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Abstract
Recent advances in phase-retrieval-based x-ray imaging methods have demonstrated the ability to reconstruct 3D distortion vector fields within a nanocrystal by using coherent diffraction information from multiple crystal Bragg reflections. However, these works do not provide a solution to the challenges encountered in imaging lattice distortions in crystals with significant defect content that result in phase wrapping. Moreover, these methods only apply to isolated crystals smaller than the x-ray illumination, and therefore cannot be used for imaging of distortions in extended crystals. We introduce multi-peak Bragg ptychography which addresses both challenges via an optimization framework that combines stochastic gradient descent and phase unwrapping methods for robust image reconstruction of lattice distortions and defects in extended crystals. Our work uses modern automatic differentiation toolsets so that the method is easy to extend to other settings and easy to implement in high-performance computers. This work is particularly timely given the broad interest in using the increased coherent flux in fourth-generation synchrotrons for innovative material research.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Bragg coherent diffraction imaging (BCDI) is a valuable tool to spatially resolve the lattice strain in crystalline materials at the scale of ~10 nm by using coherent x-rays as a probe [1–3]. Bragg ptychography, the scanning variant of BCDI, is used for extended crystalline samples larger than the beam size [4–6]. Both techniques have been applied to a variety of materials science problems, such as the performance of batteries [7–9], catalytic phenomena at metal interfaces [10–12], and characterization of quantum sensors [13]. These conventional BCDI methods enable the spatial resolution of a single component of the three-component elastic lattice distortion field. This component corresponds to the Bragg diffraction vector of momentum transfer that comes about via a specific orientation of the single crystal lattice of the sample with respect to the beam (one of the crystal’s Bragg conditions). Reorienting the crystal into other configurations with respect to the incident x-ray probe permits the independent components of the lattice distortion to be imaged. When three or more independent Bragg conditions can be satisfied and measured with BCDI, it is possible to unambiguously resolve the lattice displacement vector field within the sample.
Recent efforts to simultaneously reconstruct all components of internal lattice distortion fields with BCDI have shown promising results with simulations [14,15] as well as experiments [16,17] The experimental works have employed different combinations of well-known fixed point iterative projection algorithms [18], typically in parallel reconstruction threads. The solutions of these parallel reconstructions are analyzed for desirable properties in order to seed the next generation of randomized solutions (i.e., the ‘guided’ approach using genetic algorithms [19]). All the vector field reconstruction demonstrations referenced above explicitly minimize the discrepancy between the bulk distortion field and the computed phase from conventional phase retrieval. A recent alternate approach seeks to directly and simultaneously minimize the discrepancy in the measured and estimated coherent diffraction signals in an all-encompassing gradient descent optimization scheme [20]. However, this specific approach pertains only to full coherent illumination of compact crystals, and the method is yet to be demonstrated on the relatively data-intensive Bragg ptychography problem with extended single-crystal samples.
We address the problem of multi-peak Bragg ptychography (MPBP) in this paper through simulations of independent ptychographic data sets and subsequent reconstruction of a two-dimensional projection of the underlying crystal lattice fields; this extends upon the preliminary work first recorded in [21]. The MPBP measurement approach follows that of Bragg projection ptychography [5,22,23] where, at a given angle that fulfills the Bragg condition, data is collected at multiple overlapping beam positions. In MPBP this process is repeated at different Bragg conditions. In terms of image reconstruction, MPBP employs a joint optimization scheme similar to [20] for all the measured ptychographic data sets corresponding to the independent Bragg conditions. Methods such as MPBP are poised to take advantage of the orders of magnitude increases in coherent flux at hard x-ray energies afforded by fourth-generation synchrotron light sources to image local displacement fields in crystals.
This paper is organized in the following manner: Section 2 introduces the MPBP forward model connecting the lattice distortion to the measured coherent diffraction signal for a single crystal. Section 3 lays out specifics of the numerical optimization problem, including the gradient descent scheme, the expected phase-wrap degeneracies in the solution and their resolution. Sections 4 and 5 describe the parameters of the numerical demonstrations in this paper, and the reconstruction results respectively. Section 6 closes with a discussion of the results and concluding remarks.
2. Forward model for multi-peak Bragg ptychography (MPBP)
We define a single-crystal sample in the laboratory frame $(x_1, x_2, x_3)$ depicted in Fig. 1, with the incident beam along the $x_3$-direction. We adopt a sample geometry that mimics a crystalline free-standing membrane or a thin film affixed to a substrate with different lattice structure. Such systems are plentiful in the fields of functional materials [24] and micromechanical systems [25]. Further, we note that in such systems where the membrane/film thickness is of order of ~100 nm, lateral heterogeneities in lattice characteristics over micrometer length scales are often of interest and through-thickness variations of strain are often minor by comparison.
