Measured x-ray diffraction pattern of two clusters of gold balls.

The inversion of a diffraction pattern offers aberration-free, diffraction-limited, three-dimensional images without the resolution and depth-of-field limitations of lens-based tomographic systems. Radiation damage becomes the main limiting factor. The best-known example, crystallography, is possible owing to special conditions, namely the existence of crystalline samples and auxiliary techniques for obtaining phase information. Coherent x-ray diffraction imaging (CXDI) eliminates the need for periodic arrays of identical objects and solves the phase problem mathematically.

Three ideas developed over half a century have contributed to the development of CXDI. In 1952, David Sayre (IBM T.J. Watson Research Center) noticed that conventional crystallography, by recording discrete Bragg diffraction spots, "undersampled" the diffraction intensity relative to a theorem of the late information theorist Claude Shannon (AT&T Bell Laboratories). In the 1980s, James R. Fienup (University of Rochester) developed iterative algorithms that efficiently extracted phase information from adequately sampled diffraction patterns like the continuous patterns from nonperiodic samples. Finally, in 1999, Jianwei Miao (now at the Stanford Synchrotron Radiation Laboratory) and colleagues working at the National Synchrotron Light Source reconstructed two-dimensional images of lithographically prepared objects with a resolution of 75 nm. A number of groups have since achieved higher resolution and three-dimensional imaging with both x rays and electrons.

Imaging without a Lens

What do cameras, microscopes, and telescopes have in common? They all have lenses that collect light and form images. But lenses have limitations, and for scientists in the burgeoning realms of nanotechnology and life science at the molecular level who want to peer at three-dimensional images of the interiors of submicroscopic objects with features measured in billionths of a meter (nanometers), there is no lens that can do the job. The obvious solution is to scrap the lens altogether.

It turns out that crystallographers, with either x-ray or electron beams as their illuminating source, have been taking the lensless approach for decades, but only for samples consisting of a very large number of identical objects arrayed in a regular pattern (e.g., atoms in a crystal). In this case, it is mathematically possible to reconstruct the structure of the object (i.e., atomic positions) from the intensities and directions of beams that are deflected (diffracted) as they pass through the sample. A growing number of researchers are now taking image reconstruction from diffraction patterns a big step further by studying individual, nonrepeating objects by the new technique of coherent x-ray diffraction imaging, more colloquially known as lensless imaging. Marchesini et al. are the first to accomplish this without any supplementary information whatsoever (working with the diffraction pattern only), as shown by their two-dimensional images of clusters of gold balls 50 nm in diameter.

Image reconstruction of two clusters of gold balls showing the convergence of both the reconstructed image and the support as the number of iterations increases from 1 to 1000. For comparison, a scanning electron microscope (SEM) image is also shown. The Rayleigh resolution of the reconstructed image is 10 nm.

In Fienup's hybrid input-output algorithm, one starts with a diffraction pattern with random phases and then iteratively transforms between real space (the image) and back to reciprocal space (the diffraction pattern with phases). Each transform to reciprocal space provides improved phases for the next cycle. An essential constraint is the requirement that the intensity of the diffraction pattern be zero outside the boundary of the object (the support). The better the support is known, the faster the iterations converge to an accurate image. Most researchers have relied on x-ray microscopy or other techniques to supply this information.

Effect of noise on reconstruction error. The shrink-wrap algorithm (blue) is superior to the hybrid input-output algorithm with fixed support (all other colors), except for the case when the support is known perfectly (green). The accuracy of the supports decreases from support 1 to support 4.

The Livermore/ASU/ALS collaboration has been working to eliminate the need for supplementary experiments. Previously, the team had succeeded by preparing on a silicon nitride substrate clusters of gold balls 50 nm in diameter together with an isolated single gold ball as reference that generated the information needed to construct the support. Now, the researchers have done away with even that requirement with a new "shrink-wrap" algorithm. They use a transform of the diffraction pattern as the initial support. At intervals, they generate a new support from the transform of the current diffraction pattern. In this way, the support converges to a tight boundary around the cluster of balls, and the image also emerges.

Work on three-dimensional images from a series of diffraction patterns obtained at many illumination angles is underway. In the meantime, the researchers believe that a resolution of 10 nm will be possible for life-science samples, where radiation damage is an issue, and 2 nm for solids. A dedicated CXDI beamline at the ALS after a planned brightness upgrade could improve the figure to 1 nm, owing to an increase in imaging speed. The ultrabright, femstosecond x-ray pulses expected when SSRL's Linac Coherent Light Source comes on line around 2008 may enable atomic-resolution imaging of single molecules.

Research conducted by S. Marchesini, H.N. Chapman, S.P. Hau-Riege, and A. Noy (Lawrence Livermore National Laboratory); H. He and M. Howells (ALS); and U. Weierstall and J.C.H. Spence (Arizona State University).

Research funding: U.S. Department of Energy, Office of Basic Energy Sciences (BES), and National Science Foundation. Operation of the ALS is supported by BES.

Publication about this research: S. Marchesini, H. He, H.N. Chapman, S.P. Hau-Riege, A. Noy, M.R. Howells, U. Weierstall, and J.C.H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).