M. P. Oxley , L. J. Allen, W. McBride and N. L. O’Leary

1. Phase retrieval at atomic resolution in the presence of incohoherence Iterative wave function reconstruction M. P. Oxley, L. J. Allen, W. McBride and N. L. O’Leary School of Physics, The University of Melbourne

2. Acknowledgements • C. Kisielowski, National Center for Electron Microscopy (NCEM), Lawrence Berkeley National Laboratory: Si3N4 data and related results from the MAL algorithm. • J. Ayache, National Center for Electron Microscopy: SrTiO3 bi-crystal data. • Lian Mao Peng, Peking University, Beijing: Potassium titanate nanowire data. • P. J. McMahon, The University of Melbourne: AFM tip X-ray data. • D Paganin, Melbourne/Monash University for many useful discussions.

3. Motivation • Output microscope images are often not directly interpretable due to incoherence effects and aberrations on top of phase modulation. • Many objects in electron microscopy fall in to the category of “phase objects”, i.e. intensity measurements contain minimal information. • Wave function reconstruction allows: • Removal of coherent aberrations, • Correction for partial coherence to the extent it is present in modern FEG TEM, • Provides structural information for phase objects.

4. Motivation Why iterative wave function reconstruction? • Iterative methods are simply understood and straight forward to implement. • They are applicable to many experimental circumstances. a) Resolutions from atomic level to nano level. b) Periodic and non periodic structures. • The global nature of the method presented here makes the method robust in the presence of noise. • The method is robust in the presence of phase discontinuities.

5. Coherent Aberrations For coherent aberrations the image may be formed by the convolution of the exit surface wave function with the transfer function of the imaging system T(r). The vector r is perpendicular to the direction of propagation. This is most conveniently done in momentum space. i.e. by Fourier transforming both sides. The vector q is that conjugate to r.

6. Coherent Aberrations A(q) c(q) sample lens objective aperture c(q) =p l Df q2 + 0.5p l 3 Cs q4

7. Coherent Aberrations Advantages of coherent propagation: • Image formation is based on the propagation of the whole wave function, i.e. Intensity and Phase. • Propagation is numerically efficient using fast Fourier transforms. Problems with coherent propagation: • Many sources are NOT strictly coherent. • In particular high resolution transmission electron microscopy (HRTEM) requires careful treatment of a) Finite source size (Spatial coherence), b) Defocus spread (Temporal coherence).

8. Partial Coherence In the presence of incoherence the diffractogram (the Fourier transform of the real space intensity) is propagated via the convolution1,2: Y0(q´+q)T(q´+q) Y0*(q´)T*(q´) ´ Es(q´+q, q´) ED(q´+q,q´) dq´ Envelope function describing defocus spread due to variation in the incident wavelength. i.e. Temporal coherence Envelope function describing finite source size i.e. Spatial coherence [1] K. Ishizuka, Ultramicroscopy 5 (1980) 55-65. [2] W.M.J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 (1996) 109-135.

9. Partial Coherence Advantages of partially coherent propagation: • The intensities at each defocus plane are calculated from the incoherent addition of propagated intensities as is appropriate. • Accounts for the “blurring’’ of images due to incoherence. Problems with partially coherent propagation: • Only the intensity is propagated. There is no estimate of the quantum mechanical phase other than at the exit surface. • Propagation of intensity is based on the evaluation of a two dimensional convolution integral as the envelope functions are not in general separable: Numerically intensive, especially for large numbers of measured pixels (e.g. 1024 by 1024).

10. Spatial Coherence The general form of the envelope function for spatial coherence, based on a first order Taylor expansion of the phase transfer function c(q), is given by: b is the is the semi-angle subtended by the finite source size.

11. Spatial Coherence For modern HRTEM, using a field emission gun (FEG), b is small. In particular, for focused beams (e.g. CBED or STEM), beam convergence is due to the coherent focussing of the beam by the probe forming optics. Treating total beam convergence incoherently may lead to an overcorrection. We hence approximate the spatial coherence envelope, in the separable form, as: where

12. Temporal Coherence The general form of the envelope function for temporal coherence, based on a first order Taylor expansion of the phase transfer function c(q), is given by: Dis the 1/e value of the Gaussian distribution of the defocus spread due to variations in the incident wavelength l.

13. Temporal Coherence For a FEG, even though the spread in incident wavelengths is quite small, there can still be a substantial defocus spread D. We will use the separable approximation: where

14. "Coherent" Propagtion Assuming the form of and presented, the momentum space wave function Y(q) may be calculated using • This allows rapid calculation wave function propagation using fast Fourier transforms. • The extension to propagation between planes other than the exit surface is obvious. • The ability to rapidly calculate propagation of the wave function makes this approximation amenable to methods based upon iterative wave function reconstruction (IWFR).

15. Iterative wave function reconstruction • The method is generally based on the measurement of a through focal series (TFS) of images in real space. • May in principle use information from other than variation in defocus, for example diffraction patterns. • Based on the propagation of the entire wave function. • Works in the presence of phase discontinuities. • Requires that images be aligned. Over-sampling Astigmatic fields Time evolution in BEC Cryptography

16. Global IWFR Construct average SSE Propagate the estimated exit surface wave function back to each plane Construct estimate of exit surface wave function for the jth iteration Propagate the wave function at each plane to the exit surface Construct initial wave function at each plane n We start with N experimental images at different defocus values. A phase is guessed for each plane: usually for j=1 Calculate the sum squared error in the wave function amplitude at each plane If output If j = 1, or then j=j+ 1

17. Iterative wave function reconstruction The most appealing feature of this method is its simplicity. • In the spirit of the original Gerchberg-Saxton algorithm, the intensity is simply updated at each iteration • It can be easily modified to suit a number of experimental regimes. • It is easily understood and simple to implement.