Fig. 1. Illustration of (a) Bragg peaks of interest for the simulated thin crystal film, (b) incident ($\boldsymbol{k}_i$) and exit ($\boldsymbol{k}_f$) directions for the $[100]$ Bragg peak (labelled $\boldsymbol{G}$), (c) magnitude of the probe beam along its propagation direction, (d) phase of the probe beam, and (e) the simulated radially decaying displacement profile ($\boldsymbol{U}$), where the direction of the arrows show the direction of the lattice displacement, and the color of the arrows show the magnitude of the displacement.
The experiment that motivated the membrane model construction and the diffraction geometry considered in this work is described in [26]. In that work, a localized x-ray beam (11.8 keV energy) was oriented at near-normal incidence to a single crystal silicon membrane. By tuning the sample angle off of normal incidence by $15.9^\circ$, a Bragg reflection with $hkl$ indices of 220 from the (110) lattice planes could be measured in a Laue geometry with a diffraction vector $\boldsymbol{G}_{hkl}$ contained within the plane of the membrane. (Here, $\boldsymbol{G}_{hkl} = \boldsymbol{k}_f^{hkl} - \boldsymbol{k}_i$ with $\boldsymbol{k}_i$ and $\boldsymbol{k}_f^{hkl}$ being the wave vectors of the incident and exit x-ray beams, each with $|\boldsymbol{k}| = 1/\lambda$ where $\lambda$ is the x-ray wavelength.) A Bragg ptychography image of a single component of the lattice displacement field in the two-dimensional plane of the membrane was reconstructed from a ptychography series of 220 Bragg peak intensity patterns. In the work we present here, we conceptualize an extension of the work of Takahashi et al. by proposing multiple such measurements at different Bragg peaks with in-plane $\boldsymbol{G}_{hkl}$ vectors ($hkl$ reflections for which $l=0$) and spatially resolving a lattice displacement field $\boldsymbol{u}(\boldsymbol{r})$ in terms of components $U_1$ and $U_2$ along the $x_1$ and $x_2$ in-plane spatial coordinates via global optimization.
As shown in Fig. 1, for this work, a 3D membrane sample model with extended dimensions along $x_1, x_2$ and limited extent along $x_3$ was constructed. An inhomogeneous lattice displacement field $\boldsymbol{u}(x_1, x_2)$ was defined that was self-similar within the $x_3$ dimension of the membrane. We chose a membrane thickness of $0.1$ $\mu$m, which is a reasonable length scale over which the displacement field in a membrane can be assumed to be constant. A set of Bragg reflections with in-plane diffraction vectors were chosen with diverse in-plane orientation and different reciprocal space magnitudes. At each $hkl$ Bragg reflection, the sample has a distinct positive real-valued scattering amplitude $\boldsymbol{{\chi }}_{hkl}(\boldsymbol{r})$ and can be described as a 3D complex-valued real-space object:
In the case of Bragg ptychography from a 3D crystal, the x-ray illumination function (or probe) is also 3D and varies with different Bragg peaks due to the different angles of incidence of the beam on the sample and the different orientations of the exit beam [27]. We denote the 3D probe that satisfies each Bragg diffraction condition as ${\mathscr{P}}_{hk}$ based on a set of incident sample angles and exit beam orientations consistent with a cubic-structured crystal in a diffractometer equipped with a single rotation axis sample stage and a two-rotation-axis detector geometry typical of nanodiffraction beamlines. The 2D beam profile that was used to determine ${\mathscr{P}}_{hk}$ was one reconstructed from experimental data of a test pattern illuminated with a nanofocused beam, as described in [28]. For convenience, we then construct a diagonal matrix $\operatorname {diag}\left ({\mathscr{P}}_{hk}\right )$ with the elements of ${\mathscr{P}}_{hk}$. Since the crystal is thin in the $x_3$ dimension and we seek to reconstruct the 2D displacement field of the membrane, we do not require the information generated via an angular scan of the object about each Bragg angle. Therefore, for this experiment, the expected intensity at the far-field detector plane for each ptychographic scan position can be written as:
The forward model that we have developed has two key features. First, the phase in all the effective objects ${\mathscr{O}}_{hk}$ (Eq. (1)) is due to the shared (among all the Bragg peaks) displacement field, and this displacement field is what we aim to reconstruct. Second, the amplitudes (or magnitudes) of the objects for the different peaks differ only by a global scaling factor that varies with possible differences in the scattering strength of the sample at different Bragg peaks.
