Ptychographic coherent diffraction imaging (PCDI) is a scanning microscopy technique that has gained wide popularity during the last decade.1–3 Its ability to recover illumination and object information with phase contrast sensitivity without the need for sophisticated specimen preparation makes it suitable for both beam diagnostics4,5 and quantitative imaging6–9 in a variety of spectral ranges and experimental geometries.10–13 However, PCDI relies on accurate experimental calibration, and incorrect modeling of the data can lead to reconstruction artifacts. Here, we report on correction schemes to mitigate detector-sided systematic errors, namely axial position uncertainty, point spread, and inhomegeneous response. We derive theoretical models for the aforementioned systematic errors and present correction strategies to improve reconstruction quality. One result is that the effect of axial position uncertainty is to scale the estimated coordinates of each scan position. An algorithm is presented to detect and correct for axial position errors based on lateral position correction. A second result is that both point spread and inhomegeneous response of the detector can be mitigated by mixed state ptychography.14 The paper is organized as follows: Sec. 2 reviews details regarding PCDI and gives a theoretical description of the aforementioned systematic errors. Section 3 demonstrates experimental results on the classification and correction of axial position uncertainty and detector point spread. Concluding remarks are given in Sec. 4.
Under the thin object approximation,2 the exit wave downstream an object is modeled as the product of illumination and object transmission :14 involving a set of coherent modes: 15
Standard ptychography algorithms iterate between two constraints in reciprocal and real space. The reciprocal space constraint forces the iterate to comply with the measured intensities for each scan position, i.e.,2 The overlap in scan positions together with the factorization assumption in Eq. (1) gives the real space constraint, which is typically imposed iteratively via nonlinear optimization algorithms.2,16–18 In this report, we use the real space updates: 14,19 Assuming quasimonochromatic radiation, the reconstructed coherent modes of the probe relate to the mutual intensity, , via14 15 20 21 22
The latter step allows one to identify the orthogonalized set of probe modes with the orthogonal modes and their corresponding weights in the spectral decomposition of the mutual intensity, namely . The orthogonalization of the probe modes [Eq. (15)] does not have to be calculated at every iteration of the ptychographic algorithm. In the software implementation used here, the orthogonalization step is carried out every 10th iteration to save computational resources. The next sections discuss departures from the ideal experimental model discussed above and correction strategies.
Axial Position Correction
The first imperfection considered is axial misalignment of the object–detector distance z, which has been reported to cause nonuniqueness and artifacts in the reconstruction.23,24 The effect of axial displacement of the detector is to scale the real space coordinates attributed to each object frame . Assuming far-field diffraction, the real space pixel size and field of view are given by and , respectively, where is the detector size and is the detector pixel size. On an integer grid, a point at coordinate is converted into dimensionless pixel coordinates , where rounding is neglected. Then, the difference between the true pixel location, , and the measured pixel location, , is given byFig. 1. The sign of in Eq. (16) leads to contraction or inflation of the scan grid: If the measured object–detector distance is larger than the ground truth object–detector distance , i.e., , then and the estimated lateral positions are contracted with respect to the actual scan grid. In the other case, when , i.e., , then and the estimated scan grid is inflated. The former case is shown on the left and the latter case is shown on the right for a concentric25 and a Fermat scan grid,26 respectively.
The proportionality of and in Eq. (16) causes variable lateral position error that is most pronounced at the extremal points of the scan grid. Therefore, detector position uncertainty may be detected by the use of lateral position correction algorithms.27–29 In this report, we use a random walk lateral position correction scheme similar to algorithms reported previously.29,30 In our scheme, no underlying annealing schedule or drift model is assumed. At every iteration, the random walk position correction performs the following steps: (1) For the ’th position obtain unshifted and shifted exit wave modes, where the shift of the latter is given by in both lateral directions. (2) Update exit wave modes according to Eq. (5) and calculate the error metric:
Partial Spatial Coherence as a Convolution Operation
Assuming partial spatial coherence (PSC), the far-field diffraction intensity is given by31 For a Schell model field, the mutual intensity can be written as follows: 32 Changing to centered coordinates and , the partially coherent diffraction intensity is given by33–35 35
Detector Point Spread as a Convolution Operation
If the detector is subject to point spread, the contrast in the diffraction pattern is reduced. In the ideal case, the detector discretely samples the diffraction intensity. In practice, the measured data on each pixel is integrated over a finite area. It is now shown that the integration over a finite pixel area can be modeled as a convolution operation, where only the one-dimensional case is derived. The extension to two dimensions is straightforward. The integrated intensity is given by36,37 The main point of this section is that the detector point spread may be modeled as a convolution operation on a diffraction intensity, as described by Eq. 26.
