Multiple defocused coherent diffraction imaging: method for simultaneously reconstructing objects and probe using X-ray free-electron lasers

Makoto Hirose; Kei Shimomura; Akihiro Suzuki; Nicolas Burdet; Yukio Takahashi

doi:10.1364/OE.24.011917

1. Introduction

Coherent X-ray diffraction imaging (CXDI) was first demonstrated in 1999 [1]. Since then, CXDI using synchrotron radiation and tabletop high-harmonic generation sources has been a rapidly growing area of research [2]. Recently, CXDI has evolved into a single-shot imaging method using X-ray free electron lasers (XFELs). Highly focused XFEL pulses with extremely high peak brilliance and an ultrashort duration make it possible to capture the moment before the onset of radiation damage formation [3]. XFEL-CXDI is therefore expected to provide new insights in fields ranging from biology to materials science. In conventional XFEL-CXDI experiments with a focusing system, a large amount of far-field diffraction data is collected by scanning a membrane supporting specimens in the focal plane [4] or by injecting aerosolized samples at the focal spot [5]. Projection images of samples with various sizes, shapes, and orientations are reconstructed through phase retrieval calculation of the diffraction data [6, 7]. Almost all the XFEL-CXDI techniques based on the concept of plane-wave illumination have a significant limitation; the sample must be much smaller than the diffraction-limited spot size of the incident beam to satisfy plane-wave illumination [8]. For example, the maximum size of the sample at SACLA is ∼1 μm, corresponding to the focal spot size obtained using the Kirkpatrick–Baez(KB) mirrors installed in SACLA [9].

There are several approaches to CXDI for observing samples larger than the focal spot size. X-ray ptychography [10], which is a scanning type of CXDI, enables the visualization of both extended samples and the incident wavefront [11, 12]. X-ray ptychography is now the most promising tool for the high-resolution X-ray imaging of various specimens in biology and materials science at third-generation synchrotron facilities [13]. The core requirement of ptychography is the overlapping of the probe on the sample. Ptychography is therefore not suitable for XFEL-CXDI since a single-shot XFEL pulse destroys the sample in the overlapping area, although ptychography works by using attenuated XFEL pulses below the damage threshold [14, 15]. Fresnel CXDI [16], which is also called keyhole CXDI [17], is used to observe a specimen positioned in the defocus plane. Curved beam illumination provides a unique reconstruction upon phase retrieval calculation using paraxial Fresnel free-space propagation between the sample and the detector [18]. Fresnel CXDI requires the incident wavefront to be determined beforehand. For example, the incident wavefront has been determined using a priori knowledge of the exit pupil of the focusing device [19]. It is known that Fresnel CXDI is strongly affected by sample stability issues. It has been reported that the combination of X-ray ptychography with Fresnel CXDI provides better reconstructions than either method used separately [20].

In this paper, we propose a new approach named “multiple defocused CXDI”, in which isolated objects are sequentially illuminated with a divergent beam larger than the maximum size of the objects and the coherent diffraction pattern of each object is recorded. This approach can simultaneously reconstruct both the incident wavefield and objects in a broad range size from diffraction patterns without any a priori knowledge of the illumination optics. To assess the effectiveness of the proposal approach, we performed a computer simulation and a proof-of-principle experiment in SPring-8. We also discuss its feasibility for use with XFELs.

2. Principle of multiple defocused CXDI

Figure 1 shows a schematic drawing of multiple defocused CXDI. Isolated specimens are sequentially introduced into the defocus position. Far-field diffraction patterns are collected using a two-dimensional X-ray detector. Two-dimensional projection images of the specimens and the illumination wavefield are simultaneously reconstructed by an improved phase retrieval calculation based on an error-reduction algorithm [21] incorporating the shrink-wrap algorithm [22] and the mixed-state reconstruction algorithm [23], in which both the complex transmission functions of M samples and the wavefronts of multiple probes are reconstructed from the intensities of M diffraction patterns alone. Extra information is obtained from the zero-density region around the objects and the consistency of the probe wavefront between multiple measurements.