3. Solving the MPBP problem
To formulate the MPBP reconstruction algorithm, we first need to identify the variables we want to solve for, then define the error metric that we aim to minimize. For this work, we assume that the probe functions are known, that we obtain the intensity diffraction data at $D$ different Bragg peaks, that the lateral dimensions of interest of the membrane can be represented with $N_{x_1}\times N_{x_2}$ pixels, and that the membrane does not change along the $x_3$ direction. The variables we want to reconstruct are as follows:
- • The $2N$ displacement variables represented by the matrix $\boldsymbol{U}$ defined row-wise as $\boldsymbol{U}_n = \begin {pmatrix}u_{x_1,n},\, u_{x_2,n}\end {pmatrix}$, where each row $\boldsymbol{U}_n$ is the transpose of the displacement vector for the voxel $n$, and where $1\leq n\leq N=N_{x_1}\times N_{x_2}$ indexes the voxels.
- • The $N$ per-pixel amplitude variables for the Bragg peak chosen as reference. We represent these variables as $\boldsymbol{\chi }$ and note that $\boldsymbol{\chi }\geq 0$ elementwise. We note that any of the $D$ Bragg peaks can serve equally as the reference peak.
- • The $D-1$ global scaling factors, represented by the vector $\boldsymbol{\alpha }$ (with $\boldsymbol{\alpha } > 0$ elementwise), that we use to scale the object magnitudes for the remaining Bragg peaks. These scaling factors simultaneously account for differences in scattering power of different Bragg peaks and differences in the measurement between Bragg peak ptychography scans (i.e., choice of exposure time).
We therefore have a total $3N + D -1$ unknowns to solve for. We can write the objects (per Bragg peak) in terms of these variables as:
The nonlinear minimization problem is then
3.1 Ambiguities in the displacements: a challenging reconstruction problem
Previous works [6,28,30] have shown that the single-peak Bragg ptychographic phase retrieval problem can be solved using a gradient-descent-based update algorithm. However, the reconstruction of the crystal lattice displacement field, using either multiple such phase profiles (per Bragg peak) or directly from the diffraction datasets, is challenged by an implicit periodicity that leads to a preponderance of sub-optimal local minima in the landscape for the error metric (Eq. (4)). This necessitates an adaptation so that gradient-descent-based iterative approaches can find the true solution. In this section, we first demonstrate how the lattice displacement reconstruction problem is affected by ambiguities, then we propose a solution to address this challenge. While we focus this discussion on the MPBP setting, the arguments we develop apply to general Bragg coherent imaging of single-crystal lattice displacements. We first consider the effective object for a single Bragg peak $\boldsymbol{G}_1$, which can be written using several equivalent forms,
The true displacement field $\boldsymbol{U}_\star$ has $\boldsymbol{m}_1 = (0,0,\dots,0)$, yet the $\boldsymbol{U}$ fields arising from $\boldsymbol{m}_1$ and $\boldsymbol{m}_1'$ both produce the same diffraction patterns. Moreover, $\boldsymbol{m}_1'$ is associated with a highly unphysical spatially discontinuous strain tensor. Consequently, we cannot distinguish the true displacement $\boldsymbol{U}_\star$ from any $\boldsymbol{U}'$ using data along just the Bragg peak $\boldsymbol{G}_1$, even when $\boldsymbol{U}_\star \cdot \boldsymbol{G}_1 = \boldsymbol{1}$ so that the projection does not cause any loss of information. We illustrate this challenge in Fig. 2: Fig. 2(a) schematically depicts the family of displacements $\boldsymbol{u} \pm m \Delta \tfrac {\boldsymbol{G}_{110}}{||{\boldsymbol{G}_{110}}||^2}$ that would all lead to the same $\boldsymbol{G}_{110}$ phase for any pixel $P$, and Figs. 2(b)-(g) show the effect of this per-pixel ambiguity for the 2D thin film lattice displacement profile of the membrane model under consideration.
Fig. 2. (a) A schematic representation of the ambiguity in the displacement vector at a given pixel. Considering the $[110]$ peak for example, at any location in the reconstruction field of view $P$, the displacements $\boldsymbol{u} + m\Delta \boldsymbol{u}$ (with $m$ any integer) all give rise to the same phase value. (b) The true displacement profile (scaled by the lattice distance $a_0$) is associated with the unwrapped phase profile (c), which after wrapping gives the profile in (d). If we choose $m\in {-1,0,1}$ randomly for every point in the film, we get the field in (e). Adding the fields in (b) and (e), we get the unphysical displacement field (f) that is associated with the phase profile (g). After wrapping, the phase profile in (g) also gives the profile in (d). We therefore cannot distinguish between the displacement fields (b) and (f) based on the object phase for the $\mathcal {G}_{110}$ peak alone.
