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Ptychography and Related Diffractive Imaging Methods J.M. RODENBURG1

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Ptychography and Related Diffractive Imaging Methods J.M. RODENBURG1 aiep v.2005/11/15 Prn:6/12/2007; 13:13 F:aiep1503.tex; VTEX/JOL p. 1 aid: 3 pii: S1076-5670(07)00003-1 docsubty: REV ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 150 1 1 2 2 3 Ptychography and Related Diffractive Imaging Methods 3 4 4 5 5 6 J.M. RODENBURG1 6 7 7 8 1Department of Electronic and Electrical Engineering, University of Sheffield, Mappin Street, 8 9 S1 3JD, United Kingdom 9 10 10 11 11 12 I. Introduction . 87 12 13 II. Ptychography in Context . 88 13 A. History . 88 14 14 B. The Relative Strength of Diffractive and Imaging Methods . 93 15 C. The Definition of Ptychography . 94 15 16 D. A Qualitative Description of Ptychography . 97 16 17 E. Mathematical Formulation of Crystalline STEM Ptychography . 103 17 18 F. Illumination of a “Top Hat” or Objects with Finite Support . 109 18 G. Two-Beam Ptychography . 113 19 19 H. Commentary on Nyquist Sampling . 114 20 I. Concluding Remarks on Classical Ptychography . 117 20 21 III. Coordinates, Nomenclature, and Scattering Approximations . 119 21 22 A. Geometry of the Two-Dimensional, Small-Angle Approximation . 119 22 23 B. Three-Dimensional Geometry . 123 23 C. Dynamical Scattering . 132 24 24 IV. The Variants: Data, Data Processing, and Experimental Results . 138 25 A. The Line Scan Subset . 140 25 26 B. The Bragg–Brentano Subset: Projection Achromatic Imaging . 149 26 27 C. Ptychographical Iterative (Pi) Phase-Retrieval Reconstruction . 163 27 28 D. The Wigner Distribution Deconvolution Method . 172 28 V. Conclusions . 178 29 29 References . 180 30 30 31 31 32 32 33 I. INTRODUCTION 33 34 34 35 Ptychography is a nonholographic solution of the phase problem. It is a 35 36 method of calculating the phase relationships between different parts of a 36 37 scattered wave disturbance in a situation where only the magnitude (intensity 37 38 or flux) of the wave can be physically measured. Its usefulness lies in its ability 38 39 (like holography) to obtain images without the use of lenses, and hence to lead 39 40 to resolution improvements and access to properties of the scattering medium 40 41 (such as the phase changes introduced by the object) that cannot be easily 41 42 obtained from conventional imaging methods. Unlike holography, it does not 42 87 ISSN 1076-5670 Copyright 2008, Elsevier Inc. DOI: 10.1016/S1076-5670(07)00003-1 All rights reserved. aiep v.2005/11/15 Prn:6/12/2007; 13:13 F:aiep1503.tex; VTEX/JOL p. 2 aid: 3 pii: S1076-5670(07)00003-1 docsubty: REV 88 RODENBURG 1 need a stable reference wave; the only interferometer used in ptychography is 1 2 diffraction interference occurring in the object itself. 2 3 The concept of ptychography was originated by the late Walter Hoppe 3 4 between about 1968 and 1973 (Hoppe, 1969a, 1969b; Hoppe and Strube, 4 5 1969; Hegerl and Hoppe, 1970, 1972). In its original formulation it was 5 6 difficult to contemplate how to implement an actual experimental proof of 6 7 the idea at short (atomic scale) wavelengths, other than for a crystalline object 7 8 in a scanning transmission electron microscope (STEM). Hoppe and Strube 8 9 (1969) demonstrated the idea experimentally using visible light optics, but at 9 10 the time that Hoppe was thinking about these ideas, STEMs were simply not 10 11 sufficiently developed (particularly the detectors available) to undertake the 11 12 necessary measurements. At that time, the benefits of the technique were ill 12 13 defined and offered no clear improvements over existing imaging, diffraction, 13 14 and holographic methods. In fact, when a proof of principle was eventually 14 15 established by Nellist et al. (1995), it was discovered that ptychography did 15 16 have one uniquely and profoundly important advantage over all other phase- 16 17 retrieval or imaging techniques: it is not subject to the limitations of the 17 18 coherence envelope (the “information limit”), which today is still regarded 18 19 as the last hurdle in optimal electron imaging. 19 20 In view of the recent advances in iterative solution of the ptychographic 20 21 data set (see Section IV.C) and their experimental implementation in visible 21 22 light optics and hard X-ray imaging, a review of the neglected strengths of 22 23 the ptychographical principle is of benefit. Recent years have seen a growing 23 24 interest in iterative solutions of the phase problem. In most modern work, 24 25 Hoppe’s pioneering thoughts have been entirely neglected. This is unfortunate 25 26 for two reasons. First is the scholarly misrepresentation of the precedence of 26 27 certain key ideas in the history of the field of phase problems. Second, a true 27 28 appreciation of what ptychography encompasses, especially with respect to 28 29 “conventional” diffractive imaging methods, means that a great opportunity 29 30 has yet to be fully exploited. This Chapter attempts to demonstrate these 30 advantages are and why this Cinderella of the phase-retrieval story may be 31 31 on the brink of revolutionizing short-wavelength imaging science. 