1. Introduction

Since full-field transmission X-ray microscopy has provided high-spatial resolution images of the internal structures of complex bulk materials, it has been advanced with the use of bright synchrotron radiation X-ray sources [1–3]. By exploiting the unique property of X-rays in which the high-energy photon interacts with the inner-shell electron of atoms, X-ray spectromicroscopy has been applied as a powerful tool for the chemical and elemental analysis of various specimens in two and three dimensions [4]. Energy-varying spectromicroscopy can be facilitated if objective optics with a wide-range achromaticity are used. As a promising candidate for achromatic imaging optics, an advanced Kirkpatrick–Baez (AKB) mirror [5] has been developed. The AKB mirror follows the Wolter mirror configuration [6], leading to the reduction of coma aberration and the formation of an X-ray magnified image. While shape errors of the reflecting surfaces on the order of nanometers are required to suppress image degradation, mirrors can be fabricated with high accuracy owing to ultraprecise processing and measurement techniques for nearly planer mirrors [7–9]. Consequently, a full-field X-ray microscope based on total-reflection AKB mirrors has been demonstrated and achieved ∼50-nm-resolution observations and absorption- and fluorescence-spectroscopic imaging with achromatic properties [10,11].

However, the AKB microscope has measured only transmission X-ray images and has been not suitable for observing low-absorption samples, especially in the case of biological tissues and/or nanometer-sized materials with hard X-rays. For this issue, X-ray phase contrast imaging (PCI) has been generally utilized. The Zernike phase contrast method [12] has been commonly used for X-ray microscopes with Fresnel zone plates (FZPs). This method requires the insertion of a phase ring at the back focal plane. AKB mirrors, however, do not allow insertion of a phase ring because the back focal plane is usually located on mirrors, and the positions of the vertical and horizontal back foci never coincide. In this study, we demonstrated a propagation-based phase-contrast imaging method (PBPCI) [13,14] for a full-field X-ray microscope. PBPCI provides quantitative phase-contrast images from single or multiple defocused images without any additional optics. Among the several approaches of PBPCI, a widely used method is the transport of intensity equation (TIE) method, which exploits the fact that the derivative of the phase distribution along the image plane is connected to the derivative of the intensity distribution along the optical axis by a partial differential equation [15]. The TIE method is often discussed mainly in the field of electron microscopy, and various approximations and solutions have been reported [16–18]. Contrast transfer function (CTF) methods [19] similar to the TIE method were also proposed, which can handle longer propagation distances than the TIE method by assuming that the objects have slowly varying phases and weak absorption characteristics. Recently, iterative methods [20,21] that optimize the complex wavefield through iterative calculation processes based on wave propagation have been developed. Iterative methods are superior in robustness to noise and phase discontinuities because the complex wavefield is optimized through the process of iterative calculation based on wave propagation without a priori sample information. These methods can handle wide propagation ranges equivalent to that of CTF methods, enabling the determination of phase distributions from high to low spatial frequency components with high reliability. In the X-ray region, these lens-based applications using PBPCI are minimally studied [22] but are often applied to lens-less holography using the above methods, in which only a point source and image detector are utilized to record holograms [23]. Notably, there is essentially no algorithmic difference between lens-based applications and lens-less holography approaches, and the objective lens has no influence on the wave propagation results [24]. Detailed equations are presented in the next section. However, lens-based microscopy with PBPCI has the considerable advantage of requiring less coherence than lens-less holography methods. This is because the inclusion of an objective lens leads to the convergence of the diffracted light diverging from an object at the image plane. Therefore, the required spatial coherence should be comparable to the full width of the point spread function (PSF) of the microscope.

In addition to visualizing the phase contrast, another major objective is that PCI is strongly desired in AKB-based microscopes, which suffer from the field-curvature problem [25]. If the complex transmission function of a sample can be accurately determined by PCI, then the defocus caused by the field curvature aberration of the AKB mirrors can be corrected, and the field of view (FOV) can be expanded. Considering recent great improvements in the brightness of synchrotron radiation facilities and the FOV of X-ray cameras, there is a high demand for efficient observation through an enlarged FOV.