Next, we examine the case in which we have diffraction data generated along two Bragg peaks $\boldsymbol{G}_1$ (for $(h_1, k_1, 0)$) and $\boldsymbol{G}_2$ (for $(h_2, k_2, 0)$). If $\boldsymbol{G}_1^T \cdot \boldsymbol{G}_2=0$, then $\boldsymbol{U}' \cdot \boldsymbol{G}_2 = \boldsymbol{U}_\star \cdot \boldsymbol{G}_2$, which means we again cannot isolate the summand $\boldsymbol{U}_\star$ using just the diffraction data along these two Bragg peaks. When $\boldsymbol{G}_2^T \cdot \boldsymbol{G}_1 \neq 0$, the additional Bragg peak may reduce the set of ambiguities that are inherent to the single Bragg peak case. However, to our best knowledge, there is no formal proof that any combination of noise-free measurements about specific Bragg peaks provides an unambiguous solution for the problem. It is also likely that local minima exist for the minimization problem Eq. (5). Since gradient-based methods can only guarantee convergence to local minima, applying these to the optimization problem in Eq. (4) is prone to stagnation at these minima — we observe this in practice in the 4-peak MPBP simulation that we present subsequently.
Our analysis so far shows that the inherent ambiguities in the displacements lead to spurious global or local minima in the landscape for the error metric $f$ (Eq. (4)), and we propose the following heuristic solution to this challenge. We formulate our solution by noting that, in the MPBP forward model, we first project the pixelwise displacements $\boldsymbol{U}_n$ through the operation $\boldsymbol{\phi } (\boldsymbol{U}_n, \boldsymbol{G}_d) = 2\pi \boldsymbol{U}_n\cdot \boldsymbol{G}_d$ per Bragg peak $\boldsymbol{G}_d$. If we ignore any physics-based constraints, the elements of $\boldsymbol{U}$ can attain any value in $(-\infty,\infty )$, and the range of $\boldsymbol{\phi } (\boldsymbol{U}_n, \boldsymbol{G}_d)$ is also $(-\infty,\infty )$. In contrast, when we invert this model, we calculate these projection through the function $\boldsymbol{\phi }_1^\text {inv} ({\mathscr{O}}_{1,n}) = \arctan \left (\mathfrak {I}[{\mathscr{O}}_{1,n}] / \mathfrak {R}[{\mathscr{O}}_{1,n}]\right )$, the range of which is only $(-\pi, \pi )$. The gradient-based inversion program does not correct for this mismatch, leading to possibly incorrect $\boldsymbol{U}_n$ values. A natural solution to this mismatch would then be to somehow change the range of $\boldsymbol{\phi }_1^\text {inv} ({\mathscr{O}}_{1,n})$ to $(-\infty,\infty )$: we can achieve this through the phase unwrapping procedure [31]. Since the phase unwrapping procedure removes the $2\pi$ jumps in the phase map, they change the range of the phase map from $(-\pi, \pi )$ to $(-\infty, \infty )$. As such, incorporating the phase unwrapping procedure within the MPBP optimization algorithm should be sufficient to correctly solve for the crystal displacements, and we demonstrate this with the subsequent numerical experiments.
3.2 CDI-based strain reconstruction methods in the literature
While we are not aware of any existing literature that specifically addresses the MPBP problem, there exist two primary classes of algorithms that reconstruct crystalline displacement (or strain) by combining the diffraction data obtained from multi-peak BCDI experiments: the classical two-step reconstruction methods (which are the methods of choice in the literature), and the newly developed concurrent reconstruction methods.
In the two-step reconstruction approach, one first reconstructs the full effective objects for the individual Bragg peaks, then uses the reconstructed phase profiles to obtain the displacement and strain profiles [1,16,32,33]. This approach suffers from a number of technical challenges. First, it requires the initial solution of a total of $2ND$ magnitude and phase variables, which can be much higher than the $3N + D -1$ variables that we actually require, and is therefore a harder inversion problem. If we have insufficient or noisy data along one of the Bragg peaks, then the corresponding object reconstruction can contain artifacts or fail altogether, and thereby significantly deteriorate our desired displacement and strain reconstructions. Second, while individual BCDI reconstructions are agnostic to translations of the real-space objects, we need to ensure that these individual real-space objects coincide to perform the displacement reconstructions, and aligning these can itself be a challenging problem [14].
In the concurrent reconstruction approach, we treat the diffraction data generated from multiple Bragg peaks as one dataset and use this dataset to simultaneously reconstruct the real-space objects and the displacement fields. While the existing works that use this approach [14,15,17] do not specifically solve for the variables we cataloged earlier in this section (viz., the displacement, magnitude, and scaling variables), and solve for individual real-space objects instead, they share the information between these objects by also tracking the displacement profiles and the real-space support, thereby implicitly solving for the same $3N + D -1$ variables instead of the full $2ND$ object variables.
