Coherent Diffractive Imaging (CDI)

Recent years have witnessed two revolutionary developments in X-ray science. First, a new approach to X-ray crystallography, known as CDI, was first demonstrated in 1999 (1) that allows structural determination of non-crystalline specimens and nano-crystals with a resolution limited only by the spatial frequency of the diffracted waves. Second, large-scale coherent X-ray sources, such as XFELs and diffraction-limited storage rings, have been under rapid development worldwide. Furthermore, tabletop coherent X-ray sources based on HHG have advanced rapidly, significantly increasing access to ultrafast coherent X-ray beams for applications in nano and materials science. In recent years, the combination of powerful coherent X-ray sources and CDI methods, coupled with advanced X-ray detectors and computational algorithms, has opened up new research frontiers in the physical and biological sciences that are not attainable with conventional X-ray crystallography (2-5).

Schematic layout of five main CDI methods and iterative phase retrieval algorithms. (A) Plane-wave CDI: A plane wave illuminates a sample, and an oversampled diffraction pattern is measured by a detector. (B) Bragg CDI: The diffraction pattern surrounding a Bragg peak is acquired from a nanocrystal. (C) Ptychography: A coherent x-ray probe is generated by an aperture or focusing optics. An extended sample is scanned through the probe on a 2D grid, and diffraction patterns are collected from a series of partially overlapping regions. (D) Fresnel CDI: A sample is positioned in front of (or behind) the focal spot of a coherent x-ray wave, and the Fresnel diffraction pattern ismeasured by a detector. (E) Reflection CDI: A coherent x-ray wave is specularly reflected off a sample on a substrate, and the diffraction intensity around the reflected beam is collected by a detector. (F) Phase retrieval algorithms iterate back and forth between real and reciprocal space. In each iteration, various constraints, including support, positivity (i.e., electron density cannot be negative), or partially overlapping regions, are enforced in real space, while the measured Fourier magnitude is updated in reciprocal space. Usually, after hundreds to thousands of iterations, the correct phase information can be recovered.

In CDI, a coherent wave illuminates an object and the diffracted wave field in the far-field is proportional to the Fourier transform of the object. While the magnitude squared of the Fourier transform can be measured as an intensity by a detector, the phase information is lost, which constitutes the well-known phase problem. For a non-crystalline specimen or nano-crystal, the diffraction pattern is continuous and can be sampled at a frequency finer than the Nyquist interval. When the number of independently sampled intensity points is larger than the number of unknown variables associated with a specimen, the phase information is in principle encoded inside the diffraction intensity (6) and can be retrieved by iterative algorithms (7). Over the years, various CDI methods have been developed, including plane-wave CDI, Bragg CDI, ptychography, reflection CDI, Fresnel CDI, and sparsity CDI. These CDI methods have been used to study a broad range of samples in physics, chemistry, materials science, nanoscience, geology, and biology (2-5).

Revolution in coherent X-ray sources. Moore’s Law for the computer speed of CPU chips vs. the X-ray brilliance of coherent light sources. While the computer speed has increased by 12 orders of the magnitude in 6 decades, the X-ray brilliance has improved by 20 orders of the magnitude in 6 decades (brilliance is a measure of coherent X-ray flux). The inset shows the self-amplified spontaneous emission (SASE) process to produce extremely intense and short XFEL pulses.

Selected Publications

1. J. Miao, P. Charalambous, J. Kirz and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens", Nature 400, 342-344 (1999).

2. J. Miao, T. Ishikawa, I. K. Robinson and M. M. Murnane, "Beyond crystallography: Diffractive imaging using coherent x-ray light sources", Science 348, 530-535 (2015).

3. M. Gallagher-Jones, J. A. Rodriguez and J. Miao, "Frontier Methods in Coherent X-ray Diffraction for High-Resolution Structure Determination", Q. Rev. Biophys. 49, e20 (2016).

4. I. Schlichting and J. Miao, "Emerging opportunities in structural biology with X-ray free-electron lasers", Curr. Opin. Struct. Biol. 22, 613–626 (2012).

5. J. Miao, R. L. Sandberg and C. Song "Coherent X-ray Diffraction Imaging", IEEE J. Sel. Top. Quant. Electron. 18, 399-410 (2012).

6. J. Miao, D. Sayre and H. N. Chapman, "Phase Retrieval from the Magnitude of the Fourier transform of Non-periodic Objects", J. Opt. Soc. Am. A. A 15, 1662-1669 (1998).

7. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao and M. Segev, "Phase Retrieval with Application to Optical Imaging: A contemporary overview", IEEE Signal Processing Mag. 32, 87-109 (2015).