Topics in Applied Physics 134

Tim Salditt
Alexander Egner
D. Russell Luke   Editors

Nanoscale
Photonic
Imaging
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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,
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Tim Salditt Alexander Egner D. Russell Luke
•

•

Editors

Nanoscale Photonic Imaging



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Editors
Tim Salditt
Institut für Röntgenphysik
Universität Göttingen
Göttingen, Germany

Alexander Egner
Laser Laboratorium
University of Göttingen
Göttingen, Germany

D. Russell Luke
Institut für Numerische
und Angewandte Mathematik
Universität Göttingen
Göttingen, 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)
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Preface

The word ‘Nano’ has been around for a long time. It became a topic of significant
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,
fluorescence 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 Göttingen. 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 Göttingen 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 further 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


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vi

Preface

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!
Göttingen, Germany

Tim Salditt
Alexander Egner
D. Russell Luke

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Contents

Part I
1

2

Fundamentals and Tutorials

STED Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alexander Egner, Claudia Geisler and René Siegmund
1.1
Fundamentals of Fluorescence Microscopy . . . . . .
1.1.1 Vectorial Diffraction Theory and Intensity
Distribution Within the Focal Spot . . . . .
1.1.2 Incoherent Image Formation . . . . . . . . . .
1.1.3 Classical Resolution Limit . . . . . . . . . . . .
1.1.4 Confocal Microscopy . . . . . . . . . . . . . . .

1.2
Fundamentals of STED Microscopy . . . . . . . . . . .
1.2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Basic Photophysics of Dye Molecules . . .
1.2.3 Shaping the STED Beam . . . . . . . . . . . .
1.2.4 Resolution . . . . . . . . . . . . . . . . . . . . . . .
1.3
Imaging Examples . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Coherent X-ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tim Salditt and Anna-Lena Robisch
2.1
X-ray Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Scalar Diffraction Theory and Wave Equations .
2.1.2 Propagation in Free Space . . . . . . . . . . . . . . . .
2.1.3 The Fresnel Scaling Theorem . . . . . . . . . . . . .
2.1.4 Numerical Implementation of Free-Space
Propagation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 X-ray Propagation in Matter . . . . . . . . . . . . . .
2.1.6 Propagation by Finite Difference Equations . . .

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2.2

Coherent Image Formation . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Holographic Imaging in Full Field Setting . . . . .
2.2.2 Contrast in X-ray Holograms . . . . . . . . . . . . . . .
2.3
Solving the Phase Problem in the Holographic Regime . .
2.3.1 Single-Step Phase Retrieval . . . . . . . . . . . . . . . .
2.3.2 Iterative Phase Retrieval . . . . . . . . . . . . . . . . . .
2.4

From Two to Three Dimensions: Tomography and Phase
Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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X-ray Focusing and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tim Salditt and Markus Osterhoff
3.1
General Aspects of X-ray Optics and Focusing . . . . . . . . .
3.2
X-ray Reflectivity and Reflective X-ray Optics . . . . . . . . .
3.2.1 X-ray Reflectivity of an Ideal Single Interface . . .
3.2.2 Multiple Interfaces and Multilayers . . . . . . . . . . .
3.2.3 Interfacial Roughness . . . . . . . . . . . . . . . . . . . . .
3.3
X-ray Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Kirkpatrick-Baez Geometry . . . . . . . . . . . . . . . . .
3.3.2 Multilayer Mirrors . . . . . . . . . . . . . . . . . . . . . . .
3.4
X-ray Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Waveguide Modes: The Basics . . . . . . . . . . . . . .
3.4.2 Coupling and Propagation . . . . . . . . . . . . . . . . . .
3.4.3 Fabrication and Characterisation of X-ray
Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Advanced Waveguide Configurations . . . . . . . . . .
3.5
Diffractive Optics and Zone Plates . . . . . . . . . . . . . . . . . .
3.5.1 Basic Theory of Fresnel Zone Plates . . . . . . . . . .
3.5.2 Fabrication Techniques . . . . . . . . . . . . . . . . . . . .
3.5.3 Diffractive Optics Beyond the Projection

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Basic Coherence Theory and Simulations for X-ray Optics
3.6.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Coherence Propagation and Filtering . . . . . . . . . .
3.7
Putting It All Together: Optics and X-ray
Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Foundations of Nanoscale Photonic Imaging .
Axel Munk, Thomas Staudt and Frank Werner
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Background and Examples . . . . . . . . . . .
4.1.2 Purpose of the Chapter . . . . . . . . . . . . . .

