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 Councilof 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 ofwhich presents a comprehensive survey of a selected topic within the area ofapplied physics. Edited and written by leading research scientists in the fieldconcerned, each volume contains review contributions covering the various aspectsof the topic. Together these provide an overview of the state of the art in therespective field, extending from an introduction to the subject right up to thefrontiers of contemporary research.Topics in Applied Physics is addressed to all scientists at universities and inindustry who wish to obtain an overview and to keep abreast of advances in appliedphysics. The series also provides easy but comprehensive access to the fields fornewcomers starting research.

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Tim Salditt • Alexander Egner • D. Russell LukeEditors

Nanoscale Photonic Imaging

EditorsTim SaldittInstitut für RöntgenphysikUniversität GöttingenGöttingen, Germany

Alexander EgnerLaser LaboratoriumUniversity of GöttingenGöttingen, Germany

D. Russell LukeInstitut für Numerischeund Angewandte MathematikUniversität GöttingenGöttingen, Germany

ISSN 0303-4216 ISSN 1437-0859 (electronic)Topics in Applied PhysicsISBN 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 4.0International 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 tothe original author(s) and the source, provide a link to the Creative Commons license and indicate ifchanges were made.The images or other third party material in this book are included in the book’s Creative Commonslicense, unless indicated otherwise in a credit line to the material. If material is not included in the book’sCreative Commons license and your intended use is not permitted by statutory regulation or exceeds thepermitted 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 specific statement, that such names are exempt from therelevant 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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Preface

The word ‘Nano’ has been around for a long time. It became a topic of significantinterest in the eighties of the last century, after instruments such as the scanningtunneling microscope and the atomic force microscope had been invented. The‘nanoscale’ was probed based on electric currents through a tunneling tip or bymeasuring the forces with a cantilever. In other words, the ‘room at the bottom’ wasconquered not by ‘seeing’, but rather by ‘feeling’. Too strong was the belief thatoptical imaging was limited to the microscale due to the diffraction barrier. But theinsight that photonics and nanoscale also make a perfect match followed onlyshortly 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,fluorescence correlation spectroscopy was emerging, and above all there was a newway to turn microscopes into nanoscopes based on optical switching, as pioneeredby Stefan Hell here in Göttingen. While very few physicists cared about opticalmicroscopes 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 coherentoptics and computational phase retrieval.

Pairing up optical and short wavelength to extend the scales of ‘imaging’,research teams in Göttingen set out for new discoveries. But how to empower theirvessels? The solution was found by mathematics. Using results from inverseproblems, stochastics, and optimization theory, new and bountiful shores werediscovered, and photonic data was turned into useful information….

As we now come back from our expeditions funded for the last 12 years by theGerman Science Foundation (DFG) through SFB755 Nanoscale Photonic Imaging,we do not want to keep all the treasures for ourselves. The current book is acompilation 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 projecton 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 beenessential to keeping the expedition on track, on budget and on time—all with graceand joyful optimism.

Now, let us dive deep into the nanoscale, and not just scratch at its surface!

Göttingen, Germany Tim SaldittAlexander EgnerD. Russell Luke

vi Preface

Contents

Part I Fundamentals and Tutorials

1 STED Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Alexander Egner, Claudia Geisler and René Siegmund1.1 Fundamentals of Fluorescence Microscopy . . . . . . . . . . . . . . . 3

1.1.1 Vectorial Diffraction Theory and IntensityDistribution Within the Focal Spot . . . . . . . . . . . . . . 3

1.1.2 Incoherent Image Formation . . . . . . . . . . . . . . . . . . . 91.1.3 Classical Resolution Limit . . . . . . . . . . . . . . . . . . . . . 111.1.4 Confocal Microscopy . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Fundamentals of STED Microscopy . . . . . . . . . . . . . . . . . . . . 161.2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Basic Photophysics of Dye Molecules . . . . . . . . . . . . 181.2.3 Shaping the STED Beam . . . . . . . . . . . . . . . . . . . . . 231.2.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Imaging Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Coherent X-ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Tim Salditt and Anna-Lena Robisch2.1 X-ray Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Scalar Diffraction Theory and Wave Equations . . . . . . 362.1.2 Propagation in Free Space . . . . . . . . . . . . . . . . . . . . . 412.1.3 The Fresnel Scaling Theorem . . . . . . . . . . . . . . . . . . 442.1.4 Numerical Implementation of Free-Space

