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1 In the format provided by the authors and unedited. SUPPLEMENTARY INFORMATION DOI: /NPHOTON Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source Dennis F. Gardner 1*, Michael Tanksalvala 1, Elisabeth R. Shanblatt 1, Xiaoshi Zhang 2, Benjamin R. Galloway 1, Christina L. Porter 1, Robert Karl Jr. 1, Charles Bevis 1, Daniel E. Adams 1, Henry C. Kapteyn 1,2, Margaret M. Murnane 1,2 and Giulia F. Mancini 1 1 JILA, University of Colorado, 440 UCB, Boulder, CO , USA. 2 Kapteyn-Murnane Laboratories, 1855 S. 57th Court, Boulder, CO 80301, USA. * dennis.gardner@colorado.edu S1. Modulus Enforced Probe Simulations S1.1 Faster convergence with Modulus Enforced Probe constraint The modulus enforced probe (MEP) form of ptychography detours from the epie algorithm just after the probe update is calculated (equation (7) in Ref. [1]) 1. The implementation of MEP consists of taking the current probe guess, propagating it to the detector where the measured intensity is enforced (following equation (3) in the main text), then back propagating the new guess to the sample plane (equation (4) in the main text), where it re-enters the epie algorithm. More generally, the ptychographic probe update condition combined with the novel MEP technique form a set of two projections (although the overlap constraint is not a formal projection) applied in reciprocal spaces. This allows for the use of any of the various alternating projection algorithms proven suitable for single diffraction pattern CDI 2 5. Here we demonstrate the performance of standard epie versus error reduction MEP (ER-MEP), and relaxed averaged alternating reflections MEP (RAAR-MEP) in simulation. In ER-MEP, there are no tunable parameters other than those in epie, which we call α and β for the gain associated with the object and probe update calculations, respectively. However, when using more advanced algorithms like RAAR-MEP, there will be a relaxation parameter that can be tuned for optimal performance. In these RAAR-MEP simulations, the relaxation parameter was chosen to be a constant and equal to 0.7 for reasons we discuss later. Computationally, the MEP technique propagates the probe back and forth between the detector and sample planes to apply the amplitude constraint, so the form of the propagator, which in this paper is Fresnel propagation, dominates the calculation time. Even though RAAR-MEP requires a few more element-wise operations than ER-MEP, it uses only two projections, which means the total computational cost is still dominated by propagation. Another issue is the tendency for single diffraction pattern algorithms/techniques to display trivial ambiguities, which will also be present in MEP. In particular, images rendered using single diffraction pattern techniques can appear arbitrarily translated on the numeric grid. In MEP, this ambiguity allows the probe to reconstruct away from the center of the grid. This effect also appears in epie and has been leveraged to image various sites simultaneously with multiple incident beams 6. However, the fact that epie is a gradient descent based algorithm mitigates the issue of the probe wandering on the grid. It is worth noting that even though the probe can reconstruct away from the NATURE PHOTONICS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

2 center of the grid, this effect does not hinder convergence, and the inconvenience can be alleviated by simply circularly shifting the probe to grid center, using a whole pixel centroid calculation. In the following simulations, a grid of N 2 = 64 x 64 pixels sampled a complex image whose amplitude and phase are shown in Fig. S2a and S2b. A Gaussian beam with one Fresnel zone of phase curvature and radius w 0 = 3μm (Fig. S2c, d) was truncated at the 1/e point. At each scan position, the exit surface wave was propagated a distance of 65 mm to the detector using the Fresnel approximation. The distance between scan positions was w 0/4, and the simulated detector pixel size was 54 μm. A random shift was added to each scan position with a magnitude of 3 sample pixels which were each ~0.25μm wide. The random shifts prevent periodic artifacts in the reconstructions 7. The scanned area is depicted in Fig. S2a as a faint white outline at the 1/e radius of the probe. The first interesting result that comes out of these simulations is shown in Fig. S3e, which is a plot of the root mean square (RMS) error, given by, ℇ 1 M 2 ( 1 N 2 Ψ Gj(u) Ψ Mj (u) 2 j u ) 1/2, (S1) as function of iteration for epie, ER-MEP and RAAR-MEP. In this case, the algorithms were given the exact probe used to generate the simulated diffraction patterns and an initial guess for the object of unit amplitude with flat phase. The gain parameter for the probe was β=1. In each algorithm, the error reaches machine precision (ℇ~ ) but the MEP variants converge in fewer iterations. The error curve shown in Fig. S2e is consistent over many initial trials, with the only random aspect of the simulation being the order in which the positions are called. After initially testing the new algorithms to ensure they maintain the ability to reach machine precision in the ideal case (e.g. given the exact probe), we evaluated the robustness of each algorithm to several non-ideal scenarios starting with a relatively general, but poor guess for the probe. Fig. S2f shows the RMS error as a function of iteration, using a guess for the probe that consists of an exactly oversampled disc with unit amplitude and random phase ranging from φ [ 0.5, 0.5]. In the case where the probe gain parameter β=1, not shown, epie does not converge in a reasonable number of iterations (>10,000). Worse, often times epie switched the probe and object reconstructions essentially reaching a local minimum from which it was unable to escape. This brings up a very important point, which is that the MEP variants will likely never suffer from this effect owing to the drastic inconsistency of the probe modulus constraint with the object overlap and exit-surface-wave (ESW) modulus constraints. Accordingly, we increased the gain parameter (β=4) until epie was able to consistently converge to the correct solution for this probe guess, and allowed all algorithms to reach machine precision. Fig. S2f shows that the MEP variants converge to machine precision in fewer iterations with RAAR-MEP reaching minimum error in significantly fewer iterations than epie. This led us to consider the total range of β values that would result in convergence from each algorithm. Figure S2g shows the final RMS error after 500 iterations for a range of β [0.5, 12]. All simulation parameters were kept the same from the previous test. The algorithms were run 5 times for 500 iterations for each β value on the graph. The smallest final error value from any of the 5 trials was plotted. Due to the randomness in calling the positions and probe phases, the profiles plotted in Fig. S2g

