Quantifying Resolution in Pink Beam Dark Field X-ray Microscopy: Experiments and Simulations

Abstract

Pink-beam Dark-Field X-ray Microscopy (pDFXM) is a powerful emerging technique for time-resolved studies of microstructure and strain evolution in bulk crystalline materials. In this work, we systematically assess the performance of pDFXM relative to monochromatic DFXM when using a compound refractive lens (CRL) as the objective. Analytical expressions for the spatial and angular resolution are derived and compared with numerical simulations based on geometrical optics and experimental data. The pink-beam configuration provides an increased diffraction intensity depending on the deformation state of the sample, accompanied by a general tenfold degradation in angular resolution along the rocking and longitudinal directions. This trade-off is disadvantageous for axial strain mapping, but can be advantageous in cases where integrated intensities are needed. For a perfect crystal under parallel illumination with a pink beam, our results show that chromatic aberration is absent, whereas under condensed illumination it becomes significant. The aberration is shown to depend strongly on the local distortion of the crystal. Weak-beam imaging conditions, such as those required for resolving dislocations, are shown to remain feasible under pink-beam operation and may even provide an improved signal-to-noise ratio. The higher incident flux, enhanced by nearly two orders of magnitude, is quantified in terms of beam heating effects, and implications for optimized scanning protocols are discussed.

Figures (9)

• Figure 1: The DFXM set-up for the monochromatic illumination. The incident beam is shaped by the double multilayer monochromator (DMM), then by the channel cut monochromator (CCM). The objective is a Compound Refractive Lens (CRL). This is inserted in the line of the diffracted beam, characterized by angles $2\theta$ and $\eta$. The optical axis of the objective is shown as a dashed line with indications of the sample-to-entry-of-objective distance, $d_1$ and the exit-of-objective-to-detector distance, $d_2$.
• Figure 2: Reciprocal space resolution function for box beam (quasi-parallel illumination) with monochromatic (a) and pink (b) beam, and line beam (condensed illumination) for monochromatic (c) and pink (d) beam. In all cases, the reciprocal space is normalized, cf. Eq. \ref{['q_normalised']}. The resolution function is displayed as a point cloud in 3D in blue, while the red, yellow and cyan point clouds are projections onto 2D planes.
• Figure 3: Simulated diffraction response from ring-shaped, undeformed single crystals under different beam conditions. When the beam is both pink and condensed (D1 and D2), strong chromatic aberrations appear. In A1--D1, the rings are centered on the optical axis, while in A2--D2, they diffract toward the detector edge. Comparing D1 and D2 shows that chromatic aberrations depend on the offset from the optical axis. Chromatic aberrations are much stronger in the vertical scattering plane (detector $z$) than in the transverse plane (detector $y$). Cartesian line profiles across the ring edges are shown in E for all four pink-beam scenarios (C1--D2). Chromatic aberrations in D1--D2 are seen to cause a $1~\mu\text{m}$ blurring , while the pink-parallel beam cases (C1--C2) show no blurring.
• Figure 4: Comparison of normalized experimental data (dots) and simulations (blue curves) of the rocking and rolling direction components of the reciprocal space resolution function for the box beam case. Also shown are fits of the experimental data to Gaussian distributions (red lines). a) Resolution function in $q_{\text{rock}'}$ for monochromatic beam. b) Resolution function in $q_{\text{roll}}$ for monochromatic beam. c) Resolution function in $q_{\text{rock}'}$ for pink beam. d) Resolution function in $q_{\text{roll}}$ for pink beam.
• Figure 5: Comparison of normalized experimental data (dots) and simulations (blue curves) of the rocking and rolling direction components of reciprocal space resolution function for the line beam case. Also shown are fits of the experimental data to Gaussian distributions (red lines). a) Resolution function in $q_{\text{rock}'}$ for monochromatic beam. b) Resolution function in $q_{\text{roll}}$ for monochromatic beam. c) Resolution function in $q_{\text{rock}'}$ for pink beam. d) Resolution function in $q_{\text{roll}}$ for pink beam.