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1.5 X-ray Propagation in Matter . . . . . . . . . . . . . . . . . . . 482.1.6 Propagation by Finite Difference Equations . . . . . . . . 51

vii

2.2 Coherent Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.1 Holographic Imaging in Full Field Setting . . . . . . . . . 552.2.2 Contrast in X-ray Holograms . . . . . . . . . . . . . . . . . . . 57

2.3 Solving the Phase Problem in the Holographic Regime . . . . . . 592.3.1 Single-Step Phase Retrieval . . . . . . . . . . . . . . . . . . . . 602.3.2 Iterative Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . 60

2.4 From Two to Three Dimensions: Tomography and PhaseRetrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 X-ray Focusing and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Tim Salditt and Markus Osterhoff3.1 General Aspects of X-ray Optics and Focusing . . . . . . . . . . . . 713.2 X-ray Reflectivity and Reflective X-ray Optics . . . . . . . . . . . . 74

3.2.1 X-ray Reflectivity of an Ideal Single Interface . . . . . . 743.2.2 Multiple Interfaces and Multilayers . . . . . . . . . . . . . . 773.2.3 Interfacial Roughness . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3 X-ray Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3.1 Kirkpatrick-Baez Geometry . . . . . . . . . . . . . . . . . . . . 833.3.2 Multilayer Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4 X-ray Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.4.1 Waveguide Modes: The Basics . . . . . . . . . . . . . . . . . 893.4.2 Coupling and Propagation . . . . . . . . . . . . . . . . . . . . . 953.4.3 Fabrication and Characterisation of X-ray

Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.4.4 Advanced Waveguide Configurations . . . . . . . . . . . . . 99

3.5 Diffractive Optics and Zone Plates . . . . . . . . . . . . . . . . . . . . . 1023.5.1 Basic Theory of Fresnel Zone Plates . . . . . . . . . . . . . 1023.5.2 Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . 1053.5.3 Diffractive Optics Beyond the Projection

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.6 Basic Coherence Theory and Simulations for X-ray Optics . . . 109

3.6.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.6.2 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.6.3 Coherence Propagation and Filtering . . . . . . . . . . . . . 113

3.7 Putting It All Together: Optics and X-rayInstrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4 Statistical Foundations of Nanoscale Photonic Imaging . . . . . . . . . . 125Axel Munk, Thomas Staudt and Frank Werner4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.1.1 Background and Examples . . . . . . . . . . . . . . . . . . . . 1254.1.2 Purpose of the Chapter . . . . . . . . . . . . . . . . . . . . . . . 126

viii Contents

4.1.3 Measurement Devices . . . . . . . . . . . . . . . . . . . . . . . . 1274.1.4 Structure and Notation . . . . . . . . . . . . . . . . . . . . . . . 128

4.2 Poisson Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3 Bernoulli Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.1 Law of Small Numbers . . . . . . . . . . . . . . . . . . . . . . . 1334.4 Gaussian Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.4.1 As Approximation of the Binomial Model . . . . . . . . . 1354.4.2 As Approximation of the Poisson Model . . . . . . . . . . 1364.4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.4.4 Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.4.5 Variance Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 138

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Appendix: Poisson Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Appendix: Conditioned Poisson Processes . . . . . . . . . . . . . . . . . . . . . 141References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Thorsten Hohage, Benjamin Sprung and Frederic Weidling5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.1.1 What Is an Inverse Problem? . . . . . . . . . . . . . . . . . . . 1455.1.2 Ill-Posedness and Regularization . . . . . . . . . . . . . . . . 1465.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.1.4 Choice of Regularization Parameters

and Convergence Concepts . . . . . . . . . . . . . . . . . . . . 1485.2 Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.2.1 Variational Regularization . . . . . . . . . . . . . . . . . . . . . 1505.2.2 Iterative Regularization . . . . . . . . . . . . . . . . . . . . . . . 154

