1. Introduction

Coherent x-ray diffractive imaging (CXDI) is a form of microscopy that forms an image of a sample under investigation without optics. Rather, it recovers the exit wave leaving the sample from a measurement of its coherent diffraction pattern using phase retrieval techniques. This approach solves the “phase problem” [1], which arises due to the inability of x-ray detectors to measure complex valued wave fields. Retrieval of the missing phase starts with a guess of the sample exit wave field and iteratively corrects the guess using information known about the system. This information includes the measured diffraction intensities as well as constraints on the sample, e.g. a support constraint [2] or known positions of the illuminating wave field on the sample [3]. A solution is found when the initial guess is corrected to such a degree that it simultaneously satisfies all constraints. The recovered exit wave has been shown to be unique [4], and the achievable spatial resolution is theoretically diffraction limited. The first experimental demonstration of using iterative techniques to overcome the phase problem in x-ray microscopy was performed by Miao et al. [5]. This approach has since been expanded to many diverse samples and experimental regimes [6–10].

A typical coherent diffractive imaging experiment in the forward-scattering geometry is shown in Fig. 1 [11]. Here, we define a as the detector pixel size, λ as the x-ray wavelength, k = 2π/λ the x-ray wavenumber, ℓ as the sample to detector distance, and as the detector is placed in the far field, the array size in the sample to detector Fourier transformation as N × N. The field of view at the detector is L_D = Na, while the field of view at the sample plane is L_S = λℓN/L_D. Thus the real space pixel size at the sample plane is Δx_S = L_S/N = λℓ/Na, and the Fourier space pixel size at the detector plane is found using the relation ΔqΔx_S = 2π/N, resulting in Δq = ka/ℓ. Samples larger than the incident x-ray beam can be imaged by a scanning variant of CXDI known as ptychography. In this scheme, the sample is illuminated with overlapping regions at multiple scan positions [12], as shown in Fig. 1 by the overlapping circles. Here, a simulated x-ray wave field p(r) is generated by computing the Fresnel diffraction integral for a plane wave incident on a circular pinhole aperture, with a propagation distance of a few millimeters from the pinhole to the sample. This p(r) is incident on the sample with transmission function T(r) at some location r₁. The exit wave is defined using the projection approximation as ψ₁(r) = p(r)T(r − r₁), and is propagated to the detector by taking its Fourier transform, giving the wavefield at the detector: ℱ[ψ₁(r)], where ℱ is the spatial Fourier transform:

 

Fig. 1 Schematic of an X-ray scanning CDI measurement in the forward-scattering geometry. X-rays, defined by an illumination function p(r) are incident on a sample with a transmission function T(r). A ptychographic data set is acquired by recording a series of diffraction patterns by an X-ray area detector while scanning the sample with overlapping illumination regions, depicted here by the overlapping circles.

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The purpose of collecting diffraction from overlapping regions is that it provides a very robust constraint on the reconstruction: we have multiple diffraction measurements constraining each region on the sample. This drastically reduces image artifacts resulting from imperfect measurements of the diffraction patterns; e.g. noisy diffraction or missing low spatial frequency information due to the use of a beam stop [13]. The enantiomorph ambiguity associated with single-view diffractive imaging is also removed, because scanning the sample removes the Fourier transform symmetries from which this problem arises, allowing for vastly improved algorithmic performance. Ptychographic imaging allows for an arbitrarily large field of view imaging of extended samples. It also allows for the simultaneous determination of the sample transmission function and the x-ray wave field illuminating the sample [14, 15]. Furthermore, by using a known test sample, the ptychographic approach is a powerful and robust method for determining the full complex valued wave field of the beam used to illuminate the test sample as well as the optics upstream of it [16].

The algorithms used in CXDI to extract the sample exit wave from an initial guess using experimental constraints can be formulated in terms of gradient descent of an error metric [17–19] or as a “projections onto constraint sets” algorithm [20]. In gradient descent type algorithms, an error metric ε_j is defined:

