Coherent diffraction imaging

Coherent diffraction imaging (CDI) is a “lensless” technique for 2D or 3D imaging of nanoscale structures such as nanotubes¹, nanocrystals², defects³, potentially proteins⁴ and more⁴. In CDI, a highly coherent nano area beam of x-rays or electrons is incident on a sample to generate a diffraction pattern. This pattern is recorded electronically and then used to reconstruct an image via an iterative feedback algorithm. Effectively, the objective lens in a typical microscope is replaced with software to convert from the reciprocal space diffraction pattern into a real space image. The advantage in using no lenses is that the final image is aberration –free and so resolution is only diffraction and dose limited (dependent on wavelength, aperture size and exposure). A simple Fourier transform is insufficient for creating an image from the diffraction pattern due to the phase problem.

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The Phase Problem

Main article: phase problem

There are two relevant parameters for diffracted waves: amplitude and phase. In typical microscopy using lenses, the phase information is not lost when waves are refracted by lenses and so there is no phase problem. When a diffraction pattern is generated the amplitude information is given in the number of counts of x-ray photons or electrons into the detector but the phase of those waves is unknown. The result is that there is not a unique solution when Fourier transforming from reciprocal space into real space.

Three ideas developed that enabled the reconstruction of real space images from diffraction patterns⁴. The first idea was the realization by Sayre in 1952 that Bragg diffraction under samples diffracted intensity relative to Shannon’s theorem. If the diffraction pattern is sampled at twice the Nyquist frequency (inverse of sample size) or lower it can yield a unique real space image². The second was an increase in computing power in the 1980’s which enabled iterative “hybrid input-output” (HIO) algorithms to optimize and extract phase information using adequately sampled intensity data with feedback. This method was introduced³ by Fienup in the 1980s. Finally, the development of “phase recovery” algorithms led to the first demonstration⁴ of CDI in 1999 by Miao.

Reconstruction

In a typical reconstruction² the first step is to generate random phases and combine them with the amplitude information from the reciprocal space pattern. Then a Fourier transform is applied back and forth to move between real space and reciprocal space with the modulus squared of the diffracted wave field set equal to the measured diffraction intensities in each cycle. By applying various constraints in real and reciprocal space the pattern evolves into an image after enough iterations of the HIO process. To ensure reproducibility the process is typically repeated with new sets of random phases with each run having typically 100’s to 1000’s of cycles^2,8. The constraints imposed in real and reciprocal space typically depend on the experimental setup and the sample to be imaged. The real space constraint is to restrict the imaged object to a confined region called the “support.” For example, the object to be imaged can be initially assumed to reside in a region no larger than roughly the beam size. In some cases this constraint may be more restrictive, such as in a periodic support region for a uniformly spaced array of quantum dots².

In most cases the support constraint imposed is a priori in that it is modified by the researcher based on the evolving image. In theory this is not necessarily required and algorithms have been developed8 which impose an evolving support based on the image alone using an auto-correlation function. This eliminates the need for a secondary image (support) thus making the reconstruction autonomic.

The diffraction pattern of a perfect crystal is symmetric so the inverse Fourier transform of that pattern is entirely real valued. The introduction of defects in the crystal leads to an asymmetric diffraction pattern with a complex valued inverse Fourier transform. It has been shown⁷ that the crystal density can be represented as a complex function where its magnitude is electron density and its phase is the “projection of the local deformations of the crystal lattice onto the reciprocal lattice vector Q of the Bragg peak about which the diffraction is measured”³. Therefore, it is possible to image the strain fields associated with crystal defects in 3D using CDI and it has been reported3 in one case. On the other hand, if the diffraction geometry is insensitive to strain, such as in GISAXS, the electron density will be real valued and positive². This provides another constraint for the HIO process, thus increasing the efficiency of the algorithm and the amount of information that can be extracted from the diffraction pattern.

Coherence

Clearly a highly coherent beam of waves is required for CDI to work since the technique requires interference of diffracted waves. Coherent waves must be generated at the source (synchrotron, field emitter, etc) and must maintain coherence until diffraction. It has been shown⁸ that the coherence width of the incident beam needs to be approximately twice the lateral width of the object to be imaged. This result is due to the fact that Shannon sampling has twice the spatial period of Bragg sampling. As the coherence width is decreased, the size of the Bragg peaks in reciprocal space grows and they begin to overlap leading to decreased image resolution.

