Storage Ring-Based Inverse Compton X-ray Sources: Cavity Design, Beamline Development and X-ray Applications (Springer Theses) 9783031177415, 9783031177422, 303117741X

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Springer Theses Recognizing Outstanding Ph.D. Research

Benedikt Sebastian Günther

Storage Ring-Based Inverse Compton X-ray Sources Cavity Design, Beamline Development and X-ray Applications

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Benedikt Sebastian Günther

Storage Ring-Based Inverse Compton X-ray Sources Cavity Design, Beamline Development and X-ray Applications Doctoral Thesis accepted by Technical University of Munich, Garching, Germany

Author Dr. Benedikt Sebastian Günther Department of Physics Technical University of Munich Garching, Germany

Supervisor Prof. Franz Pfeiffer Biomedical Physics Department of Physics School of Natural Sciences Technical University of Munich Garching, Germany Munich School of BioEngineering Technical University of Munich Garching, Germany Department of Diagnostic and Interventional Radiology School of Medicine and Klinikum rechts der Isar Technical University of Munich München, Germany

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-17741-5 ISBN 978-3-031-17742-2 (eBook) https://doi.org/10.1007/978-3-031-17742-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific 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, expressed 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 affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

X-ray imaging has been one of the most important tools in clinical diagnostics and non-destructive testing. Many modern X-ray technologies developed at synchrotron facilities have not yet evolved into day-to-day clinical routines or industrial screening techniques. Mostly, because many of these techniques require a high brilliance of the X-ray beam only available at synchrotrons. Nevertheless, the development of high-intensity short-pulse laser systems opened up a new route for the generation of brilliant X-ray beams, namely inverse Compton scattering. The advantage of this process compared to X-ray generation with a classical synchrotron is that the same X-ray energies and similar X-ray beam properties can be achieved at much smaller electron energies. This shrinks the electron accelerator and in turn significantly reduces the machine’s size as well as the costs for construction and operation. Consequently, these sources enable the transfer of the synchrotron techniques into clinical practise or industrial quality control. However, several challenges, namely regarding the total X-ray flux and machine stability as well as reliability have inhibited the use of inverse Compton sources in these settings so far. Benedikt Günther addresses multiple of these issues in his thesis which makes this work an outstanding contribution to research on and with inverse Compton scattering X-ray sources. Inverse Compton scattering was proposed as an elegant way to produce synchrotron-like radiation already shortly after the invention of the laser in the 1960’s. Although it was successfully demonstrated shortly thereafter, the total X-ray flux remained low due to the rather limited laser technology at that time. This changed once compact (sub-) picosecond high-power laser systems and high-finesse optical resonators became widely available. Consequently, two different routes have been explored. In the first concept, high-energy pulses from a low-repetition ultrashort laser collide with low-emittance electron bunches produced by a linear accelerator with a low repletion rate. In the second option, pulses from a picosecond laser with a high repetition rate and a moderate power are recirculated inside a high-finesse cavity. This produces a high average laser power which interacts with an electron beam circulating inside a miniature storage ring. To date, only the latter approach has matured into a compact X-ray source providing synchrotron-like radiation at v

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flux rates compatible with user applications. This inverse Compton X-ray source, the Lyncean Compact Light Source, developed by Lyncean Technologies Inc., was acquired by the Technical University of Munich in 2015. Nevertheless, the X-ray source’s stability, both in terms of X-ray flux and source position stability was initially unsatisfactory for many applications. Furthermore, no X-ray diagnostic existed which enabled machine optimisation by an operator while X-ray experiments were conducted. Benedikt Günther addressed these issues with a novel X-ray beam monitoring and stabilisation system which is specifically optimised for the X-ray flux densities available at inverse Compton X-ray sources. This system determines the X-ray flux, source size and source position in parallel to experiments and stabilises them via a feedback acting on the laser beam. Its design is easily transferable to other inverse Compton sources and may hence benefit the whole community. Moreover, Benedikt Günther designed a deformable cavity optic and evaluated its impact on the performance of the enhancement cavity in his thesis. Combined with a measurement of the transverse mode range, thermal deformations of the cavity optics can be actively compensated using this optic. This is a key prerequisite for scaling the laser power and in turn X-ray flux. These developments on their own represent a significant leap closer towards clinical applications of inverse Compton X-ray sources for which a stable X-ray beam delivery and a high total X-ray flux is indispensable. Furthermore, a beamline and experimental endstations are required before the X-ray beam can be used for any applications. Benedikt Günther evaluates the X-ray beam properties provided by the Lyncean Compact Light Source and presents our versatile beamline in his thesis. This may help users of classical synchrotrons to judge whether their experiments may be conducted at inverse Compton X-ray sources, thereby growing the user basis for these sources. Benedikt Günther did not only contribute to the improvement of the X-ray source, but he developed a novel microscopy technique as well, which employs structured illumination to speed up data acquisition while maintaining a high resolution. This technique allows imaging of large fields-of-view at high resolution. Therefore, it is especially suited for inverse Compton X-ray sources since they offer wide fields-ofview as well as the required coherence. This thesis will serve as a handbook of storage ring-based inverse Compton Xray sources, providing a comprehensive introduction to the principles of inverse Compton X-ray scattering, laser enhancement cavities and a detailed review on the different kinds of inverse Compton X-ray sources realised so far. Subsequently, this thesis provides possible solutions to various common challenges in the development of inverse Compton X-ray sources and presents a versatile X-ray beamline as well as X-ray applications for such an instrument. The high quality of the results and their

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detailed presentation on the following pages speak for themselves. Accordingly, this work will be highly valuable not only for researchers developing an inverse Compton source, but also serve as a reference for future users of such sources. Garching, Germany February 2022

Prof. Dr. Franz Pfeiffer

Abstract

The development of modern compact X-ray sources aims for brilliant X-ray radiation from systems compatible with laboratory environments. Backscattering of laser photons at a counter-propagating beam of relativistic electrons, commonly referred to as inverse Compton scattering, generates narrow-band hard X-rays. Compared to synchrotrons, the electrons’ energy can be orders of magnitude smaller while producing radiation with similar properties which downsizes the whole instrument. The Munich Compact Light Source (MuCLS) is one of the few facilities based on such a compact inverse Compton X-ray source. On the one hand, this source emits synchrotron-like, i.e. quasi-monochromatic, low-divergence and partially coherent, radiation, but on the other hand, its flux is orders of magnitude lower. While the former X-ray properties enable the transfer of various methods to the home laboratory, the reduced flux density may complicate some experimental campaigns or even inhibit certain types of studies. Therefore, measures to improve the X-ray flux cover a large part of this work, focussing on the interaction laser system. It consists of a solid-state laser which seeds an enhancement cavity. A new seed laser system increased the laser power inside the enhancement cavity by a factor of three. Owing to the increased thermal load on the cavity’s optics and its housing, two closed-loop feedbacks were implemented: an ambient thermal compensation system and an X-ray source position stabilisation system. The latter serves as a monitor for X-ray flux and X-ray source size as well. It was specifically designed for the low flux density at inverse Compton X-ray sources compared to synchrotrons while operating in parallel to experiments. Independently, a prototype for a deformable X-ray exit optic was developed. Its radius of curvature can be adjusted by inhibiting laser focus growth during thermal equilibration which improves the extractable X-ray flux. Another aspect that has been addressed is rapid electron beam-based X-ray energy switching which enables changing X-ray energies up to a few kiloelectron volt on a second scale, a feature that is desirable, e.g. in K-edge subtraction imaging. Apart from work on the inverse Compton X-ray source, this thesis presents the versatile X-ray beamline of the MuCLS and the underlying design considerations.