The problem of reconstructing displacement fields that result in phase wraps has been handled in different ways in BCDI literature. In the two-step reconstruction approach, these phase maps have been first unwrapped before the displacement calculations, but, even so, the displacements so calculated are themselves often error-prone [19]. In a recent work, Hofmann et al. [16] refine the basic two-step approach by using carefully designed phase offsets and working with phase gradients to directly reconstruct the strain tensors. On the one hand, this method is agnostic to any phase wrap issues. On the other hand, it solves for the much larger set of $2ND$ real-space object variables, introduces even more variables in the strain reconstruction step, and also requires a careful alignment of the real-space object reconstructions. Finally, Maddali et al. [20] have recently developed a concurrent reconstruction approach that uses a median filter to minimize any distortions in the displacement field, but this approach may not resolve phase-wrap-associated artifacts at longer length scales and may also deteriorate the spatial resolution of the reconstruction.
3.3 MPBP reconstruction algorithm
In this section, we outline a MPBP reconstruction algorithm that, in contrast to the approaches described above (Section 3.2), directly solves for the desired displacement ($\boldsymbol{U}$), magnitude ($\boldsymbol{{\chi }}$), and scaling ($\boldsymbol{\alpha }$) variables, in a manner analogous to that developed in [20], but for the case considered here of ptychography data measured at multiple Bragg conditions, with specific adaptations aimed at the phase wrap problem.
To begin with, we note that magnitude and scaling variables should be constrained so that the $\boldsymbol{{\chi }} \geq 0$ and $\boldsymbol{\alpha } >0$; ignoring these constraints in the optimization can lead to stagnation in the minimization. While it is certainly possible to apply these constraints (through projection steps, for example), we implement this constraint indirectly by defining the auxiliary variables $\tilde {\boldsymbol{{\chi }}}\in \mathbb {R}^N$ and $\tilde {\boldsymbol{\alpha }}\in \mathbb {R}^{D-1}$ that satisfy the relations
Solving for these auxiliary variables naturally enforces the constraints, and we can use a small threshold at the end of the optimization procedure to set $\boldsymbol{{\chi }}=0$ in regions not illuminated by the probe.
We now define a $2\times D$ matrix $\boldsymbol{H}$ columnwise as $\boldsymbol{H}_{\bullet,d} = \begin {pmatrix}\boldsymbol{G}_{d}\cdot \boldsymbol{e}_1\\ \boldsymbol{G}_d\cdot \boldsymbol{e}_2\end {pmatrix}$, where $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$ are the unit vectors along the $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ directions respectively. We can then simultaneously calculate the phases for all the real-space objects as
Under ideal circumstances, this projector takes the displacement field, which could contain the periodicity-induced discontinuities, calculates the associated phase maps, removes the discontinuities via the phase unwrapping procedure, then calculates the now artifact-free displacement fields, thereby addressing the minimization challenges we discussed in Section 3.1. We note that this is a heuristic operation that depends greatly on the properties of the chosen phase unwrapping algorithm, and could be error-prone in practical applications. However, we find that this projector, used within the reconstruction algorithm outlined below (Algorithm 1), works quite well in our numerical simulations.
Algorithm 1. Single iteration of the minibatch MPBP reconstruction algorithm
We present the steps taken within a single iteration of our MPBP reconstruction approach in Algorithm 1 and note that this is a variation of the basic minibatch ptychography algorithm presented in [30] [Algorithm 1]. However, the use of the $\boldsymbol{U}$, $\tilde {\boldsymbol{{\chi }}}$, and $\tilde {\boldsymbol{\alpha }}$ variables, instead of the $(\mathfrak {R}[{\mathscr{O}}], \mathfrak {I}[{\mathscr{O}}])$ variables in [30], for the gradient calculations requires extra coordinate transformations and other mathematical manipulation. Indeed, this adds to the already significant algebra that would be required to calculate the gradients just for the basic MPBP model. Therefore, by using the automatic differentiation (AD) framework (which is implicit in Line 3 of Algorithm 1), we not only avoid such tedious and error-prone algebra, but also take advantage of the state-of-the-art parallelized implementations of the mathematical operations (including the complex interpolations) required for both the MPBP forward model and gradient calculations [20,30,34–36].