18. Iterative wave function reconstruction • The method is global. The exit surface wave function is calculated using equally weighted information from all images in the TFS. • Because of its global nature of this method copes well with noise. • Because the SSE is calculated at each plane, for each iteration, “faulty” data planes can be removed or re-measured. • Fast due to the use of fast Fourier transforms. • Produces consistent results from independent image sets.

19. Iterative wave function reconstruction • Case 1: b phase Si3N4 • Phillips CM30/FEG/UT microscope at NCEM with a resolution • of less than one Angstrom. • Sample ~ 100 Å thick with thin amorphous carbon layer to allow for • determination of defocus and spherical aberration. • [0001] zone axis orientation. • 20 images in total. • Ziegler, C. Kisielowski, R.O. Ritchie, Acta • Materialia 50 (2002) 565-574.

20. The Silicon Nitride Structure Silicon • Si3N4 is a light, hard engineering ceramic with many industrial applications due to its strength. • [0001] Zone axis orientation. • It has a hexagonal structure. Nitrogen Unit cell

21. Df = -2831.7 Å Df = -2754.5 Å Df = -2522.9 Å Df = -2677.3 Å Df = -2600.1 Å Df = -2812.4 Å Df = -2735.2 Å Df = -2658.0 Å Df = -2503.6 Å Df = -2580.8 Å Df = -2793.1 Å Df = -2715.9 Å Df = -2638.7 Å Df = -2561.5 Å Df = -2484.3 Å Df = -2773.8 Å Df = -2696.6 Å Df = -2619.4 Å Df = -2542.2 Å Df = -2465.0 Å

22. Iterative wave function reconstruction • After alignment the images were reduced in size to 902 by 940 pixels and padded back to 1024 by 1024. • Only 18 of 20 images were used. This will be expanded on later. • Results are compared to the MAL algorithm using the same parameters (all 20 images used in MAL reconstruction).

23. Iterative wave function reconstruction Standard Deviation IWFR 0.275 MAL 0.304 IWFR 0.202 MAL 0.201

24. Iterative wave function reconstruction • Excellent quantitative agreement is achieved between the two methods. • While IFWR uses a coherent treatment of temporal coherence, MAL uses a more exact formulation. • The close agreement between the two methods suggests the coherent approximation is good in this experimental regime. • The close agreement shows that damping down of the image and phase is due to the nature of the data set (Stobbs?) and not an artifact of the method.

25. Atomic locations in silicon nitride 0.8Å Unit cell Silicon Nitrogen 2.75Å • As expected for a nominal weak phase object, the projected structure is seen in the phase. • The hexagonal symmetry is obvious. • The Si-N pairs are easily seen. • N locations are not well resolved.

26. Rejection of "faulty" data Df = -2580.8 Å Df = -2561.5 Å Df = -2677.3 Å Df = -2542.2 Å Df = -2658.0 Å Df = -2638.7 Å Defocus step 19.3 Å

27. How many images are required? • Wave function reconstruction may be done with as few as two images (from experiencethree may be required for uniqueness). • Alignment of images however requires closely spaced defocus steps. • With fewer images the noise level on each image has a greater effect on the result. • For small numbers of images, the presence of “faulty” data will have a greater effect on the result.

28. How many images are required? • Here we compare the average SSE for differing numbers of images. • For few images the SSE is small due to the weak constraint in the presence of noise. • For N ³ 5 the value of the average SSE has stabilized.

29. Series A Df = -2754.5 Å Df = -2677.3 Å Df = -2600.1 Å Df = -2831.7 Å Df = -2522.9 Å Series B Df = -2812.4 Å Df = -2735.2 Å Df = -2658.0 Å Df = -2580.8 Å Df = -2503.6 Å Series C Df = -2793.1 Å Df = -2715.9 Å Df = -2638.7 Å Df = -2561.5 Å Df = -2484.3 Å Series D Df = -2773.8 Å Df = -2696.6 Å Df = -2619.4 Å Df = -2542.2 Å Df = -2465.0 Å

30. Comparison of IWFR use four independent data sets

31. Comparison of IWFR use four independent data sets Standard deviations about average value

32. Strontium titanate bi-crystal Image Phase

33. Potassium titanate nanowires -217 Å -361 Å -506 Å -651 Å -795 Å Intensity Image Phase Map (Data provided by Lian Mao Peng)

34. Soft X rays: AFM tip B.E. Allman et al. J Opt. Soc. Am. A17 (2000) 1732-1743

35. Soft X rays: AFM tip

36. Conclusions • Advantages • Based on direct propagation of the wave function. Both phase • and intensity are found • The global nature of the algorithm assures robustness in the • presence of noise and discontinuities. • Straight forward to implement. • Applicable to a wide range of experimental conditions. • Suitable for periodic and non-periodic samples. • Monitoring of convergence provides valuable information about • the data sets.