3 were smoothed with a 3-pixel median filter and plotted as solid lines to more clearly represent the trends. The first relevant point to consider is that the MEP variants start to converge to machine precision at lower values of β. The second feature clearly visible in Fig. S2g is the broad range of β values over which the MEP algorithms converge. The explanation for this comes from the fact that the probe amplitude constraint essentially clamps the total energy in the probe to a value determined by a direct measurement rather than indirectly through the ESW diffraction. This means the probe gain for the MEP techniques can be much higher than for epie due to the renormalization at each iteration. In fact, after β~>5, epie does not converge but the MEP algorithms are stable to β values greater than 8. S2.2 Simulations in the presence of Gaussian and Poisson noise. Additive white Gaussian noise (AWGN) is most common in experiments where thresholding is not possible due to lower photon energies. Above a cutoff photon energy, Gaussian noise can be eliminated entirely leaving the raw data limited by Poisson noise, commonly referred to as shot-noise. Fig. S3a shows the error after 500 iterations as a function of signal to noise ratio (SNR) quoted in decibels, using SNR = 10 log 10 ( μ ), where μ is the average value of the image and σ is the standard σ deviation of the random Gaussian distribution. Again, each algorithm was run 5 times with β=4 with the best trial run for each algorithm is reported here. As expected, we found that epie outperformed the MEP algorithms when the SNR was the same for the probe diffraction modulus and diffraction from the ESW. This is due to the fact that enforcing the modulus of the probe using noisy data will simply add more noise to the simulation. However, it is quite common for the SNR on the direct probe measurement to be much greater than the diffracted ESW intensity because the probe is more concentrated spatially and, depending on the reflectivity or transmissivity of the sample, much brighter. Therefore, Fig.S3a shows a simulation where we allowed the SNR of the probe to be twice that of the diffraction. In this case, all of the algorithms behave approximately the same. The jumps annotated by the black arrow in Fig. S3a occur when epie switches the probe and object reconstructions and gets stuck in a local minimum. In the last simulation, the number of photons in the incident probe is varied and Poisson noise is added using a MATLAB routine that is based on the Monte-Carlo rejection method. Again, convergence of each algorithm is tracked using the final, minimum error after 5 trials of 500 iterations each. Fig. S3b shows that in the presence of Poisson noise, the MEP algorithms out performs epie, which in this case is expected, both because the diffracted probe intensity is much less susceptible to degradation from shotnoise and because β was kept low at a value of 2. The black arrow in Fig. S3b annotates the incident photon numbers where epie switched the probe and object reconstruction in the presence of shotnoise. Somewhat paradoxically, epie performed better when the incident photon number was decreased and in the presence of shot-noise. This happens because shot-noise on the diffraction patterns prevents epie from switching the object and probe reconstruction, by making the probe more inconsistent with the object reconstruction. We also ran simulations suffering from shot-noise with a higher value of β=4. In this case, all the algorithms behaved approximately the same.

