Topics in Applied Physics 134 Tim Salditt Alexander Egner D. Russell Luke Editors Nanoscale Photonic Imaging Topics in Applied Physics Volume 134 Series Editors Young Pak Lee, Physics, Hanyang University, Seoul, Korea (Republic of) Paolo M. Ossi, NEMAS - WIBIDI Lab, Politecnico di Milano, Milano, Italy David J. Lockwood, Metrology Research Center, National Research Council of Canada, Ottawa, ON, Canada Kaoru Yamanouchi, Department of Chemistry, The University of Tokyo, Tokyo, Japan Topics in Applied Physics is a well-established series of review books, each of which presents a comprehensive survey of a selected topic within the area of applied physics. Edited and written by leading research scientists in the fi eld concerned, each volume contains review contributions covering the various aspects of the topic. Together these provide an overview of the state of the art in the respective fi eld, extending from an introduction to the subject right up to the frontiers of contemporary research. Topics in Applied Physics is addressed to all scientists at universities and in industry who wish to obtain an overview and to keep abreast of advances in applied physics. The series also provides easy but comprehensive access to the fi elds for newcomers starting research. Contributions are specially commissioned. The Managing Editors are open to any suggestions for topics coming from the community of applied physicists no matter what the fi eld and encourage prospective book editors to approach them with ideas. 2018 Impact Factor: 0.746 More information about this series at http://www.springer.com/series/560 Tim Salditt • Alexander Egner • D. Russell Luke Editors Nanoscale Photonic Imaging Editors Tim Salditt Institut f ü r R ö ntgenphysik Universit ä t G ö ttingen G ö ttingen, Germany Alexander Egner Laser Laboratorium University of G ö ttingen G ö ttingen, Germany D. Russell Luke Institut f ü r Numerische und Angewandte Mathematik Universit ä t G ö ttingen G ö ttingen, Germany ISSN 0303-4216 ISSN 1437-0859 (electronic) Topics in Applied Physics ISBN 978-3-030-34412-2 ISBN 978-3-030-34413-9 (eBook) https://doi.org/10.1007/978-3-030-34413-9 © The Editor(s) (if applicable) and The Author(s) 2020. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adap- tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book ’ s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book ’ s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi- cation does not imply, even in the absence of a speci fi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional af fi liations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface The word ‘ Nano ’ has been around for a long time. It became a topic of signi fi cant interest in the eighties of the last century, after instruments such as the scanning tunneling microscope and the atomic force microscope had been invented. The ‘ nanoscale ’ was probed based on electric currents through a tunneling tip or by measuring the forces with a cantilever. In other words, the ‘ room at the bottom ’ was conquered not by ‘ seeing ’ , but rather by ‘ feeling ’ . Too strong was the belief that optical imaging was limited to the microscale due to the diffraction barrier. But the insight that photonics and nanoscale also make a perfect match followed only shortly after the advent of the scanning tunneling and atomic force microscopes. Around the turn of the millennium it became broadly accepted that plenty of ‘ nano ’ can be done with photons: Single molecule spectroscopy had been established, fl uorescence correlation spectroscopy was emerging, and above all there was a new way to turn microscopes into nanoscopes based on optical switching, as pioneered by Stefan Hell here in G ö ttingen. While very few physicists cared about optical microscopes before, a time of rapid development had now set in. At the same time, a long-standing dream to realize X-ray microscopy was empowered by coherent optics and computational phase retrieval. Pairing up optical and short wavelength to extend the scales of ‘ imaging ’ , research teams in G ö ttingen set out for new discoveries. But how to empower their vessels? The solution was found by mathematics. Using results from inverse problems, stochastics, and optimization theory, new and bountiful shores were discovered, and photonic data was turned into useful information ... As we now come back from our expeditions funded for the last 12 years by the German Science Foundation (DFG) through SFB755 Nanoscale Photonic Imaging , we do not want to keep all the treasures for ourselves. The