4. Numerical experiment
To test our proposed reconstruction algorithm, we use a numerical experiment with the basic experimental parameters listed in Table 1, wherein we simulate a thin film with a cubic lattice structure, which has a radially decaying 2D lattice distortion field emanating from a point in the field of view. This would be consistent with the displacement field associated with a local stress concentration in a crystal membrane, for example. For convenience, we define an orthonormal “lab frame” with the axes $(\boldsymbol{x}_1, \boldsymbol{x}_2, \boldsymbol{x}_3)$ and with the voxel size $5.84 \times 5.84 \times 5.84$ cubic nanometers, in which the simulated membrane is centered at the origin, and is aligned with its surface along the $\boldsymbol{x}_1$–$\boldsymbol{x}_2$ plane so that it has the dimensions of $85 \times 85\times 17$ voxels. As discussed in [20], the voxellation in the lab frame needs to be fine enough to obtain faithful rendering of the object after interpolation and projection to the detector frames of each Bragg peak.
Table 1. Experimental details
To create a simulated data set from this membrane, we perform numerical ptychography experiments at each of the $[100]$, $[110]$, $[120]$, and $[210]$ Bragg peaks for the cubic lattice. (In an experiment setting, the choice of available peaks is dictated by the symmetry of the underlying crystal.) The magnitudes of the real-space objects ($|{\mathscr{O}}_{hk}|$) are scaled from unity to 0.02, 0.0175, 0.0105 and 0.005 for the $[100]$, $[110]$, $[120]$, and $[210]$ peaks respectively. This set of scaling parameters was chosen arbitrarily and has the effect of creating differing levels of low count rate diffraction intensity patters that yield appreciable shot noise when used to generate consistent Poisson counting statistics, as was done here. Using different scaling parameters also necessitates that the relative scaling factors be reconciled by the reconstruction algorithm via the variable $\boldsymbol{\alpha }$. We show the orientations of the chosen $\boldsymbol{G}_{hk}$ vectors and the radial profile of simulated lattice displacement field in Fig. 1, the individual components of the displacement in Fig. 3, the phase maps along each of the Bragg peaks in Fig. 4, and the magnitude maps in Fig. 5.
In our ptychography simulations, we use a 2D probe profile experimentally measured from a Fresnel zone plate x-ray focusing optic at the HXN beamline at NSLS-II [37], of size $100\times 100$ pixels at focus (with each pixel of size 1 nm). We threshold to zero any pixels in the probe below an intensity cutoff of $2{\% }$ of the maximum intensity to reduce aliasing in the simulated diffraction patterns. Using the HXN diffractometer geometry, we calculated the set of sample incident angles and detector position exit angles that satisfy the Laue condition for each of the Bragg peaks of interest. For each peak, we interpolate the probe to produce the 3D illumination volume at the appropriate incident angle, with the beam profile set to be constant along the propagation direction through the numerical window. We then scan the thin film using a $9\times 9$ grid, with a step size of 6 pixels, along the $\boldsymbol{x}_1$–$\boldsymbol{x}_2$ plane. At every beam position, a diffraction pattern is generated by applying the forward model in (2), the scaling values discussed above were applied, and Poisson noise was added. The mean count rates of the diffraction patterns at each Bragg peak were ~ 126, 78, 32, and 7 photons per pixel respectively.
During the reconstruction, we solved for an $\boldsymbol{x}_1$–$\boldsymbol{x}_2$ area of $95\times 95$ pixels, stacked appropriately to produce the 3D membrane structure, initialized as an array of zeros, and placed centrally in the numerical window. The film support along the $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ directions are set to be 5 pixels larger than the actual film dimensions to allow for limited resolution effects due to the low photon count [38]. We performed reconstructions of the simulated data set using the following reconstruction approaches in order to enable a direct comparison of results:
- • TS: the two-step approach (Section 3.2), where we first solve for the individual real-space objects (for each Bragg peak), unwrap the phases calculated for these objects, then calculate the displacements from the unwrapped phases. We use the AD-based approach described in Algorithm 1, but with the $4N$ phase variables (denoted as $\boldsymbol{\phi }_{hk}$) instead of the $2N$ displacement variables, and with the initial Adam update step sizes set to 0.01, 0.1 and 0.1 for the $\boldsymbol{\phi }_{hk}$, $\tilde {\boldsymbol{{\chi }}}$, and $\tilde {\boldsymbol{\alpha }}$ variables respectively. We thereby solve for each of the real-space objects while utilizing the shared magnitude information between the objects.
- • C-W: the concurrent-wrapped approach that uses the basic algorithm in Algorithm 1 but without any phase unwrapping, i.e., without the projection step (in Line 5), and with the Adam update sizes set to 0.01, 0.1 and 0.1 for the $\boldsymbol{U}$, $\tilde {\boldsymbol{{\chi }}}$, and $\tilde {\boldsymbol{\alpha }}$ variables respectively. In this test, no additional strategies such as local smoothing filters aimed to mitigate phase wrap discontinuities were implemented to represent baseline performance of this algorithm.
