We outline how ptychographic imaging can be performed without the need for discrete scan positions. Through an idealized experiment, we demonstrate how a discrete-position scan regime can be replaced with a continuously scanned one with suitable modification of the reconstruction scheme based on coherent modes. The impact of this is that acquisition times can be reduced, significantly aiding ptychographic imaging with x rays, electrons, or visible light.
© 2014 Optical Society of America
Ptychography  is a microscopy technique that permits high-resolution imaging with x rays [2,3], electrons [4,5], and visible light  in two and three dimensions  on biological  and materials science samples . Ptychography works by collecting a series of diffraction measurements from a sample, , that is rastered through an illumination or probe, , at discrete points , with adjacent positions sharing some overlap. Due to large redundancy in the datasets, both the probe [3,10] and object can be reconstructed simultaneously using phase retrieval. Recent algorithmic advances have extended the technique to reconstruct multiple probe and object modes ; for example, when the illumination is partially coherent, the sample is dynamic  or there is significant detector point spread . Additionally, the algorithms can be modified to account for multiple scattering from thick specimens  or when the data are under sampled on the detector .
The experimental setup of ptychography shares a number of similarities with scanning probe techniques, such as scanning transmission x-ray microscopy (STXM) and scanning electron microscopy (SEM), in that a probe is scanned across the sample with the transmitted (or diffracted) wave-field intensity recorded. However, for rapid data acquisition, the techniques such as STXM or SEM can acquire data while the sample (or probe) is scanned continuously across the sample without collecting data at discrete positions. This has the benefit that collection times can be reduced since no time is wasted waiting for motors to move and settle. Current state-of-the art setups still require motor settling times (0.15 s) that are similar to the exposure time (0.2 s)  for each position. Due to the large number of diffraction patterns collected for ptychography (typically thousands), a reduction in acquisition time would be highly desirable. In this Letter we demonstrate that a continuous scan regime (or “fly scan”) is equivalent to a degradation in spatial coherence. We approximate the continuous scanning by summing diffraction from adjacent positions and show that the deleterious effects of this can be ameliorated through postprocessing. The reduction in collection time using this scheme should enable studies of dynamics and in situ processes as well as reducing stability requirements on experimental components.
For a quasi-monochromatic, spatially coherent probe, the intensity at a position will take a form of1(a)], is a reciprocal space position coordinate, and is the Fresnel free-space propagator for a distance . If we consider the case of a continuously scanned sample with a detector that has finite detection time , then the recorded intensity will now contain additional contributions coming from the positions around the previously discrete scan positions. The intensity can now be written as 1(b)]. We now select a sufficiently small time step , so that we can discretize Eq. (2) and sum over discrete time steps to get 3) with sample translations of , as illustrated in Fig. 1. The case of a vibrating sample has been shown to be equivalent to a degradation in spatial coherence  and reconstructible  (for both discrete and continuous motion) using recent algorithmic advances  by absorbing the sample motion into the probe modes Eq. (4). For constant velocity, the integration time of the detector will dictate the apparent degree of partial coherence. This observation leads us to a simple heuristic for the scaling between the apparent coherence length, , and scanning, via . It should be noted that the illumination modes appearing in Eq. (4) can be written as an orthogonal set of modes (with ) that are themselves fully coherent but mutually incoherent . The implications of this, along with the assertion that the integration time of the detector determines the apparent coherence length, means that the number of relevant modes does not scale with our arbitrarily assigned number , but instead scales with . This can be seen in Fig. 1(d), which shows the mode power for the first five orthogonal modes (, obtained using singular value decomposition) calculated using different numbers of simulated horizontally translated probes () (but with the same maximum translation, i.e., coherence length). The mode power is relatively insensitive to the number of translated probes, but instead depends primarily on the maximum translation. The probe function used here was a Gaussian with maximum translation of one probe width (full width at half-maximum).
To test the ability of operating ptychography in a continuous scan regime, experiments were carried out at beamline 34 ID-C at the Advanced Photon Source in Chicago. A Kirk-Patrick Baez (KB) mirror system  was used to focus 9 keV x rays onto a sample that was placed at the approximate focus, 100 mm (200 mm) from the vertical (horizontal) mirror. Horizontal and vertical slits prior to the KB mirrors were used to adjust the spatial coherence entering the optics and resulted in a probe of 700 nm (horizontally) by 1000 nm (vertically) at the sample position. The sample was a lithographed 1.5 μm thick tungsten test pattern. Scanning of the sample stage perpendicular to the x-ray direction was achieved by a nPoint piezo scanner NPXY100Z25. Diffraction was recorded 2 m downstream using a TimePix Si pixel detector , consisting of pixels of side length 55 μm, which allows detection of individual photons. At each position, five exposures of 0.5 s duration were collected. To improve the dynamic range of the data, a partially attenuating beamstop  was placed over the central region of the diffraction pattern. The beamstop consisted of a 200 μm thick silicon square that covered pixels. The central region of the beamstop was scaled by the calculated absorption using tabulated values . To reduce absorption from air between the sample and detector, a 1 m evacuated flight tube was installed (with Kapton windows).
