of the former (with the noise level of NL2) and the latter (with NL3) are helpful for understanding the influence of Poisson noise on recorded diffraction patterns. (d) Three missing areas (Cut2, Cut5, and Cut7) are shown for the calculated diffraction pattern in the small-angle region, which is indicated by the red box in panel (b). (e) Examples of masked diffraction patterns used in the DFPR and Fs-DFPR calculations.

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We prepared Poisson-noise-smeared diffraction patterns of PDI with three different incident X-ray intensities of 9.39 × 10²⁷, 9.39 × 10²⁶, and 9.39 × 10²⁵ photons/μm²/pulse, which were designated Noise Level 1 (NL1), NL2, and NL3 [Fig. 3(c)], respectively. In addition, for each NL1, NL2, and NL3 pattern, we prepared three diffraction patterns (Cut2, Cut5, and Cut7) with areas of missing data with different sizes in the small-angle region [Fig. 3(d)]. The Cut2, Cut5, and Cut7 patterns lacked speckle peaks in the small-angle region, up to approximately the 2nd, 5th, and 7th layers from the center.

For each of the nine different patterns, we applied four types of PR protocols. One protocol was the HIO-SW protocol applied to diffraction patterns without masks. The protocol was already implemented in ZOCHO and started with a random electron density map. Hereafter, we will refer to this protocol as PRZ. The other three protocols were BFPR calculations, for which the initial electron density maps were dark images given by the DFPR method, the Fs-DFPR method using the symmetry mask, and the Fs-DFPR method using the annular mask [Figs. 2 and 3(e)]. The χ values of the mask functions in Eqs. (1), (4), and (6) were fixed at 5.9 Å. For the single and symmetry masks, $α = (α_{x}, α_{y})$ in Eqs. (1) and (4) was set to be (−0.20, 0.20) Å⁻¹. For the annular mask, $| α |$ in Eq. (6) was set to be 0.28 Å⁻¹. To determine which PR protocols most efficiently reconstructed the correct electron density maps from the given diffraction patterns, we prepared ten different random electron density maps as starting models and subjected them to each PR protocol.

In the PRZ, DFPR, and Fs-DFPR calculations, the support was initially given by the autocorrelation function, which was calculated from the diffraction pattern. In the BFPR calculations, the support was initially provided by the dark images. In each PR calculation, we updated the support by the SW in every set of 100 HIO cycles. After updating the support 100 times, the electron density map was refined with the final updated support with 1000 HIO iterations [28]. Because the dark image is not positive definite, we set a number of HIO iterations in this study larger than that in our previous study [31], for the convergence to a correct shape.

All calculations were performed on a supercomputer system called SACLA-HPC, which is composed of 960 Intel Xeon CPU X5690 cores (3.47 GHz/core). The process of conducting 360 different PR calculations spanned six days when 60 cores were used independently.

4. Experimental procedure

To apply the Fs-DFPR method to experimental diffraction patterns, we collected diffraction data from gold colloidal particles. Because gold colloidal particles have sufficient scattering cross-sections to provide diffraction patterns with a good signal-to-noise (S/N) ratio [17], intense diffraction from the aggregates frequently causes the saturation of detector pixels in the small-angle region. In this section, we describe the details of sample preparation, diffraction data collection, and data analysis procedures.

4-1. Sample preparation

Gold colloidal particles with 250-nm diameters (British BioCell International Solutions, U.K.) were dispersed at a number density of approximately 10 particles/10 μm² on a carbon membrane with a thickness of 25 to 30 nm [Fig. 1(a)]. The membrane was glued to a stainless disk with a 3-mm diameter (Okenshoji, Japan) to cover the 300-μm pinhole [16]. Before conducting the diffraction experiments, we collected images of the specimen disks using scanning electron microscopy (SEM), TM3000 (Hitachi High-Tech, Japan).