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4.1.3 Measurement Devices . . . . . . . . . . . . . . .
4.1.4 Structure and Notation . . . . . . . . . . . . . .
4.2
Poisson Modeling . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Bernoulli Modeling . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Law of Small Numbers . . . . . . . . . . . . . .
4.4
Gaussian Modeling . . . . . . . . . . . . . . . . . . . . . . .

4.4.1 As Approximation of the Binomial Model
4.4.2 As Approximation of the Poisson Model .
4.4.3 Comparison . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Thinning . . . . . . . . . . . . . . . . . . . . . . . .
4.4.5 Variance Stabilization . . . . . . . . . . . . . . .
4.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix: Poisson Thinning . . . . . . . . . . . . . . . . . . . . . .
Appendix: Conditioned Poisson Processes . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5

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Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thorsten Hohage, Benjamin Sprung and Frederic Weidling
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 What Is an Inverse Problem? . . . . . . . . . . . . . .
5.1.2 Ill-Posedness and Regularization . . . . . . . . . . .
5.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Choice of Regularization Parameters
and Convergence Concepts . . . . . . . . . . . . . . .
5.2

Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Variational Regularization . . . . . . . . . . . . . . . .
5.2.2 Iterative Regularization . . . . . . . . . . . . . . . . . .
5.3
Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 General Error Bounds for Variational
Regularization . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Interpretation of Variational Source Conditions
5.3.3 Error Bounds for Poisson Data . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Proximal Methods for Image Processing . . . . . . . . . . . . . . . . .
D. Russell Luke
6.1
All Together Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 What Seems to Be the Problem Here? . . . . . . . .
6.1.2 What Is an Algorithm? . . . . . . . . . . . . . . . . . . .

6.1.3 What Is a Proximal Method? . . . . . . . . . . . . . . .
6.1.4 On Your Mark. Get Set... . . . . . . . . . . . . . . . . .
6.2
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Model Category I: Multi-set Feasibility . . . . . . .
6.2.2 Model Category II: Product Space Formulations .

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6.2.3


Model Category III: Smooth Nonconvex
Optimization . . . . . . . . . . . . . . . . . . . .
6.3
ProxToolbox—A Platform for Creative Hacking .
6.3.1 Coffee Break . . . . . . . . . . . . . . . . . . . .
6.3.2 Star Power . . . . . . . . . . . . . . . . . . . . . .
6.3.3 E Pluribus Unum . . . . . . . . . . . . . . . . .
6.4
Last Word . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part II
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Progress and Perspectives

Quantifying Molecule Numbers in STED/RESOLFT
Fluorescence Nanoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jan Keller-Findeisen, Steffen J. Sahl and Stefan W. Hell
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Molecular Contribution Function (MCF) . . . .
7.2
STED Nanoscopy with Coincidence Photon Detection
7.2.1 Statistical Model . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Intrinsic Molecular Brightness Calibration . . .
7.2.3 Counting Transferrin Receptors
in HEK293 Cells . . . . . . . . . . . . . . . . . . . . .

7.3
Mean and Variance in RESOLFT Nanoscopy . . . . . . .
7.3.1 Cumulants of the Fluorescence of Switchable
Fluorophores . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Statistical Model . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Counting rsEGFP2 Fused a-tubulin Units
in Drosophila Melanogaster . . . . . . . . . . . . .
7.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Metal-Induced Energy Transfer Imaging . . . . . . . . . . . . . .
Alexey I. Chizhik and Jörg Enderlein
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Basic Principle and Theory . . . . . . . . . . . . . . . . . . . .
8.3
The MIET-GUI Software . . . . . . . . . . . . . . . . . . . . . .
8.4
Metal-Induced Energy Transfer for Biological Imaging
8.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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

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9.3

Photoswitching Mechanisms of RSFPs . . . . . . . . . . . .
9.3.1 Negative Switching Mode . . . . . . . . . . . . . . .
9.3.2 Positive Switching Mode . . . . . . . . . . . . . . .
9.3.3 Decoupled Switching Mode . . . . . . . . . . . . .
9.4
RSFP Properties Important for RESOLFT Nanoscopy .
9.4.1 Brightness . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Ensemble Switching Speed . . . . . . . . . . . . . .
9.4.3 Residual Fluorescence in the Off-State . . . . . .
9.4.4 Switching Fatigue . . . . . . . . . . . . . . . . . . . . .

