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We report that reference objects, strong scatterers neighboring weak phase objects, enhance the phase retrieval and spatial resolution in coherent x-ray diffraction imaging (CDI). A CDI experiment with Au nano-particles exhibited that the reference objects amplified the signal-to-noise ratio in the diffraction intensity at large diffraction angles, which significantly enhanced the image resolution. The interference between the diffracted x-ray from reference objects and a specimen also improved the retrieval of the phase of the diffraction signal. The enhancement was applied to image NiO nano-particles and a mitochondrion and confirmed in a simulation with a bacteria phantom. We expect that the proposed method will be of great help in imaging weakly scattering soft matters using coherent x-ray sources including x-ray free electron lasers.
© 2014 Optical Society of America
1. Introduction
Among recent efforts to develop optical microscopic techniques with nano-meter scale resolution including near field and super-resolution microscopies, x-ray microscopy is one of the most straightforward approaches complying the fundamental principle of Abbe’s diffraction limit which states that structures much smaller than wavelength cannot be resolved. Coherent diffraction imaging (CDI) [1, 2], especially using hard x-rays with wavelength λ in angstrom scale, has been expected to provide an extremely high spatial resolution [3, 4]. In CDI, an image is reconstructed from oversampled diffraction intensity without employing lenses. Various CDI techniques including plane-wave CDI, Fresnel-CDI, and ptychography have been introduced to achieve a desired resolution under specific experimental constraints [5–11]. Together with intense coherent x-ray pulses from x-ray free electron lasers (XFEL), structures of single biomolecules were expected to be resolved by CDI [12].
The practical resolution of hard x-ray CDI, however, has mostly been limited by weak diffraction intensity from a specimen overwhelmed by instrumental noise which constraints the effective numerical aperture, rather than the wavelength of x-rays. At small diffraction angles, diffracted intensity is high and the effect of instrumental noise is negligible. However, at large diffraction angles (or high momentum transfer Qs), which are critical for resolution, the diffracted intensity decreases rapidly and instrumental noise becomes dominant, which applies to all diffraction measurements including CDI since the scattering cross section decays rapidly with increasing Q. Currently in most CDI experiment, the resolution is limited by the lack of the diffracted x-ray signal and the resolution might be set at a Q-value above which the instrumental noise becomes dominant over the photon counting noise. Low signal-to-noise ratio (SNR) in diffraction intensity at large diffraction angles has placed a practical limit of around 10 nm on the image resolution of CDI [13, 14]. Fundamentally, the scattering cross-section of matter to x-rays becomes extremely small as the wavelength decreases below 1 nm.
Intrinsic SNR due to photon counting shot noise following the Poisson statistics can in principle be improved by increasing exposure. In practice, however, exposure cannot be indefinitely increased for objects prone to radiation damage such as biological objects or polymers [15–17]. Furthermore, inherent instrumental noises associated with light sources, optical elements, and detectors, impose limits on improving SNR regardless of exposure. Although obtaining diffraction signal prior to radiation damage using an intense XFEL pulse within a few tens of femtoseconds has been suggested and being applied, limited SNR dominated by instrumental noise is still a major limiting factor [18–21].
Recently overcoming limited SNR by placing or tagging reference scatterers has been discussed in both CDI and digital holography [22–24]. Amplified holographic images with enhanced resolution were reported using a known reference object [22]. Shintake elaborated on the improvement of the SNR in CDI but finally concluded that there is no gain in intrinsic SNR by reference scatterers [23]. However, improving SNR overcoming instrumental noise, which is more critical in many cases, was not discussed in detail. In addition, there has been no report on the experimental verification of the effects of reference objects in CDI using hard x-rays.
In this paper, we report CDI image reconstructions of weakly scattering objects with enhanced spatial resolution utilizing the interference with relatively strong signal from neighboring but unknown objects, namely ‘reference objects’. We noticed that the SNR limited by instrumental noise was amplified by the reference objects significantly. The proposed method of overcoming instrumental noise was applied to image core-shell structure of NiO as well as morphology of a mitochondrion. The experimental verification of the amplification was substantiated in a simulation of a bacteria phantom.