4-2. Diffraction data collection

We conducted an XFEL-CXDI experiment for gold colloidal samples using the KOTOBUKI-1 apparatus at SACLA [Fig. 1(a)] [17]. X-rays with a wavelength of 2.25 Å were focused at the specimen position with a spot size of approximately 2 × 2 μm² at the full width at half maximum [17, 32]. Then, the incident X-ray intensity was approximately 10¹⁰ photons/2 × 2 μm²/pulse with a 10-fs duration. We collected diffraction patterns with raster scans for each specimen disk at a step size of 50 μm/pulse using the IDATEN software suite to control the goniometer [33]. Diffraction patterns were recorded by the two MPCCD detectors [34]. Placed 1.6 m downstream from the sample, the MPCCD-Octal detector recorded diffraction patterns in the resolution range of 210 to 7 nm. The MPCCD-Dual detector 1.6 m downstream from the MPCCD-Octal detector covered the resolution range of 500 to 210 nm. An aluminum attenuator with a 50-μm thickness (with a transmittance of 14.8% for 2.25-Å X-rays) and a direct beam stopper with dimensions of 2 × 2 mm² were placed in front of the MPCCD-Dual detector.

4-3. Data processing

Using our custom-made data processing software suite, G-SITENNO [26, 33], we merged diffraction patterns recorded by the pair of the MPCCD detectors. The Friedel symmetry of a diffraction pattern was evaluated using the following correlation function:

\begin{array}{l} C_{sym} = \frac{E - O}{E + O}, \\ E = \sum_{x, y} {[I_{0} (x, y) + I_{sym} (- x, - y)]}^{2}, \\ O = \sum_{x, y} {[I_{0} (x, y) - I_{sym} (- x, - y)]}^{2}, \end{array}

where

I_{0} (x, y)

is the diffraction intensity in the region of interest (ROI) with 100 × 100 pixels and

I_{sym} (- x, - y)

is the diffraction intensity of the Friedel mate. The

C_{sym}

value of a diffraction pattern with ideal Friedel symmetry is 1.

The electron density maps of sample particles projected along the direction of the incident X-ray beam were reconstructed using the Fs-DFPR method with an annular mask followed by the BFPR method. For PR, the diffraction patterns processed by G-SITENNO were trimmed to a size of 512 × 512 pixels. The resolution of the trimmed pattern was 29 nm at the edge. The reconstructed electron density map was compared with the SEM image to determine the success or failure of the reconstruction.

5. Results

5-1. Reconstruction of dark images in the model simulations

Composed of four globular domains, PDI has a characteristic overall shape resembling the letter “J” [Fig. 3(a)]. Based on this structural characteristic, we classified the dark images into Type A, Type B, Type C, and Type D [Fig. 4(a)]. Type-A images display molecular shapes consistent with the shape of PDI. Type-B images are similar to the shape of PDI, but the sizes of the images vary up to ± 10% from the size of PDI. Type-C images have overall shapes resembling the PDI model, but these images deviate from the J-shape because of significant electron densities appearing in the cleft surrounded by the four globular domains. The overall shape of Type-D images is ellipsoid with the dimensions of PDI. We assessed the efficiency of DFPR and Fs-DFPR by inspecting how many correct dark images were retrieved [Figs. 4(b)-(d)].

Fig. 4 Results of a set of model simulations that were used to compare the efficiencies of the DFPR and Fs-DFPR protocols. (a) Representative Types-A to -D dark images retrieved from the simulated diffraction patterns by the DFPR and Fs-DFPR calculations. These images are distinguished by the color of the name labels. The scale bars indicate 20 Å. In panels (b), (c), and (d), the results of PR calculations for diffraction patterns from Cut2, Cut5, and Cut7 are summarized, respectively. As shown in panels (b)-(d), each protocol (DFPR, Fs-DFPR using a symmetry mask, and Fs-DFPR using an annular mask) is applied to the diffraction patterns with noise levels of NL1, NL2, and NL3. The number of retrieved Types-A to -D dark images is indicated by the length of the histogram bars. The histogram bars for Types-A to -D images are distinguished according to the color scheme of the name labels in panel (a).

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All protocols retrieved Type-A images from Cut2 diffraction patterns with noise levels of NL1 and NL2. For the Cut2 pattern at NL3, DFPR reconstructed no dark images of Type A and Type B, while Fs-DFPR produced Type-B dark images [Fig. 4(b)]. From Cut5 patterns at NL2 and NL3 and from all Cut7 patterns, the DFPR method provided only Type-D dark images [Figs. 4(c) and (d)]. In contrast, Type-B dark images were retrieved with Fs-DFPR for Cut5 patterns at NL2 and NL3. For Cut7 patterns at NL1 and NL2, Fs-DFPR reconstructed Type-A and Type-B dark images with an 82.5% probability. Even for the Cut7 pattern at NL3, which was the most difficult to retrieve among the simulated patterns, the Fs-DFPR method gave Type-C dark images with a 50% probability.