The ability to quickly and robustly converge to a solution for T(r) and p(r) is drastically degraded when errors in the assumed scan positions accumulate. Errors in the assumed scan positions r_j can be caused by thermal drift, vibrations and other mechanical errors in the experimental equipment when undertaking experiments. If using a beam on the order of tens of nanometers in size, knowledge of the scan positions can be compromised significantly if vibrations are not damped adequately. The effect of scan position errors on ptychography due to vibration and drift and various schemes to correct for them have been addressed recently [21–26]. The schemes devised in those references include the use of genetic algorithms, simulated annealing, transmission function correlation methods, and model-based drift correction. The common feature of these methods is to find some configuration of scan positions r_j which minimize an error metric of a similar form to Eq. (2), and they explore various ways of accomplishing this. In this paper, we quantify the improvements that can be achieved using conjugate gradient (CG) descent methods to minimize Eq. (2) and so determine the scan positions r_j. In contrast to reference [19], which uses CG methods to solve for T(r), p(r), and the scan positions r_j simultaneously, we introduce a novel method showing how to combine existing and well-established ptychographic algorithms, the enhanced ptychographic iterative engine (ePIE) [15] and the difference map (DM) [14], with a CG approach to correct for insufficiently known scan positions. This is important as using only a gradient descent approach to ptychographically find T(r), p(r), and the r_j simultaneously can become trapped in local minima easily; combining CG correction of the scan positions r_j with ePIE and DM updates for T(r) and p(r) can overcome this. We also introduce the concept of a critical scan position error in ptychography and then determine it using the combination of established methods to recover T(r) and p(r) and a GC recovery of the scan positions r_j.

We performed a simulated ptychography experiment to explore the effects of scan position errors on image quality. A series of diffraction patterns was generated by raster scanning a sample T(r) with an illumination function p(r), which had a diameter of ≃ 280Δx_S, in the forward scattering geometry shown in Fig. 1. The simulated sample T(r) is an image of several cells, and is represented in an HSV colorspace, with the hue as the phase (ranging from 0 to 2π) of the sample, the brightness as the magnitude (ranging from 0 to 1), and the saturation is set to 1. The illumination p(r) is generated by Fresnel propagating a simulated perfectly circular aperture over a small distance. The overlap between adjacent illumination regions is 75%, which corresponds to having the adjacent scan positions separated by a distance of 70Δx_S. In this way, 49 diffraction patterns are created by scanning a 7×7 square grid. A thin rectangular portion of each of these diffraction patterns is removed to simulate the effect of a beam stop, also as seen in Fig. 1. When no errors in the scan positions are present, we reconstruct p(r) and T(r) as shown in Fig. 2(a) and 2(b). When scan position errors are present, here random scan position errors of up to 10Δx_S added to the x and y components for each of the 49 scan positions, we see significant degradation of the reconstructions of p(r) and T(r), as seen in Fig. 2(c) and 2(d). The reconstructions shown in Fig. 2 illustrate how sensitive ptychography is to errors in the assumed scan positions.

 

Fig. 2 (a–b) Reconstructions of p(r) and T(r) for when there is no error in scan position constraint. (c–d) Reconstructions of p(r) and T(r) for when random errors up to 10Δx_S are added to the x and y components of each of the scan locations. (e–f) Line cuts of T(r) along the red line in (b); (e) is the magnitude of T(r) along this line while (f) is the phase. (g–h) Line cuts of p(r) along the red line in (a); (g) is the magnitude of p(r) along this line while (h) is the phase. In Fig. 2(e) to 2(h), the solid green line is the true T(r) or p(r), the solid black line is the line cut from reconstructions for T(r) or p(r) assuming no scan position errors, while the dotted red line is the line cut from reconstructions for T(r) or p(r) with scan position errors. The white horizontal scale bars in frames (a) and (b) are both 100Δx_S.

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2. Iterative refinement of the overlap constraint

Ptychographic diffraction measurements are often information-rich enough to allow us to extract, in addition to the sample transmission function T(r) and the illumination function p(r), the scan positions r_j. One conceptually simple way of doing this is to take the gradient of Eq. (2) with respect to the scan positions r_j and use an iterative process to correct for scan position errors [19]. Just as we are able to take the gradient of Eq. (2) with respect to T(r) and p(r) and update our guesses of these quantities in an iterative fashion, we can use a gradient descent scheme to refine our initial guesses for the r_j so that we iteratively find the true locations.

