Concurrent phase retrieval for imaging strain in nanocrystals

Coherent diffraction imaging is a form of microscopy that permits high resolution imaging of atomic displacements from equilibrium where the use of conventional optics is not feasible. Approaches to date for the recovery of atomic displacements from equilibrium and subsequently strain information occur after phase reconstruction of the complex real-space images from at least three independent Bragg diffraction amplitude measurements. While this is a more accessible and effective approach to recover strain information, there is potential for erroneous results if the recovered phase information is not carefully treated. Here we present a strategy for imaging strain with coherent x rays that eliminates the technical challenges that exist in conventional approaches by constructing the strain field concurrently during the phase retrieval process of recovering phase information.

Figure 1

Diffraction experimental geometry and concurrent phase retrieval method. (a) Multiple Bragg reflections with wave vector ${\widehat{k}}_{f_N}$ are excited from a single nanocrystal and each speckle pattern acquired (only two are displayed, for clarity). The incident wave vector is denoted as ${\widehat{k}}_i$. The rocking curve method is used to acquire two-dimensional slices of the three-dimensional speckle pattern by rotating the nanocrystal through the Bragg condition. (b) Flow chart of concurrent multiple Bragg reflection method. Enclosed region shows retrieval processes that are common for a single ($N=1$) Bragg reflection.

Figure 2

Reconstructed phase information for each Bragg reflection. Phase is shown in the $xy$ plane at half distance along the $z$ direction. (a)–(d) Ideal phase profile. (e)–(h) Phase reconstructed concurrently shows good agreement with the ideal case and has few artefacts. (i)–(l) Phase reconstructed independently for each Bragg reflection. The phase is recovered sufficiently well but contains a greater number of phase reconstruction artefacts as this approach does not benefit from the normalizing process inherent in the concurrent approach.

Figure 3

Two-dimensional slices through the Strain tensor information shown for the ideal, concurrent and independent phase reconstruction methods. Strain is shown in the $xy$ plane at half distance along the $z$ direction. (a) Ideal strain tensor information. (b) Strain tensor information reconstructed concurrently shows good agreement with the ideal case. (c) Strain tensor information reconstructed independently for each Bragg reflection shows reasonable agreement with the ideal case. Artefacts such as ripples in the phase which can occur as a results of sub-optimal alignment are visible in the independently reconstructed phase information.

Figure 4

Error residual plotted for the concurrent and conventional independent phase reconstruction methods. The error plotted is the mean error for each Bragg reflection at a given iteration. The concurrent approach clearly shows stronger convergence down to $5\times {10}^{-4}$ while the independent phase reconstruction method was found to stagnate at just above ${10}^{-3}$.