2. Sample preparation and experimental setup
To investigate the effect of reference objects in CDI, we employed two test samples: with and without reference objects. We first prepared assemblies of Au nano-crystals of about 10 to 40 nm in size on a Si3N4 membrane, and considered them as weakly scattering objects. We patterned three asymmetric shapes of about 500 nm in size and deposited 3 nm thick Au film which was then annealed at 700°C to form self-assemblies of Au nano-crystals as revealed in a scanning electron micrograph (SEM) in Fig. 1(a). As for the sample with reference objects, we prepared three 70 nm thick solid Au references of about 500 nm in size together with the nano-particle assemblies on a single membrane as shown in Fig. 1(b).
Fig. 1 (a, c, e) SEM image, measured diffraction intensity, and reconstructed image of Au nano-crystal assemblies respectively. (b, d, f) SEM image, measured diffraction intensity, and reconstructed image of Au nano-crystal assemblies together with neighboring Au reference objects respectively. Insets in (a), (b), (e), and (f) illustrate a view of the nano-particles magnified by a factor of 6.5. The scale bars represent 500 nm (a, b, e, f) and 100 μm−1 (c, d).
The hard x-ray CDI measurements were performed at the Taiwan undulator beamline 12XU at the SPring-8. The x-ray wavelength λ was fixed to 2.27 Å by a Si (111) monochromator. A collimating mirror was employed to increase x-ray flux and to reject high harmonics. A 10 μm aperture defined the x-ray beam within the coherent volume at a flux of about 6 × 109 photons/sec. Samples and a CCD detector were placed 180 mm and 1.9 m downstream of the aperture respectively, which yielded an oversampling ratio of about 10 in each dimension. A beam-stop was employed to block the intense direct beam transmitting through a sample, and the blocked data were patched by those obtained at opposite Q values using the centro-symmetric property of a weak phase object. Due to the limited dynamic range of the CCD, data in the low and high-Q region were separately measured with exposure time of 50 sec and 25 min respectively, and patched together.
3. Results and discussion
3.1. Experimental demonstration of the resolution enhancement
Coherent x-ray diffraction patterns of two test specimens, Au nano-assemblies without and with reference objects shown in Fig. 1(a) and (b) respectively, are illustrated in Fig. 1(c) and (d). The diffraction intensity of the Au nano-assemblies shown in Fig. 1(c) was much weaker than the one with the reference objects shown in Fig. 1(d), especially in the high-Q region with large diffraction angles. In contrast, the sample with the reference objects showed strong intensity out to high-Q region providing much better resolved speckles.
The reference objects enhanced the spatial resolution of the reconstructed image greatly. The modified hybrid-input-output (mHIO) algorithm [25] was applied to retrieve the phase of the diffracted wave and reconstruct sample images. At the start of our mHIO algorithm, a single large rectangular support estimated from the speckle size was applied to confine the real space electron density. As the shape of reference objects appeared after a few ten iterations, we refined the support size to include only the area of meaningful amplitude in real space. Once the reference objects were clearly defined, separated rectangular boxes surrounding each reference objects and later the areas of non-fluctuating amplitude, were used as updated supports. More than ten different initial conditions were used to check the consistency.
The resolution in the reconstructed image of the Au assemblies alone shown in Fig. 1(e) was not enough to resolve each individual Au nano-particles due to the low SNR. However, the reconstructed image with reference objects was much clearer as shown in Fig. 1(f). Au nano-crystals as small as 10 nm were clearly identified with the help of reference particles. The magnified view in the inset showed that the size and position of individual Au particles matched well with those in the SEM image although the size of single pixel of the CDI image was as large as 14 nm. In case of the sample without the references, however, individual particles were not distinguishable due to the low SNR as shown in the inset in Fig. 1(e).