current book is a compilation of tutorials, experiments and experiences, and a compendium for fur- ther reading. In addition to the contributing authors and Angela Lehee at Springer, we are grateful to Leon Lohse, Shahroz Shahjahan for helping to keep this project on track. Above all we would like to express our deepest gratitude to Eva Hetzel v who has been with this collaborative research center for the duration and has been essential to keeping the expedition on track, on budget and on time — all with grace and joyful optimism. Now, let us dive deep into the nanoscale, and not just scratch at its surface! G ö ttingen, Germany Tim Salditt Alexander Egner D. Russell Luke vi Preface Contents Part I Fundamentals and Tutorials 1 STED Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Alexander Egner, Claudia Geisler and Ren é Siegmund 1.1 Fundamentals of Fluorescence Microscopy . . . . . . . . . . . . . . . 3 1.1.1 Vectorial Diffraction Theory and Intensity Distribution Within the Focal Spot . . . . . . . . . . . . . . 3 1.1.2 Incoherent Image Formation . . . . . . . . . . . . . . . . . . . 9 1.1.3 Classical Resolution Limit . . . . . . . . . . . . . . . . . . . . . 11 1.1.4 Confocal Microscopy . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Fundamentals of STED Microscopy . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 Basic Photophysics of Dye Molecules . . . . . . . . . . . . 18 1.2.3 Shaping the STED Beam . . . . . . . . . . . . . . . . . . . . . 23 1.2.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3 Imaging Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Coherent X-ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Tim Salditt and Anna-Lena Robisch 2.1 X-ray Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Scalar Diffraction Theory and Wave Equations . . . . . . 36 2.1.2 Propagation in Free Space . . . . . . . . . . . . . . . . . . . . . 41 2.1.3 The Fresnel Scaling Theorem . . . . . . . . . . . . . . . . . . 44 2.1.4 Numerical Implementation of Free-Space Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.1.5 X-ray Propagation in Matter . . . . . . . . . . . . . . . . . . . 48 2.1.6 Propagation by Finite Difference Equations . . . . . . . . 51 vii 2.2 Coherent Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.1 Holographic Imaging in Full Field Setting . . . . . . . . . 55 2.2.2 Contrast in X-ray Holograms . . . . . . . . . . . . . . . . . . . 57 2.3 Solving the Phase Problem in the Holographic Regime . . . . . . 59 2.3.1 Single-Step Phase Retrieval . . . . . . . . . . . . . . . . . . . . 60 2.3.2 Iterative Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . 60 2.4 From Two to Three Dimensions: Tomography and Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3 X-ray Focusing and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Tim Salditt and Markus Osterhoff 3.1 General Aspects of X-ray Optics and Focusing . . . . . . . . . . . . 71 3.2 X-ray Re fl ectivity and Re fl ective X-ray Optics . . . . . . . . . . . . 74 3.2.1 X-ray Re fl ectivity of an Ideal Single Interface . . . . . . 74 3.2.2 Multiple Interfaces and Multilayers . . . . . . . . . . . . . . 77 3.2.3 Interfacial Roughness . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3 X-ray Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Kirkpatrick-Baez Geometry . . . . . . . . . . . . . . . . . . . . 83 3.3.2 Multilayer Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4 X-ray Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.1 Waveguide Modes: The Basics . . . . . . . . . . . . . . . . . 89 3.4.2 Coupling and Propagation . . . . . . . . . . . . . . . . . . . . . 95 3.4.3 Fabrication and Characterisation of X-ray Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.4.4 Advanced Waveguide Con fi gurations . . . . . . . . . . . . . 99 3.5 Diffractive Optics and Zone Plates . . . . . . . . . . . . . . . . . . . . . 102 3.5.1 Basic Theory of Fresnel Zone Plates . . . . . . . . . . . . . 102 3.5.2 Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . 105 3.5.3 Diffractive Optics Beyond the Projection Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.6 Basic Coherence Theory and Simulations for X-ray Optics . . . 109 3.6.1 Basic De fi nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.6.2 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.6.3 Coherence Propagation and Filtering . . . . . . . . . . . . . 113 3.7 Putting It All Together: Optics and X-ray Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4 Statistical Foundations of Nanoscale Photonic Imaging . . . . . . . . . . 