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Abstract

With this X-ray beamline, several X-ray applications have been explored and evaluated. First, a novel approach for full-field structured-illumination X-ray transmission microscopy is introduced and demonstrated using synchrotron radiation. Simulations indicate that a transfer to the MuCLS is possible, despite its reduced spatial coherence. The existence of a sufficiently modulated structured-illumination pattern at the MuCLS could be proven. An overview of various other applications which are available at the MuCLS to date follows. Those are mainly: X-ray (phase-contrast) imaging, X-ray absorption spectroscopy and micro-beam radiation therapy. This versatility demonstrates that the MuCLS can accomplish a wide range of synchrotron applications after the laser upgrades described earlier.

Acknowledgments

Here I would like to acknowledge the various types of assistance and support from my supervisors, collaborators, colleagues and friends without whom this endeavour would not have been possible. First of all, I would like to thank my supervisor Franz Pfeiffer most sincerely for the opportunity to pursue my doctorate at your Chair of Biomedical Physics. When I initially applied for my doctorate, I had the idea to conduct research at laser-driven X-ray sources. Nevertheless, you gently guided me towards the “hybrid” inverse Compton X-ray source, the Lyncean Compact Light Source (CLS), which had been recently installed in one of your labs at the time I started my doctorate and has been remaining a unique source of X-rays up to now. I never regret this slight shift of focus to this exciting new type of X-ray source and its application in biomedical problems. Thank you for providing a positive working environment and an excellent research infrastructure throughout all these years of my doctorate. Finally, I would like to thank you for supporting my work in both, improvements of our Munich Compact Light Source and X-ray imaging applications, as well as my research exchange at Lyncean Technologies Inc. I am extremely thankful to my mentor Martin Dierolf. You always took the time to discuss any problem which I encountered throughout my doctoral studies and provided good advice for solving these problems every time. Your profound knowledge on the CLS is amazing. Without you, solving the many smaller and larger issues that occurred at the CLS over the years would have taken much longer. Your detailed knowledge on X-ray imaging and -detectors was important for the development of the experimental set-ups at the MuCLS. Thank you for operating the CLS during my experiments which often lasted deep into the night. Your indepth feedback on various manuscripts. as well as my thesis. was highly valuable. Moreover, you made me aware of the BaCaTec-programme, which funded large parts of my research exchange. Last but not least, thank you for the fun times at coffee breaks and conferences, the exciting beamtimes at various synchrotrons and motivating words when I needed them. Thank you, Rod Loewen, for patiently answering my many questions about the CLS as well as providing tips and tricks for its operation. Moreover, I am very xi

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grateful for making my research exchange at Lyncean Technologies Inc. possible. Living in the Bay Area and working at your company for six months was a fantastic experience in which I learned a lot. In particular, I would like to thank you for partially supervising my research throughout my stay at Lyncean Technologies Inc. and for the fruitful discussions. You were also a big help in finding a great place for living during this exchange. Finally, I thank you, your family, Peggy and Shelly for a wonderful Thanksgiving experience. I would like to thank Regine Gradl for all the time we have spent together at the lab, in front of a computer, at conferences and working out–mostly running around campus or swimming in lake Eching in the summer. I enjoyed building the set-up in Hutch 1 and I lively remember the time when you told me that we had to “urgently upgrade” our detector tower. I am grateful to you for proofreading some of my manuscripts or conference proceedings. Performing multiple studies with you and Kaye Morgan was a pleasure. Thank you, Kaye, for showing me around Monash University and Melbourne when I visited the city. Furthermore, I am thankful to Lorenz Hehn, for sharing his Python knowledge with me and for many valuable discussions. Thank you for your support at our beamtimes and all the fun at the E17-soccer matches or coffee breaks, well, in your case, Cola or Spezi breaks. I will always remember the road trip to our beamtime at the ESRF, you made it very entertaining. Thank you, Johannes Brantl, for designing many of the mechanical parts for the structured-illumination experiment at the MuCLS and the filter-wheel system for rapid filter-based K-edge subtraction imaging. Moreover, I would like to thank you for supporting me during my structured-illumination experiments at the MuCLS, for detailed feedback on some of my manuscripts and large parts of my thesis and for all the fun. I am grateful for all the effort you put into designing the LEGO-models of the CLS. It was a lot of fun to build the big one after work with colleagues from the chair. Thank you for designing a second smaller one as well. This model will always remind me of the great time I had here during my doctorate! Thank you, Stephanie Kulpe, for proofreading some of my manuscripts and the exciting K-edge subtraction set-up development as well as -experiments which we performed together. Thank you for keeping me motivated to write my thesis during corona times. Ivan Kokhanovskyi optimised the electron beam-based fast X-ray energy switching during his master thesis. I am glad that you enjoyed converting my idea and initial codes into a stable and user-friendly procedure. It was fun to perform the experiments with you and to see the nice results which were achieved. Thank you very much. I owe a big thank you to Juanjuan Huang for showing me many tricks in PowerPoint rendering presentation slides much neater, like the process to include 3D models. The XAS experiments with you were always a pleasure and always returned very rewarding results. Building the set-ups inside Hutch 2 with you was an adventure, Christoph Jud. One event that has been particularly memorable is setting-up the optical table inside