Diffraction Imaging Techniques

Coherent x-ray diffraction imaging (CXDI or CXD) uses x-rays (typically .5-4kV)⁴ to form a diffraction pattern which may be more attractive for 3D applications than electron diffraction since x-rays typically have better penetration. For imaging surfaces, the penetration of X-rays may be undesirable, in which case a glancing angle geometry may be used such as GISAXS2. A typical x-ray CCD is used to record the diffraction pattern. If the sample is rotated and two perpendicular CCD’s are used, a 3-Dimensional image may be reconstructed³.

Due to radiation damage4, resolution is limited (for continuous illumination set-ups) to about 10 nm for frozen-hydrated biological samples but resolutions of as high as 1 to 2 nm should be possible for inorganic materials less sensitive to damage (using modern synchrotron sources). It has been proposed⁴ that radiation damage may be avoided by using ultra short pulses of x-rays where the time scale of the destruction mechanism is longer than the pulse duration. This may enable higher energy and therefore higher resolution CXDI of organic materials such as proteins. However, without the loss of information “the linear number of detector pixels fixes the energy spread needed in the beam”⁸ which becomes increasingly difficult to control at higher energies.

In a 2006 report³, resolution was 40 nm using the Advanced Photon Source (APS) but the authors suggest this could be improved with higher power and more coherent X-ray sources such as the X-ray free electron laser.

Coherent electron diffraction imaging works the same as CXDI in principle only electrons are the diffracted waves and an imaging plate is used to detect electrons rather than a CCD. In one published report¹ a double walled carbon nanotube (DWCNT) was imaged using nano area electron diffraction (NAED) with atomic resolution. In principle, electron diffraction imaging should yield a higher resolution image because the wavelength of electrons can be much smaller than photons without going to very high energies. Electrons also have much weaker penetration so they are more surface sensitive than X-rays. However, typically electron beams are more damaging than x-rays so this technique may be limited to inorganic materials.

In Zuo’s approach¹, a low resolution electron image is used to locate a nanotube. A field emission electron gun generates a beam with high coherence and high intensity. The beam size is limited to nano area with the condenser aperture in order to ensure scattering from only a section of the nanotube of interest. The diffraction pattern is recorded in the far field using electron imaging plates to a resolution of 0.0025 1/Å. Using a typical HIO reconstruction method an image is produced with Å resolution in which the DWCNT chirality (lattice structure) can be directly observed. Zuo found that it is possible to start with non-random phases based on a low resolution image from a TEM to improve the final image quality.

See also

Diffraction

Diffraction Tomography

Nanotechnology

Surface Physics

Synchrotron

List of materials analysis methods

References

[1] Atomic Resolution Imaging of a Carbon Nanotube from Difraction Intensities, JM Zuo, I Vartanyants, M Gao, R Zhang, LA Nagahara, Science 300, 1419 (2003)

[2] Coherent x-ray diffraction from Quantum dots, IA Vartanyants, IK Robinson, JD Onken, MA Pfeifer, GJ Williams, F Pfeiffer, H Metzger, Z Zhong, G Bauer, Phys. Rev. B 71, 245302 (2005)

[3] Three-dimensional mapping of a deformation field inside a nanocrystal, M Pfeifer, GJ Williams, IA Vartanyants, R Harder, IK Robinson, Nature Lett. Vol 442, 63-66 (2006)

[4] Coherent X-ray diffractive imaging: applications and limitations, S Marchenini, HN Chapman, SP Hau-Riege, RA London, A Szoke, H He, MR Howells, H Padmore, R Rosen, JCH Spence, U Weierstall, Optics Express 2344, Vol. 11, No. 19, (2003)

[5] Reconstruction of a complex-valued object from the modulous of its Fourier transform using a support constraint, JR Fineup, J. Opt. Soc. Am. A 4, 118-123 (1987)

[6] Extending the methodology of x-ray crystallography to allow imaging of micromere-sized non-crystalline specimens, J Miao, P Charalambous, J Kirz, D Sayre, Nature 400, 342-344 (1999)

[7] Partial coherence effects on the imaging of small crystals using coherent X-ray diffraction, IA Vartanyants, IK Robinson, J. Phys. Condensed Matter 13, 10593-10611, (2001)

[8] Coherence and sampling requirements for diffractive imaging, JCH Spence, U Weierstall, M Howells, Ultramicroscopy 101, 149-152, (2004)

[9] X-ray image reconstruction from a diffraction pattern alone, S Marchesini, H He, HN Chapman, SP Hau-Riege, A Noy, MR Howells, Weierstall, JCH Spence, Phys. Rev. B 68, 140101, (2003)