Partial Spatial Coherence versus Detector Point Spread Ambiguity
If both partial coherence and detector point spread are non-negligible, the observed diffraction intensity is given by
Under the above discussed approximations (Schell model beam and space-invariant detector point spread), ptychography’s capability to reconstruct individual terms in the orthogonal mode decomposition has principally no means to distinguish the effects of partial coherence and detector point spread. However, it can be tested whether the terms in the orthogonal mode decomposition are due to partial coherence or detector point spread by changing the coherence defining aperture of the optical system. This is demonstrated in Sec. 3.2.
Mixed-State Ptychography in the Presence of Inhomogeneous Detector Response
In Ss. 2.2 and 2.4, it was discussed that the effect of PSC (for the case of a Schell model fields) and a space invariant detector point spread are mathematically modeled through a convolution operation with an a priori unknown kernel. Instead of estimating the kernel, the mixed state formalism allows one to incorporate convolution effects such as partial coherence,14,38 sample vibrations,39,40 stage movement during exposure41,42 (fly scan effects), and point spread of the detector14,40 into a mixed state probe. Mathematically, this is justified by Mercer’s theorem, which states that a non-negative definite, hermitian, and square integrable kernel may be decomposed into a series of orthogonal modes.43 A general cross-spectral density satisfies the latter conditions,15 including the special case of Schell model fields. In addition, it was observed that the mixed state algorithm can mitigate static detector imperfections such as imhomogeneous response.38 It is easily seen that an inhomogeneous detector response obeys the conditions for Mercer’s theorem to apply. Let a detector responce be given by a real-valued function:
Detection and Correction of Axial Detector Position
To test the effect of axial detector position uncertainty, a visible light ptychographic scan was acquired. In the experiment, a He-Ne laser beam () was focused onto a 12 bit CMOS detector (IDS UI-3370CP-M-GL, , pixel size) by a lens with a focal length of 100 mm, as depicted in Fig. 2(a). The object was placed a distance of 41 mm downstream the lens and upstream the detector. Under the paraxial approximation in this configuration, the detector measures the scaled Fourier transform of the exit wave behind the object emulating far-field diffraction.44 The probe was approximately Gaussian with a standard deviation of as calculated after reconstruction by4519 with the random walk position correction scheme described in Sec. 2.2 for distances , as shown in Fig. 3. It is seen that for (panel a) and for (panel c), the initial scan grids are inflated and contracted, respectively, as compared to the corrected scan grids. The scaling of the corrected scan grids with respect to the initial scan grid is in agreement with Eq. (16) and may be used to correct for axial position uncertainty. This step may be automated or carried out manually as done here.
Detector Point Spread and Static Detector Imperfections
To test whether the coherent mode structure of the illuminating beam can be attributed to PSC or detector point spread, experiments were carried out at the MAXYMUS end station at the UE46-PGM2 beam line at the BESSY II synchrotron radiation facility.46 A kinoform spiral zone plate of diameter and 400 nm outer zone width was placed 3 m downstream crossed exit slits to generate a charge one vortex beam with a spot size of to critically illuminate a binary test target at a photon energy of 800 eV. The experimental setup is depicted in Fig. 2(b). The region of interest on the test target was in each lateral dimension with a scan step size of 150 nm. The high linear overlap47 of 90% ensured stable recovery of the higher coherent mode structure () of the beam. A total of 1600 diffraction patterns were recorded on a CCD (cropped to , pixel size) placed 15 cm downstream the object resulting in a real space pixel size and field of view per object patch of and , respectively.
The ptychographic reconstructions are shown in Fig. 4 for exit slit sizes of (top) and (bottom). For both exit slit openings, 1000 iterations were carried out with one (a, d) and nine (b, c, e, f) probe modes, four of which are shown (c, f). For the single mode reconstruction, the objects show artifacts at the outer region indicating that the tails of the probe were not reconstructed properly. For the multimode reconstruction, both objects show stronger similarity than for the single-mode reconstruction. Closer inspection of the multimode probe structure reveals that the degree of coherence did not significantly change between the two scans. The beam purities for the and slits are 63.9% and 60.5%, respectively. If partial coherence had been the cause for the coherent mode structure of the probe, then increasing the slit size by a factor of 2 would have resulted approximately in a twofold decrease in beam purity. From this, we conclude that the probe mode structure can mainly be attributed to detector point spread and static detector imperfections.