5.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3.1 General Error Bounds for Variational

Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3.2 Interpretation of Variational Source Conditions . . . . . 1585.3.3 Error Bounds for Poisson Data . . . . . . . . . . . . . . . . . 162

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6 Proximal Methods for Image Processing . . . . . . . . . . . . . . . . . . . . . 165D. Russell Luke6.1 All Together Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.1.1 What Seems to Be the Problem Here? . . . . . . . . . . . . 1666.1.2 What Is an Algorithm? . . . . . . . . . . . . . . . . . . . . . . . 1696.1.3 What Is a Proximal Method? . . . . . . . . . . . . . . . . . . . 1766.1.4 On Your Mark. Get Set... . . . . . . . . . . . . . . . . . . . . . 177

6.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.2.1 Model Category I: Multi-set Feasibility . . . . . . . . . . . 1786.2.2 Model Category II: Product Space Formulations . . . . . 182

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6.2.3 Model Category III: Smooth NonconvexOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.3 ProxToolbox—A Platform for Creative Hacking . . . . . . . . . . . 1926.3.1 Coffee Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.3.2 Star Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1966.3.3 E Pluribus Unum . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.4 Last Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Part II Progress and Perspectives

7 Quantifying Molecule Numbers in STED/RESOLFTFluorescence Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Jan Keller-Findeisen, Steffen J. Sahl and Stefan W. Hell7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.1.1 Molecular Contribution Function (MCF) . . . . . . . . . . 2077.2 STED Nanoscopy with Coincidence Photon Detection . . . . . . 208

7.2.1 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2097.2.2 Intrinsic Molecular Brightness Calibration . . . . . . . . . 2127.2.3 Counting Transferrin Receptors

in HEK293 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.3 Mean and Variance in RESOLFT Nanoscopy . . . . . . . . . . . . . 215

7.3.1 Cumulants of the Fluorescence of SwitchableFluorophores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.3.2 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.3.3 Counting rsEGFP2 Fused a-tubulin Units

in Drosophila Melanogaster . . . . . . . . . . . . . . . . . . . 2207.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8 Metal-Induced Energy Transfer Imaging . . . . . . . . . . . . . . . . . . . . 227Alexey I. Chizhik and Jörg Enderlein8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.2 Basic Principle and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.3 The MIET-GUI Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.4 Metal-Induced Energy Transfer for Biological Imaging . . . . . . 2338.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

9 Reversibly Switchable Fluorescent Proteins for RESOLFTNanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Nickels A. Jensen, Isabelle Jansen, Maria Kamper and Stefan Jakobs9.1 Overcoming the Diffraction Barrier . . . . . . . . . . . . . . . . . . . . 2419.2 RSFPs for Live-Cell RESOLFT Nanoscopy . . . . . . . . . . . . . . 242

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9.3 Photoswitching Mechanisms of RSFPs . . . . . . . . . . . . . . . . . . 2439.3.1 Negative Switching Mode . . . . . . . . . . . . . . . . . . . . . 2459.3.2 Positive Switching Mode . . . . . . . . . . . . . . . . . . . . . 2459.3.3 Decoupled Switching Mode . . . . . . . . . . . . . . . . . . . 246

9.4 RSFP Properties Important for RESOLFT Nanoscopy . . . . . . . 2479.4.1 Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.4.2 Ensemble Switching Speed . . . . . . . . . . . . . . . . . . . . 2489.4.3 Residual Fluorescence in the Off-State . . . . . . . . . . . . 2489.4.4 Switching Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