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Hutch 2. You also were a great support during beamtimes. The E17-cooking events that you organised were a lot of fun as well as the E17-soccer games. I would like to thank Elena Eggl for introducing me to the MuCLS and showing me how to run the CLS during my first few months at the chair. You designed the grating interferometer and stabilised it together with Christoph Jud. Without your contribution, we would not have such a nice instrument today. Thank you, Madleen Busse for the support in developing the solid iodine filter, for patiently answering my chemical questions, for the many valuable discussions and for being a great office mate. Working with our mechanical technician, Alen Begic, was a pleasure. You fabricated all mechanical parts which I designed for our set-ups. Moreover, you often had good ideas on how to improve my current design. Thank you very much for all your support throughout my doctorate. Thank you, Johannes Melcher, for thoroughly proofreading several chapters of my thesis. Moreover, I would like to thank Klaus Achterhold and Bernhard Gleich for their large amount of administrative work to keep the MuCLS running. In this respect, I especially thank Klaus for making sure that the MuCLS fulfils all the radiation safety regulations and Bernhard for his support on electrical installations or general issues as well as his administration of all the construction work at the MuCLS lab. Finally, I would like to thank all the other members of the MuCLS-team which I have not mentioned before and without whom this doctorate would not have been such an enjoyable endeavour: Jessica Böhm, Thomas Buchner, Karin Burger, Eva Braig, David Cont, Lisa Heck and Sandra Resch. I am thankful to Bruce Borchers for performing the finite element simulations, designing the mechanical components of the proposed new exit optic and the fruitful discussions. I thank Martin Gifford for helping me with the autocorrelation measurement of the laser system and for your diligent and continued support and service of the CLS laser system. The soccer breaks with you in the parking lot behind the building of Lyncean Technologies Inc. were a lot of fun. I am thankful to the whole team of Lyncean Technologies Inc. for your commitment and the enjoyable dinners during your service visits, often together with Chris Juan, Matt Mezzata, John Khaydarov and Bryan Woo who were frequently on site for the CLS service visits as well. I would like to thank Michael Feser, the CEO of Lyncean Technologies Inc., for supporting my research exchange, organising the administrative work accompanying it and the interesting discussions. Furthermore, I am grateful to the BaCaTec-programme and the TUM-Graduate School for funding my research exchange at Lyncean Technologies Inc. Thank you Nelly de Leiris and Veronica Bodek for your support regarding administrative work, especially concerning working contracts, business trips and my research exchange. To the many people at our chair that keep our nice IT-infrastructure up and running: Thank you, your efforts are indispensable to our research here at the Chair for Biomedical Physics. Moreover, I thank the people at the chair for creating such

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a nice working atmosphere. Working with all of you was a pleasure as well as our seminar days, our conference visits and our leisure activities. Lastly–and most importantly–, I would like to thank my family for their continuous support and motivation in every aspect of my life.

Contents

1

Inverse Compton X-ray Sources—A Revolution or a Complement? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

1 5

The Physics of Inverse Compton Scattering X-ray Sources

X-ray Generation by Laser-Electron Interaction . . . . . . . . . . . . . . . . . 2.1 A Short Glance on Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Einstein’s Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Lorentz-Transformation of Coordinates . . . . . . . . . . . . . . 2.1.3 The Invariant Interval and Proper Time . . . . . . . . . . . . . . 2.1.4 4-Vectors in Special Relativity . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Relativistic Doppler-Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 From Covariant Electrodynamics to Liénard-Wiechert Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Covariant Formulation of the Inhomogeneous Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Solution of the Wave-Equation . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The Liénard-Wiechert Fields . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Undulator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Liénard-Wiechert Spectrum . . . . . . . . . . . . . . . . . . . . 2.3.2 Particle Trajectory and Wiggling Strength Parameter K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Effective Undulator Parameters of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Undulator Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Spatial and Spectral Distribution of Undulator Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discussion of the Undulator Line Spectrum . . . . . . . . . . . . . . . . . . 2.5 Comparison of a MuCLS-Type Laser Undulator with a Permanent Magnet One . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 10 11 12 14 15 15 16 19 21 21 23 25 27 28 30 33

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2.6

3

4

5

X-ray Generation by Inverse Compton Scattering . . . . . . . . . . . . . 2.6.1 The Scattered Photon’s Frequency . . . . . . . . . . . . . . . . . . . 2.6.2 The Klein-Nishina Differential Cross-Section . . . . . . . . . 2.6.3 The Compton Scattered Spectrum . . . . . . . . . . . . . . . . . . . 2.6.4 Effects of Electron Beam Parameters on the Inverse Compton Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 36 38 40

Scalar Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Wavefields in Free-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 From the Maxwell- to the Helmholtz Equations . . . . . . . 3.1.2 Rayleigh-Sommerfeld Diffraction Theory . . . . . . . . . . . . 3.1.3 The Paraxial Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Free-Space Propagation of Paraxial Waves . . . . . . . . . . . 3.2 Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Fundamental-Mode Gaussian Beam . . . . . . . . . . . . . 3.2.2 Higher-Order Gaussian Modes . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Propagation of Gaussian Beams Through Optical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 47 50 52 53 53 56

Enhancement Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stable Optical Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Stability Criteria of Resonators . . . . . . . . . . . . . . . . . . . . . 4.1.2 Axial Resonances of Passive Resonators . . . . . . . . . . . . . 4.1.3 Misaligned Optical Elements . . . . . . . . . . . . . . . . . . . . . . . 4.2 Enhancement Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Mode Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Phase Shift at a Multilayer Dielectric Mirror . . . . . . . . . . 4.2.3 Steady State Power Enhancement and Finesse of a Passive External Cavity . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Locking a Laser Oscillator to a Passive Resonator . . . . . 4.2.5 Determination of the Circulating Power . . . . . . . . . . . . . . 4.2.6 Thermally Induced Mode Coupling . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 72 74 74 74

41 42

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76 80 86 90 91

Fundamentals of X-ray Imaging and Spectroscopy . . . . . . . . . . . . . . . 93 5.1 X-ray Interactions with Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.1.1 Differential Cross Section and Complex Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1.2 Interaction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 K-Edge Subtraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 X-ray Phase Contrast Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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5.4.1 Propagation-Based Phase-Contrast Imaging . . . . . . . . . . 5.4.2 Grating-Based Phase Contrast Imaging . . . . . . . . . . . . . . 5.5 X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 6

105 107 111 112

R&D at the Inverse Compton X-ray Source of the MuCLS

Overview on Inverse Compton X-ray Sources . . . . . . . . . . . . . . . . . . . . 6.1 From Compton Scattering to Modern Inverse Compton Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Modern Inverse Compton Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Linear Accelerator-Based ICSs . . . . . . . . . . . . . . . . . . . . . 6.2.2 Storage Ring Based ICSs . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The CLS Installed at the MuCLS . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The CLS Laser Upgrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Interaction Laser System of the MuCLS . . . . . . . . . . . . . . . . . 7.2 Characterisation of the CLS Laser . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The M2 Beam Propagation Factor . . . . . . . . . . . . . . . . . . . 7.2.2 Laser Pulse Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Laser Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The CLS Enhancement Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 The CLS Cavity System . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Transverse Mode Range Measurement . . . . . . . . . . . . . . . 7.3.3 Stored Power and Thermal Effects . . . . . . . . . . . . . . . . . . 7.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 152 152 154 155 159 160 164 167 173 174

8

Development of a Deformable Exit Optic . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Implications of the Current Interaction Geometry and Pressure Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Pressure Correction of a Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Steady State Model of the Enhancement Cavity . . . . . . . . . . . . . . . 8.4 Simulation Tool: Capabilities and Limitation . . . . . . . . . . . . . . . . . 8.5 Expected Effect of Pressure Correction on Diffraction Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Pressure Correction of an Existing Optic . . . . . . . . . . . . . 8.5.2 Pressure Correction of an Optimised Optic . . . . . . . . . . . 8.5.3 Evaluation of the Two Deformable Exit Optic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Effect of a Rayleigh-Range Reduction . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Fundamental Limitation for an Ideal OC . . . . . . . . . . . . . 8.6.2 Limitation on the Rayleigh Length Imposed by a Infra-red Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