9.5
Overview of RSFPs for RESOLFT Nanoscopy . . . . . .
9.5.1 RSFPs Emitting in the Green . . . . . . . . . . . .
9.5.2 RSFPs Emitting in the Yellow . . . . . . . . . . . .
9.5.3 RSFPs Emitting in the Red . . . . . . . . . . . . . .
9.6
Applications of RESOLFT Nanoscopy . . . . . . . . . . . .
9.6.1 Other Fluorophores for RESOLFT Nanoscopy
9.7
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 A Statistical and Biophysical Toolbox to Elucidate Structure
and Formation of Stress Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . .
Benjamin Eltzner, Lara Hauke, Stephan Huckemann, Florian Rehfeldt
and Carina Wollnik
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Live Cell Imaging-Opportunities and Challenges . . . . . . . . . .
10.3 Automated Unbiased Binarization of Filament Structure . . . .
10.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 The FilamentSensor and the Benchmark Dataset . . . .
10.3.3 Detecting Slightly Bent Filaments . . . . . . . . . . . . . .
10.4 Orientation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Orientation Field Evolution . . . . . . . . . . . . . . . . . . .
10.4.2 Backward Nested Descriptor Analysis . . . . . . . . . . .
10.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Photonic Imaging with Statistical Guarantees: From Multiscale
Testing to Multiscale Estimation . . . . . . . . . . . . . . . . . . . . . . . . .
Axel Munk, Katharina Proksch, Housen Li and Frank Werner
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Testing on an Image . . . . . . . . . . . . . . . . . . . . . . .
11.2.4 Testing Multiple Hypotheses . . . . . . . . . . . . . . . . .
11.2.5 Connection to Extreme Value Theory . . . . . . . . . .

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11.2.6 Scanning . . . . . . . . . . . . . . . . . . . . . . .
11.2.7 Theory for the Multiscale Scanning Test
11.2.8 Deconvolution and Scanning . . . . . . . . .
11.2.9 FDR Control . . . . . . . . . . . . . . . . . . . .
11.3 Statistical Multiscale Estimation . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Efficient, Quantitative Numerical Methods for Statistical Image
Deconvolution and Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. Russell Luke, C. Charitha, Ron Shefi and Yura Malitsky
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Abstract Problem . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.2 Saddle Point and Dual Formulations . . . . . . . . . . .
12.2.3 Statistical Multi-resolution Estimation . . . . . . . . . .
12.3 Alternating Directions Method of Multipliers
and Douglas Rachford . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 ADMM for Statisitcal Multi-resolution Estimation
of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Primal-Dual Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.4.1 EPAPC for Statisitcal Multi-resolution Estimation
of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Randomized Block-Coordinate Primal-Dual Methods . . . . .
12.5.1 RBPD for Statisitcal Multi-resolution Estimation
of STED Images . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Holographic Imaging and Tomography of Biological Cells and
Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tim Salditt and Mareike Töpperwien
13.1 Propagation-Based Phase-Contrast Tomography . . . . . . . .
13.2 Nano-CT Using Synchrotron Radiation: Optics,
Instrumentation and Phase Retrieval . . . . . . . . . . . . . . . . .
13.2.1 Cone-Beam Holography . . . . . . . . . . . . . . . . . . .
13.2.2 Waveguide Optics and Imaging . . . . . . . . . . . . . .
13.2.3 Dose-Resolution Relationship . . . . . . . . . . . . . . .
13.2.4 Phase Retrieval Algorithms . . . . . . . . . . . . . . . . .
13.3 CTF-based Reconstruction and Its Limits . . . . . . . . . . . . .
13.4 Laboratory µ-CT: Instrumentation and Phase Retrieval . . .
13.5 Novel Tomography Approaches . . . . . . . . . . . . . . . . . . . .
13.5.1 Combined Phase Retrieval and Tomographic
Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.2 Tomographic Reconstruction Based
on the 3D Radon Transform (3DRT) . . . . . . . . . .

