General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging. Part I

Abstract

This two‐part article series provides a generalized description of the scattering geometry of Bragg coherent diffraction imaging (BCDI) experiments, the shear distortion effects inherent in the 3D image obtained from presently used methods and strategies to mitigate this distortion. Part I starts from fundamental considerations to present the general real‐space coordinate transformation required to correct this shear, in a compact operator formulation that easily lends itself to implementation with available software packages. Such a transformation, applied as a final post‐processing step following phase retrieval, is crucial for arriving at an undistorted, correctly oriented and physically meaningful image of the 3D crystalline scatterer. As the relevance of BCDI grows in the field of materials characterization, the available sparse literature that addresses the geometric theory of BCDI and the subsequent analysis methods are generalized here. This geometrical aspect, specific to coherent Bragg diffraction and absent in 2D transmission CDI experiments, gains particular importance when it comes to spatially resolved characterization of 3D crystalline materials in a reliable nondestructive manner. This series of articles describes this theory, from the diffraction in Bragg geometry to the corrections needed to obtain a properly rendered digital image of the 3D scatterer. Part I of this series provides the experimental BCDI community with the general form of the 3D real‐space distortions in the phase‐retrieved object, along with the necessary post‐retrieval correction method. Part II builds upon the geometric theory developed in Part I with the formalism to correct the shear distortions directly on an orthogonal grid within the phase‐retrieval algorithm itself, allowing more physically realistic constraints to be applied. Taken together, Parts I and II provide the X‐ray science community with a set of generalized BCDI shear‐correction techniques crucial to the final rendering of a 3D crystalline scatterer and for the development of new BCDI methods and experiments.

Annealing of focused ion beam damage in gold microcrystals: an in situ Bragg coherent X-ray diffraction imaging study

“…If the CXDP is oversampled by at least twice the Nyquist frequency (Miao & Sayre, 2000), at least 4 pixels per fringe period, iterative phase retrieval algorithms that apply constraints in real and reciprocal space can be used to recover the phase (Fienup, 1982;Robinson et al, 2001). The amplitude and phase in reciprocal space are related to the real-space object via an inverse Fourier transform (Miao & Sayre, 2000;Robinson et al, 2001;Clark et al, 2012) followed by a space transformation from detector conjugated space to orthogonal lab or sample space (Yang et al, 2019;Li et al, 2019;Maddali et al, 2020). The resulting amplitude, (r), where r is the position vector, is proportional to the effective electron density of the crystalline volume associated with the particular crystal reflection.…”

Detector Tilt Considerations in Bragg Coherent Diffraction Imaging: A Simulation Study

“…The directionk 3 is perpendicular to the imaging plane along the nominal exit beam direction. This is identical to the frame B det from [23].…”

“…The tilt-distorted 3D diffraction pattern is obtained using a customized Fourier transform-based forward propagation model that explicitly takes into account an arbitrary detector tilt. This method draws from the geometric theory of BCDI developed in [23,24] and is demonstrated here with simulations.…”

“…As a recap of the representative experimental schematic [23], we refer to Figure 2, which depicts the following features:…”

“…The azimuth and elevation angles δ and γ in Figure 2 are specific to the configuration at the 34-ID-C end station of the Advanced Photon Source and denote typical diffractometer-based parameterizations of the detector position. In BCDI, the column vectors of B recip are not mutually orthogonal owing to the geometry of q k ( [23] contains a derivation of this fact). However, the detector is typically arranged with its face normal to the exit beam such that q i k 1 and q j k 2 , as seen in Figure 2.…”