In this way, we have a very simple, and as will soon be seen, very effective method for incorporating iterative scan position correction into DM and ePIE, the standard ptychographic reconstruction algorithms [14, 15]. The steepest descent method for correcting scan positions using an analytical calculation of the gradient of Eq. (2) with respect to the scan positions r_j is given by:

We integrate this CG correction of the scan positions into the standard ptychographic reconstruction algorithms by using the r_j,0 positions initially, and update T(r − r_j) and p(r) using either the DM or ePIE for some tens of iterations. Once this is done, we use these newly obtained T(r − r_j) and p(r) in the gradient calculation of Eq. (2), and update the r_j using Eq. (4). We choose the step length α_n by rescaling Λ_j,n so that it is a unit vector, and either simply pick a value of α_n (say 1 or 2 pixels along the Λ_j,n direction), or perform a line search along the Λ_j,n direction by evaluating the error metric Eq. (2) at a few trial values of α_n:

3. Experimental demonstration

Here we demonstrate experimentally that the scan position correction scheme given in the preceding section works robustly and allows us to recover T(r), p(r) and the scan positions r_j even when large scan position errors are present. The experiment was performed at beamline 2-ID-B [28] at the Advanced Photon Source in the transmission geometry shown in Fig. 1. The pinhole aperture used to define p(r) was circular in shape, 10 μm in diameter and placed a few millimeters upstream of the sample. A 13×13 square grid was scanned, with scanning steps of 3 μm to give an overlap of 70%. The sample used was a magnetic multilayer which exhibits maze-like ordering of the magnetic domains [29–32]. In these multilayer systems, depositing alternating layers of a transition metal and a rare-earth metal causes formation of an artificial ferrimagnet with perpendicular magnetic anisotropy. Previous experiments and measurements on multilayer samples similar to the one used here have shown that the ferrimagnetic domain widths range between 200 nm to 1 μm, with the Bloch domain walls ∼ 50 nm [8, 33–36]. The magnetization of the domains in this type of sample is primarily out of plane. X-ray magnetic circular dichroism is the primary magnetic contrast mechanism in the sample transmission function T(r) at L and M resonances in the multilayer materials.

Due to scan stage misalignment, rather than scanning a set of positions on a square grid, a region on the sample can instead inadvertently be scanned on a parallelogram shaped grid. Because of this, the larger the field of view scanned, the more that scan position errors can accumulate. Horizontal and vertical scanning stage misalignments of as little as a few degrees can cause scan position errors of up to a few microns to accumulate in scans a few tens of μm in size. The effect this has on reconstructions is seen in Fig. 3(a) and 3(b). Here, the reconstructed T(r) (in Fig. 3(a)) and p(r) (in Fig. 3(b)) show significant artifacts. For this experimental geometry with the multilayer plane oriented normal to the incident x-ray beam as shown in Fig. 1, the domains are oriented parallel and antiparallel to the wavefield propagation direction [33, 37]. A reconstruction of T(r) should therefore be approximately a binary structure with values of only ±M_s, where M_s is the out of plane saturation magnetization value [8]. However in Fig. 3(a), there are many regions in the T(r) reconstruction which are intermediate to the ±M_s out of plane magnetizations, which is unphysical. Also the reconstruction of p(r) is far from what is expected of a wavefield a few millimeters downstream of a plane wavefield exiting a circular pinhole aperture. Fresnel fringes within the circular region should be evident but are absent; instead other numerical artifacts are shown.

 

Fig. 3 (a) Reconstructed T(r) and (b) p(r) when incorrect scan positions are used from experimentally collected diffraction at the Advanced Photon Source. (c) Improved reconstructions for T(r) and (d) p(r) after application of the scan position correction method presented in Section 2. (e) The recovered scan positions r_j (green dots) starting from the incorrect scan positions (black dots). The distance between the black dots on the square scanning grid is 3 μm.

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The effectiveness of the scan position scheme discussed in the previous section can be seen in Fig. 3(c) to 3(e), where it is shown that we can recover from quite severe errors in the scan positions. After its application, we have corrected the scan positions r_j with the result that the reconstructions for T(r) and p(r) are of much higher quality with the magnetic moment directions predominantly having the expected values of only ±M_s. We also can see for the reconstruction of p(r) the expected clearly defined Fresnel fringing. The recovered scan positions r_j, shown in Fig. 3(e), can be seen to be off from the incorrectly assumed scan positions by up to almost 3 μm in some locations, which is almost equal to the scanning step sizes and is about 30% of the diameter of the illumination function p(r).