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Tomography of Biological Tissues: Applications

and Benchmarks . . . . . . . . . . . . . . . . . . . . . . . .
13.6.1 3D Structure of Cochlea . . . . . . . . . . . .
13.6.2 Small Animal Imaging . . . . . . . . . . . . .
13.6.3 3D Virtual Histology of Nerves . . . . . . .
13.6.4 Macrophages in Lung Tissue . . . . . . . . .
13.6.5 Neuron Locations in Human Cerebellum
13.6.6 Outlook: Time-Resolved Phase-Contrast
Tomography . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Constrained Reconstructions in X-ray Phase Contrast Imaging:
Uniqueness, Stability and Algorithms . . . . . . . . . . . . . . . . . . . . .
Simon Maretzke and Thorsten Hohage
14.1 Forward Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.1 Physical Model and Preliminaries . . . . . . . . . . . . .
14.1.2 Forward Operators for XPCI . . . . . . . . . . . . . . . . .
14.1.3 Forward Operators for XPCT . . . . . . . . . . . . . . . .
14.2 Uniqueness Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 Preliminary Results and Counter-Examples . . . . . .
14.2.2 Sources of Non-uniqueness—The Phase Problem . .
14.2.3 Relation to Classical Phase Retrieval Problems . . . .
14.2.4 Holographic Nature of Phase Retrieval in XPCI . . .
14.2.5 General Uniqueness Under Support Constraints . . .
14.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Lipschitz-Stability and its Meaning . . . . . . . . . . . .
14.3.2 Stability for General Objects and one Hologram . . .
14.3.3 Homogeneous Objects and Multiple Holograms . . .
14.3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Regularized Newton Methods for XPCI . . . . . . . . . . . . . . .
14.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4.2 Reconstruction Method . . . . . . . . . . . . . . . . . . . . .
14.4.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . .
14.5 Regularized Newton-Kaczmarz-SART for XPCT . . . . . . . .
14.5.1 Efficient Computation by Generalized SART . . . . .
14.5.2 Parallelization and Large-Scale Implementation . . .
14.5.3 Reconstruction Example . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Scanning Small-Angle X-ray Scattering and Coherent X-ray
Imaging of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Tim Salditt and Sarah Köster
15.1 X-ray Structure Analysis of Biological Cells:
A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405


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15.2

Methods: X-ray Optics and Sample Environment
15.2.1 Focusing Optics and Imaging Modalities
15.2.2 X-ray Compatible Microfluidic Sample
Environments for Cells . . . . . . . . . . . . .
15.3 Scanning Small-Angle X-ray Scattering of Cells .
15.4 Coherent X-ray Imaging of Cells . . . . . . . . . . . .
15.4.1 Ptychography . . . . . . . . . . . . . . . . . . . .
15.4.2 Holography . . . . . . . . . . . . . . . . . . . . .
15.5 Correlative Microscopy . . . . . . . . . . . . . . . . . . .
15.6 From Cells to Tissues . . . . . . . . . . . . . . . . . . . .
15.7 Outlook: FEL Studies of Cells . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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16 Single Particle Imaging with FEL Using Photon Correlations . .
Benjamin von Ardenne and Helmut Grubmüller
16.1 The Single Molecule Scattering Experiment . . . . . . . . . . . .
16.2 Structure Determination Using Few Photons . . . . . . . . . . . .
16.2.1 Theoretical Background on Three-Photon
Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.2 Bayesian Structure Determination . . . . . . . . . . . . .
16.2.3 Reduction of Search Space Using Two-Photon
Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.4 Optimizing the Probability Using Monte Carlo . . . .
16.3 Method Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3.1 Resolution Scaling with Photon Counts . . . . . . . . .
16.3.2 Impact of the Photon Counts per Image . . . . . . . . .
16.3.3 Structure Results in the Presence of Non-Poissonian
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 Structure Determination from Multi-Particle Images . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 Development of Ultrafast X-ray Free Electron Laser Tools in
(Bio)Chemical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simone Techert, Sreevidya Thekku Veedu and Sadia Bari
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 The Concept: Filming Chemical Reactions in Real Time
Utilizing Ultrafast High-Flux X-ray Sources . . . . . . . . . . . .