4 S2. Phase reconstructions with and without the novel MEP constraint Figure S4 shows the phase images corresponding to the intensity images shown in the main text Figs. 2a and 2c. Without the MEP constraint, Fig. S4b, suffers from artifacts such as ringing and striping compared to the MEP reconstruction (Fig. S4a). The phase of the PMMA is smooth and continuous. In between the zone plate features there are discontinues in the phase image corresponding phase wrapping. S3. Full ptychographic reconstruction of the zone plate Figure S5 shows the full ptychographic reconstruction from all 121 diffraction patterns covering an area of 11.3µm x 11.3µm. Only the reconstruction with the MEP constraint is shown here. The area in the upper right region of the intensity image shows degraded zone plate features. This is not an artifact of the reconstruction, since the same degradation is confirmed by the SEM image (see inset of Fig. 1a, main text). S4. Transmission values obtained for the Fresnel Zone Plate We analyzed the transmission values in the main text Fig. 2b inset. The average value of the 50nm thick Si 3N 4 region of the reconstruction is This is lower than the tabulated values reported in Ref. [8]. 8 However, the difference is not large and can be explained by the incomplete liftoff of the PMMA. Incomplete liftoff of the PMMA can be seen in the SEM images in main text Fig. 1 inset and Fig. 2a. The discrepancy between the 150nm thick PMMA tabulated transmission values, 0.46, 8 and the reconstructed values, , are large. However, the reconstructed transmission of PMMA is consistent with a transmission measurement of the ZP using the exact same setup for the ptychographic data collection. The transmission measurement was done on the central disk of the ZP feature. We took an exposure with and without the ZP in the beam, keeping all detector settings the same. After background subtraction of the readout noise, the transmission of the ZP is the ratio of the sum of the image with the ZP in the beam over the sum of the image without the ZP. This yielded 0.10 average transmission of the zone plate. Using an SEM image as template for the PMMA regions and the Si 3N 4 we were able calculate the transmission of only the PMMA features to be The calculated transmission of the PMMA from the measurement agrees well with our reconstructed values. One possible reason for the low transmission of the PMMA is its age. The ZP sample we used was purchased in 2004 and was stored in atmosphere and exposed to light and contaminates from the laboratory environment. Further material characterization of the sample is beyond the scope of the paper. Instead, we highlight that the MEP methodology yields reasonable values for the transmission, consistent with experimental measurement. On the other hand, without the MEP constraint, the reconstructed values are unphysical.

5 S5. Assignment of CCD counts to the MEP un-diffracted beam We retrieve the un-diffracted beam, by isolating the mean of the DC peak from all of the diffraction patterns within the ptychographic dataset, and thresholding at 30% of the maximum value. It is important to note that spatial isolation of the DC peak is not necessary in the case of a weakly scattering object. In this case, the DC peak may be extracted simply by thresholding. Weakly scattering objects can be used to attenuate the beam in situations with very high photon flux that may cause radiation damage to the detector. While taking the dataset used in this work, we did not take a measurement of the raw beam without the sample. It was only in reconstructing the image from the data that we discover the need for an additional constraint to improve convergence and image quality. Fortunately, we did have a maximum count of the raw beam taken before the data collection. We used this maximum count number to scale the thresholded DC peak of the diffraction pattern used for the MEP constraint, P(u). We frequently monitor the flux of our HHG beam by recording the maximum count on the same detector used for data collection without any samples in the beam path. The measurement of the max count had the following settings: 2x2 binning, 1x accumulation, and 0.1s exposure. This resulted in a maximum count value of 23,550 after subtracting the offset from camera s the analogy-to-digital converter. The maximum count in P(u) was scaled to match the maximum count of our flux measurement. To do so we had to adjust for differences in exposure time and accumulations because the data was taken with 4.5second exposure and 2x accumulations. This resulted in a maximum count of 2, 119, 500 for P(u).

6 References 1. Maiden, A. & Rodenburg, J. An improved ptychographical phase retrieval algorithm for diffractive imaging. Ultramicroscopy 109, (2009). 2. Fienup, J. Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett. 3, (1978). 3. Fienup, J. R. Phase retrieval algorithms: a comparison. Appl. Opt. 21, (1982). 4. Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. New J. Phys. 14, 1 20 (2012). 5. Luke, D. R. Relaxed Averaged Alternating Reflections for Diffraction Imaging. Inverse Probl. 37, 13 (2004). 6. Karl, R. et al. Spatial, spectral, and polarization multiplexed ptychography. Opt. Express 23, (2015). 7. Thibault, P., Dierolf, M., Bunk, O., Menzel, A. & Pfeiffer, F. Probe retrieval in ptychographic coherent diffractive imaging. Ultramicroscopy 109, (2009). 8. Henke, B. L., Gullikson, E. M. & Davis, J. C. X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 ev, Z = At. Data Nucl. Data Tables 54, (1993).

7 Supplementary Figures Figure S1 Experimental methodology for Modulus Enforced Probe. The sample is removed from the beam path and one additional measurement of the direct EUV beam is collected on the detector. This single, additional measurement is used as a constraint which enables for high-fidelity imaging of periodic objects with faster convergence of the algorithm.

8 Figure S2 Simulations in the presence of a very poor initial probe guess. a, Amplitude and b, phase of the complex image. The white outline in (a) is the area scaned by the probe to simulate diffraction patterns. c, Probe amplitude and d, phase of a Gaussian beam with one Fresnel zone of phase curvature. e, RMS error as a function of iteration for epie, ER-MEP and RAAR-MEP algorithms when given the exact probe used to generate the simulated diffraction patterns. f, RMS error using a guess for the probe which consists of an exactly oversampled disc with unit amplitude and random phase ranging from -0.5 to 0.5 radians. g, Final RMS error after 500 iterations for a range of β [0.5, 12]. All simulation parameters were kept the same from the previous test.

9 Figure S3 Simulations in the presence of a, Gaussian and b, Poisson noise.

10 Figure S4 Phase image of the Fresnel zone plate sample a, with and b, without the MEP constraint. The scale bar in (a) is shared with (b).

11 Figure S5 Full ptychographic image of the zone plate sample. a, Intensity and b, phase. The area highlighted in (a) is the region in main text Fig. 2a and supplementary Fig. S4a. The scale bar in (b) is shared with (a).