4. Maximum recoverable scan position error

Here we address in simulated experiments the limits to our ability to solve for T(r), p(r) and the scan positions r_j using the methods of Section 2 when the diffraction signal is degraded in an experimentally realistic way. The primary cause of signal degradation encountered in coherent imaging experiments at synchrotron light sources is the limited dynamic range of the detectors typically used, resulting in information loss at high spatial frequencies due to Poisson noise. Also, it may be necessary to use a beam stop to prevent damage to the detector. As a result, low spatial frequency information may be altogether missing (the rectangular black region in Fig. 4(a)). Another experimental parameter that greatly affects the performance of ptychographic reconstruction algorithms and scan position recovery is the overlap of the scan positions. For example, in Fig. 4(b) and 4(c) a 2×2 square grid is scanned; the overlap in Fig. 4(b) is significantly lower than in Fig. 4(c), which means that the four diffraction patterns one would obtain in Fig. 4(b) contain less spatial information redundancy than the four diffraction patterns one would obtain in Fig. 4(c). As it is this redundant information content in the diverse diffraction that allows us to solve for T(r), p(r) and the scan positions r_j, we expect to be able to tolerate larger scan position errors with greater overlap. As will be seen, the overlap plays a crucial role in the maximum recoverable scan position error e_j,max. For example with the loosely scanned region, it might not be possible to recover the true scan positions (shown as the black circles with centers at the black dots) from the incorrect scan positions (the white dots). If we use a higher overlap like that shown on the right, it becomes much more likely that we are able to recover the true scan positions.

 

Fig. 4 To explore the limits of recoverable scan position error we look at two parameters: the photon fluence, and correspondingly the SNR of a diffraction pattern, and the amount of overlap, which determines the degree to which regions on the sample are overdetermined by information in the diffraction. (a) A simulated diffraction pattern. A beamstop, used in experiments to prevent damage to the area detector, is also assumed to be present in all cases. (b) A relatively low amount of overlap, and (c) a relatively high amount of overlap. (d) The different average diffraction intensities I(q) integrated azimuthally versus spatial frequency used for the simulations in this letter, and (e) the SNR of the diffraction intensities in (d). Here, $\text{SNR}=10{\text{log}}_{10}\left(I(\mathbf{q})/\sqrt{I(\mathbf{q})}\right)$.

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We present simulation results exploring the performance of the CG scan position correction method versus varying the diffraction signal to noise as a function of spatial frequency as well as the overlap of the scanned region on the sample. We look at different values of diffraction integrated intensities; in Fig. 4(d) the integrated intensity I(q) is varied between 10⁵ and 10¹³ (arbitrary units) in steps of 10², and to show the intensity versus spatial frequency decay, is then integrated azimuthally at each spatial frequency. The SNR, defined as $\text{SNR}=10{\text{log}}_{10}\left(I(\mathbf{q})/\sqrt{I(\mathbf{q})}\right)$, for these different integrated intensities is shown in Fig. 4(e). In all cases, a portion of the diffraction intensity is next removed (as in Fig. 4(a)) to simulate the effects of a beamstop. We also vary the overlap of the scan positions. We scan a simulated sample on a square 7 × 7 grid with ptychographic overlap of 65%, 75%, and 85%, and add random errors e_j to each scan position so that the actual (and assumed unknown) scan positions used to generate diffraction are r_j + e_j. The procedure given in Sec. 2 is then performed with the aim of determining just how large of maximum random errors e_j,max we can tolerate, and still recover the positions r_j.

The results of varying SNR and overlap versus average scan position error are shown in Fig. 5. Remarkably, there appears to be a point at which increasing the SNR versus spatial frequency of a diffraction pattern has no further effect on the maximum recoverable scan position error. For the 85% overlap square 7 × 7 grid (corresponding to Fig. 5(a)), integrated intensities of 10⁷, 10⁹, 10¹¹, and 10¹³ (which correspond to the black, blue, green, and magenta curves respectively in Fig. 4(d) and 4(e)) all have a maximum recoverable error of about ${\mathbf{e}}_{j,\mathit{max}}\simeq 90\sqrt{2}\mathrm{\Delta }{x}_S$, which we define as where the final average scan position error $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}$ ceases to be roughly flat with increasing $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ and begins to increase with increasing $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$. However when the integrated intensity is 10⁵ (corresponding the the red curves in Fig. 4(d) and 4(e)), the maximum recoverable error is reduced to $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}\simeq 53.6\mathrm{\Delta }{x}_S$, corresponding to ${\mathbf{e}}_{j,\mathit{max}}\simeq 70\sqrt{2}\mathrm{\Delta }{x}_S$. What this indicates is that there is a strong SNR and overlap dependency on the maximum recoverable error for lower SNR diffraction, but at some point the SNR dependency is lost and the maximum recoverable error becomes fixed for a particular amount of overlap. Similar behavior is seen for the 75% and 65% overlap square 7 × 7 grids, as shown in Fig. 5(b) and 5(c). This means that we can recover errors up to 45% of the diameter of p(r) for 85% overlap, up to 30% for 75% overlap, and only 20% for 65% overlap at integrated intensities greater than 10⁷ (the diameter of p(r) is ≃ 280Δx_S). For the integrated intensity of 10⁵, Fig. 5 shows a decrease in e_j,max of about 20Δx_S for all overlap cases when compared to the other intensities.