- • C-UW: the concurrent-unwrapped approach outlined in Algorithm 1. For the phase unwrapping step, we use the algorithm [31] implemented within the “scikit-image” platform [39]. The Adam update sizes are set to 0.01, 0.1 and 0.1 for the $\boldsymbol{U}$, $\tilde {\boldsymbol{{\chi }}}$, and $\tilde {\boldsymbol{\alpha }}$ variables respectively.
All the reconstructions used a minibatch size of 50 diffraction patterns, for a total of 1000 iterations of reconstruction. The update step sizes and the minibatch sizes were obtained through the trial and error approach outlined in [30]. The calculations were performed using the Tensorflow AD platform [40].
5. Results
In Figs. 3 to 5 we present the reconstructions obtained for the lattice displacement, the phase maps, and the magnitude maps respectively. First, we examine the TS case, where Figs. 4 and 5 show that the individual real-space object reconstructions are highly inaccurate, with artifacts for both phase and magnitude maps that are most pronounced in regions with phase discontinuities. The locations of these phase discontinuities are different for the real-space objects from each of the Bragg conditions. In these regions, the magnitude maps show "gaps" (regions of low magnitude)—a reconstruction artifact that has been observed previously in the literature [15,41,42]. As a consequence, the displacement calculated from the real-space phase maps are also discontinuous and inaccurate, as shown in Fig. 3(b),(f).
Fig. 3. Reconstructed displacement components. (a),(e) Simulated (true) values. The reconstructions obtained through the (b),(f) two-step (TS) reconstruction method and the (c),(g) concurrent method without the phase unwrapping step (C-W) show significant artifacts. The (d),(h) concurrent-unwrapped (C-UW) reconstructions show close agreement with the true values.
Next, we can see that the C-W reconstructions, where we directly reconstructed the displacement and magnitude maps without accounting for phase wrapping, show significant discontinuities in the displacement reconstructions (Fig. 3(c),(g)). Similarly, the phase maps (Fig. 4(c),(g),(k),(o)) and the reconstructed amplitudes projected to the $\boldsymbol{G}_{hk}$ vectors (Fig. 5(c),(g),(k),(o)) both show prominent artifacts. The similarity of the artifacts in the magnitudes of the C-W reconstruction arise because in this approach the magnitude is scaled and shared among all Bragg peaks.
Fig. 4. Reconstructed phase maps for each Bragg peak. (a),(e),(i),(m) Simulated (true) values. The (b),(f),(j),(n) TS phase maps and the (c),(g),(k),(o) C-W phase maps show significant artifacts primarily at locations coinciding with discontinuous phase jumps (for both the TS and C-W cases), and other unrelated locations (for the C-W case). The C-UW phase maps agree closely with the true values.
Finally, we can see that C-UW reconstructions algorithm accurately reproduces the simulated displacement and magnitude maps. In particular, while the magnitude maps (Fig. 5) still contain a gap in the location of the stress concentration, which we discuss below, they do not have either the strong phase artifacts (seen in the TS reconstructions). The reconstructed displacement maps, and consequently the phase maps, are relatively free of artifacts. We quantify the reconstruction quality by calculating the normalized root mean square error
Fig. 5. Reconstructed magnitude maps for each Bragg peak. (a),(e),(i),(m) Simulated (true) values, where the black triangle indicates the location of the point distortion in the lattice displacements. The (b),(f),(j),(n) TS magnitude maps, which are reconstructed individually for each Bragg peak, are inaccurate primarily at locations coinciding with discontinuous phase jumps. Among the (c),(g),(k),(o) C-W and (d),(h),(l),(p) C-UW methods, which both reconstruct a reduced number of magnitude variables, the C-W results contain major artifacts while the C-UW results maps agree closely with the true values.
Table 2. Normalized RMSEs for the TS, C-W and C-UW reconstruction algorithms
The C-UW method, though it showed the lowest real-space error with respect to the ground truth displacement fields and object amplitude compared to the other methods, still contains artifacts in the reconstruction. These arise from the fact that low-count-rate Poissonian counting statistics were introduced into the simulated diffraction patterns prior to reconstruction. As in all coherent diffraction imaging, the presence of shot noise in the input diffraction has the effect of introducing uncertainty in the image reconstructions especially in regions with high spatial frequency spectral content. In our study, this manifests most prominently as a region of low amplitude in the C-UW reconstructions in Fig. 5. This location represents the stress concentration center in our simulated membrane where $U_x$ and $U_y$ change rapidly. In diffraction, these changes in the sample over short distances are encoded in the weakly-scattering high-spatial-frequency areas of the pattern, which is precisely where low-count-rate shot noise is most prominent. The effect of shot noise can also be seen when comparing the results of the phase maps in Fig. 4. The phase maps from the 120 and 210 peaks, which had mean photon count rates per diffraction pattern of 32 and 7 respectively, have more artifacts than the phase maps from the 100 and 110 Bragg peaks, which had higher mean photon count rates of 126 and 78.