A grid scan of 81 points horizontally (with 100 nm step size) and 11 points vertically (with 200 nm step size) was performed on a region of the sample that contained numerals and bars. Both the probe and object were reconstructed simultaneously using a multiresolution approach where initial low-resolution reconstructions were obtained (using cropped data), which were then used to seed the next higher-resolution reconstruction [12,24]. Three levels of resolution were used (, , and pixels), each lasting 500 iterations where propagation was achieved using a fast Fourier transform. The object was initialized with an array of random numbers and the probe from the KB mirror system was modeled as a rectangular lens. Position correction  was used for the highest resolution only, implemented using a local search (restricted to the nine closest pixels) and selecting the position that minimized the difference between the measured and calculated diffraction.
Figure 2(a) shows a reconstruction using all grid positions and a single sample and probe mode that amounts to a regular ptychography experiment (assuming full coherence), with intensity model given by Eq. (1). The sample features are well reproduced, with bars clearly visible along with some numerals at a resolution of 40 nm as determined from the phase-retrieval transfer function . To test the equivalence of a continuously scanned sample and partial coherence, a new dataset was created by summing intensity measurements (in the horizontal direction only) from adjacent positions into bins to form a new set of diffraction measurements [Fig. 1(c)]. Two new datasets were created by summing diffraction from 4 and 7 adjacent horizontal positions [resulting in an intensity model of Eq. (3)]. The new datasets created from the 4 and 7 position summing now contained 20 and 11 points in the horizontal direction with step sizes of 400 and 700 nm, respectively. The last columns of the scan in the horizontal direction were omitted appropriately to obtain an integer number of points in the new scan. The increasing number of positions is equivalent to an increasing integration time or scan speed.
Figures 2(b) and 2(c) show reconstructions from the new datasets using a single illumination mode for the summing of four and seven positions, respectively. There is a clear degradation in the image quality [compared to Fig. 2(a)], with some distortion of the bars and numbers and an increased nonuniformity in the feature-free regions. This failure is attributed to the fact that using a single illumination mode does not adequately describe the forward process given by Eq. (3). With increased summing we see a degradation in the pertinent features of the image with the numbers becoming very hard to identify. The increased degradation with the size of the summation can be understood through the equivalence of the summing with partial spatial coherence, i.e., the larger the distance the summing occurs over, the smaller the (apparent) coherence length. To ameliorate the deleterious effects introduced by the summing, a second reconstruction was performed on the original and summed datasets using five illumination modes (initiated using random perturbations of the ideal probe ). The reconstructions for the datasets are shown in Figs. 2(a) and 2(d) (right-hand column, five modes) and the probes in Fig. 3. The reconstruction now reproduces the relevant features well, with the bars and numerals easily recognizable and represent a significant improvement over the single-mode reconstructions, supporting the thesis that continuous scanning can be modeled as partial spatial coherence. The original reconstruction [Fig. 2(a)] has seen some improvement, indicating that the illuminating wave field was not quite fully coherent. Figure 3 shows the relative orthogonal probe mode power from the reconstructions. It can be seen that the increased summing has resulted in a commensurate decrease in the primary mode power and increase in higher-order mode () powers, a clear indication of partial coherence [11 –13,19,26]. An outstanding question is how many modes are required and how should this number be determined? Although a clear answer remains elusive and beyond the scope of this work, one strategy that could be employed would be to gradually incorporate more modes into the iterative procedure until the power in the highest (orthogonal) mode falls below some threshold value, for example, 1%.
Finally, we note that there are still some small-scale artifacts present in the reconstructions approximating the continuous scans. Possible explanations could be that there has been an apparent increase in step size, reducing the average overlap  from 0.92 for the original dataset 0.56 for the seven summed position dataset. However, a more likely explanation comes from noting that there is a requirement that the velocity be identical (but not necessarily constant) around each scan position Eq. (2). In reality, any positional errors that are present before summing and creating the new datasets will cause the effective velocity to be different for every scan position, violating the previously stated requirement. By comparing the original positions with those after iterative refinement for the original data without summation, we found that the average error for the horizontal positions was 3% and for the vertical, 12%. The actual positions (particularly the vertical) have enough error that when summing the data over adjacent positions the relative translations for each summed region are no longer identical, meaning Eq. (4) is not entirely true, even if position refinement is performed on the summed data.
We have demonstrated the equivalence of a continuous scanning regime in ptychography with partial spatial coherence. This scanning regime eliminates the need for discrete positions, avoiding time overheads associated with moving the object or probe. We anticipate this scheme will be adopted for rapid data collection across different ptychographic imaging modalities using x rays, electrons, and visible light, enabling a plethora of new and exciting science.
This work was supported by an FP7 advanced grant from the European Research Council under grant no. EP/I022562/1. X. H. is supported by the Department of Energy, Office of Basic Energy Sciences under Contract No. DE-AC-02- 98CH10886. The experimental work was carried out at the Advanced Photon Source Beamline 34-ID-C built with funds from the U.S. National Science Foundation under Grant No. DMR- 9724294 and operated by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-06CH11357. The authors gratefully acknowledge Graeme Morrison for design of the test pattern.
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