Advances in Ptychography Tega Edo A Thesis submitted for the degree of Doctor of Philosophy Department of Electronics and Electrical Engineering University of Sheffield England September 2011 Abstract Ptychography aims to completely revolutionise imaging in visible light, X-rays and Electron wavelengths by providing a robust platform for sub-Nyquist high-resolution real-time imaging. This thesis explores the framework of the very promising implementation of ptychography called the Ptychographic Iterative Engine (PIE). The PIE algorithm provides an elegant solution to the phase problem that facilitates recovery of complex representations of both the illuminating wave and the object of interest. The aim of this thesis is to present work done on the study of the machinery behind the PIE algorithm. This thesis formulates the solution provided by the PIE algorithm in reciprocal space; this shows the exact minimisation routine implemented by the PIE update function and provides a unified framework for quantifying the performance of the PIE algorithm. This work is timely because it highlights aspects of the PIE algorithm that permits practical implementation of bandwidth extrapolation of a specimen from a small detector and demonstrates the uniqueness of the corresponding solution provided by the PIE algorithm. This thesis also presents a viable scheme that utilises the redundancy of the ptychographic dataset to greatly reduce the sampling requirement on the detector; thus optimising the dataset size employed in real-time high-resolution reconstruction of the specimen over a wide field of view. ii Declaration I confirm that this thesis is my own work based on the research carried out in the Ultimate Imaging Centre, Department of Electronic and Electrical Engineering, University of Sheffield, United Kingdom. The materials in this thesis have not been submitted in part, or in whole for any other degree or qualification and references are made to materials that are not the work of the author. Tega Edo st 31 of October 2011 iii Acknowledgements I would to express my thanks to my supervisor, Prof. John Rodenburg, for his guidance and support throughout the course my research. I am very grateful for the opportunity to work in the frontier of diffractive imaging and feel very privileged to be have been exposed to his research and thinking style. I would also like to take this opportunity to thank my group manager, Grainne Riley, for all the invaluable assistance and advice offered during the course of my research. I also want to thank the staff of Phase Focus Ltd for providing the experimental equipment used in investigating the resolution improvement scheme of ptychography and I am grateful to Andy Maiden and Martin Humphry for their assistance in these experiments. I also want to express my gratitude to the members of my research group for the stimulating conversations that impacted on the direction of my research and for the support of Andy Maiden, Aaron Hurley, especially Darren Batey and John Rodenburg during the write up stage. I am indebted to the late Andy Hurst for his invaluable assistance, both academic and non-academic, during the course of my research. He is greatly missed. Publications Edo, T.B., Zhang, F. and Rodenburg, J.M. (2010) 'Resolution improvement in coherent diffractive imaging (ptychography)', Proc. SPIE 7729, 77291H Edo, T.B., Sweeney, F., Lui, C. and Rodenburg, J.M. (2010) 'Noise limit on practical electron ptychography', J. Phys.: Conf. Ser. 241 012065 iv Contents Abstract ........................................................................................................................... ii Declaration ..................................................................................................................... iii Acknowledgements ......................................................................................................... iv Publications..................................................................................................................... iv Contents .......................................................................................................................... v List of Figures ................................................................................................................ viii 1 Introduction ............................................................................................................... 1 2 Theory of Imaging ....................................................................................................... 7 2.1 Wave Propagation ............................................................................................... 8 2.1.1 Scalar wave approximation of light propagation .................................................... 9 2.2 Abbe theory of imaging ...................................................................................... 15 2.3 Limitations of the lens ........................................................................................ 16 2.4 Aberrations in imaging systems .......................................................................... 18 2.5 Overcoming the limitations of the lens ............................................................... 21 2.6 Diffractive Imaging ............................................................................................. 24 2.6.1 Ptychography ........................................................................................................ 27 2.6.2 Iterative phase retrieval ........................................................................................ 29 2.6.3 Benefits of diversity in iterative phase retrieval ................................................... 34 2.6.4 The Ptychographic Iterative Engine (PIE) ............................................................ 35 2.7 Sampling ............................................................................................................ 41 2.8 Resolution in diffractive imaging ........................................................................ 44 2.9 Resolution improvement with ptychographic diffraction patterns ...................... 45 2.10 Summary ......................................................................................................... 47 3 The PIE update function ............................................................................................ 48 3.1 PIE update function with different experimental setups ..................................... 49 3.1.1 Results .................................................................................................................. 54 3.1.2 Discussion ............................................................................................................. 56 3.2 Mathematical Concepts and Nomenclature ........................................................ 58 3.2.1 Solution to the phase problem via interference .................................................... 59 3.2.2 Diffractive imaging with plane-wave and curved-wave illuminations ................. 61 3.2.3 Diffractive imaging and sampling ......................................................................... 63 3.3 Derivation of the PIE iterative deconvolution kernel (IDK) .................................. 65 3.3.1 IDK magnitude calculation ................................................................................... 70 3.3.2 IDK phase calculation ........................................................................................... 71 3.4 The coupling function of the PIE algorithm ......................................................... 75 v 3.5 Summary ........................................................................................................... 78 4 Impact of experimental constraints on the PIE update function ................................. 79 4.1 Impact of object type on the update function ..................................................... 80 4.1.1 Results .................................................................................................................. 83 4.1.2 Discussion ............................................................................................................. 84 4.2 Impact of Counts on the update function............................................................ 85 4.2.1 Results .................................................................................................................. 89 4.2.2 Discussion ............................................................................................................. 92 4.3 Optimisation of illumination parameters ............................................................ 95 4.4 Impact of lens aperture size on the PIE update function...................................... 96 4.4.1 Results .................................................................................................................. 99 4.4.2 Discussion ........................................................................................................... 101 4.5 Impact of defocus on the update function ........................................................ 105 4.5.1 Results ................................................................................................................ 109 4.5.2 Discussion ........................................................................................................... 117 4.6 Impact of defocus error on the PIE update function .......................................... 120 4.6.1 Results