17.3 X-ray Diffraction and Crystallography for Condensed State
Chemistry Studies—Crystallography with Ultrahigh
Temporal and Ultrahigh Spatial Resolution . . . . . . . . . . . . .
17.4 Applications in Energy Research . . . . . . . . . . . . . . . . . . . .
17.5 From Local to Global: Ultrafast Multidimensional Soft
X-ray Spectroscopy and Ultrafast X-ray Diffraction
Shake Their Hands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xv

17.6

Applications in Bimolecular Reaction Studies and
Photocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7 Applications in Unimolecular Liquid Phase Reaction
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.8 Applications in Bioelectronics, Aqueous and Prebiotics
Reaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9 Applications in Biophysics and Gas Phase Biomolecules . . .

17.10 Ultrafast Imaging of Unimolecular Gas-Phase Reactions . . .
17.11 Applications in Nanoscience and Multiphoton-Ionisation . . .
17.12 Applications in Unimolecular Gas Phase Dynamics . . . . . . .
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 . . . . . . . . . . . . .
17.14 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18 Polarization-Sensitive Coherent Diffractive Imaging
Using HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sergey Zayko, Ofer Kfir and Claus Ropers
18.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . .
18.2 Phase Retrieval of Experimental Data . . . . . . . . .
18.3 Experimental Results . . . . . . . . . . . . . . . . . . . . .
18.4 Polarization Dependence . . . . . . . . . . . . . . . . . .
18.5 Magneto-Optical Imaging Using High-Harmonic
Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.6 Dichroic Imaging . . . . . . . . . . . . . . . . . . . . . . .
18.7 Signal Enhancement Mechanism . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19 Nonlinear Light Generation in Localized Fields Using Gases
and Tailored Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Murat Sivis and Claus Ropers
19.1 Plasmonic Enhancement for EUV Light Generation . . .
19.2 High-Harmonic Generation and Imaging in Tailored
Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 Wavefront and Coherence Characteristics of Extreme UV
and Soft X-ray Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bernd Schäfer, Bernhard Flöter, Tobias Mey and Klaus Mann
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 Wavefront Metrology and Beam Characterization
with Hartmann Sensors . . . . . . . . . . . . . . . . . . . . . . .
20.2.1 Hartmann Wavefront Sensing . . . . . . . . . . . .

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20.2.2 EUV Wavefront Sensor for FEL
Characterization . . . . . . . . . . . . . . . . . . . . . . . .
20.2.3 Beam Characterization of High-Harmonic
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.4 Thermal Lensing of X-ray Optics . . . . . . . . . . .
20.3 Wigner Distribution for Diagnostics of Spatial Coherence
20.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . .
20.3.3 4D Wigner Measurements . . . . . . . . . . . . . . . . .
20.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 Laboratory-Scale Soft X-ray Source for Microscopy
and Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . .
Matthias Müller and Klaus Mann
21.1 Table-Top Soft X-ray Source Using a Pulsed Gas Jet
21.2 Soft X-ray Microscopy . . . . . . . . . . . . . . . . . . . . . .
21.3 X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . .
21.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 Multilayer Zone Plates for Hard X-ray Imaging . . . .
Markus Osterhoff and Hans-Ulrich Krebs
22.1 From Focusing to Imaging . . . . . . . . . . . . . . . .

22.2 Let There be an Ideal World . . . . . . . . . . . . . .
22.3 Back to the Real World: Fabrication Challenges
22.3.1 Pulsed Laser Deposition . . . . . . . . . . .
22.3.2 FIB Processing . . . . . . . . . . . . . . . . . .
22.3.3 From MLL to MZP . . . . . . . . . . . . . .
22.3.4 Material and Parameter Studies . . . . . .
22.3.5 Summary . . . . . . . . . . . . . . . . . . . . . .
22.4 The World of Synchrotron Instrumentation . . . .
22.4.1 Hard X-rays Near 14 keV . . . . . . . . . .
22.4.2 High Energies: From 60 to 101 keV . .
22.4.3 Sampler Scanner . . . . . . . . . . . . . . . . .
22.4.4 Improvements of the GINIX Setup . . .
22.5 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.5.1 Ptychography . . . . . . . . . . . . . . . . . . .
22.5.2 Holography and Scanning SAXS . . . . .
22.5.3 Scanning WAXS . . . . . . . . . . . . . . . .
22.5.4 Correlative Scans . . . . . . . . . . . . . . . .
22.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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www.pdfgrip.com
Contents