 

Fig. 5 The dependence on the maximum recoverable scan position error e_j,max (and corresponding average error over all scan positions $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$) on the diffraction SNR and overlap. Here, we look at when a square 7 × 7 grid is scanned with 65%, 75%, and 85% overlap between adjacent scan positions, and then scan position errors e_j, up to some maximum given by e_j,max, are added to each of the j scan positions. The procedure given in Section 2 is performed and we determine final scan position configurations with average error $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}$. The results shown here are those with the lowest final error metric value averaged over all j scan positions, as defined by Eq. (2), out of a total of twenty independent reconstruction trials. The maximum average scan position error $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ the CG method is able to handle is determined by where $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}$ stops being roughly flat with increasing $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ and begins to increase with increasing $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$. For the 65% case, this cutoff is $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}\simeq 30.6\mathrm{\Delta }{x}_S$ with ${\mathbf{e}}_{j,\mathit{max}}\simeq 40\sqrt{2}\mathrm{\Delta }{x}_S$, for the 75% case, this cutoff is $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}\simeq 45.9\mathrm{\Delta }{x}_S$ with ${\mathbf{e}}_{j,\mathit{max}}\simeq 60\sqrt{2}\mathrm{\Delta }{x}_S$, while for the 85% case, this cutoff is $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}\simeq 68.9\mathrm{\Delta }{x}_S$ with ${\mathbf{e}}_{j,\mathit{max}}\simeq 90\sqrt{2}\mathrm{\Delta }{x}_S$. Also included are results for perfect data when there is no simulated beamstop or Poisson noise, to establish a baseline for the performance of the method presented in Sec. 2. The primary differences for these perfect data baseline cases are greatly decreased total computation times (about a factor of 2–3), in addition to a modest increase in tolerable $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$.

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We see the impact on the SNR versus spatial frequency for integrated intensities between 10⁵ and 10⁷: no photons are detected near the spatial frequency limit of the detector, defined here as q_max = NΔq = Nka/ℓ (we used an array size N = 512). For example, the average intensity for the 10⁵ integrated intensity case in Fig. 4(d) is less than unity for spatial frequencies greater than ≈ 35Δq. This corresponds to a SNR of ≈ 2 dB, about where the signal becomes lost in the noise. Because of this, some dependence of $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ on the integrated intensity is both expected and observed. Combined with the effects of the missing spatial frequencies due to the beamstop, the diffraction data for integrated intensities between 10⁵ and 10⁷ appear to have insufficient information content for position error recovery comparable to the higher integrated intensities. For an integrated intensity of 10⁷ the signal becomes lost in the noise at ≈ 10²Δq, still far from q_max = NΔq/2, yet we are able to solve for approximately the same e_j for the higher integrated intensity cases, indicating these data have sufficient information content. Starting above an integrated intensity of 10⁷, the SNR is greater than ≈ 2 dB for all spatial frequencies. From these simulations we find that the recoverable position errors e_j,max becomes independent of the integrated intensity when the SNR approaches this value for pixels at the spatial frequency limit of the detector.

Some representative recovered scan positions r_j as well as the recovered T(r) and p(r) are shown in Fig. 6. As the maximum scan position error e_j,max becomes large, corresponding to when $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}$ begins to increase with increasing $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$, the scan positions at the periphery of the scanned region become more and more difficult to solve for, as seen in Fig. 6(a). The reason for this is that these peripheral regions have fewer independent diffraction patterns constraining possible solutions for T(r), p(r), and r_j while scan positions in the center of the scanned region have relatively many diffraction patterns. What this means is that there is not enough information content in the ptychographic diffraction data at these peripheral regions to allow us to effectively converge to simultaneous solutions for T(r), p(r), and r_j. As $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ becomes even larger, even some central scan positions become intractable, as seen in Fig. 6(b), and at some point none of the scan positions are recoverable (Fig. 6(c)). This critical $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ at which the peripheral scan positions become difficult to recover corresponds to when the average scan position error, defined as

 

