The influence of strain on image reconstruction in Bragg coherent X-ray diffraction imaging and ptychography

Abstract

A quantitative analysis of the effect of strain on phase retrieval in Bragg coherent X-ray diffraction imaging is reported. It is shown in reconstruction simulations that the phase maps of objects with strong step-like phase changes are more precisely retrieved than the corresponding modulus values. The simulations suggest that the reconstruction precision for both phase and modulus can be improved by employing a modulus homogenization (MH) constraint. This approach was tested on experimental data from a highly strained Fe-Al crystal which also features antiphase domain boundaries yielding characteristic π phase shifts of the (001) superlattice reflection. The impact of MH is significant and this study outlines a successful method towards imaging of strong phase objects using the next generation of coherent X-ray sources, including X-ray free-electron lasers.

Keywords: Bragg ptychography; anti-phase domain boundary; coherent X-ray diffraction imaging; modulus homogenization; strong phase object.

Figure 1

Two different numerical models employed for quantitative analyses of the effect of strain. (a, b) General phase model (GPM) which assigns both a modulus (a) and a phase (b) value to each pixel. (c, d) Real atomistic model (RAM) which contains only modulus values. Here the phase shift in Bragg diffraction originates from lattice distortions. L and ΔL in (d) correspond to half of the lattice constant (a/2) in the (10) crystal plane direction and the step-like displacement, respectively. L was fixed to 32 pixels and ΔL was varied for the different simulations (see Table 1 ▸).

Figure 2

(a, b, c) Calculated diffraction amplitudes of the GPM with three different phase steps, π/8 (a), π/2 (b), and π (c). (d, e, f) Calculated diffraction amplitudes of the RAM with phase steps of π/8 (d), π/2 (e), and π (f).

Figure 3

(a, b, c) Reconstructed modulus (left column) and phase (right column) of GPM with π/8 (a), π/2 (b), and π (c) phase steps. (d, e, f) Reconstructed modulus (left column) and phase (right column) of RAM with π/8 (d), π/2 (e), and π (f) phase steps. Only the central 160 × 160 array is used for image reconstruction. One representative image out of 30 independent reconstructions is shown for each simulation.

Figure 4

(a) SI value of the calculated diffraction amplitudes (Fig. 2 ▸) as a function of phase shift. (b) Averaged R _err after 30 independent reconstructions as a function of phase shift. (c, d) R _ρ value (c) and R _ϕ value (d) which were calculated as an average of the five best images out of 30 independent reconstructions.

Figure 5

(a, b) Reconstructed modulus (a) and phase (b) of the RAM object with a π phase step without the MH constraint applied. (c, d) Line profile of the reconstructed modulus (c) and phase (d), shown by blue dotted lines in (a) and (b), respectively. The five best reconstructions were averaged for the line profile analysis and the black solid lines show the original modulus and phase. (e, f) Reconstructed modulus (e) and phase (f) of the RAM object with π phase step using the MH constraint. (g, h) Line profile of the reconstructed modulus (g) and phase (h), shown by red dotted lines in (e) and (f), respectively. The five best reconstructions are averaged for the line profile analysis where the black solid lines show the original modulus and phase.

Figure 6

(a, b) Central pixel slice (sectioned) of the reconstructed (001) phase image (voxel) of an Fe–Al alloy crystal (B2 phase) without the MH constraint at different magnifications. (c, d) Same as above but with the MH constraint applied. The insets in (b) and (d) show reconstructed modulus images. (e) Line profiles of the reconstructed phases marked by a blue dotted line in (b) and a red dotted line in (d). The phase profiles along the lines denoted A and B in (a) and (c) are plotted in Fig. 7 ▸.

Figure 7

Phase line profiles from Figs. 6 ▸(a) and 6(c). A is the line profile across an ADB while B corresponds to a monodomain, see white dotted lines in Figs. 6 ▸(a) and 6(c). In the former case the step is sharpened by the MH constraint while in the latter case a smoothening is the result.