Coherent Diffraction Imaging - Reconstruction

Reconstruction

In a typical reconstruction 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 hundreds to thousands of cycles. 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 dots. Other researchers have investigated imaging extended objects, that is, objects that are larger than the beam size, by applying other constraints.

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 developed 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 shown 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”. Therefore, it is possible to image the strain fields associated with crystal defects in 3D using CDI and it has been reported in one case. Unfortunately, the imaging of complex-valued functions (which for brevity represents the strained field in crystals) is accompanied by complementary problems namely, the uniqueness of the solutions, stagnation of the algorithm etc. However, recent developments that overcame these problems (particularly for patterned structures) were addressed. On the other hand, if the diffraction geometry is insensitive to strain, such as in GISAXS, the electron density will be real valued and positive. 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.