125 Axel Munk, Thomas Staudt and Frank Werner 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.1.1 Background and Examples . . . . . . . . . . . . . . . . . . . . 125 4.1.2 Purpose of the Chapter . . . . . . . . . . . . . . . . . . . . . . . 126 viii Contents 4.1.3 Measurement Devices . . . . . . . . . . . . . . . . . . . . . . . . 127 4.1.4 Structure and Notation . . . . . . . . . . . . . . . . . . . . . . . 128 4.2 Poisson Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3 Bernoulli Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.3.1 Law of Small Numbers . . . . . . . . . . . . . . . . . . . . . . . 133 4.4 Gaussian Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.4.1 As Approximation of the Binomial Model . . . . . . . . . 135 4.4.2 As Approximation of the Poisson Model . . . . . . . . . . 136 4.4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.4.4 Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.4.5 Variance Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 138 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Appendix: Poisson Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Appendix: Conditioned Poisson Processes . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Thorsten Hohage, Benjamin Sprung and Frederic Weidling 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.1.1 What Is an Inverse Problem? . . . . . . . . . . . . . . . . . . . 145 5.1.2 Ill-Posedness and Regularization . . . . . . . . . . . . . . . . 146 5.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.1.4 Choice of Regularization Parameters and Convergence Concepts . . . . . . . . . . . . . . . . . . . . 148 5.2 Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.2.1 Variational Regularization . . . . . . . . . . . . . . . . . . . . . 150 5.2.2 Iterative Regularization . . . . . . . . . . . . . . . . . . . . . . . 154 5.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3.1 General Error Bounds for Variational Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3.2 Interpretation of Variational Source Conditions . . . . . 158 5.3.3 Error Bounds for Poisson Data . . . . . . . . . . . . . . . . . 162 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6 Proximal Methods for Image Processing . . . . . . . . . . . . . . . . . . . . . 165 D. Russell Luke 6.1 All Together Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.1.1 What Seems to Be the Problem Here? . . . . . . . . . . . . 166 6.1.2 What Is an Algorithm? . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.3 What Is a Proximal Method? . . . . . . . . . . . . . . . . . . . 176 6.1.4 On Your Mark. Get Set... . . . . . . . . . . . . . . . . . . . . . 177 6.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2.1 Model Category I: Multi-set Feasibility . . . . . . . . . . . 178 6.2.2 Model Category II: Product Space Formulations . . . . . 182 Contents ix 6.2.3 Model Category III: Smooth Nonconvex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.3 ProxToolbox — A Platform for Creative Hacking . . . . . . . . . . . 192 6.3.1 Coffee Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.3.2 Star Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.3.3 E Pluribus Unum . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.4 Last Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Part II Progress and Perspectives 7 Quantifying Molecule Numbers in STED/RESOLFT Fluorescence Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Jan Keller-Findeisen, Steffen J. Sahl and Stefan W. Hell 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.1.1 Molecular Contribution Function (MCF) . . . . . . . . . . 207 7.2 STED Nanoscopy with Coincidence Photon Detection . . . . . . 208 7.2.1 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.2.2 Intrinsic Molecular Brightness Calibration . . . . . . . . . 212 7.2.3 Counting Transferrin Receptors in HEK293 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.3 Mean and Variance in RESOLFT Nanoscopy . . . . . . . . . . . . . 215 7.3.1 Cumulants of the Fluorescence of Switchable Fluorophores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3.2 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.3.3 Counting rsEGFP2 Fused a -tubulin Units in Drosophila Melanogaster . . . . . . . . . . . . . . . . . . . 220 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8 Metal-Induced Energy Transfer Imaging . . . . . . . . . . . . . . . . . . . . 227 Alexey I. Chizhik and J ö rg Enderlein 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.2 Basic Principle and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.3 The MIET-GUI Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.4 Metal-Induced Energy Transfer for Biological Imaging . . . . . . 