117 121 121 132 137 141

177 179 182 188 192 192 195 196 198 199 200

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Contents

8.6.3

Limitation on the Rayleigh Length Imposed by a Visible-Light Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Expected Heat Deformation and Its Consequences for the Design of the Deformable OC90-Optic . . . . . . . . . . . . . . . . 8.8 First Evaluation of a Deformable OC90 Prototype . . . . . . . . . . . . . 8.8.1 The Test Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Alignment of the Optical Cavity . . . . . . . . . . . . . . . . . . . . 8.8.3 Performance of a Deformable Exit Optic Prototype . . . . 8.8.4 Feedback System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

207 210 213 213 215 216 222 224 225

Fast X-ray Energy Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Necessity of Fast Energy Switching . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Fast Spectral Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Synthesis of the Solid State Filter . . . . . . . . . . . . . . . . . . . 9.2.2 Characterisation of the Solid State Filter . . . . . . . . . . . . . 9.2.3 A Filter System for Rapid X-ray Energy Switching . . . . 9.3 Rapid Electron Energy Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Protocol for the Generation of Magnet Configurations for Rapid Electron Energy Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Demonstration of Electron Beam-Based Rapid X-ray Energy Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 228 229 230 234 235

10 X-ray Beam Position Monitoring and Stabilisation . . . . . . . . . . . . . . . 10.1 Necessity of Source Position Stabilisation . . . . . . . . . . . . . . . . . . . 10.2 An X-ray Beam Monitor (XBM) for Inverse Compton Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Design Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 X-ray Beam Monitor Design . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Characterization of the X-ray Beam Monitor . . . . . . . . . . 10.3 Closed-Loop X-ray Source Stabilization . . . . . . . . . . . . . . . . . . . . . 10.3.1 Correction of Source Position Drift . . . . . . . . . . . . . . . . . . 10.3.2 Evaluation of the XBM and Its Performance in Beam Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Significance of X-ray Source Position Stabilization . . . . . . . . . . . . 10.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 243

236 237 241 242

244 244 245 246 249 249 250 251 252 252

Contents

xix

Part III X-ray Imaging and Spectroscopy at the MuCLS 11 The MuCLS Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Evaluation of the MuCLS in Comparison to Synchrotrons and Advanced X-ray Tube Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Comparison of the X-ray Sources’ Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Discussion of Typical Facility Implementations of Aforementioned X-ray Sources . . . . . . . . . . . . . . . . . . . 11.2 The MuCLS Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The MuCLS Front-End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 MuCLS End-Station 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 MuCLS End-Station 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Full-Field Structured-Illumination Super-Resolution X-ray Transmission Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 A New Super-Resolution X-ray Transmission Microscopy Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Proof-of-Principle Study at a Synchrotron . . . . . . . . . . . . . . . . . . . 12.2.1 The Experimental Endstation P05 at Petra III at DESY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Demonstration of Structured Full-Field Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Demonstration of Super-Resolution Imaging . . . . . . . . . . 12.2.4 Evaluation of and Perspective on the Proposed Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Transfer to Inverse Compton X-ray Sources . . . . . . . . . . . . . . . . . . 12.3.1 Simulation of the Expected Structured-Illumination Produced by 1Dand 2D-Gratings at the MuCLS . . . . . . . . . . . . . . . . . . . . . 12.3.2 Demonstration of Structured-Illumination at the MuCLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Conclusion for Structured-Illumination Super-Resolution X-ray Transmission Microscopy at the MuCLS . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 A Brief Outlook on Data Analysis in Frequency-Space . . . . . . . . 12.4.1 Reconstruction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 257 258 260 262 263 265 266 270 270 273 273 276 276 280 281 287 291

293 299

303 305 308 309 310

13 X-ray Techniques and Applications at the MuCLS . . . . . . . . . . . . . . . 313 13.1 X-ray Imaging at the MuCLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13.1.1 X-ray Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

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Contents

13.1.2 Contrast Enhanced and K-Edge Subtraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Propagation-Based Phase-Contrast Imaging . . . . . . . . . . 13.1.4 Grating-Based Phase-Contrast Imaging . . . . . . . . . . . . . . 13.1.5 X-ray Vector Radiography and X-ray Tensor Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Microbeam Radiation Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Planned Upgrades to the MuCLS Beamline . . . . . . . . . . . . . . . . . . 13.4.1 Grating-Based Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Replacement of the Second End-Station . . . . . . . . . . . . . . 13.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

316 318 322 325 325 327 328 328 329 329 330

Part IV Conclusion 14 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Performance Improvements of the Compact Light Source . . . . . . 14.2 Beamline Instrumentation and Experimental Techniques . . . . . . . 14.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 335 336 338 339

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Abbreviations

ACM ATC CLS CPA EOM EXAFS FFT FSR GBI ICS KES MTF MuCLS PBI PDH PSF PVP rms SBT STXM TEM TMR TUM TXM WP XANES XAS

All-curved mirror (bow-tie resonator) (Align) thermal compensation system Compact light source Chirped pulse amplification Electro-optic modulator Extended X-ray Absorption Fine Structure Fast Fourier transform Free spectral range Grating-based imaging Inverse Compton scattering K-edge subtraction Modulation transfer function Munich compact light source Propagation-based imaging Pound-Drever-Hall (locking scheme for a laser to a optical cavity) Point-spread-function Polyvinylpyrrolidone Root-mean-square Standard bow-tie (resonator) Scanning transmission X-ray microscopy Transverse electro-magnetic mode Transverse mode range Technical University of Munich Transmission X-ray microscopy Wave plate X-ray absorption near edge structure X-ray absorption spectroscopy

xxi

xxii

Abbreviations

Constants α 0 nˆ μ0 c re

Fine structure constant Dielectric permittivity of free space Unit vector Magnetic permeability of free space Speed of light Classical electron radius

Symbols β¯ β β β˙ B E j k δ ωFSR dσscat. d

γ λL λu E ω ρ ρe σa τ GVD P P(ω) Refl(ω) ˜ ψ(r) u˜ sph (r ⊥ , z) ϕG a a0 Jl (u) K

Average velocity of the particle parallel to the undulator Absorption index Normalised relativistic velocity Temporal derivative of β Magnetic field Electric field Current density Wave number or spatial frequency Refractive index decrement Free spectral range Differential (scattering) cross section Relativistic γ-factor Laser wavelength Undulator wavelength/period Energy Angular frequency Charge- or mass density Electron density Absorption cross section Proper time Group velocity dispersion Power enhancement Power enhancement of an enhancement cavity Reflectivity of an enhancement cavity’s input coupler Scalar wave-field expressed in the angular frequencies ω Paraxial (spherical) wave Gouy-phase of a Gaussian beam Amplitude of the transverse electron oscillation in an undulator Dimensionless normalised peak vector potential of a laser Bessel’s function The wiggler strength parameter or K-parameter

Abbreviations

n(E ) Nu v vph vg w Z zR I L U

xxiii

Refractive index Number of undulator periods Velocity Phase velocity Group velocity Beam waist, 1/e beam radius Atomic number Rayleigh length Intensity Cavity perimeter (resonator length for one roundtrip) Overlap integral

Chapter 1

Inverse Compton X-ray Sources—A Revolution or a Complement?