We have discussed two common detector-sided errors relevant in ptychographic diffraction imaging. The first error, axial misalignment, causes scaling of the correct lateral scan positions. It was shown that lateral position correction algorithms can be used to detect and correct for axial position uncertainty. The second detector-sided error discussed was detector point spread. We showed that this error can be identified by changing the spatial coherence defining element in the optical system, here the exit slit of a synchrotron beam line. If a change in exit slit size causes no response in the coherent mode structure of the illumination beam, the decreased diffraction pattern contrast can be attributed to detector point spread and static detector imperfections rather than PSC. However, while detector point spread is not attributable to decoherence in the beam, spectral mode decomposition, typically used for the representation of partially coherent beams, can be used to increase ptychographic reconstruction quality in the presence of detector point spread. Not shown in this work are initial tests on deconvolution strategies that we found to be less robust as compared to the mixed state algorithm. We believe that this is due to the dependence of the particular deconvolution ansatz on the specific underlying model. By contrast, the mixed state algorithm is conveniently applied since no a priori model of the detector error is required. Work on deconvolution of ptychographic data with explicit recovery of the detector point spread may be found elsewhere.48 We believe that the results presented are important for improving reconstructions as well as rigorous quantification of partially coherent beams by means of ptychography.
L. L. gratefully acknowledges help from D. Treffer in collecting the Siemens star data set and thanks I. Besedin for useful discussions. Part of the content of this paper was presented at the SPIE Computational Optics 2018 Conference (contribution 10694-9).
Lars Loetgering is currently enrolled as a PhD student at the Institute of X-Optics at the University of Applied Science, Koblenz, and Technical University Berlin. His research interests include computational imaging, ptychography, and phase retrieval.
Max Rose is PhD student at Deutsches Elektronen-Synchrotron Hamburg in the group of I. A. Vartanyants. His research interests include coherent x-ray diffraction imaging with phase retrieval and ptychography using synchrotron and x-ray free-electron laser sources.
Kahraman Keskinbora is currently leading the Micro/Nano Optics Group in the Department of Modern Magnetic Systems at Max-Planck-Institute for Intelligent Systems in Stuttgart. His research is focused on developing novel nanofabrication routes for innovative, high-efficiency and high-resolution x-ray optics using direct-write lithography techniques. The optics that he and his team develop are routinely used at the scanning transmission x-ray microscopy and soft x-ray ptychography beamline MAXYMUS at BESSY II.
Margarita Baluktsian is a PhD student in the Micro/Nano Optics Group under the Department of Modern Magnetic Systems of the Max Planck Institute for Intelligent Systems in Stuttgart. In 2014, she graduated from the University of Stuttgart with a masters degree in physics. During her masters thesis and in the subsequent research stay at Oklahoma University in Norman in 2015, she was working in the field of atomic and optical physics. Since 2015, she works on the development, fabrication, and application of novel x-ray optics within the scope of her PhD thesis for various scientific needs.
Gül Dogan is currently a student associate in Kahraman Keskinbora’s group in the Department of Modern Magnetic Systems at Max-Planck-Institute for Intelligent Systems in Stuttgart.
Umut Sanli is currently a PhD candidate at the Max Planck Institute for Intelligent Systems. His PhD research is focused on developing x-ray optics using innovative approaches for advancing x-ray microscopy. He is skilled in various nanoengineering and characterization techniques including atomic layer deposition, focused ion beam lithography, multiphoton lithography, microscopy, and spectroscopy using electrons and x-rays.
Iuliia Bykova is a PhD student in Max Planck Institute for Intelligent Systems in the group of Professor Schütz. Her research interests include soft x-ray microscopy and ptychography with magnetic and chemical contrast.
Markus Weigand is part of the Department of Modern Magnetic Systems at the Max-Planck-Institute for Intelligent Systems, leading the x-ray Microscopy Group. He is working locally at the BESSY II synchrotron operated by the Helmholtz-Zentrum Berlin as beamline/endstation scientist for the MAXYMUS x-ray microscope. His research is focused on implementing and improving advanced imaging methods such as ptychography or pump-and-probe microscopy.
Gisela Schütz is a director at the Max Planck Institute for Intelligent Systems. Her research interests cover the application of synchrotron radiation in x-ray spectroscopy and microscopy and the development of advanced spintronic/magnonic systems and new supermagnets.