Fig. 1 Schematic drawing of multiple defocused CXDI. Focused X-rays are sequentially irradiated to isolated samples in the defocus plane. Multiple diffraction data are collected in the far field using a two-dimensional detector.

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The exit wave ψⁱ(r) of the ith sample under single-mode illumination can be expressed as

ψ^{i} (r) = T^{i} (r) \times P (r),

where r denotes the real-space coordinate vector, Tⁱ(r) is the complex transmission function of the ith sample, and P(r) is the complex probe function. The exit wave propagates and then produces a diffraction pattern in the far field. The diffraction intensity from the ith sample is described as

I^{i} (q) = {| ℱ [ψ^{i} (r)] |}^{2} = {| Ψ^{i} (q) |}^{2},

where q is the reciprocal-space coordinate vector, Ψ(q) denotes the wavefront in the detector plane, and ℱ represents the Fourier transform operator.

Figure 2 shows a flowchart of the phase retrieval procedure. First, initial estimates of the M sample functions $T_{0}^{1} (r), T_{0}^{2} (r), \dots, T_{0}^{M} (r)$ and probe functions P₀(r) are input. The probe functions are divided into K probe modes P_0,1(r), P_0,2(r), ··· ,P_0,K(r), which are themselves fully coherent but mutually incoherent and are identical for all exposures. The mixed-state reconstruction algorithm can deal with the decoherence of diffraction patterns arising from partially coherent illumination and/or sample displacement during exposure [23, 24]. The iterative process sequentially enforces consistency with the reciprocal and real-space constraints. In the reciprocal space at the nth cycle, the modulus of the current estimate of the wavefront in the detector plane $Ψ_{n, k}^{i'} (q)$ is replaced by the square root of the recorded intensity. Given the multiple probe modes, the reciprocal-space constraint is given as

Ψ_{n, k}^{i'} (q) = \sqrt{\frac{I^{i} (q)}{\sum_{k = 1}^{k = K} {| Ψ_{n, k}^{i} (q) |}^{2}}} Ψ_{n, k}^{i'} (q) .

The inverse Fourier transformation of

Ψ_{n, k}^{i'} (q)

gives a corrected

ψ_{n, k}^{i'} (r)

in the real space. For

r \in S^{i} (r) \cap \arg [T_{n}^{i} (r)] \leq 0

, where Sⁱ(r) represents the support function for the ith sample, the update of the transmission function for the ith sample is given by

T_{n + 1}^{i} (r) = T_{n}^{i} (r) + \frac{\sum_{k = 1}^{k = K_{mode}} P_{n, k}^{i *} (r) [ψ_{n, k}^{i'} (r) - ψ_{n, k}^{i} (r)]}{\sum_{k = 1}^{k = K_{mode}} {| P_{n, k}^{i} (r) |}^{2}} .

Otherwise,

T_{n + 1}^{i} (r)

is 1 + i0, corresponding to the vacuum region. On the other hand, the update of the probe function is given as

P_{n, k}^{i + 1} (r) = P_{n, k}^{i} (r) + \frac{T_{n}^{i *} (r) [ψ_{n, k}^{i'} (r) - ψ_{n, k}^{i} (r)]}{{| T_{n}^{i} (r) |}^{2}} .

After the update of the ith sample and the probe functions, the same procedure is applied to the (i + 1)th sample. When all the samples and probe functions are updated, the nth cycle of the phase retrieval has been completed. This cycle is repeated iteratively until suitable convergence is obtained for all the sample and probe functions. In this algorithm, the probe wavefront is recovered owing to the consistency for all exposures, which is the same concept as the probe reconstruction in ptychography [11, 12]. It is known that more scanning points are necessary in ptychographic measurement to increase the dataset redundancy. It is therefore assumed that the convergence of the phase retrieval calculation is improved by increasing the number of diffraction patterns in multiple defocused CXDI.