Visual perception greatly influences human experience, emotion and evolution. The ability to observe and draw conclusions from the observed process is a key capacity for scientific advancement as well. However, this constrained observations to structures larger than the naked eye’s resolution for a long time. This limit was only overcome with the invention of the visible light microscope in the 17th century which opened up a new regime for exploration. Yet, unlike for transparent specimens, it could only show the surface of opaque ones, which limited area of application significantly. The interior of opaque objects became only accessible when Röntgen discovered that an electron tube produces unknown radiation, therefore named “X-Strahlung” or “X-ray” by him, capable of penetrating thick opaque objects (Röntgen 1895). The benefit of this radiation for medical diagnostic was immediately recognised (Russo 2018, p. 141) and the development of crystallography followed soon thereafter (Willmott 2019, p. 8 f.). While the former benefited from the large divergence of the X-rays, the latter suffered from the broadband highly-divergent radiation. Nevertheless, electron impact tubes were the common way to produce X-rays for approximately the next 50 years until synchrotron radiation was observed first in circular particle accelerators in 1947 (Willmott 2019, p. 13).1 As synchrotrons at that time had electron energies around or below 1 GeV, they were capable of producing radiation in the visible, ultra-violet or soft X-ray regime. Relativistic effects contract the naturally isotropic radiation into a narrow forward cone with respect to the electron motion. The benefits of the resulting collimated broadband high-intensity radiation for applications such as spectroscopy were recognised early on Tombou1 The term synchrotron for this particular type of circular particle accelerator originates from the acceleration method. The particles are accelerated by linear accelerator modules and kept on the circular trajectory by a set of magnetic deflector and focussing magnets. Since the particles’ kinetic energy increases every turn, magnetic field strength has to be synchronously raised in order to keep the particle trajectory constant.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_1

1

2

1 Inverse Compton X-ray Sources—A Revolution or a Complement?

lian and Hartman (1956). The properties X-ray bandwidth, photon flux, source size and divergence angle characterising an X-ray beam can be combined into a single figure of merit, the brilliance B. It prescribes the number of photons within 0.1% bandwidth emitted by the X-ray source divided by the radiation’s solid angle and its transverse source sizes, see e.g. Als-Nielsen and McMorrow (2011, p. 32): B=

(mrad)2

(mm2

photons/second . source area) (0.1% bandwidth)

(1.1)

Hard X-ray radiation (up to 125 keV) from synchrotrons became only available with the 6 GeV Deutsches Elektronen-Synchrotron (DESY) in 1964 (Willmott 2019, p. 16). Around the same time, rotating anode X-ray generators became widely available (Als-Nielsen and McMorrow 2011, p. 30) which combined with the development of computed tomography by Hounsfield and Cormack started to revolutionise diagnostic medical imaging (Als-Nielsen and McMorrow 2011, p. 307). In contrast, research at synchrotron facilities focussed on applications that require highly-intense narrow bandwidth X-rays, such as X-ray diffraction, -spectroscopy or -microscopy. A very important leap forward in the development of synchrotrons was a significant reduction of the electron beam phase space due to improved designs of the electron storage ring’s magnet lattice in combination with the introduction of insertion devices (Willmott 2019, p. 16). The most common insertion devices can be classified into “wigglers” and “undulators”. Since wigglers consist of a periodic array of very strong dipole magnets, they reach higher X-ray energies while increasing the X-ray flux by the number of dipole magnets compared to a normal bending magnet. Undulators are constructed similarly, but with weak dipole magnets. As a result, radiation emitted in the individual dipoles interferes. This generates intrinsically narrowband X-rays at twice the Lorentz-contracted magnet period and harmonics thereof. The process boosts brilliance and spatial coherence but at reduced X-ray energies compared to wigglers (Attwood and Sakdinawat 2017, p. 216). The first of these optimised “third generation” facilities was the European Synchrotron Radiation Facility (ESRF) commissioned in 1994 (Willmott 2019, p. 16). The much higher spatial coherence provided by this new generation of synchro trons paved the way to multiple phase-sensitive imaging methods, e.g. propagationbased phase-contrast imaging, grating-interferometry or coherent diffraction imaging. The inherent advantages of phase-sensitive imaging techniques for soft-tissue contrast (Willmott 2019, p. 375 f.) would be beneficial for diagnostic medical imaging as well. However, setting-up a synchrotron at hospitals is infeasible due to the space requirements and financial investments. Accordingly, alternative methods capable of producing synchrotron radiation with compact and affordable instruments would be desirable. Inverse Compton scattering can produce synchrotron-like X-rays from relatively low-energy electrons thereby reducing the instrument’s dimensions significantly. This is possible by replacing the centimetre long period of the magnet undulator by the electromagnetic field of a compact short-pulse high-power laser. Since the latter

1 Inverse Compton X-ray Sources—A Revolution or a Complement?

3

became available only in recent years, the interest in employing inverse Compton scattering for X-ray generation has been renewed.2 Albeit inverse Compton sources can provide X-ray radiation similarly coherent to synchrotrons, their X-ray brilliance is significantly lower owing to their lower photon flux and larger divergence. On the one hand, the latter implies a larger field-of-view which is important, if large specimens are to be investigated, e.g. in medical imaging. On the other hand, the large divergence angle at lower brilliance strongly reduces the usable transverse coherence length at inverse Compton sources compared to the ones at synchrotrons. Moreover, the brilliance of inverse Compton sources is still significantly higher than the one of X-ray tubes outside their fixed characteristic lines. Consequently, inverse Compton sources bridge the gap between synchrotrons and classical X-ray generators. One of the first “modern” inverse Compton sources that actually delivered sufficient flux to perform experiments was the Lyncean Compact Light Source (CLS). At the CLS-prototype, first investigations regarding crystallography (Abendroth et al. 2010), beam-hardening in computed tomography (Achterhold et al. 2013) and grating-interferometry (Bech 2009; Bech et al. 2012; Schleede et al. 2012; Eggl et al. 2015) were performed. Despite these successful demonstrations, significant drifts of the X-ray source position could be observed, which negatively affect, e.g. grating interferometry. In addition, the X-ray flux of the CLS-prototype as well as the final commercial product installed here at the Munich School of BioEngineering right before the start of this thesis was around 1 × 1010 ph/s (Eggl et al. 2016) which is much less than the one available at synchrotrons. Accordingly, science at inverse Compton sources benefits from research on potential applications, development of new techniques as well as improvement of the inverse Compton source itself. This thesis aims to contribute to all three areas. After the CLS had been installed by the manufacturer, various contributions have been made as part of this thesis project to the initial design and continued development of the versatile beamline of the Munich Compact Light Source (MuCLS).3 This presentation is complemented by an overview over of all the methods and applications that are currently available at the MuCLS. Furthermore, a new technique for structured-illumination high-resolution X-ray microscopy is introduced. Finally, this thesis aims to contribute developments for inverse Compton scattering sources, especially for the CLS, that improve their source position stability and their overall X-ray flux or brilliance, respectively. The thesis is structured into four parts, first, the theory part, the second one describing research and development at the CLS, the third one presenting beamline development and applications at the MuCLS and finally a short conclusion and outlook. The 2