The enhancement originates from the interference between the diffracted wave from a sample and reference objects in the measured intensity I,
where IS(IR) is the diffraction intensity from sample (reference) in terms of photon numbers, ϕS(ϕR) is the phase of diffracted wave from sample (reference), and N is total noise including photon counting noise and instrumental noise NI. Instrumental noise here includes all noises other than photon counting noise in principle. In practice, it includes mainly noises from detecting electronics and fluctuations in background signal caused from x-ray sources and optical elements. When IR is much larger than IS and NI is larger than photon counting noise as in large diffraction angles, third interference term in Eq. (1) is the dominant one that retains sample information. The effective SNR is then, , which can be rewritten as, Equation (2) shows that the SNR is amplified by the amplification factor indicated above as compared to the SNR without any reference objects IS/NI, and enhancement in resolution is expected. When is much larger than NI, the SNR approaches to pure photon counting noise . This shows that instrumental noise can be overcome by strongly scattering reference objects.The interference term, , also carries information on the phase ϕS. This greatly aids the retrieval of ϕS which is essential in CDI image reconstruction. In the presence of strongly diffracting reference objects, we expect ϕR be determined in the early stage of phase retrieval iterations, which consequently helps the retrieval of ϕS greatly. This should be clearly distinguished from Fourier transform holography with a priori information on the reference object was reported previously [22]. In our CDI, the reference objects scatter strongly but unknown, and we found that they are still very helpful in retrieving phase.
3.2. Application of reference objects to image biological and nano oxide specimens
Figure 2 shows two CDI images obtained with the aid of the reference objects that illustrate the usefulness of the amplification in SNR in practical nano-bio applications. Shown in Fig. 2(a) and 2(b) are an SEM and CDI image of a nickel oxide (NiO) nano-particle assembly which was prepared by annealing a Ni nano crystal assembly in air. The NiO specimen was prepared by depositing a Ni film on Si3N4 substrate and annealing at 700°C in N2 environment to form nano-particles followed by oxidation for 5 minutes at 500°C in air. During oxidation process, NiO nano-shell was reported to form as inner Ni atoms diffuse out to leave empty-core inside.
Fig. 2 (a, b) SEM and CDI image of NiO nano-particles respectively. The core-shell structure of the small NiO particles was clear in the CDI image as illustrated in the inset. (c, d) SEM and CDI image of a mitochondrion together with Au reference particles. The mitochondrion is indicated by an arrow. The scale bars represent 500 nm.
In this case, relatively large particles with size over 200 nm acted as reference objects and improved the image resolution, which helped imaging smaller particles. The resolution in the retrieved CDI image was about 12 nm as estimated by the phase retrieval transfer function [26]. The CDI image provides the core-shell structure of NiO nano-particles vividly that was difficult to observe in the SEM image. The inset in Fig. 2(b) shows that the size of the core and the thickness of the shell were about 30 nm each, which is consistent with the report claiming that inner Ni atoms diffuse out to form empty-core inside during oxidation [27, 28]. The CDI image exhibited detailed structure of the Ni nano assembly including this core-shell structure that was difficult to observe in the SEM image.
Figure 2(c) and 2(d) show an SEM and CDI image of a mitochondrion (indicated by an arrow) fixed to a Si3N4 substrate together with five Au reference particles. To prepare the mitochondrion specimen, NIH 3T3 cells were grown on a petri dish and submerged to culture media (Dulbecco’s modified Eagle’s medium with 10 % fetal bovine serum and 1% penicillin/streptomycin), and incubated in a humidified incubator. The cells were then detached and homogenized by bounce homogenizer. The homogenized media including mitochondria were extracted by sucrose-gradient centrifugation.
The internal structure of the weakly scattering mitochondrion was well reconstructed thanks to the reference particles. We found that placing more than three reference objects surrounding specimen was advantageous to define the support in the CDI reconstruction. In addition, it is desirable to distribute them asymmetrically to avoid stagnation due to multiple solutions in the phase retrieval.
In the biological applications of the CDI like mitochonrion imaging, three-dimensional(3D) structural determination is much more informative than two-dimensional(2D) projection imaging. The 2D resolution enhancement presented here can be straightforwardly expanded to the 3D phase retrieval as we apply a computed tomography (CT) process, since one can enhance the resolution of each 2D frame required for 3D tomography. When placing reference objects, it is recommended arrange to minimize possible overlapping with the sample at most rotation angles.