233 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9 Reversibly Switchable Fluorescent Proteins for RESOLFT Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Nickels A. Jensen, Isabelle Jansen, Maria Kamper and Stefan Jakobs 9.1 Overcoming the Diffraction Barrier . . . . . . . . . . . . . . . . . . . . 241 9.2 RSFPs for Live-Cell RESOLFT Nanoscopy . . . . . . . . . . . . . . 242 x Contents 9.3 Photoswitching Mechanisms of RSFPs . . . . . . . . . . . . . . . . . . 243 9.3.1 Negative Switching Mode . . . . . . . . . . . . . . . . . . . . . 245 9.3.2 Positive Switching Mode . . . . . . . . . . . . . . . . . . . . . 245 9.3.3 Decoupled Switching Mode . . . . . . . . . . . . . . . . . . . 246 9.4 RSFP Properties Important for RESOLFT Nanoscopy . . . . . . . 247 9.4.1 Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.4.2 Ensemble Switching Speed . . . . . . . . . . . . . . . . . . . . 248 9.4.3 Residual Fluorescence in the Off-State . . . . . . . . . . . . 248 9.4.4 Switching Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.5 Overview of RSFPs for RESOLFT Nanoscopy . . . . . . . . . . . . 249 9.5.1 RSFPs Emitting in the Green . . . . . . . . . . . . . . . . . . 250 9.5.2 RSFPs Emitting in the Yellow . . . . . . . . . . . . . . . . . . 252 9.5.3 RSFPs Emitting in the Red . . . . . . . . . . . . . . . . . . . . 253 9.6 Applications of RESOLFT Nanoscopy . . . . . . . . . . . . . . . . . . 253 9.6.1 Other Fluorophores for RESOLFT Nanoscopy . . . . . . 255 9.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10 A Statistical and Biophysical Toolbox to Elucidate Structure and Formation of Stress Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Benjamin Eltzner, Lara Hauke, Stephan Huckemann, Florian Rehfeldt and Carina Wollnik 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.2 Live Cell Imaging-Opportunities and Challenges . . . . . . . . . . . 266 10.3 Automated Unbiased Binarization of Filament Structure . . . . . 267 10.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.3.2 The FilamentSensor and the Benchmark Dataset . . . . . 269 10.3.3 Detecting Slightly Bent Filaments . . . . . . . . . . . . . . . 270 10.4 Orientation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.4.1 Orientation Field Evolution . . . . . . . . . . . . . . . . . . . . 274 10.4.2 Backward Nested Descriptor Analysis . . . . . . . . . . . . 277 10.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 11 Photonic Imaging with Statistical Guarantees: From Multiscale Testing to Multiscale Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Axel Munk, Katharina Proksch, Housen Li and Frank Werner 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11.2 Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.2.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . 286 11.2.3 Testing on an Image . . . . . . . . . . . . . . . . . . . . . . . . . 288 11.2.4 Testing Multiple Hypotheses . . . . . . . . . . . . . . . . . . . 291 11.2.5 Connection to Extreme Value Theory . . . . . . . . . . . . 295 Contents xi 11.2.6 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 11.2.7 Theory for the Multiscale Scanning Test . . . . . . . . . . 302 11.2.8 Deconvolution and Scanning . . . . . . . . . . . . . . . . . . . 303 11.2.9 FDR Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.3 Statistical Multiscale Estimation . . . . . . . . . . . . . . . . . . . . . . . 308 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 12 Ef fi cient, Quantitative Numerical Methods for Statistical Image Deconvolution and Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 D. Russell Luke, C. Charitha, Ron She fi and Yura Malitsky 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 12.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 12.2.1 Abstract Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 12.2.2 Saddle Point and Dual Formulations . . . . . . . . . . . . . 319 12.2.3 Statistical Multi-resolution Estimation . . . . . . . . . . . . 321 12.3 Alternating Directions Method of Multipliers and Douglas Rachford . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 12.3.1 ADMM for Statisitcal Multi-resolution Estimation of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 12.4 Primal-Dual Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 12.4.1 EPAPC for Statisitcal Multi-resolution Estimation of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12.5 Randomized Block-Coordinate Primal-Dual Methods . . . . . . . 