These results indicate that Fs-DFPR improves the convergence of dark images to the correct shape, even for noise-smeared diffraction patterns with data missing in a large small-angle region. In particular, Fs-DFPR using an annular mask is the most powerful method for retrieving correct dark images.

5-2. BFPR reconstruction of electron density maps in model simulations

The BFPR method was applied to diffraction patterns without masks using the dark images provided by Fs-DFPR or DFPR as the starting electron density maps. The number of correct J-shaped electron density maps produced by the BFPR and PRZ calculations is listed in Table 1. PRZ reconstructed the J-shaped maps only for diffraction patterns with data missing in the Cut2 area. The BFPR calculations starting with DFPR dark images failed to retrieve correct electron density maps for both Cut5 and Cut7 patterns. For Cut5 patterns, the number of J-shaped maps retrieved by BFPR calculations decreased in the following order: Fs-DFPR using the annular mask, Fs-DFPR using the symmetry mask, and DFPR. For Cut7 patterns, J-shaped maps appeared only in the BFPR calculation starting with dark images reconstructed by the Fs-DFPR method.

Table 1. Number of J-shaped Electron Density Maps Reconstructed Using PRZ and BFPR

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Therefore, the quality of BFPR maps strongly depend on the dark images used as starting models. The quality also depends on the quantity of missing data, and the noise level of diffraction patterns. The best results were achieved when BFPR was applied to the dark images retrieved by Fs-DFPR using an annular mask. The probability of reconstructing correct electron density maps decreased when the noise level of diffraction patterns increased. It was also noteworthy that BFPR retrieved correct electron density maps, even when starting from Types-C and -D dark images. This finding likely resulted because both types of dark images restrained the size of support regions to the dimensions of PDI.

5-3. Reconstruction from experimental diffraction data

Many diffraction patterns collected in XFEL-CXDI experiments display good S/N ratios up to a 29-nm resolution but miss data in a large area of the small-angle regions, mainly because of detector saturation. The representative diffraction patterns shown in Fig. 5 miss all the small-angle regions recorded by the MPCCD-Dual detector and also the saturated regions from the MPCCD-Octal detector up to a resolution of approximately 9 μm⁻¹ (Table 2). The number of missing speckle layers in the patterns is comparable to the number of missing speckle layers in Cut5 or Cut7 patterns in the model simulation (Table 2).

Fig. 5 Fs-DFPR with an annular mask followed by BFPR successfully retrieved electron density maps from experimental diffraction patterns with data missing in small-angle regions. Diffraction patterns result from aggregates of (a) three, (b) four, and (c) eight gold colloidal particles. For each experimental diffraction pattern shown in the left column, the two white dotted circles indicate the area used for calculating $R_{S/N}$ (see Table 3). The panels in the second left column are diffraction patterns multiplied by annular masks. In the third column, the upper panel shows the reconstructed dark image. The lower panel is the electron density map retrieved by the BFPR calculation using the dark image. In the SEM image shown in the right column, the aggregates of gold colloidal particles observed before XFEL irradiation are indicated by red dotted circles.

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Table 2. Statistics for Experimental Diffraction Patterns and Scores for Phase Retrieval

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Fs-DFPR with annular masks was applied to diffraction patterns displaying $C_{sym}$ values better than 0.89 (Table 2). Considering the S/N ratios of the diffraction patterns, the annular mask was defined by a $χ$ value of 0.13 μm and an $| α |$ value of 20 μm⁻¹ to extract a diffraction pattern predominantly in the resolution range of 13 to 27 μm⁻¹ [Figs. 2(a) and 5]. Each dark image retrieved by Fs-DFPR was composed of clusters with globular densities that had approximate diameters of 250 nm (Fig. 5).

In the subsequent BFPR calculation, the OS ratio for the diffraction patterns ranged from 104 to 324 (Table 2). The BFPR images (Fig. 5) retained the shapes of clusters in the dark images and provided structural details regarding both the shape of and the variation in electron density. In the dark images and BFPR density maps, the numbers and arrangements of particles with approximate sizes of 250 nm were consistent with the features observed in the SEM images before the X-ray exposure (Fig. 5). The $γ$ [4] and $R_{F}$ [8] values were described in Table 2.