23 Convergence Analysis of Iterative Algorithms for Phase
Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. Russell Luke and Anna-Lena Martins
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.2 Phase Retrieval as a Feasibility Problem . . . . . . . . .
23.3 Notation and Basic Concepts . . . . . . . . . . . . . . . . .
23.3.1 Projectors . . . . . . . . . . . . . . . . . . . . . . . . .
23.3.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . .
23.3.3 Fixed Points and Regularities of Mappings
23.4 A Toolkit for Convergence . . . . . . . . . . . . . . . . . .
23.5 Regularities of Sets and Their Collection . . . . . . . .
23.6 Analysis of Cyclic Projections . . . . . . . . . . . . . . . .
23.7 Application to Phase Retrieval Algorithms . . . . . . .
23.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


xvii

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24 One-Dimensional Discrete-Time Phase Retrieval . . . . . . . .
Robert Beinert and Gerlind Plonka
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.2 The Discrete-Time Phase Retrieval Problem . . . . . . .
24.3 Trivial and Non-trivial Ambiguities . . . . . . . . . . . . .
24.4 Non-negative Signals . . . . . . . . . . . . . . . . . . . . . . . .
24.5 Additional Data in Time-Domain . . . . . . . . . . . . . . .
24.5.1 Using an Additional Signal Value . . . . . . . .
24.5.2 Using Additional Magnitude Values
of the Signal . . . . . . . . . . . . . . . . . . . . . . .
24.6 Interference Measurements . . . . . . . . . . . . . . . . . . . .
24.6.1 Interference with a Known Reference Signal
24.6.2 Interference with an Unknown Reference
Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24.6.3 Interference with the Modulated Signal . . . .
24.7 Linear Canonical Phase Retrieval . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629



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Contributors

Sadia Bari FS-Strukturdynamik (bio)chemischer Systeme, Deutsches ElektronenSynchrotron DESY, Hamburg, Germany
Robert Beinert Institute for Mathematics and Scientific Computing, University of
Graz, 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 the
Biosciences, 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 for
Biophysical Chemistry, Göttingen, Germany;
Department of Optical Nanoscopy, Max Planck Institute for Medical Research,
Heidelberg, Germany

xix

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www.pdfgrip.com
xx

Contributors

Thorsten Hohage Institute for Numerical and Applied Mathematics, Universität
Göttingen, Göttingen, Germany
Stephan Huckemann Felix-Bernstein-Institute for Mathematical Statistics in the
Biosciences, Universität Göttingen, Göttingen, Germany
Stefan Jakobs Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Isabelle Jansen Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Nickels A. Jensen Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Maria Kamper Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Jan Keller-Findeisen Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Ofer Kfir IV. Physical Institute - Solids and Nanostructures, Universität
Göttingen, 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, Universität

Göttingen, Göttingen, Germany
Yura Malitsky Institute for Numerical and Applied Mathematics, Universität
Göttingen, Göttingen, Germany
Klaus Mann Laser-Laboratorium Göttingen e.V., Göttingen, Germany
Simon Maretzke Institute for Numerical and Applied Mathematics, Universität
Göttingen, 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


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Contributors

xxi

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, Universität
Göttingen, 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
Göttingen, Germany


Institute for X-ray Physics, Universität Göttingen,

Claus Ropers IV. Physical Institute - Solids and Nanostructures, Universität
Göttingen, Göttingen, Germany
Steffen J. Sahl Department of NanoBiophotonics, Max Planck Institute for
Biophysical Chemistry, Göttingen, Germany
Tim Salditt
Germany

Institute for X-ray Physics, Universität Göttingen, Göttingen,

Bernd Schäfer Laser-Laboratorium Göttingen eV., Göttingen, Germany
Ron Shefi Institute for Numerical and Applied Mathematics, Universität
Göttingen, Göttingen, Germany
René Siegmund Laser-Laboratorium Göttingen, Göttingen, Germany
Murat Sivis IV. Physical Institute - Solids and Nanostructures, Universität
Göttingen, Göttingen, Germany
Benjamin Sprung Institute for Numerical and Applied Mathematics, Universität
Göttingen, Göttingen, Germany
Thomas Staudt Institute for Mathematical Stochastics, Universität Göttingen,
Göttingen, Germany
Simone Techert FS-Strukturdynamik (bio)chemischer Systeme, Deutsches
Elektronen-Synchrotron DESY, Hamburg, Germany
Sreevidya Thekku Veedu FS-Strukturdynamik (bio)chemischer
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
Mareike Töpperwien
Göttingen, Germany