332 12.5.1 RBPD for Statisitcal Multi-resolution Estimation of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13 Holographic Imaging and Tomography of Biological Cells and Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Tim Salditt and Mareike T ö pperwien 13.1 Propagation-Based Phase-Contrast Tomography . . . . . . . . . . . 339 13.2 Nano-CT Using Synchrotron Radiation: Optics, Instrumentation and Phase Retrieval . . . . . . . . . . . . . . . . . . . . 341 13.2.1 Cone-Beam Holography . . . . . . . . . . . . . . . . . . . . . . 341 13.2.2 Waveguide Optics and Imaging . . . . . . . . . . . . . . . . . 343 13.2.3 Dose-Resolution Relationship . . . . . . . . . . . . . . . . . . 345 13.2.4 Phase Retrieval Algorithms . . . . . . . . . . . . . . . . . . . . 346 13.3 CTF-based Reconstruction and Its Limits . . . . . . . . . . . . . . . . 347 13.4 Laboratory μ -CT: Instrumentation and Phase Retrieval . . . . . . 349 13.5 Novel Tomography Approaches . . . . . . . . . . . . . . . . . . . . . . . 354 13.5.1 Combined Phase Retrieval and Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 13.5.2 Tomographic Reconstruction Based on the 3D Radon Transform (3DRT) . . . . . . . . . . . . . 356 xii Contents 13.6 Tomography of Biological Tissues: Applications and Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 13.6.1 3D Structure of Cochlea . . . . . . . . . . . . . . . . . . . . . . 358 13.6.2 Small Animal Imaging . . . . . . . . . . . . . . . . . . . . . . . 360 13.6.3 3D Virtual Histology of Nerves . . . . . . . . . . . . . . . . . 363 13.6.4 Macrophages in Lung Tissue . . . . . . . . . . . . . . . . . . . 363 13.6.5 Neuron Locations in Human Cerebellum . . . . . . . . . . 365 13.6.6 Outlook: Time-Resolved Phase-Contrast Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 14 Constrained Reconstructions in X-ray Phase Contrast Imaging: Uniqueness, Stability and Algorithms . . . . . . . . . . . . . . . . . . . . . . . 377 Simon Maretzke and Thorsten Hohage 14.1 Forward Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 14.1.1 Physical Model and Preliminaries . . . . . . . . . . . . . . . 378 14.1.2 Forward Operators for XPCI . . . . . . . . . . . . . . . . . . . 380 14.1.3 Forward Operators for XPCT . . . . . . . . . . . . . . . . . . 382 14.2 Uniqueness Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 14.2.1 Preliminary Results and Counter-Examples . . . . . . . . 383 14.2.2 Sources of Non-uniqueness — The Phase Problem . . . . 384 14.2.3 Relation to Classical Phase Retrieval Problems . . . . . . 384 14.2.4 Holographic Nature of Phase Retrieval in XPCI . . . . . 385 14.2.5 General Uniqueness Under Support Constraints . . . . . 386 14.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 14.3.1 Lipschitz-Stability and its Meaning . . . . . . . . . . . . . . 387 14.3.2 Stability for General Objects and one Hologram . . . . . 388 14.3.3 Homogeneous Objects and Multiple Holograms . . . . . 391 14.3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 14.4 Regularized Newton Methods for XPCI . . . . . . . . . . . . . . . . . 394 14.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 14.4.2 Reconstruction Method . . . . . . . . . . . . . . . . . . . . . . . 395 14.4.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . . . . 396 14.5 Regularized Newton-Kaczmarz-SART for XPCT . . . . . . . . . . 396 14.5.1 Ef fi cient Computation by Generalized SART . . . . . . . 398 14.5.2 Parallelization and Large-Scale Implementation . . . . . 399 14.5.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . . . . 400 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 15 Scanning Small-Angle X-ray Scattering and Coherent X-ray Imaging of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Tim Salditt and Sarah K ö ster 15.1 X-ray Structure Analysis of Biological Cells: A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Contents xiii 15.2 Methods: X-ray Optics and Sample Environment . . . . . . . . . . 408 15.2.1 Focusing Optics and Imaging Modalities . . . . . . . . . . 408 15.2.2 X-ray Compatible Micro fl uidic Sample Environments for Cells . . . . . . . . . . . . . . . . . . . . . . . 409 15.3 Scanning Small-Angle X-ray Scattering of Cells . . . . . . . . . . . 413 15.4 Coherent X-ray Imaging of Cells . . . . . . . . . . . . . . . . . . . . . . 418 15.4.1 Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 15.4.2 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 15.5 Correlative Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 15.6 From Cells to Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 15.7 Outlook: FEL Studies of Cells . . . . . . . . . . . . . . . . . . . . . . . . 