As expected based on the simulations, reconstructing correct electron density maps from these diffraction patterns was a difficult task for either PRZ or DFPR followed by BFPR. Indeed, these two methods produced non-convergent support shapes and, therefore, failed to retrieve electron density maps interpretable as gold colloidal particles.

6. Discussion

In the present study, we have proposed using the Fs-DFPR method to retrieve electron density maps from diffraction patterns with data missing in small-angle regions because of saturated detector pixels and a beam stopper. We examined the effectiveness of the Fs-DFPR method for practical use in XFEL-CXDI experiments with a set of model simulations and by applying this method to experimental diffraction data. In this section, we discuss the merit, limitations, and outlook of the Fs-DFPR method.

6-1. Merit of Fs-DFPR with an annular mask

We first consider the reasons why, compared with DFPR (Fig. 4), Fs-DFPR using a symmetry mask produced better dark images (Fig. 4). The quantity of diffraction data extracted with a symmetry mask is twice that with a single mask. However, because of the redundancy of the two-fold symmetry in symmetry mask, the data quantity is almost equivalent to that in the single mask. Therefore, the net quantity of data in the masks would contribute to only minor differences between DFPR and Fs-DFPR with a symmetry mask. Rather than the data quantity, the symmetry constraint in Fs-DFPR is likely a greater factor. Because of this constraint, Fs-DFPR treats each dark image as a real number, while DFPR uses a complex number (see Theoretical background section). Consequently, in the Fs-DFPR method, the possible number of solutions is reduced to half the number in DFPR. From this perspective, successful PR for real-valued objects was much easier than PR for complex-valued objects [1].

With respect to the efficiency to retrieve correct dark images, the Fs-DFPR method with an annular mask produced better dark images than Fs-DFPR with a symmetry mask (Fig. 4). Because both Fs-DFPR calculations used the same real-space constraint, the quantity of diffraction data would be a major factor contributing to the difference in their efficiencies. For experimentally obtained diffraction patterns, among the PR protocols examined, only the Fs-DFPR calculations with annular masks successfully provided dark images that could be interpreted as the shapes of gold colloidal particles (Fig. 5). Martin et al. reported that dark images with better quality are expected in PR using a larger mask [27]. Therefore, a large quantity of masked diffraction data is advantageous for obtaining dark images that approximate the overall shapes of sample particles.

In subsequent BFPR calculations, dark images retrieved by Fs-DFPR with annular masks most frequently produced J-shaped electron density maps for PDI (Table 1). In addition, the BFPR method had the potential to reconstruct J-shaped electron density maps, even from Type-C and Type-D dark images with the approximate size of PDI, and this finding was typically observed in the simulation for the Cut7 pattern at NL3 [Fig. 3(d) and Table 1]. Therefore, the best way to apply the PR protocols examined in this study is BFPR that started with dark images retrieved by Fs-DFPR with an annular mask.

A dark image appears as a convolution of a point-spread function and the electron density map of targeted particle [Eq. (2)]. Therefore, rigorously speaking, electron density map calculated through a de-convolution of the dark image is suitable as the initial model for the subsequent BFPR. However, in our trials, the de-convoluted images frequently include artifacts caused by reduction of low spatial-frequency information even when the dark image displays correct shape. When starting from those images as initial models, BFPR calculations tend to fail. Taking the scores to get correct BFPR images, dark images are suitable as initial models for BFPR better than the de-convoluted images.

Consequently, this protocol dramatically improves the probability of obtaining correct electron density maps for non-crystalline particles from diffraction patterns with data missing in the small-angle region.

6-2. Limitation of Fs-DFPR depending on Poisson noise

For diffraction patterns with the noise level of NL3, even the Fs-DFPR method with an annular mask failed to reconstruct Type-A dark images (Fig. 4). This result suggests that the Fs-DFPR method is limited by the noise level. Even under the geometrical condition in Eq. (3), Poisson noise causes the breakdown of the Friedel symmetry though the symmetry is strictly necessary to formulate the theory of Fs-DFPR. On the other hand, although the symmetry in the diffraction patterns is violated in the simulation and in the experimental diffraction patterns, dark images approximating the true electron density maps were retrieved in the Fs-DFPR calculations using the annular masks (Figs. 2, 3, and 5).