Fig. 6 Recovered T(r), p(r), scan positions r_j and reconstructions for three values of e_j,max. (a) ${\mathbf{e}}_{j,\mathit{max}}\simeq 90\sqrt{2}\mathrm{\Delta }{x}_S$, (b) ${\mathbf{e}}_{j,\mathit{max}}\simeq 110\sqrt{2}\mathrm{\Delta }{x}_S$, and (c) ${\mathbf{e}}_{j,\mathit{max}}\simeq 120\sqrt{2}\mathrm{\Delta }{x}_S$. All reconstructions correspond to an integrated intensity of 10⁹, corresponding to the blue curves in Fig. 4(d) and 4(e) and Fig. 5, and the 85% overlap square 7 × 7 grid (the black circles with distance between them of 40Δx_S), with the red dots corresponding to the true scan positions, and the green dots corresponding to the recovered scan positions.

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When we are in the vicinity of, but still below the critical $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$, it is possible to further refine those scan positions at the periphery which have not been correctly solved for. Take for example the scan position configuration in Fig. 7(a). Here, the central scan positions are correctly solved for while some of those located at the periphery have not been. What can be done next is to use the recovered scan positions r_j in Fig. 7(a) as well as the recovered illumination p(r), but start with a new initial transmission function T(r) consisting of complex valued random numbers. Running the method given in Section 2 again a number of times, here twenty, and at the end averaging the results for the recovered r_j gives almost perfectly recovered scan positions, with a $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}\simeq 0.1\mathrm{\Delta }{x}_S$, as seen in Fig. 7(b). It should be noted in Fig. 7(b) that we have recovered the scan positions only to integer pixel accuracy. If positioning errors exist on sub-pixel length scales, the recovered positions in Fig. 7(b) can be used as a starting point for a correlation method [21, 38] combined with sub-pixel registration methods [39].

 

Fig. 7 When below the critical maximum scan position error e_j,max, it is possible to further refine incorrectly recovered peripheral scan positions shown in (a). By keeping the previously recovered probe p(r) and scan positions r_j but resetting T(r) to be initially consisting of complex valued random numbers, we perform the method given in Section 2 a number of subsequent times, here twenty, and average the results for all the recovered r_j. This gives almost perfectly recovered scan positions, as seen in (b), with a $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{final}}\simeq 0.1\mathrm{\Delta }{x}_S$. All values for Δr are in units of Δx_S.

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We also note that the results shown in Fig. 5 are independent on the scan trajectory used. Spiral and concentric circular scan patterns are popular for avoiding the so-called “raster pathology” that can arise with scans performed on a periodic lattice [40–42]. However, as long as the overlap between adjacent scan positions r_j is the same to within a few percentage points, the critical initial average error $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ is about the same for both raster and non-raster scans. The only significant difference between the two cases is that the starting probe position guesses are different. The CG method still determines the true scan positions from different guessed positions if $\mathrm{\Delta }{r}_{\mathit{avg}}^{\mathit{initial}}$ is about the same. This situation is analogous to using different initial guesses for T(r): using DM or ePIE, we can still recover the sample transmission function from these different starts.

5. Conclusions

We have shown in numerical simulations that the conjugate gradient method can be used to correct scan position errors in ptychography robustly and quickly when used with established phase retrieval algorithms, even when the diffraction patterns have been degraded in ways similar to those we see in experiments. Using this method we found in simulation that we can recover errors of up to approximately 45% of the illumination function diameter and up to almost 300% of the scan step size when the initial overlap of the illumination function between adjacent ptychographic scan steps is 85%. We show experimentally that this method can recover errors of approximately 30% of the illumination function diameter and almost 100% of the scan step size. We further observe that the integrated intensity of the diffraction data (and thus the signal-to-noise ratio versus spatial frequency), if increased past some cutoff, ceases to play a role in the ability of the method to recover severe scan position errors compared to lower integrated intensities, and that the degree of overlap plays the dominant role in the ability to recover large scan position errors at larger integrated intensities. From these simulations we obtain an upper bound on the recoverable random errors at each scan location - a critical scan position error - as a function of overlap and diffraction integrated intensity. This critical scan position error is specific to the use of the conjugate gradient method used to solve for scan position errors in combination with the DM and ePIE methods. This work shows that both the signal-to-noise ratios of the diffraction patterns, and the overlap between adjacent scan positions, play crucial roles in determining the magnitude of scan position errors can be recovered before the conjugate gradient method fails. We anticipate that these results can be extended to experiments to set quantitative limits on tolerable scan errors and the source brightness or measurement time required for weakly scattering samples.