In this paper, Section 2 describes the PBPCI method. Sections 3 and 4 report the results of a PBPCI demonstration experiment. By applying PBPCI with an image registration algorithm to our developed mirror-based X-ray microscope, phase-contrast imaging of siemens star charts, polystyrene spheres, and chemically fixed cells was demonstrated. Additionally, the field curvature aberration of AKB mirrors was digitally corrected by locally propagating the determined complex wavefield on the image. In Section 5, the spatial resolution, sensitivity, and effects of the wavefront aberration are discussed.

2. Propagation-based phase-contrast imaging method (PBPCI) for AKB mirror-based microscopy

2.1 Iterative phase retrieval algorithm with multiple defocus images

In a coherent imaging system, we set $\psi ({x,\; y;z} )$ as the complex wavefield observed at the image plane when the sample is on focus ($z = 0$) . Then, the wavefield $\psi ({x,\; y;\Delta z} )$ observed at the image plane when the sample is defocused ($z = \Delta z$) can be expressed as follows:

 

Fig. 1. Schematic and flow of the PBPCI method. (a) Schematic of the PBPCI method for capturing multiple defocused images. (b) Schematic for each plane in the iterative phase retrieval process. (c) Flow of the iterative phase retrieval algorithm.

Download Full Size | PDF

The phase distribution of the wavefield ${\theta _j}({x,\; y;{z_j}} )$ is updated by the gradient descent method, a nonlinear optimization technique, to minimize this error function in each plane.

The derivative of $Erro{r_j}$ is expressed as follows [20]:

Figure 1(c) shows the flow of the iterative phase retrieval algorithm to optimize ${\theta _j}$. First, the initial values of the wavefield in the first plane are determined as follows:

The initial value of the phase distribution ${\theta _{initial}}$ is a matrix of ones. The phase distribution of the wavefield is updated using Eqs. (3) and (4). The amplitude distribution of the updated wavefield is replaced by the square root of the captured image (real-space constraint). After several iterations ($\textrm{N}\sim 10$ times) among updates of the phase distributions and real-space constraints, the updated wavefield on the first plane is propagated to the second plane. Then, the wavefield on the second plane is updated in the same way and is propagated to the next plane. After updating the last wavefield on the $J$th plane, the wavefield is back-propagated to the first plane, and the same procedures are repeated until the sum of the loss functions at each plane converges to a sufficiently low value (∼30 loops). Finally, the closest on-focus wavefield among the optimized wavefields near the focus is propagated to the optimal focus.

2.2 Image registration

Generally, in PBPCI, the sample position along the sample plane unintentionally varies due to the motion error of the scanning stage when a focal stack is captured. It is necessary to completely eliminate this misalignment to reconstruct clear phase contrast images. In recent years, a cross-correlation-based algorithm that corrects sample mispositioning during iterative phase retrieval has been proposed for ptychography [26]. In this paper, this strategy was applied to register the captured focal stacks. We consider the registration of the image on the $j$th plane. The cross-correlation function $C({\Delta x,\; \Delta y} )$ between the amplitude of the wavefield on the $j$th plane propagated from the ($j - 1$)th plane and the square-rooted image captured in the $j$th plane is expressed as follows:

The $\tilde {\Delta x},\tilde {\; \Delta y}$ with maximum $|{C({\Delta x,\; \Delta y} )} |$ is defined as the estimated misalignment. $\tilde {\Delta x},\tilde {\; \Delta y}$ can be determined with subpixel accuracy [27]. Then, the $j\; $th captured image is registered using the estimated $\tilde {\Delta x},\tilde {\; \Delta y}$.

By performing this image registration before the real-space constraints on each plane, the wavefield can be updated, and the misalignments of the sample are corrected simultaneously.