Regarding what noise level limits the application of the Fs-DFPR with annular masks, we quantitatively estimated the tolerance of the Fs-DFPR protocol when applied to noise-smeared diffraction patterns. We calculated the S/N ratio of the masked region (Figs. 2 and 5) using the parameter $R_{S/N}$ defined in Table 3. The $R_{S/N}$ value for the simulated diffraction pattern of NL3 is 3.17. For the experimental diffraction patterns examined thus far, $R_{S/N}$ values of 4 may suggest a limit for successfully applying the Fs-DFPR method with an annular mask. This value could serve as a good criterion for judging whether diffraction patterns with data missing in the small-angle region are worthy of analysis. In practice, intense diffraction patterns, which saturate the detector in small-angle regions, probably display good $R_{S/N}$ values, exceeding 4. Therefore, the Fs-DFPR protocol should be chosen for structural analyses of such diffraction patterns before discarding them.

Table 3. $R_{S/N}$ Values for the Annular Masked Regions in the Diffraction Patterns of the Model Simulation and the Experimental Data

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In the present study, we used the HIO-SW phase-retrieval method for BFPR. For further refining the electron density map, we can apply other phase-retrieval algorithms such as noise-robust HIO [7] and oversampling smoothness [8]. To examine whether those algorithms are effective, we will perform CXDI experiments for sample particles with well-defined internal structures in the future. In addition, we will carefully take the influences of the beam intensity profiles, which also smear the retrieved electron density maps in our experiments.

6-3. Outlook for practical use of the Fs-DFPR method

In the present study, we carried out rigorous simulations for the atomic model of protein PDI and a very short wavelength. Although the simulations seem to be non-realistic, the results are useful to estimate the conditions of real experiments through the considerations described below. The zero-angle diffraction intensity $I (S = 0)$ is related to the incident intensity of X ray ( $I_{incident}$ ), the volume of the sample particle ( $V$ ), and the dimension of the particle in the direction of the incident X-ray ( $d$ ) [28] as

I (S = 0) = I_{incident} r_{e}^{2} \frac{λ^{2}}{σ} V d {\tilde{ρ}}^{2},

where

r_{e}

is the classical electron radius,

λ

is the wavelength,

σ

is the OS ratio and

\tilde{ρ}

is the average electron density of the sample particle. In the case of PDI with

\tilde{ρ}

of approximately 0.42 e/Å³,

V

of 1.7 × 10⁴ Å³ and

d

of 20 Å,

I_{incident}

of 9.39 × 10²⁵ photons/μm²/pulse is necessary to record the diffraction pattern. According to this procedure, we can estimate

I_{incident}

necessary to record a diffraction pattern from a biological particle with spherical shape of 2.0 μm diameter when using X-ray with the wavelength of 2.25 Å, and the detector pixel size that gives

σ

of 16. To record

I (S = 0)

at the level of the NL 3,

I_{incident}

is approximately 6.7 × 10¹⁰ photons/2 × 2 μm²/pulse. The value is comparable with the incident intensity used in our experiments at SACLA. Thus, the results in the present simulation would be useful for discuss on experimental conditions.

Because metal colloidal particles tend to form aggregates on membranes, the amount of diffraction patterns with data missing in small-angle regions, which result from detector saturation, reaches more than 20% of the collected diffraction patterns, as reported in our literature [26]. Furthermore, in diffraction patterns from sub-micrometer-sized cellular organelles, a lot of the speckle peaks are blocked by a beam stopper exceeding the 3rd speckle layer in our experimental condition [Fig. 1(a)]. Therefore, for structural analyses of metal colloidal particles and large biological samples, the Fs-DFPR method with annular masks is a means of providing overall structures (dark images) and detailed structures (BFPR images).

Using the KOTOBUKI-1 apparatus, thousands of diffraction patterns worthy of analysis are collected within a few hours in our XFEL-CXDI experiments [17, 25, 26]. We plan to implement an automatic Fs-DFPR procedure (followed by BFPR) in the automated data processing program, G-SITENNO [26, 33]. To perform the Fs-DFPR method, the regions for annular masks must be automatically determined by calculating the S/N ratio in diffraction patterns with $R_{S/N}$ values in resolution shells. Because the size and position of an annular mask are defined by only two scalar parameters in Eq. (6), the automated procedure will be implemented in our program ZOCHO in an effort to provide high-performance and high-throughput XFEL-CXDI structural analyses.