9.5 Overview of RSFPs for RESOLFT Nanoscopy . . . . . . . . . . . . 2499.5.1 RSFPs Emitting in the Green . . . . . . . . . . . . . . . . . . 2509.5.2 RSFPs Emitting in the Yellow. . . . . . . . . . . . . . . . . . 2529.5.3 RSFPs Emitting in the Red . . . . . . . . . . . . . . . . . . . . 253

9.6 Applications of RESOLFT Nanoscopy . . . . . . . . . . . . . . . . . . 2539.6.1 Other Fluorophores for RESOLFT Nanoscopy . . . . . . 255

9.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10 A Statistical and Biophysical Toolbox to Elucidate Structureand Formation of Stress Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Benjamin Eltzner, Lara Hauke, Stephan Huckemann, Florian Rehfeldtand Carina Wollnik10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.2 Live Cell Imaging-Opportunities and Challenges . . . . . . . . . . . 26610.3 Automated Unbiased Binarization of Filament Structure . . . . . 267

10.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26810.3.2 The FilamentSensor and the Benchmark Dataset . . . . . 26910.3.3 Detecting Slightly Bent Filaments . . . . . . . . . . . . . . . 270

10.4 Orientation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27210.4.1 Orientation Field Evolution . . . . . . . . . . . . . . . . . . . . 27410.4.2 Backward Nested Descriptor Analysis . . . . . . . . . . . . 277

10.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

11 Photonic Imaging with Statistical Guarantees: From MultiscaleTesting to Multiscale Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Axel Munk, Katharina Proksch, Housen Li and Frank Werner11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28411.2 Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . 285

11.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.2.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.2.3 Testing on an Image . . . . . . . . . . . . . . . . . . . . . . . . . 28811.2.4 Testing Multiple Hypotheses . . . . . . . . . . . . . . . . . . . 29111.2.5 Connection to Extreme Value Theory . . . . . . . . . . . . 295

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11.2.6 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.2.7 Theory for the Multiscale Scanning Test . . . . . . . . . . 30211.2.8 Deconvolution and Scanning . . . . . . . . . . . . . . . . . . . 30311.2.9 FDR Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.3 Statistical Multiscale Estimation . . . . . . . . . . . . . . . . . . . . . . . 308References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

12 Efficient, Quantitative Numerical Methods for Statistical ImageDeconvolution and Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313D. Russell Luke, C. Charitha, Ron Shefi and Yura Malitsky12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31312.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

12.2.1 Abstract Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 31812.2.2 Saddle Point and Dual Formulations . . . . . . . . . . . . . 31912.2.3 Statistical Multi-resolution Estimation . . . . . . . . . . . . 321

12.3 Alternating Directions Method of Multipliersand Douglas Rachford . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32212.3.1 ADMM for Statisitcal Multi-resolution Estimation

of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.4 Primal-Dual Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

12.4.1 EPAPC for Statisitcal Multi-resolution Estimationof STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

12.5 Randomized Block-Coordinate Primal-Dual Methods . . . . . . . 33212.5.1 RBPD for Statisitcal Multi-resolution Estimation

of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

13 Holographic Imaging and Tomography of Biological Cells andTissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Tim Salditt and Mareike Töpperwien13.1 Propagation-Based Phase-Contrast Tomography . . . . . . . . . . . 33913.2 Nano-CT Using Synchrotron Radiation: Optics,

Instrumentation and Phase Retrieval . . . . . . . . . . . . . . . . . . . . 34113.2.1 Cone-Beam Holography . . . . . . . . . . . . . . . . . . . . . . 34113.2.2 Waveguide Optics and Imaging . . . . . . . . . . . . . . . . . 34313.2.3 Dose-Resolution Relationship . . . . . . . . . . . . . . . . . . 34513.2.4 Phase Retrieval Algorithms . . . . . . . . . . . . . . . . . . . . 346

13.3 CTF-based Reconstruction and Its Limits . . . . . . . . . . . . . . . . 34713.4 Laboratory µ-CT: Instrumentation and Phase Retrieval . . . . . . 34913.5 Novel Tomography Approaches . . . . . . . . . . . . . . . . . . . . . . . 354