425 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 16 Single Particle Imaging with FEL Using Photon Correlations . . . . 435 Benjamin von Ardenne and Helmut Grubm ü ller 16.1 The Single Molecule Scattering Experiment . . . . . . . . . . . . . . 436 16.2 Structure Determination Using Few Photons . . . . . . . . . . . . . . 437 16.2.1 Theoretical Background on Three-Photon Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 16.2.2 Bayesian Structure Determination . . . . . . . . . . . . . . . 441 16.2.3 Reduction of Search Space Using Two-Photon Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 16.2.4 Optimizing the Probability Using Monte Carlo . . . . . . 443 16.3 Method Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 16.3.1 Resolution Scaling with Photon Counts . . . . . . . . . . . 445 16.3.2 Impact of the Photon Counts per Image . . . . . . . . . . . 448 16.3.3 Structure Results in the Presence of Non-Poissonian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 16.4 Structure Determination from Multi-Particle Images . . . . . . . . 451 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 17 Development of Ultrafast X-ray Free Electron Laser Tools in (Bio)Chemical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Simone Techert, Sreevidya Thekku Veedu and Sadia Bari 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 17.2 The Concept: Filming Chemical Reactions in Real Time Utilizing Ultrafast High-Flux X-ray Sources . . . . . . . . . . . . . . 461 17.3 X-ray Diffraction and Crystallography for Condensed State Chemistry Studies — Crystallography with Ultrahigh Temporal and Ultrahigh Spatial Resolution . . . . . . . . . . . . . . . 462 17.4 Applications in Energy Research . . . . . . . . . . . . . . . . . . . . . . 466 17.5 From Local to Global: Ultrafast Multidimensional Soft X-ray Spectroscopy and Ultrafast X-ray Diffraction Shake Their Hands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 xiv Contents 17.6 Applications in Bimolecular Reaction Studies and Photocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 17.7 Applications in Unimolecular Liquid Phase Reaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 17.8 Applications in Bioelectronics, Aqueous and Prebiotics Reaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 17.9 Applications in Biophysics and Gas Phase Biomolecules . . . . . 476 17.10 Ultrafast Imaging of Unimolecular Gas-Phase Reactions . . . . . 480 17.11 Applications in Nanoscience and Multiphoton-Ionisation . . . . . 481 17.12 Applications in Unimolecular Gas Phase Dynamics . . . . . . . . . 481 17.13 Outlook and Conclusion: First High-Repetition Frequency, Ultrafast Hard and Soft X-ray Studies of Chemical Reactions at the European X-ray Free Electron Laser . . . . . . . . . . . . . . . 484 17.14 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 18 Polarization-Sensitive Coherent Diffractive Imaging Using HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Sergey Zayko, Ofer K fi r and Claus Ropers 18.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 18.2 Phase Retrieval of Experimental Data . . . . . . . . . . . . . . . . . . . 503 18.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 18.4 Polarization Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 18.5 Magneto-Optical Imaging Using High-Harmonic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 18.6 Dichroic Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 18.7 Signal Enhancement Mechanism . . . . . . . . . . . . . . . . . . . . . . 518 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 19 Nonlinear Light Generation in Localized Fields Using Gases and Tailored Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Murat Sivis and Claus Ropers 19.1 Plasmonic Enhancement for EUV Light Generation . . . . . . . . 523 19.2 High-Harmonic Generation and Imaging in Tailored Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 20 Wavefront and Coherence Characteristics of Extreme UV and Soft X-ray Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Bernd Sch ä fer, Bernhard Fl ö ter, Tobias Mey and Klaus Mann 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 20.2 Wavefront Metrology and Beam Characterization with Hartmann Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 20.2.1 Hartmann Wavefront Sensing . . . . . . . . . . . . . . . . . . 532 Contents xv 20.2.2 EUV Wavefront Sensor for FEL Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 20.2.3 Beam Characterization of High-Harmonic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 20.2.4 Thermal Lensing of X-ray Optics . . . . . . . . . . . . . . . 