Overcoming resolution and contrast limits in x-ray tomography

X-ray tomography is a workhorse tool for nondestructive imaging. It is used to probe 3D structures (across a wide range of length scales) of objects that offer good absorption contrast to x-rays. In recent years, extremely-high-resolution imaging (on the order of tens of nanometers) has become possible because of technological advances in x-ray optics. This has made the technique an excellent option for probing samples with complicated 3D structures on micron and submicron scales (see Figure 1). However, probing of (sub)nanometer scales and examination of samples that offer poor absorption requires a new approach.

Figure 1. Reconstruction of 26.4keV x-ray-tomographic data taken at beamline 2-BM at the Advanced Photon Source. The image shows a rendered section of human femoral bone with osteocyte lacunae in gold and pore structure in red. Voxel (3D pixel) size: 1.4μm. (Sample and image courtesy of J. Clement and C. D. L. Thomas, School of Dental Science, University of Melbourne, Australia.)

To overcome the likely limitations that will be reached by x-ray optics, a method—generically referred to as coherent diffractive imaging (CDI)¹—that draws on ideas from the field of crystallography has been developed. CDI can phase diffraction patterns from nonperiodic samples. Ultimately, the technique is limited only by the wavelength of the x-ray radiation used to image the sample. It also has the advantage that x-ray-transparent structures can produce high-contrast diffraction patterns, thus making high-resolution imaging of biological samples a reality.

In its simplest form, CDI works by measuring the far-field diffraction pattern from a sample at a rate of twice that suggested by the Nyquist-Shannon sampling theorem. If the sample is finite in extent or illuminated with a known finite beam, in all practical cases, sufficient information is available to solve for the phase of the diffraction pattern. Once its phase and intensity are known, it can be inverted by Fourier transformation to obtain (under certain conditions) a map of the projected density of the sample. This approach has now been demonstrated by a number of groups.

Since the strength of the measured diffraction pattern falls off rapidly with distance from the sample, highly intense and coherent sources are required. Much of the work to date has been performed at third-generation synchrotron sources. The science that can be performed with CDI is one of the drivers behind the application of so-called fourth-generation (free-electron-laser) sources. We have examined ways in which more reliable phasing of the diffraction pattern can be obtained, thus allowing more efficient light use. This has resulted in a number of recent advances.

Figure 2. Diffraction pattern produced using curved-beam illumination at beamline 2-ID-B of the Advanced Photon Source. The central holographic region (blue) shows the spoke structure of the test sample, imaged using 2.535keV x-rays. The surrounding pattern is diffraction beyond the numerical aperture of the zone plate used to create the illumination. This can be used to reconstruct the sample at a resolution better than what could be obtained using the zone plate as a lens in a conventional microscope. (Image courtesy of J. Clark, La Trobe University, Melbourne, Australia.)

We have shown that using a curved beam results in a faster and more reliable solution for the phase that is robust to partial coherence in the illumination. Part of the reason for this is that, in addition to the high-angle scatter that provides high-spatial-frequency information about the sample, curved-beam illumination produces a Gabor-type, in-line hologram. This provides low-spatial-frequency information that would otherwise be swamped by the intense, undiffracted beam (see Figure 2). We have also developed a method² to obtain the structure of the illuminating beam as part of our measurement process (see Figure 3).

Figure 3. Intensity reconstruction in the focal region of a beam produced by a 100nm outer-zone-width zone plate of diameter 160μm at an x-ray energy of 9.9keV, obtained at the x-ray-fluorescence microscopy beamline at the Australian Synchrotron. The width of the image is approximately 180nm. (Image courtesy of A. Carroll, La Trobe University.)

By generalizing our understanding of the diffraction processes that create the final measured diffraction pattern, we have also formulated a new approach to the inversion techniques currently used. Typically, the resultant diffraction pattern can be thought of as an incoherent sum of various modes of illumination. These could be different wavelengths (for polychromatic sources), coherent modes (for partially coherent sources), sample positions (when a sample is moving during an experiment), or illuminations (when phase-diversity methods are used). By correctly combining the effects of these modes and applying the resultant diffraction pattern appropriately as a constraint to the inversion algorithm, we have significantly improved the recovered sample density.

Our novel approaches have improved the images we obtain using CDI, so that the technique may be rightly considered a true microscopy method. The challenges ahead now lie in further applying our methods to biological samples at high resolution. To avoid problems associated with high imaging doses, this might include the application of cryogenic sample cooling or through rapid imaging using a free-electron laser.

We acknowledge support of the Australian Research Council's Centre of Excellence for Coherent X-ray Science. Parts of this research were undertaken at the x-ray-fluorescence microprobe beamline at the Australian Synchrotron (Victoria) and at the Advanced Photon Source. Use of the latter is supported by the US Department of Energy, Office of Science, and Office of Basic Energy Sciences, under contract DE-AC02-06CH11357.