13.5.1 Combined Phase Retrieval and TomographicReconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

13.5.2 Tomographic Reconstruction Basedon the 3D Radon Transform (3DRT) . . . . . . . . . . . . . 356

xii Contents

13.6 Tomography of Biological Tissues: Applicationsand Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35713.6.1 3D Structure of Cochlea . . . . . . . . . . . . . . . . . . . . . . 35813.6.2 Small Animal Imaging . . . . . . . . . . . . . . . . . . . . . . . 36013.6.3 3D Virtual Histology of Nerves . . . . . . . . . . . . . . . . . 36313.6.4 Macrophages in Lung Tissue . . . . . . . . . . . . . . . . . . . 36313.6.5 Neuron Locations in Human Cerebellum . . . . . . . . . . 36513.6.6 Outlook: Time-Resolved Phase-Contrast

Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

14 Constrained Reconstructions in X-ray Phase Contrast Imaging:Uniqueness, Stability and Algorithms . . . . . . . . . . . . . . . . . . . . . . . 377Simon Maretzke and Thorsten Hohage14.1 Forward Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

14.1.1 Physical Model and Preliminaries . . . . . . . . . . . . . . . 37814.1.2 Forward Operators for XPCI . . . . . . . . . . . . . . . . . . . 38014.1.3 Forward Operators for XPCT . . . . . . . . . . . . . . . . . . 382

14.2 Uniqueness Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38314.2.1 Preliminary Results and Counter-Examples . . . . . . . . 38314.2.2 Sources of Non-uniqueness—The Phase Problem . . . . 38414.2.3 Relation to Classical Phase Retrieval Problems . . . . . . 38414.2.4 Holographic Nature of Phase Retrieval in XPCI . . . . . 38514.2.5 General Uniqueness Under Support Constraints . . . . . 386

14.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38714.3.1 Lipschitz-Stability and its Meaning . . . . . . . . . . . . . . 38714.3.2 Stability for General Objects and one Hologram . . . . . 38814.3.3 Homogeneous Objects and Multiple Holograms . . . . . 39114.3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

14.4 Regularized Newton Methods for XPCI . . . . . . . . . . . . . . . . . 39414.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39414.4.2 Reconstruction Method . . . . . . . . . . . . . . . . . . . . . . . 39514.4.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . . . . 396

14.5 Regularized Newton-Kaczmarz-SART for XPCT . . . . . . . . . . 39614.5.1 Efficient Computation by Generalized SART . . . . . . . 39814.5.2 Parallelization and Large-Scale Implementation . . . . . 39914.5.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . . . . 400

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

15 Scanning Small-Angle X-ray Scattering and Coherent X-rayImaging of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405Tim Salditt and Sarah Köster15.1 X-ray Structure Analysis of Biological Cells:

A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Contents xiii

15.2 Methods: X-ray Optics and Sample Environment . . . . . . . . . . 40815.2.1 Focusing Optics and Imaging Modalities . . . . . . . . . . 40815.2.2 X-ray Compatible Microfluidic Sample

Environments for Cells . . . . . . . . . . . . . . . . . . . . . . . 40915.3 Scanning Small-Angle X-ray Scattering of Cells . . . . . . . . . . . 41315.4 Coherent X-ray Imaging of Cells . . . . . . . . . . . . . . . . . . . . . . 418

15.4.1 Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41815.4.2 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

15.5 Correlative Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42215.6 From Cells to Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42415.7 Outlook: FEL Studies of Cells . . . . . . . . . . . . . . . . . . . . . . . . 425References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

16 Single Particle Imaging with FEL Using Photon Correlations . . . . 435Benjamin von Ardenne and Helmut Grubmüller16.1 The Single Molecule Scattering Experiment . . . . . . . . . . . . . . 43616.2 Structure Determination Using Few Photons . . . . . . . . . . . . . . 437

16.2.1 Theoretical Background on Three-PhotonCorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

16.2.2 Bayesian Structure Determination . . . . . . . . . . . . . . . 44116.2.3 Reduction of Search Space Using Two-Photon

Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44216.2.4 Optimizing the Probability Using Monte Carlo . . . . . . 443

16.3 Method Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44416.3.1 Resolution Scaling with Photon Counts . . . . . . . . . . . 44516.3.2 Impact of the Photon Counts per Image . . . . . . . . . . . 44816.3.3 Structure Results in the Presence of Non-Poissonian

Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44916.4 Structure Determination from Multi-Particle Images . . . . . . . . 451References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

17 Development of Ultrafast X-ray Free Electron Laser Tools in(Bio)Chemical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Simone Techert, Sreevidya Thekku Veedu and Sadia Bari17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45717.2 The Concept: Filming Chemical Reactions in Real Time

Utilizing Ultrafast High-Flux X-ray Sources . . . . . . . . . . . . . . 46117.3 X-ray Diffraction and Crystallography for Condensed State

Chemistry Studies—Crystallography with UltrahighTemporal and Ultrahigh Spatial Resolution . . . . . . . . . . . . . . . 462

17.4 Applications in Energy Research . . . . . . . . . . . . . . . . . . . . . . 46617.5 From Local to Global: Ultrafast Multidimensional Soft

X-ray Spectroscopy and Ultrafast X-ray DiffractionShake Their Hands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

xiv Contents

17.6 Applications in Bimolecular Reaction Studies andPhotocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

17.7 Applications in Unimolecular Liquid Phase ReactionDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

17.8 Applications in Bioelectronics, Aqueous and PrebioticsReaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

17.9 Applications in Biophysics and Gas Phase Biomolecules . . . . . 47617.10 Ultrafast Imaging of Unimolecular Gas-Phase Reactions . . . . . 48017.11 Applications in Nanoscience and Multiphoton-Ionisation . . . . . 48117.12 Applications in Unimolecular Gas Phase Dynamics . . . . . . . . . 48117.13 Outlook and Conclusion: First High-Repetition Frequency,

Ultrafast Hard and Soft X-ray Studies of Chemical Reactionsat the European X-ray Free Electron Laser . . . . . . . . . . . . . . . 484

17.14 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

18 Polarization-Sensitive Coherent Diffractive ImagingUsing HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Sergey Zayko, Ofer Kfir and Claus Ropers18.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50118.2 Phase Retrieval of Experimental Data . . . . . . . . . . . . . . . . . . . 50318.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50718.4 Polarization Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51118.5 Magneto-Optical Imaging Using High-Harmonic

Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51318.6 Dichroic Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51718.7 Signal Enhancement Mechanism . . . . . . . . . . . . . . . . . . . . . . 518References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

19 Nonlinear Light Generation in Localized Fields Using Gasesand Tailored Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523Murat Sivis and Claus Ropers19.1 Plasmonic Enhancement for EUV Light Generation . . . . . . . . 52319.2 High-Harmonic Generation and Imaging in Tailored

Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

20 Wavefront and Coherence Characteristics of Extreme UVand Soft X-ray Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531Bernd Schäfer, Bernhard Flöter, Tobias Mey and Klaus Mann20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53120.2 Wavefront Metrology and Beam Characterization

with Hartmann Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53220.2.1 Hartmann Wavefront Sensing . . . . . . . . . . . . . . . . . . 532

Contents xv

20.2.2 EUV Wavefront Sensor for FELCharacterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

20.2.3 Beam Characterization of High-HarmonicSources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

20.2.4 Thermal Lensing of X-ray Optics . . . . . . . . . . . . . . . 53920.3 Wigner Distribution for Diagnostics of Spatial Coherence . . . . 540

20.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54020.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 54220.3.3 4D Wigner Measurements . . . . . . . . . . . . . . . . . . . . . 543

20.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

21 Laboratory-Scale Soft X-ray Source for Microscopyand Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549Matthias Müller and Klaus Mann21.1 Table-Top Soft X-ray Source Using a Pulsed Gas Jet . . . . . . . 54921.2 Soft X-ray Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55221.3 X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 55321.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