32 32 33 33 34 II. PTYCHOGRAPHY IN CONTEXT 34 35 35 36 This section places ptychography in the context of other phase-retrieval 36 37 methods in both historical and conceptual terms. 37 38 38 39 A. History 39 40 40 41 Recovery of the lost phase (or relative phase) by some calculation method 41 42 is still considered a problem because when these issues were first addressed 42 aiep v.2005/11/15 Prn:6/12/2007; 13:13 F:aiep1503.tex; VTEX/JOL p. 3 aid: 3 pii: S1076-5670(07)00003-1 docsubty: REV PTYCHOGRAPHY AND RELATED DIFFRACTIVE IMAGING METHODS 89 1 (before World War II), it appeared that in certain scattering geometries 1 2 (especially those most easily accessible experimentally) a solution could not 2 3 exist, and if it did exist, it would be very ill conditioned mathematically. X-ray 3 4 crystallography was the first experimental discipline to admit to the conse- 4 5 quences of lost phase. In fact, given that large quantities of stereochemical a 5 6 priori information is often available, the crystalline diffraction phase problem 6 7 is much more tractable than the generalized phase problem. Indeed, it could 7 8 be argued that Watson and Crick’s discovery of the structure of DNA (Watson 8 9 and Crick, 1953) was itself an ad hoc solution to a phase problem – they had 9 10 access to the intensity of the scattered X-ray diffraction pattern from a crystal 10 11 composed of DNA molecules and essentially found a “solution” (a physical 11 12 distribution of atoms) that was consistent with that intensity and with the 12 13 known stereochemistry constraints of the constituent amino acids. This use 13 14 of a priori information is an important characteristic of all solutions to the 14 15 phase problem. 15 16 The key advantage of phase-retrieval methods in the context of imaging is 16 17 that image formation can be unshackled from the difficulties associated with 17 18 the manufacture of high-quality lenses of large numerical aperture. In some 18 19 fields, such as stellar interferometry (Michelson, 1890), it is only possible to 19 20 record a property related to the intensity of the Fourier transform of the object. 20 21 The creation of an image from these data is then a classic phase problem 21 22 (in this case, rather easily solvable given that the a priori information that 22 23 the object in question is finite, of positive intensity, and is situated on an 23 24 empty background). In the field of visible wavelength optical microscopy, 24 25 little need exists to solve the phase problem because very good lenses of very 25 26 large numerical aperture can be relatively easily manufactured (although a 26 27 phase problem still exists in the sense that the image itself is recorded only in 27 28 intensity). For short-wavelength microscopies (using X-rays or high-energy 28 29 electrons), good-quality lenses are not easily manufactured. For example, the 29 30 field of electron imaging has recently seen great progress in the manufacture 30 31 of non-round lenses correctors (for which spherical aberration need not be 31 32 positive, as with a round magnetic lens), a scheme first proposed by Scherzer 32 33 (1947) but which has only recently become feasible with the advent of 33 34 powerful inexpensive computers that can be interfaced to very high-stability 34 35 power supplies. However, the cost of such systems is high (typically of the 35 36 order of $1000 per corrector), and the gains in resolution over existing round 36 37 magnetic lenses are only of an order of a factor of 2 or so. In the case of 37 38 X-ray microscopy, the zone plate remains the predominant lens technology, 38 39 but this technology requires manufacturing precision of at least the final 39 40 requisite resolution of the lens itself; making a sufficiently thick zone plate 40 41 with very high lateral definition (< 50 nm) difficult, especially for hard X- 41 42 rays. For example, compare the best resolution obtainable from such a lens 42 aiep v.2005/11/15 Prn:6/12/2007; 13:13 F:aiep1503.tex; VTEX/JOL p. 4 aid: 3 pii: S1076-5670(07)00003-1 docsubty: REV 90 RODENBURG 1 (also 50 nm) with the “resolution” of conventional X-ray diffraction using 1 ∼ 2 the same wavelength; notwithstanding the fact that the object of interest (the 2 3 unit cell) must be amenable to crystallization, the effective resolution of the 3 4 “image” (an estimate of that unit cell) is roughly the same as the wavelength: 4 5 0.1 nm.
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