3.3. Simulation with a bacteria phantom
Results of a simulation on a bacteria phantom substantiated the effect of reference objects in overcoming instrumental noise. Figure 3 show simulated diffraction intensities and reconstructed images of the phantom using the parameters similar to those used in our experiment. The exposure was set to a value corresponding to the radiation damage limit (∼ 1×109 Gy) of biomaterials for a few nanometer spatial resolution [15, 17]. While only photon counting noise was included in Fig. 3(a), both photon counting and the CCD instrumental noise (dark current & readout noise estimated with the experimental parameters) were included in Fig. 3(b). While the read noise is independent of data acquisition time, the dark current is proportional to the data acquisition time. A reasonable image was obtained in Fig. 3(a) that showed detailed structures such as ribosomes and flagella which were not reconstructed at all when the instrumental noise was included as shown in Fig. 3(b). This illustrated that the instrumental noise was more critical than photon counting shot noise in deteriorating image resolution. With the aid of reference objects, three Au disks with a diameter (thickness) of 150 nm (120 nm), the image resolution was much enhanced as shown in Fig. 3(c) and 3(d), and details of the phantom shown in Fig. 3(e) were reconstructed successfully even in the presence of the instrumental noise. The behavior of the SNR shown in Fig. 3(f) explains the enhancement of the image resolution by the reference objects. In the presence of the instrumental noise, the reference objects amplified the SNR significantly, especially at high-Q values. The SNR was then improved to a level close to the value free of instrumental noise.
Fig. 3 (a, b) Simulated diffraction intensity IS, and the reconstructed image of a Bacteria phantom. (c, d) Simulated diffraction intensity subtracted by IR and the reconstructed image of the phantom together with the reference objects. While only photon counting noise was included in (a) and (c), both photon counting and instrumental noise were included in (b) and (d). The scale bar in the diffraction pattern represents 100 μm−1. (e) Bacteria phantom designed for the simulation. (f) SNR in diffraction profiles shown in (a), (b), (c), and (d) as a function of Q in the horizontal direction. The scale bars represent 200 nm.
For quantitative analysis of the effect of the reference objects, we evaluated the reconstruction error, namely Rerr which is defined as the difference between the reconstructed electron density and true density divided by the sum of them [29, 30]. Reasonable image reconstruction is expected when Rerr drops below 0.05. Figure 3(a) and (b) show the behavior of Rerr in the absence and presence of the reference objects respectively. In each figure, three curves representing Rerr with no noise, photon noise, and both photon & instrumental noise, are illustrated. As shown in Fig. 4(a) that exhibits Rerr without the reference objects, it takes about 300 iterations for Rerr to drop below 0.05 when only photon noise was included. However, Rerr remained above 0.1 in the presence of additional instrumental noise indicating that the reconstruction would not be successful. In contrast, Rerr in the presence of the reference objects dropped below 0.05 in all cases within 30 iterations as shown in Fig. 4(b). This analysis exhibits that the aid of the reference objects is especially critical in practical experiments where instrumental noise is often dominant.
Fig. 4 (a) Rerr during the reconstruction iterations in the absence of the reference objects. Results of three simulations, including no noise, photon counting noise, and photon & CCD noise, were illustrated. (b) Rerr in the presence of the reference objects. (c) Saturated R value as a function of the electron density of the reference objects in the unit of the density of the bacteria specimen.
Shown in Fig. 4(c) is Rerr after 500 iterations as a function of the electron density of the reference objects in the presence of both photon and instrumental noise. As the density of the references increases over about five times that of the bacteria phantom, Rerr decreases below 0.05 resulting reasonable reconstructions. It is also desirable to use reference objects whose size is smaller than the specimen under investigation to provide meaningful interference signal and magnification in high-Q region.
4. Conclusion
In summary, we demonstrated both in experiments and simulation that reference objects are effective in amplifying low SNR overcoming instrumental noises and improving image resolution in CDI. With the aid of the reference objects, it was possible to identify Au particles as small as 10 nm close to the dimension of a single pixel in image plane. The applicability of the reference objects were demonstrated by imaging NiO nano-shell structure and a mitochondrion. Our simulation showed that the reference objects are useful in imaging biological specimens susceptible to low SNR due to weak scattering power and inevitable instrumental noise. The reference objects with atomic density over five times the biological specimen were predicted to be effective. We expect that biological materials tagged by reference objects [31, 32] can be applied for the high resolution bio-imaging at synchrotron and XFEL facilities.
Acknowledgments
We thank T. K. Lee, N. Hiraoka, and C. Chen for their discussions and help at the SPring-8. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) through NCRC (No. 2008-0062606, NCRC-CELA) and PAL XFEL project. We also acknowledge the GSG Project by GIST in 2014 and the support by Institute for Basic Science (IBS). The BL12XU was supported by the NSRRC with funding from the National Science Council (NSC) of Taiwan.
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