Acknowledgments

The authors are grateful to Dr. Masaki Yamamoto of the RIKEN SPring-8 Center for his kind help during these XFEL-CXDI experiments. This study was supported by the grant for X-ray Free Electron Laser Priority Strategy Program from the MEXT to M.N. Sample preparation and software development were also supported by grants from the JSPS to M.N. (No. 24654140) and to T.O. (No. 26800227) and by grants from the MEXT to M.N. (Nos. 23120525 and 25120725) and to T.O. (Nos. 24113723 and 26104535). For software development, we used diffraction data collected in XFEL-CXDI experiments performed at SACLA (proposal Nos. 2013A8043, 2013B8049, and 2014A8033).

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Diffraction patterns	PRZ	BFPR
		Method for retrieving dark images
		DFPR	Fs-DFPR with symmetry mask	Fs-DFPR with annular mask
Cut2 of NL1 / NL2 / NL3	9 / 6 / 3	10 / 10 / 9	10 / 10 / 10	10 / 10 / 10
Cut5 of NL1 / NL2 / NL3	0 / 0 / 0	5 / 4 / 0	7 / 6 / 4	10 / 9 / 7
Cut7 of NL1 / NL2 / NL3	0 / 0 / 0	0 / 0 / 0	6 / 5 / 3	9 / 8 / 5

Diffraction pattern	Figure 5(a)	Figure 5(b)	Figure 5(c)
Statistics for diffraction patterns
C_sym/the resolution of the center pixel in the ROI (μm⁻¹)	0.89/15.2	0.90/15.2	0.94/14.6
Maximum resolution of missing area in the small-angle region (μm⁻¹)	9.32	8.55	8.50
Layers of missing speckles	up to 5th layer	up to 6th layer	up to 7th layer
Scores for phase retrieval
OS ratio^a	324	226	104
γ ^b	0.011	0.007	0.008
R _F ^c	0.241	0.240	0.245

	Model simulation [see Fig. 3(c)]			Experimental data
Diffraction patterns	NL1	NL2	NL3	Figure 5(a)	Figure 5(b)	Figure 5(c)
Resolution shell for calculation	0.14-0.35 (Å⁻¹)			13-27 (μm⁻¹)
$R_{S/N}$ ^a	29.45	9.38	3.17	4.13	5.29	6.36

Diffraction patterns	PRZ	BFPR
		Method for retrieving dark images
		DFPR	Fs-DFPR with symmetry mask	Fs-DFPR with annular mask
Cut2 of NL1 / NL2 / NL3	9 / 6 / 3	10 / 10 / 9	10 / 10 / 10	10 / 10 / 10
Cut5 of NL1 / NL2 / NL3	0 / 0 / 0	5 / 4 / 0	7 / 6 / 4	10 / 9 / 7
Cut7 of NL1 / NL2 / NL3	0 / 0 / 0	0 / 0 / 0	6 / 5 / 3	9 / 8 / 5

Diffraction pattern	Figure 5(a)	Figure 5(b)	Figure 5(c)
Statistics for diffraction patterns
C_sym/the resolution of the center pixel in the ROI (μm⁻¹)	0.89/15.2	0.90/15.2	0.94/14.6
Maximum resolution of missing area in the small-angle region (μm⁻¹)	9.32	8.55	8.50
Layers of missing speckles	up to 5th layer	up to 6th layer	up to 7th layer
Scores for phase retrieval
OS ratio^a	324	226	104
γ ^b	0.011	0.007	0.008
R _F ^c	0.241	0.240	0.245

Dark-field phase retrieval under the constraint of the Friedel symmetry in coherent X-ray diffraction imaging

Abstract

1. Introduction

2. Theoretical background

2-1. Theory of DFPR

2-2. The DFPR method under the constraint of Friedel symmetry

2-3. Implementation of Fs-DFPR

3. Setup of model simulation

4. Experimental procedure

4-1. Sample preparation

4-2. Diffraction data collection

4-3. Data processing

5. Results

5-1. Reconstruction of dark images in the model simulations

5-2. BFPR reconstruction of electron density maps in model simulations

5-3. Reconstruction from experimental diffraction data

6. Discussion

6-1. Merit of Fs-DFPR with an annular mask

6-2. Limitation of Fs-DFPR depending on Poisson noise

6-3. Outlook for practical use of the Fs-DFPR method

Acknowledgments

References and links

Cited By

Figures (5)

Tables (3)

Equations (9)

Optics Express