Inverse Compton scattering was demonstrated already right after the first lasers were invented in the 1960s, cf. Fiocco and Thompson (1963), Kulikov et al. (1964), Bemporand et al. (1965). Nevertheless, the low X-ray flux that was achievable with lasers of that time inhibited major interest, especially because undulators produced much denser virtual photon fields (Madey 1971). A detailed history and overview over the various inverse Compton sources is provided in Chap. 6. 3 The term MuCLS denotes the CLS developed by Lyncean Technologies Inc. including the beamline developed in-house by staff of the Technical University of Munich.

4

1 Inverse Compton X-ray Sources—A Revolution or a Complement?

first part, “The physics of inverse Compton scattering X-ray sources”, focusses on the main theoretical concepts employed in the framework of this thesis. This section was written with the intent to provide new students an in-depth introduction into the important concepts to understand X-ray generation as well as laser enhancement cavities which are the key technology for high-power, high-repetition rate laser systems like the one at the CLS. Therefore these chapters intentionally contain a level of detail similar to textbooks. Chapter 2, “X-ray Generation by Laser-Electron Interaction”, discusses the X-ray generation by a relativistic charge in an electromagnetic field. Chapter 3, “Scalar Wave Theory”, provides the basics of Rayleigh-Sommerfeld diffraction theory, free-space propagation of paraxial waves and introduces Gaussian beams and their propagation through optical systems. This is required to understand and model, e.g. enhancement cavities. Chapter 4, “Enhancement Cavities”, first describes stable optical resonators in general, before presenting several important aspects necessary to understand the working principle of enhancement cavities, their characterisation and some limitations. Chapter 5, “Fundamentals of X-ray Imaging and Spectroscopy”, the final chapter of the theory part, discusses in brevity the interactions of X-rays with matter, coherence, and the physics behind the various techniques available at the MuCLS. The second part, “R&D at the inverse Compton X-ray source of the MuCLS”, presents research on several aspects of the X-ray source. Chapter 6, “Overview on Inverse Compton X-ray Sources”, contains a short walk through the historic developments of inverse Compton X-ray sources followed by short descriptions of the various modern inverse Compton sources that have been built so far. It concludes with a short introduction to the CLS. Chapter 7, “The CLS Laser Upgrade”, presents the current interaction laser system of the CLS, its characterisation and upgrades to the CLS enhancement cavity. All of these have been implemented into the CLS at the MuCLS. Chapter 8, “Development of a Deformable Exit Optic”, presents research on a new mirror assembly for more efficient X-ray extraction at the CLS. Chapter 9, “Fast X-ray Energy Switching”, introduces two techniques, a filter-based approach and an electron-beam based one, to rapidly switch the X-ray energy. These techniques were mainly developed for dynamic K-edge subtraction imaging. Chapter 10, “X-ray Beam Position Monitoring and Stabilisation”, describes a beam monitor system for X-ray flux, -source position and -source size. It has been combined with a closed loop feedback adjusting the laser orbit to keep the X-ray source position constant. The third and final main part, “X-ray Imaging and Spectroscopy at the MuCLS”, presents the current capabilities at the MuCLS and a structured-illumination Xray microscopy technique. Chapter 11, “The MuCLS Beamline”, showcases the beamline developed at the Technical University of Munich (TUM) for the CLS. Chapter 12, “Full-Field Structured-Illumination Super-Resolution X-ray Transmission Microscopy”, explains a novel grating-based structured-illumination superresolution technique, demonstrates its feasibility at synchrotrons and evaluates its transfer to the MuCLS. The final Chap. 13, “X-ray Techniques and Applications at the MuCLS”, evaluates and summarises all the techniques and applications that have been demonstrated at the MuCLS and are currently available.

References

5

References Abendroth J et al (2010) X-ray structure determination of the glycine cleavage system protein H of Mycobacterium tuberculosis using an inverse Compton synchrotron X-ray source. J Struct Funct Genom 11:91–100 (2010) Achterhold K et al (2013) Monochromatic computed tomography with a compact laser-driven X-ray source. Sci Rep 3:1313 Als-Nielsen J, McMorrow D (2011) Elements of modern X-ray physics, 2nd edn. Wiley, Chichester. ISBN: 978-0-470-97394-3 Attwood D, Sakdinawat A (2017) X-rays and extreme ultraviolet radiation: principles and applications, 2nd edn. Cambridge University Press, Cambridge. ISBN: 9781107477629 Bech M (2009) X-ray imaging with a grating interferometer. PhD-thesis, University of Kopenhagen Bech M et al (2012) Experimental validation of image contrast correlation between ultra-smallangle X-ray scattering and grating-based dark-field imaging using a laser-driven compact X-ray source. Photon Lasers Med 1:47–50 Bemporand C et al (1965) High-energy photons from Comyton scattering of light on 6.0-GeV electrons. Phys Rev 138:1546–1549 Eggl E et al (2016) The Munich compact light source: initial performance measures. J Synchrotron Radiat 23:1137–1142 Eggl E et al (2015) X-ray phase-contrast tomography with a compact laser-driven synchrotron source. Proc Natl Acad Sci 112:5567–5572 Fiocco G, Thompson E (1963) Thomson scattering of optical radiation from an electron beam. Phys Rev Lett 10:89–92 Kulikov OF et al (1964) Compton effect on moving electrons. Sovjet J Exp Theor Phys 47:1591– 1594 Madey JM (1971) Stimulated emission of bremsstrahlung in a periodic magnetic field. J Appl Phys 42:1906–1913 Röntgen WC (1895) Ueber eine neue Art von Strahlen. Sitzungsberichte der Würzburger physik.medi. Gesellschaft 137:132–141 Russo P (ed) Handbook of X-ray imaging: physics and technology. CRC Press, Boca Raton. ISBN: 978-1-4987-4152-1 Schleede S et al (2012) Emphysema diagnosis using X-ray dark-field imaging at a laser-driven compact synchrotron light source. Proc Natl Acad Sci 109:17880–17885 Tomboulian DH, Hartman PL (1956) Spectral and angular distribution of ultraviolet radiation from the 300-MeV Cornell synchrotron. Phys Rev 102:1423–1447 Willmott P (2019) An introduction to synchrotron radiation, 2nd edn. Wiley, Chichester. ISBN: 978-1-119-280378