3. Demonstration experiment

Demonstration experiments were performed at SPring-8. X-rays (14.5 keV) produced by the BL29XU undulator were monochromatized by an Si (111) double-crystal monochromator (ΔE/E$ \approx 1.3 \times {10^{ - 4}}$). The experimental setup was constructed in the second experimental hutch (distance from the undulator to the sample ∼100 m). The experimental setup consisted of a slit, a diffuser (rotating sandpaper), samples, AKB mirrors, and an image detector (Fig. 2). Because PBPCI assumes coherent imaging conditions, illumination optics were not used. The sample and AKB mirrors were placed inside a chamber filled with helium to prevent attenuation of X-rays by air and contamination of the mirror surface. The AKB mirrors consisted of an AKB type-I mirror with an NA of 1.00 × 10⁻³ for vertical imaging and an AKB type-III mirror with an NA of 1.04 × 10⁻³ for horizontal imaging; these mirrors were designed and fabricated to be capable of diffraction-limited imaging [28]. The indirect imaging detector was composed of a CMOS camera (2048 × 2048 pixels, 6.5 × 6.5 µm²), a × 5 lens, and a scintillator. The scintillator consisted of Ce-doped Lu₃Al₅O₁₂ (LuAG:Ce, 5 µm thick) bonded onto a nondoped LuAG (1 mm thick) by solid-state diffusion [29]. The siemens star test chart (XRESO-50HC, NTT Advanced Technology Corporation), polystyrene spheres (Polybead Microspheres 6.00 µm, Polysciences), and chemically fixed CHO-K1 (Chinese Hamster ovary) cells were used as test samples. The star chart is a 500 nm thick sample made of tantalum with a minimum line width of 50 nm in the innermost part.

 

Fig. 2. Experimental setup. (a) Schematic of the setup. (b) Photograph inside the He chamber.

Download Full Size | PDF

After tuning the AKB mirrors, the focal stacks of the three samples were obtained by capturing images of each sample at different defocus positions. The imaging conditions for each sample are listed in Table 1. To reduce artifacts associated with the reconstruction, each sample was exponentially defocused (see Supplement 1, Section 1) [30]. In addition, the images of the chemically fixed CHO-K1 cells were captured with an exposure time of 10 s/image because its phase shift value was close to the expected detection limit. Before acquiring each focal stack measurement, an image without samples (background) was captured. Flat-field correction was performed by dividing each image of the samples with the background image. These images with/without samples were subtracted dark before division.

Table 1. Acquisition Conditions of Each Sample

View Table

4. Results

The effect of the misalignment in the focal stack of the captured star chart was corrected by image registration (see Visualization 1). The intensity and phase distributions of the wavefield reconstructed from the focal stack of the star chart are shown in the first row of Fig. 3(a). As a result, the 50-nm line-and-space feature, which is the smallest structure in the star chart, was clearly observed in the reconstructed on-focus phase distribution. Additionally, the phase shift value of the star chart was 0.42 rad, which corresponded to its designed thickness of 500 nm.

 

Fig. 3. PBPCI results. (a) Reconstructed amplitude and phase images. Polystyrene spheres and CHO cells were also observed with a visible-light differential interference contrast (DIC) microscope. All the scale bars represent 5 µm. (b) FRC results of the reconstructed phase in the star chart. The red line indicates the intersection of the FRC and the 1-bit threshold curve, i.e., the cutoff frequency. (c) Line profile of the small structure in chemically fixed CHO-K1 cells.

Download Full Size | PDF

The results of the iterative phase retrieval calculation for the focal stack of polystyrene spheres are shown in the second row of Fig. 3(a). The phase distribution showed a high-contrast image reflecting the edge of the spheres. In addition, small string-like structures were observed; these structures were not observed in the image captured by visible-light differential interference microscopy (DIC image) or the on-focus image. We speculated that these structures were the surfactant agglomeration component in the polystyrene sphere dispersion solution. Additionally, complicated stripe-like contrast appeared due to an error in the flat-field correction caused by a slight drift of the optical system; this artifact could be eliminated by further stabilizing the entire optical system.

Similar to the polystyrene spheres, the chemically fixed CHO-K1 cells also exhibited nonnegligible stripe-like artifacts caused by the error of the flat field correction. The stripe noise component could be selectively removed from the obtained focal stack in Fourier space because the noise was monotonous in the case of fixed CHO-K1 cells. The edge structure of cells and microstructures were clearly observed in the phase distribution (the third row of Fig. 3(a)).