Systeme,


Institute for X-ray Physics, Universität Göttingen,

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xxii

Contributors

Benjamin von Ardenne Department of Theoretical and Computational
Biophysics, Max Planck Institute for Biophysical Chemistry Göttingen, Göttingen,
Germany
Frederic Weidling Institute for Numerical and Applied Mathematics, Universität
Göttingen, 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, Universität
Göttingen, Göttingen, Germany


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Part I

Fundamentals and Tutorials


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Chapter 1

STED Nanoscopy
Alexander Egner, Claudia Geisler and René Siegmund

The very first step of every understanding of the Microscope is ...
to become familiar with the idea that it is a thing sui generis
– Ernst Abbe [1]

1.1 Fundamentals of Fluorescence Microscopy
This section will present the basics of fluorescence microscopy. Starting from the
intensity distribution within the focal spot of an objective lens, we will discuss the
image formation and derive the classical formula of the resolution limit. Furthermore,
we will introduce the principle of confocal detection.

1.1.1 Vectorial Diffraction Theory and Intensity Distribution
Within the Focal Spot
The complex electric vector field E (r) in the focal region of an optical system
can be expressed in terms of a modified Huygens-Fresnel principle as the coherent
superposition of secondary plane waves at the exit pupil [2, 3]
A. Egner (B) · C. Geisler · R. Siegmund
Laser-Laboratorium Göttingen, Hans-Adolf-Krebs-Weg 1, 37077 Göttingen, Germany
e-mail:
C. Geisler

e-mail:
R. Siegmund
e-mail:
© The Author(s) 2020
T. Salditt et al. (eds.), Nanoscale Photonic Imaging, Topics in Applied Physics 134,
/>
3


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4

A. Egner et al.

(a)

g0

(b)
dS0

g*0
dSp
gp

G

g*p

s


y
s
f

x
z

W

Fig. 1.1 Secondary plane waves as well as strength and polarization of a focused wavefront. a
A secondary plane wave can be defined for each point G of a wave front W leaving the pupil of
an optical imaging system. The wave front of the secondary plane wave is tangential to W in G.
b When focusing through a lens, the projection of the electric vector field onto the meridional plane
changes its direction of polarization (vectors g0 and g p ). The part of the vector field orthogonal
to the meridional plane retains its polarization direction (vectors g∗0 and g∗p ). Due to the transition
from a plane to a spherical wavefront, the strength of the electric field within associated surface
segments (d S0 and d S p ) changes such that the energy passing through the surface elements remains
constant

E (r) = −

ik
2π

R1 R2 E p (s)eiksr dΩ,

(1.1)

Ω


where Ω is the solid angle, E p is the complex amplitude of the secondary plane waves,
R1 and R2 are the principle radii of curvature of the wavefront at the exit pupil and
s is the unity vector in the respective direction of propagation, see Fig. 1.1a. The
wave number k is given by k = 2π n/λ0 , with n being the refractive index and λ0
the vacuum wavelength. Note that the geometric focus is at position r = (0, 0, 0).
In order to derive E (r) for an aplanatic, i.e. axially stigmatic and obeying the sine
condition, imaging system such as the objective lens of a microscope, E p (s), R1 and
R2 have to be determined [4].
The typical scenario when focusing with an infinitely corrected lens is shown in
Fig. 1.1b. Without loss of generality, we first assume that the light at the entrance
pupil of the objective is linear polarized in the x-direction
E0 (x0 , y0 ) = E 0 e x ei P(x0 ,y0 ) ,

(1.2)

where E 0 is the (real-valued) amplitude of the electric field, e x is the unit vector in
x-direction and P is the pupil function which encodes the phase distribution of the
electric field [5]. For arbitrary polarization states, E (r) can then be calculated by
the coherent superposition of several solutions of E (r) for correspondingly linear

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