Part I

The Physics of Inverse Compton Scattering X-ray Sources

Chapter 2

X-ray Generation by Laser-Electron Interaction

This section intends to introduce the undulator concept in general, and more specifically the physical principles of X-ray generation at the MuCLS. As the electrons travel at velocities close to the speed of light while undergoing a wiggling motion, the observed radiated frequency is subject to relativistic effects. Therefore, the Sect. 2.1 presents necessary ingredients of special relativity first, before the interaction process of relativistic electrons and a photon beam is described in detail. This chapter is mainly based on content from Jackson’s “Classical Electrodynamics”, Chaps. 11, 12 and 14; Zangwill’s “Modern Electrodynamics”, Chaps. 22 and 231 ; Wiedemann’s “Particle Accelerator Physics”, Chaps. 25 and 26 as well as “Synchrotron Radiation and Free Electron Lasers”, Chap. 2, by Kwang-je Kim, Zhirong Huang and Ryan Lindberg.

2.1 A Short Glance on Special Relativity Special relativity describes how different observers, moving at a constant velocity relative to each other, perceive the same physical process. It is deeply related to electromagnetism, thus to Maxwell’s equations of the electromagnetic field. This connection has been emphasised by Einstein in the introduction of his annus mirabilis paper “Zur Elektrodynamik bewegter Körper” (Einstein 1905): “Daß die Elektrodynamik Maxwells - wie dieselbe gegenwärtig aufgefaßt zu werden pflegt - in ihrer Anwendung auf bewegte Körper zu Asymmetrien führt, welche den Phänomenen nicht anzuhaften scheinen, ist bekannt. Man denke z.B. an die elektrodynamische Wechselwirkung zwischen einem Magneten und einem Leiter. Das beobachtbare Phänomen hängt hier nur ab von der Relativbewegung von Leiter und

1

Zangwill uses the invariant interval r 2 − c2 t 2 , while throughout this thesis c2 t 2 − r 2 is used.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_2

9

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2 X-ray Generation by Laser-Electron Interaction

Magnet, während nach der üblichen Auffassung die beiden Fälle, daß der eine oder der andere dieser Körper der bewegte sei, streng voneinander zu trennen sind.” Einstein resolved this issue by introducing his concept of “relativity of simultaneity” and a theory of space and time.

2.1.1 Einstein’s Postulates As a consequence of this asymmetry Einstein postulated that (formulation as in Zangwill (2012, p. 825)): 1. The laws of physics take the same form in every inertial frame. 2. The speed of light in vacuum is constant. The second postulate follows from light experiments, in which effects expected under Galilean relative motion were absent. A severe consequence of these two postulates is that absolute time does not exist and consequently neither does universal temporal simultaneity.

2.1.2 Lorentz-Transformation of Coordinates This section is based on (Zangwill 2012, p. 826 ff.). Two different inertial frames K and K' have aligned coordinate axes and move at a constant velocity v relative to one another as depicted in Fig. 2.1. Without loss of generality, the two coordinate systems coincide at their origin at time t = t ' = 0 and K' moves along the positive z-axis relative to K. Considering homogeneity and isotropy of space and time, which is implied in the first postulate, the relation between the two coordinate systems is linear and in its most general form: x ' = C x,

y ' = C y, z ' = Az + Bt, t ' = Dz + Et.

(2.1)

Because z ' = 0 coincides with z = vt and B = −v At, C cannot depend on the direction of motion of the two systems relative to each other since both, x- and y-axes, are orthogonal to the direction of motion. Thus, x = C x ' and y = C y ' supplement Eq. 2.1 by symmetry and C = 1 is the sole valid conclusion. At t = t ' = 0, a spherical wave of radiation is emitted. Applying postulate 2 results in c2 t 2 − x 2 − y 2 − z 2 = 0

(2.2)

c2 t '2 − x '2 − y '2 − z '2 = 0.

(2.3)

and

2.1 A Short Glance on Special Relativity

11

Fig. 2.1 Two reference frames K and K' which move relative to another in the “standard configuration”. The origins of the two coordinate systems coincide at t = t ' = 0 and the axes are oriented identically. They move at a relative velocity v along the z-axis. A plane wave with a wave vector k is schematically depicted as well

c is the speed of light. Inserting Eq. 2.1 into 2.3 and enforcing the reproduction of Eq. 2.2 generates the required constraints to solve the resulting system of equations. The Lorentz-transformation for this case is: x ' = x,

y ' = y, z ' = γ(z − βct), ct ' = γ(t − βz).

(2.4)

Similarly, the following equations for arbitrary relative motion between two frames can be obtained (Zangwill 2012, p. 834): r ' ⊥ = r ⊥ , r ' ∥ = γ(r ∥ − βct) ct ' = γ(ct − β · r ∥ ) r ⊥ = r ' ⊥ , r ∥ = γ(r ' ∥ + βct ' ) ct = γ(ct ' + β · r ' ∥ ),

(2.5)

v 1 and β = . γ=√ 2 c 1−β

(2.6)

where

Time dilation and length contraction follow directly from these transformations.

2.1.3 The Invariant Interval and Proper Time This section follows (Jackson 1999, p. 527 f.). Nevertheless, a quantity can be defined that has the same numerical value in every inertial frame, i.e. which is invariant under Lorentz-transformations. This so-called invariant interval Δs has the dimension of a distance and distinguishes past events from future ones as well as causes from effects. It combines the distance in space of two events with the time lapse separating them temporally into a single value (Jackson 1999, p. 527)

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2 X-ray Generation by Laser-Electron Interaction

Δs 2 = c2 (t1 − t2 )2 − |r 1 − r 2 |2 .

(2.7)

If the square of the invariant interval is smaller than zero, the two events are spaced farther apart than the distance travelled by light in the time interval between the events. Such events have a space-like separation, because an inertial system exists, in which these two events happen simultaneously. Consequently, they cannot be causally connected. In contrast, events with a time-like separation (Δs 2 > 0), can be causally connected as an inertial frame exists in which the two events occur at the same location in space. A signal travelling at the speed of light can connect events with Δs 2 = 0 (null separation). If those events are infinitesimally close, the incremental invariant interval is (Jackson 1999, p. 528) | ) | 2 | d r |2 | = c2 dt 2 (1 − β 2 ) = c2 dt . 1 − || | 2 γ cdt

( ds = c dt − |d r| = c dt 2

2

2

2

2

2

(2.8)

Consequently, a new invariant time can be defined as dτ =

dt , γ

(2.9)

where dτ is the time increment in the instantaneous rest frame system (Jackson 1999, p. 528). Therefore, τ is called the proper time of the system.