6. Discussion
To apply the proposed algorithm to general experiment datasets, we need to be mindful about a few different aspects of the algorithm. First, we note that phase unwrapping is itself a difficult problem, particularly for objects with steep phase changes. Moreover, while our 2D phase unwrapping problem can be solved efficiently [31], its 3D extension can be significantly more time consuming [43,44] and difficult to solve. In this context, it could be preferable to replace the phase unwrapping procedure altogether with a direct physics-based solution to the periodicity-driven model mismatch problem. Second, the reconstruction could contain unexpected artifacts if the datasets along the different peaks have significantly different photon counts (as we observe for the C-W case in the current work), in which case reweighing the diffraction datasets could produce more optimal results. Third, registry of the fields of view scanned at the different Bragg angles is imperative for this approach and has been realized in Bragg scanning nanodiffraction experiments by use of fluorescing fiduciary marks intentionally deposited on the sample surface. Finally, the reconstruction procedure is sensitive to the choice of the optimization hyperparameters, which therefore require careful tuning. We hope to address this in the future by designing more robust second order minibatch optimization procedures [35].
Since the proposed algorithm is simple in concept, it can be either incorporated within existing concurrent multi-peak BCDI workflows, or extended (via the AD framework) for applications in general BCDI or Bragg ptychography experimental models. We can also envision straightforward applications of the proposed method to simpler experimental models, such as the recently demonstrated “simultaneous” multi-Bragg peak CDI experiments [45].
In single-particle imaging with multiple Bragg peaks, it has been established that the minimum number of independent Bragg peaks needed to reconstruct a vector displacement field must be equal to the dimensionality of that vector. Thus, for 3D particle reconstructions for which displacement fields with components varying along all three dimensions are sought, at least three independent Bragg peaks are needed. In these studies, utilizing more than three Bragg peaks is preferred (4-6 typically) in order to assure a robust reconstruction. Though conceptually there is no upper limit on the number of Bragg peaks one can use to constrain the displacement field reconstruction problem, practical considerations do come into play, such as the total measurement time as well as the fact that typically not more than 6 Bragg reflections that can be reached from a given crystal at coherent diffraction beamlines. In this work, we constructed a model in which the $U$ field is composed of only two components ($U_x$, $U_y$), such that the minimum number of Bragg peaks needed to solve for this field is two, and we over constrained the problem by considering four Bragg peaks.
In the context of imaging lattice displacement with this method, single crystal membranes are of interest in that they provide a platform for engineering novel heterointerfaces in materials such as diamond and oxides. The distribution of lateral lattice displacement fields are also of interest in functional thin films affixed to substrates, and we speculate that with sufficiently high energy x-rays and x-ray-transmitting substrates, the MPBP method could potentially be used to characterize deposited films without releasing them from their substrate. In this paper, we examined the limit of significant buildup of lattice displacement and with it the challenge of reconciling phase wraps. However, in cases where very subtle lattice displacement fields are of interest, the MPBP method will also be of utility, especially when high-HKL-index Bragg peaks are selected that have more sensitivity to lattice displacement.
7. Conclusion
In summary, we propose a multi-peak Bragg ptychography reconstruction framework to combine diffraction ptychographic datasets generated at multiple Bragg reflections to image the full lattice displacement and strain profiles. The results in Figs. 3 to 5 demonstrate that our proposed approach can reconstruct both the displacement and the real-space magnitude profiles measured in the MPBP experiment. Compared to the TS procedure, this algorithm: i) significantly reduces the number of variables of interest, which makes for a more robust reconstruction, and ii) efficiently utilizes diffraction datasets that could be insufficient for individual real-space object reconstructions. By incorporating the phase unwrapping step, this algorithm also addresses the periodicity-driven model mismatch in the MPBP problem. We expect that our algorithm marks a concrete step in connecting the ever-improving coherence of synchrotron sources to characterization of functional materials under real-world working conditions.
Funding
Office of Science, Basic Energy Science, Argonne National Laboratory, Laboratory Directed Research and Development, Brookhaven National Laboratory.
Acknowledgments
This work (concept development, forward model development, algorithm design) was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Algorithm testing was supported by the Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. We gratefully acknowledge the computing resources provided on Swing, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. This research uses the resources of the Advanced Photon Source, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract No. DE-AC02-06CH11357. This research used the 3-ID Hard X-ray Nanoprobe (HXN) beamline of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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