Fig. 2 Flowchart of the iterative phase retrieval algorithm for multiple Fresnel CXDI. At each sample position, both the object and probe functions are updated. This process is repeated iteratively until suitable convergence is obtained.

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3. Simulation

We investigated the number of isolated samples required for the reliable reconstruction of both samples and the probe by a computational simulation. In this simulation, the incident X-ray energy was set to 6.5 keV (wavelength λ = 0.1907 nm) and the X-ray was two-dimensionally focused to a spot size of 500 nm (FWHM) using a pair of KB mirrors. All the samples were assumed to be composed of 200-nm-thick tungsten. The size of each sample was ∼2×2 μm², i.e., four times larger than the focal spot size. The samples were placed 10 mm downstream of the focal plane so that the size of the illuminated beam was comparable to the sample size. It was assumed that fully coherent X-rays were irradiated to the static samples, i.e., K_mode = 1. The phase retrieval calculation described above was applied to the diffraction patterns. The initial estimates of the functions of the samples were given by random numbers and the support functions for all samples were given by a square region and were fixed during the iteration. On the other hand, the modulus of the initial estimate of the probe function was given by a Gaussian function with an FWHM of ∼2 μm and the phase was set to zero. In addition, the weak-phase object approximation [25] was applied to improve the convergence. The convergence was quantitatively evaluated using the root-mean-square (RMS) error between the input phase and the reconstructed phase of the objects as follows:

{RMS}_{error} (M) = \frac{1}{M} \sum_{i = 1}^{M} \sqrt{\frac{1}{X Y} \sum_{x = 1, y = 1}^{x = X, y = Y} {[\arg [T_{rec}^{i} (r)] - \arg [T_{input}^{i} (r)]]}^{2}},

where X and Y denote the pixel numbers of the object function in the horizontal and vertical directions, respectively.

Figure 3(a) shows the sample-number dependence of the RMS error after 5000 iterations. As expected, the RMS error decreased with increasing number of samples. Good convergence was obtained when the number of samples was more than 30. Figure 3(b) shows images of one of the original samples and the probe used in this simulation. Figure 3(c) shows the reconstructed images when the number of samples was 20, 28, and 35. When the number of samples was 20, neither the samples nor the probe could be recognized. The quality of the reconstruction markedly improved with increasing number of samples. When the number of samples reached 30, both the samples and the probe were successfully reconstructed. Note that the number of samples required for the reconstruction strongly depends on the initial estiamte for the probe. If the initial probe can be anticipated with high accuracy, fewer samples and iterations are required. In addition, since the samples are illuminated with a curved beam, each object image is uniquely recovered regarding both the position and the relative phase, which is also a feature of Fresnel CDI. When the object images are reconstructed using Fraunhofer diffraction patterns, phase images with phase offsets are derived. However, by using a phase reference, i.e., the vacuum region around isolated objects, as known information, the phase images can be uniquely recovered.

Fig. 3 (a) Sample-number dependence of the RMS error in the reconstruction. (b) Images of one of the original samples (upper) and the probe (lower) on the sample in the simulation. The scale bar is 2 μm. (c) Reconstructed phase images (upper) and the probe (lower) when the number of samples is 20 (left), 28 (middle), and 35 (right).

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4. Experiment

A proof-of-principle experiment on multiple defocused CXDI was carried out at BL29XUL in SPring-8. The experimental conditions, such as the X-ray energy and the parameters of the focusing device, were the same as those for the simulation. We prepared 32 isolated samples on a SiN membrane using focused ion beam (FIB) chemical vapor deposition. A Ga⁺ FIB was used to decompose W(CO)₆ seed gas. The sample size was ∼2×2 μm², which was approximately four times larger than the diffraction-limited spot size. The incident X-ray was monochromatized using a Si(111) double-crystal monochromator. A secondary slit was placed ∼48 m upstream of the KB mirrors. The slit opening was adjusted to 10×10 μm², which produced spatially fully coherent X-rays [26]. Thirty-two diffraction patterns were collected using an in-vacuum front-illuminated CCD detector (Princeton Instruments PyLoN 1300) with a pixel size of 20×20 μm², which was positioned 1219 mm downstream of the focus. In this configuration, the one-dimensional oversampling ratio of the diffraction patterns was ∼6, which fully satisfies the criteria for reconstruction from the diffraction patterns [27]. The X-ray exposure time at each sample position was 40 s. To increase the effective dynamic range of the diffraction patterns, 10-μm-thick Ta of size 640×640 μm² was positioned as a semitransparent central stop [28] in front of the CCD detector. To suppress the effect of fluctuation of the incident X-ray, all the measured diffraction patterns were normalized using the average intensity in the bright field.

