General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging: Part 2

TL;DR

This paper introduces new mathematical methods for shear-correcting coordinate transformations in Bragg coherent diffraction imaging, enabling more accurate 3D imaging of crystalline particles by performing the entire inversion in an orthogonal frame.

Contribution

It presents two novel transformation strategies that allow phase retrieval to be performed directly in an orthogonal frame, improving accuracy and incorporating experimental geometry information.

Findings

Sample images are correctly interpreted in a shear-free frame.

Physically realistic constraints are more easily incorporated into phase retrieval.

The methods work for both evenly and non-evenly sampled data.

Abstract

X-ray Bragg coherent diffraction imaging has been demonstrated as a powerful three-dimensional (3D) microscopy approach for the investigation of sub-micrometer-scale crystalline particles. It is based on the measurement of a series of coherent diffraction intensity patterns that are numerically inverted to retrieve an image of the spatial distribution of relative phase and amplitude of the Bragg structure factor of the scatterer. This 3D information, which is collected through an angular rotation of the sample, is necessarily obtained in a non-orthogonal frame in Fourier space that must be eventually reconciled. To deal with this, the currently favored approach (detailed in Part I) is to perform the entire inversion in conjugate non-orthogonal real and Fourier space frames, and to transform the 3D sample image into an orthogonal frame as a post-processing step for result analysis. In this article, a direct follow-up of Part I, we demonstrate two different transformation strategies that enable the entire inversion procedure of the measured data set to be performed in an orthogonal frame. The new approaches described here build mathematical and numerical frameworks that apply to the cases of evenly and non-evenly sampled data along the direction of sample rotation (the rocking curve). The value of these methods is that they rely on and incorporate significantly more information about the experimental geometry into the design of the phase retrieval Fourier transformation than the strategy presented in Part I. Two important outcomes are 1) that the resulting sample image is correctly interpreted in a shear-free frame, and 2) physically realistic constraints of BCDI phase retrieval that are difficult to implement with current methods are easily incorporated. Computing scripts are also given to aid readers in the implementation of the proposed formalisms.