22 Multilayer Zone Plates for Hard X-ray Imaging . . . . . . . . . . . . . . . 561Markus Osterhoff and Hans-Ulrich Krebs22.1 From Focusing to Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 56122.2 Let There be an Ideal World . . . . . . . . . . . . . . . . . . . . . . . . . 56322.3 Back to the Real World: Fabrication Challenges . . . . . . . . . . . 565

22.3.1 Pulsed Laser Deposition . . . . . . . . . . . . . . . . . . . . . . 56522.3.2 FIB Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56522.3.3 From MLL to MZP . . . . . . . . . . . . . . . . . . . . . . . . . 56622.3.4 Material and Parameter Studies . . . . . . . . . . . . . . . . . 56622.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

22.4 The World of Synchrotron Instrumentation . . . . . . . . . . . . . . . 56922.4.1 Hard X-rays Near 14 keV . . . . . . . . . . . . . . . . . . . . . 56922.4.2 High Energies: From 60 to 101 keV . . . . . . . . . . . . . 57022.4.3 Sampler Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57022.4.4 Improvements of the GINIX Setup . . . . . . . . . . . . . . 572

22.5 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57322.5.1 Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57322.5.2 Holography and Scanning SAXS . . . . . . . . . . . . . . . . 57422.5.3 Scanning WAXS . . . . . . . . . . . . . . . . . . . . . . . . . . . 57622.5.4 Correlative Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

22.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

xvi Contents

23 Convergence Analysis of Iterative Algorithms for PhaseRetrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583D. Russell Luke and Anna-Lena Martins23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58323.2 Phase Retrieval as a Feasibility Problem . . . . . . . . . . . . . . . . . 58423.3 Notation and Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 585

23.3.1 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58623.3.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58723.3.3 Fixed Points and Regularities of Mappings . . . . . . . . 587

23.4 A Toolkit for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 58923.5 Regularities of Sets and Their Collection . . . . . . . . . . . . . . . . 59123.6 Analysis of Cyclic Projections . . . . . . . . . . . . . . . . . . . . . . . . 59323.7 Application to Phase Retrieval Algorithms . . . . . . . . . . . . . . . 59623.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

24 One-Dimensional Discrete-Time Phase Retrieval . . . . . . . . . . . . . . . 603Robert Beinert and Gerlind Plonka24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60424.2 The Discrete-Time Phase Retrieval Problem . . . . . . . . . . . . . . 60624.3 Trivial and Non-trivial Ambiguities . . . . . . . . . . . . . . . . . . . . 60724.4 Non-negative Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61124.5 Additional Data in Time-Domain . . . . . . . . . . . . . . . . . . . . . . 614

24.5.1 Using an Additional Signal Value . . . . . . . . . . . . . . . 61424.5.2 Using Additional Magnitude Values

of the Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61624.6 Interference Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

24.6.1 Interference with a Known Reference Signal . . . . . . . 61824.6.2 Interference with an Unknown Reference

Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62024.6.3 Interference with the Modulated Signal . . . . . . . . . . . 622

24.7 Linear Canonical Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . 623References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Scientific Computing, University ofGraz, Graz, Austria

C. Charitha Indian Institute of Technology Indore, Indore, India

Alexey I. Chizhik Third Institute of Physics - Biophysics, Universität Göttingen,Göttingen, Germany

Alexander Egner Laser-Laboratorium Göttingen, Göttingen, Germany

Benjamin Eltzner Felix-Bernstein-Institute for Mathematical Statistics in theBiosciences, Universität Göttingen, Göttingen, Germany

Jörg Enderlein Third Institute of Physics - Biophysics, Universität Göttingen,Göttingen, Germany

Bernhard Flöter Laser-Laboratorium Göttingen eV., Göttingen, Germany

Claudia Geisler Laser-Laboratorium Göttingen, Göttingen, Germany

Helmut Grubmüller Department of Theoretical and Computational Biophysics,Max Planck Institute for Biophysical Chemistry Göttingen, Göttingen, Germany