Next, we reconstructed both the samples and the probe using the phase retrieval calculation, in which two independent probe modes were considered. The initial objects $T_{0}^{1} (r), T_{0}^{2} (r), \dots, T_{0}^{i} (r), \dots, T_{0}^{M} (r)$ consisted of a random number array. The initial probe $P_{0, 1}^{1} (r)$ for the first mode was set as the probe expected from the present experimental setup, while a random number array was input for the probe for the second mode $P_{0, 2}^{1} (r)$ . The initial support was a square of 122×122 pixels. To further improve the convergence, the shrink-wrap algorithm [22], which enables the support function to be updated during iteration, was applied every 100 iterations up to 2000 iterations. Figure 4(a) shows phase images of the samples after 2500 iterations. The pixel size of the images is 20 nm. The 32 reconstructed images of the samples are in good agreement with each SEM image. Figure 4(b) shows the intensity distributions of the two probe modes. The population is 84.2% for the first mode and 15.8% for the second mode. Additional modes did not result in good convergence of the iteration, which may be due to the small number of measurements. It is considered that the origin of the second mode is a dominant vibration mode of the sample or the illumination optics during the measurement.

Fig. 4 (a) Reconstructed phase images of the 32 samples. Inset is each SEM image. (b) Reconstructed intensities of two probe modes. The scale bar is 2 μm.

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5. Summary and conclusions

We have proposed multiple defocused CXDI, which can simultaneously reconstruct both isolated samples larger than the diffraction-limited spot size and the incident wavefront from diffraction patterns alone. We investigated the number of samples required for this technique by computer simulation and then demonstrated the technique in a synchrotron experiment. In this method, it is necessary to increase the number of measured samples to increase the dataset redundancy. Therefore, high-throughput measurement is required for the practical use of this method. In the present synchrotron experiment, the measurement throughput was very low; it took more than 24 h to complete the measurement including the time required to find the samples on the membrane. On the other hand, high-throughput measurement was realized in the XFEL experiment. For example, it took ∼10 s to collect 32 diffraction patterns at SACLA, in which the repetition rate is 30 Hz [29] and hit ratio was ∼10%. Intensity fluctuation is typically kept within 10 % at SACLA [30]. Multiport CCD [31] with a semitransparent central stop [28] can measure both the fluctuation and the central hologram patterns in a direct beam in a shot-by-shot manner. Sample stability issues due to drift and vibration are eliminated in the case of single-shot measurement at XFELs since the pulse duration is ∼10 fs. Multiple defocused CXDI is therefore a promising tool for observing a broad range of samples and characterizing the wavefront at XFELs. We believe that the present method will promote single-shot imaging using XFELs.

Acknowledgments

This work was supported by KAKENHI (Grant Nos. 25709057, 26600143, and 26106515), the X-ray Free Electron Laser Priority Strategy Program of MEXT, and the SENTAN of JST. We thank T. Ishikawa and K. Yamauchi for many stimulating discussions, as well as Y. Kohmura for help in the experimental setup.

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Multiple defocused coherent diffraction imaging: method for simultaneously reconstructing objects and probe using X-ray free-electron lasers

Abstract

1. Introduction

2. Principle of multiple defocused CXDI

3. Simulation

4. Experiment

5. Summary and conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

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