We corrected the field curvature aberration using the reconstructed star chart’s complex wavefield. The defocus distribution caused by the field curvature was estimated from the reconstructed complex wavefield (Fig. 4(a)). Based on the defocus information, one-dimensional wave propagations in the vertical and horizontal directions were performed (Fig. 4(b)). Although structures outside the FOV (e.g., the pink square) before the field curvature aberration correction were blurred, they were clearly visualized after the correction. This correction expanded the FOV from 12.4 µm × 7.6 µm [28] to 25 µm (horizontal) × 30 µm (vertical). Since the expanded area was limited by the maximum range of the acquisition area, the FOV can be further expanded using state-of-the-art large-format image detectors. The detailed method is described in Supplement 1, Section 3.

 

Fig. 4. Correction of the field curvature aberration. (a) Estimated field curvature of the AKB mirror for the horizontal (H) and vertical (V) directions. (b) Phase distribution of the complex wavefield before and after the field curvature aberration correction. The inhomogeneous area in the flat regions is due to the poor reconstruction of the low spatial frequency information and the slight errors in the flat-field correction. These scale bars denote 5 µm.

Download Full Size | PDF

5. Discussion

The spatial resolution of the on-focus phase-contrast images was evaluated by Fourier ring correlation (FRC) [31]. The FRC was obtained using two phase-contrast images of the star chart obtained from two different datasets. In this paper, a 1-bit curve was used as the threshold [32]. The FRC results showed that the 73-nm full-period (36.5-nm half-period) line width could be resolved (Fig. 3(b)). This result was consistent with the theoretical diffraction limit of 70.2-nm full-period (35.1-nm half-period) obtained with Abbe’s spatial resolution of $0.82\lambda /NA$ [33], confirming that the phase-contrast images could be obtained at the almost diffraction-limited performance of the AKB mirror imaging optics.

The phase sensitivity detected by the PBPCI was evaluated by the experimental results. Figure 3(c) shows a line profile of the reconstructed phase distribution of chemically fixed CHO-K1 cells; PBPCI could visualize the phase distribution with at least 0.03 rad of phase shifts. The evaluated phase sensitivity was nearly equivalent to that of general ptychography (0.01 rad) [34] and the Zernike phase-contrast method (0.02 rad) [35]. A longer exposure time could provide higher phase sensitivity.

From Fig. 3(a), image degradation in the on-focus intensity distribution with respect to the reconstructed phase distribution was observed due to a slight wavefront aberration caused by residual shape error on the mirror surfaces. The effect of wavefront aberration was evaluated by a simulation (Fig. 5). The sample function was assumed to be a star chart made of Ta with a thickness of 500 nm, and wavefront error with various peak-to-valley (PV) values was introduced. Under the coherent imaging condition induced by plane-wave illumination, the intensity distribution could be degraded even if the wavefront aberration of the mirror was below λ/4; this is often used as the threshold for mirror fabrication. In contrast, the reconstructed phase distribution was notably more robust to wavefront aberrations (see Supplement 1, Section 2).

 

Fig. 5. Simulation to investigate the effect of wavefront aberration. The wavefront errors were artificially constructed for these simulations. The scale bars denote 1 µm.

Download Full Size | PDF

In this paper, although we assume full spatial coherence, the actual coherence length should be estimated as 2–3 µm considering the undulator source size, the distance from the source, and the presence of the diffuser. Therefore, in this experiment, the coherence length approximately covered the width of the PSF at the defocus position, indicating that the microscope can function as a coherent imaging system. However, at the defocus positions of 1.6 and 1.2 mm for the polystyrene spheres and CHO cells, the assumption of full coherence may be incorrect. Nevertheless, the reconstruction error was small, which may be because the results were reconstructed using five images. More details are discussed in Supplement 1, Section 4.

6. Conclusion

We demonstrated that PBPCI using a full-field X-ray microscope based on AKB mirrors could observe samples with low absorption, such as polystyrene spheres and fixed CHO-K1 cells at SPring-8. Furthermore, we showed that the reconstructed wavefield can be used for the field curvature aberration correction of AKB mirrors. Future PBPCI research will be expanded to new applications, such as phase CT based on AKB mirror-based full-field X-ray microscopy.