539 20.3 Wigner Distribution for Diagnostics of Spatial Coherence . . . . 540 20.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 20.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 542 20.3.3 4D Wigner Measurements . . . . . . . . . . . . . . . . . . . . . 543 20.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 21 Laboratory-Scale Soft X-ray Source for Microscopy and Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Matthias M ü ller and Klaus Mann 21.1 Table-Top Soft X-ray Source Using a Pulsed Gas Jet . . . . . . . 549 21.2 Soft X-ray Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 21.3 X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 553 21.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 22 Multilayer Zone Plates for Hard X-ray Imaging . . . . . . . . . . . . . . . 561 Markus Osterhoff and Hans-Ulrich Krebs 22.1 From Focusing to Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 22.2 Let There be an Ideal World . . . . . . . . . . . . . . . . . . . . . . . . . 563 22.3 Back to the Real World: Fabrication Challenges . . . . . . . . . . . 565 22.3.1 Pulsed Laser Deposition . . . . . . . . . . . . . . . . . . . . . . 565 22.3.2 FIB Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 22.3.3 From MLL to MZP . . . . . . . . . . . . . . . . . . . . . . . . . 566 22.3.4 Material and Parameter Studies . . . . . . . . . . . . . . . . . 566 22.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 22.4 The World of Synchrotron Instrumentation . . . . . . . . . . . . . . . 569 22.4.1 Hard X-rays Near 14 keV . . . . . . . . . . . . . . . . . . . . . 569 22.4.2 High Energies: From 60 to 101 keV . . . . . . . . . . . . . 570 22.4.3 Sampler Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 22.4.4 Improvements of the GINIX Setup . . . . . . . . . . . . . . 572 22.5 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 22.5.1 Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 22.5.2 Holography and Scanning SAXS . . . . . . . . . . . . . . . . 574 22.5.3 Scanning WAXS . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 22.5.4 Correlative Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 22.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 xvi Contents 23 Convergence Analysis of Iterative Algorithms for Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 D. Russell Luke and Anna-Lena Martins 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 23.2 Phase Retrieval as a Feasibility Problem . . . . . . . . . . . . . . . . . 584 23.3 Notation and Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 585 23.3.1 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 23.3.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 23.3.3 Fixed Points and Regularities of Mappings . . . . . . . . 587 23.4 A Toolkit for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 589 23.5 Regularities of Sets and Their Collection . . . . . . . . . . . . . . . . 591 23.6 Analysis of Cyclic Projections . . . . . . . . . . . . . . . . . . . . . . . . 593 23.7 Application to Phase Retrieval Algorithms . . . . . . . . . . . . . . . 596 23.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 24 One-Dimensional Discrete-Time Phase Retrieval . . . . . . . . . . . . . . . 603 Robert Beinert and Gerlind Plonka 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 24.2 The Discrete-Time Phase Retrieval Problem . . . . . . . . . . . . . . 606 24.3 Trivial and Non-trivial Ambiguities . . . . . . . . . . . . . . . . . . . . 607 24.4 Non-negative Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 24.5 Additional Data in Time-Domain . . . . . . . . . . . . . . . . . . . . . . 614 24.5.1 Using an Additional Signal Value . . . . . . . . . . . . . . . 614 24.5.2 Using Additional Magnitude Values of the Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 24.6 Interference Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 24.6.1 Interference with a Known Reference Signal . . . . . . . 618 24.6.2 Interference with an Unknown Reference Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 24.6.3 Interference with the Modulated Signal . . . . . . . . . . . 622 24.7 Linear Canonical Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . 623 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Contents xvii Contributors Sadia Bari FS-Strukturdynamik (bio)chemischer Systeme, Deutsches Elektronen- Synchrotron DESY, Hamburg, Germany Robert Beinert Institute for Mathematics and Scienti fi c Computing, University of Graz, Graz, Austria C. Charitha Indi