Lara Hauke Third Institut of Physics - Biophysics, Universität Göttingen,Göttingen, Germany

Stefan W. Hell Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany;Department of Optical Nanoscopy, Max Planck Institute for Medical Research,Heidelberg, Germany

xix

Thorsten Hohage Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Stephan Huckemann Felix-Bernstein-Institute for Mathematical Statistics in theBiosciences, Universität Göttingen, Göttingen, Germany

Stefan Jakobs Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Isabelle Jansen Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Nickels A. Jensen Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Maria Kamper Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Jan Keller-Findeisen Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Ofer Kfir IV. Physical Institute - Solids and Nanostructures, UniversitätGöttingen, Göttingen, Germany

Sarah Köster Institute for X-ray Physics, Universität Göttingen, Göttingen,Germany

Hans-Ulrich Krebs Institute for Material Physics, Universität Göttingen,Göttingen, Germany

Housen Li Institute for Mathematical Stochastics, Universität Göttingen,Göttingen, Germany

D. Russell Luke Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Yura Malitsky Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Klaus Mann Laser-Laboratorium Göttingen e.V., Göttingen, Germany

Simon Maretzke Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Anna-Lena Martins Institute for Numerical and Applied Mathematics,Universität Göttingen, Göttingen, Germany

Tobias Mey Laser-Laboratorium Göttingen eV., Göttingen, Germany

Matthias Müller Laser-Laboratorium Göttingen e.V., Göttingen, Germany

xx Contributors

Axel Munk Institute for Mathematical Stochastics, Universität Göttingen,Göttingen, Germany;Max Planck Institute for Biophysical Chemistry, Göttingen, Germany

Markus Osterhoff Institute for X-ray Physics, Universität Göttingen, Göttingen,Germany

Gerlind Plonka Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Katharina Proksch Institute for Mathematical Stochastics, Universität Göttingen,Göttingen, Germany

Florian Rehfeldt Third Institut of Physics - Biophysics, Universität Göttingen,Göttingen, Germany

Anna-Lena Robisch Institute for X-ray Physics, Universität Göttingen,Göttingen, Germany

Claus Ropers IV. Physical Institute - Solids and Nanostructures, UniversitätGöttingen, Göttingen, Germany

Steffen J. Sahl Department of NanoBiophotonics, Max Planck Institute forBiophysical Chemistry, Göttingen, Germany

Tim Salditt Institute for X-ray Physics, Universität Göttingen, Göttingen,Germany

Bernd Schäfer Laser-Laboratorium Göttingen eV., Göttingen, Germany

Ron Shefi Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

René Siegmund Laser-Laboratorium Göttingen, Göttingen, Germany

Murat Sivis IV. Physical Institute - Solids and Nanostructures, UniversitätGöttingen, Göttingen, Germany

Benjamin Sprung Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Thomas Staudt Institute for Mathematical Stochastics, Universität Göttingen,Göttingen, Germany

Simone Techert FS-Strukturdynamik (bio)chemischer Systeme, DeutschesElektronen-Synchrotron DESY, Hamburg, Germany

Sreevidya Thekku Veedu FS-Strukturdynamik (bio)chemischer Systeme,Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

Mareike Töpperwien Institute for X-ray Physics, Universität Göttingen,Göttingen, Germany

Contributors xxi

Benjamin von Ardenne Department of Theoretical and ComputationalBiophysics, Max Planck Institute for Biophysical Chemistry Göttingen, Göttingen,Germany

Frederic Weidling Institute for Numerical and Applied Mathematics, UniversitätGöttingen, Göttingen, Germany

Frank Werner Institute for Mathematical Stochastics, Universität Göttingen,Göttingen, Germany;Max Planck Institute for Biophysical Chemistry, Göttingen, Germany

Carina Wollnik Third Institut of Physics - Biophysics, Universität Göttingen,Göttingen, Germany

Sergey Zayko IV. Physical Institute - Solids and Nanostructures, UniversitätGöttingen, Göttingen, Germany

xxii Contributors