2.1.4 4-Vectors in Special Relativity This section follows (Jackson 1999, p. 543 ff.). It is convenient to write the invariant interval ds 2 as a contraction of two four vectors and a metric tensor g μν of rank two generating the desired difference between the space and time coordinates (Jackson 1999, p. 542). ds 2 = dr · dr = drα dr α = dr α drα = gαβ dr α dr β = dr α gαβ dr β ,

(2.10)

where (Jackson 1999, p. 543 f.) ⎛

⎛ ⎞ ⎛ ⎞ cdt cdt 1 ⎜−d x ⎟ ⎜ dx ⎟ ⎜0 α αβ ⎜ ⎟ ⎜ ⎟ drα = ⎜ ⎝−dy ⎠ , dr = ⎝ dy ⎠ , gαβ = g = ⎝0 −dz dz 0

0 0 −1 0 0 −1 0 0

⎞ 0 0⎟ ⎟. 0⎠ −1

(2.11)

drα is called covariant vector and dr α contravariant. Linearity of the Lorentz-transformation relating the space-time 4-vector r α in one inertial frame with r 'α in another

2.1 A Short Glance on Special Relativity

13

allows for a matrix representation of this notation. The elements of this matrix are calculated using the transformation rule (Jackson 1999, Eq. 11.61, p. 540) by r 'β =

∂r 'β α ∂r β α β α β r = L r and r = r = [L −1 ]βα r α , α ∂r α ∂r 'α

(2.12)

where Lαβ is the matrix for a general Lorentz-transformation and [L −1 ]βα its inverse (Zangwill 2012, p. 960). Similarly, 4-vectors for the velocity V =

( ) dr d dr = γ(v) (ct, r) = γ(v) c, = γ(v)(c, v) = (V0 , V ) dτ dt dt

(2.13)

and momentum p = mV = (E/c, p)

(2.14)

are defined (Zangwill 2012, p. 961). Their invariant lengths are (Zangwill 2012, p. 961)2 V · V = c2 and p · p = m 2 c2 . (2.15) 2 2 2 Combining the second equation with √ p · p = E /c − p results in the well 2 2 2 4 known energy-momentum relation E = p c + m c . Other important quantities in electrodynamics are derivatives with respect to space and time, since they interconnect electric and magnetic fields and relate them to their potentials. A very straight forward approach is to manually calculate the derivative of the contravariant space-time vector in the frame K ' using

∂ r∥ ∂ ∂ r⊥ ∂ ∂t ∂ ∂ = ' + + ' =γ ∂ r ∥ ∂t ∂ r∥ ∂ r∥ ∂ r '∥ ∂ r ⊥ ∂ r '∥

(

∂ β ∂ + ∂ r∥ c ∂t

)

∂ ∂ ' = ∂r⊥ ∂ r⊥ ( ) ∂ ∂ r∥ ∂ ∂t ∂ ∂ ∂ = + ' = γ cβ · + . ∂t ' ∂t ∂t ∂ r∥ ∂t ∂t ' ∂ r ∥

(2.16)

Comparing this result with Eq. 2.5 reveals that the contravariant derivatives transform like the covariant space-time vector in Eq. 2.12. Consequently, the partial derivative can be written as a 4-vector (Jackson 1999, p. 543) ∂ ∂α ≡ α = ∂r

2

(

1 ∂ ,∇ c ∂t

)

∂ and ∂ ≡ = ∂rα α

(

1 ∂ , −∇ c ∂t

)

= g αβ ∂β .

(2.17)

The negative sign in D.26 originates from the opposite definition of the metric tensor used by Zangwill.

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The 4-divergence of a space-time vector r is the invariant ∂α r α = ∂ α rα =

1 ∂r0 +∇ ·r c ∂t

(2.18)

and the invariant length associated with the operator ∂α ∂ α =

1 ∂2 − ∇2 = ◻ c2 ∂t 2

(2.19)

is the wave equation operator (Jackson 1999, p. 543).

2.1.5 Relativistic Doppler-Shift This section is inspired by (Zangwill 2012, p. 845 f.) and (Jackson 1999, p. 529 f.). These tools at hand, the relativistic Doppler-shift can be derived. Consider a plane wave in the inertial frame K propagating in vacuum at an angular frequency ω and a wave number k (dispersion relation ω = ck), cf. Fig. 2.1, E(r, t) = E 0 ei (ωt−k·r) .

(2.20)

The Lorentz invariance of the wave operator provides a mathematical argument for the invariance of the phase of a plane wave. It implies that the wave’s form is the same regardless of the inertial frame in which it is observed. An observer in another inertial frame K ' moving at a velocity of v relative to K will observe the plane wave in the coordinates of his frame: ' '

'

'

E ' (r ' , t ' ) = E '0 ei(ω t −k ·r ) .

(2.21)

The prime denotes the quantities as observed in the moving frame K ' . They can be calculated by Lorentz-transformation of Eq. 2.20 with the relations in Eq. 2.5. This procedure results in E ' (r ' , t ' ) = E '0 e

] [ iωγ (t ' +β·r '∥ /c)−i k· r '⊥ +γ ( r '∥ +βct ' )

.

(2.22)

Now, the contributions in the exponential are rearranged into pure space-parts and pure time-parts applying k = ∥k∥nˆ = k nˆ and the dispersion relation: E ' (r ' , t ' ) = E '0 e

] [ ˆ ' −i k⊥ ·r '⊥ +r '∥ ·γ ( k∥ −βk ) iωγ(1−β·n)t

.

(2.23)

A comparison of Eqs. 2.21 and 2.23 yields ( ) ( ) ω ' = ωγ 1 − β · nˆ , k'∥ = γ k∥ − βk , k'⊥ = k⊥ .

(2.24)

2.2 From Covariant Electrodynamics to Liénard-Wiechert Fields

15

The inverse transformation from K ' to K is realised by interchanging primed and unprimed quantities in Eq. 2.24 and reversing the sign of β: ) ( ) ( ω = ω ' γ 1 + β · nˆ , k∥ = γ k'∥ + βk ' , k⊥ = k'⊥ .

(2.25)

Equations 2.24 and 2.25 describe the relativistic Doppler-shift, which plays a crucial role in understanding the frequency shift of the radiation emitted from magnetic undulators in synchrotrons or in the interaction between relativistic electrons and laser light.

2.2 From Covariant Electrodynamics to Liénard-Wiechert Fields This section is mainly based on (Jackson 1999, Sects. 11.9, 12.11 and 14.1). This section discusses the implications of the relativistic effects presented in the previous section on the theory of electromagnetism. Its covariant form is used to derive the Liénard-Wiechert potential and its corresponding fields, which determine the motion of electrons in external fields, e.g. a dipole magnet or an undulator.

2.2.1 Covariant Formulation of the Inhomogeneous Maxwell Equations As far as we know, electric charge is absolutely conserved, i.e. invariant under Lorentz transformations. In other words, the charge of an object is independent of its state of motion. Moreover, the first postulate of special relativity states that the laws of physics are independent of the inertial frame. Therefore, this is valid for conservation of charge described by the continuity equation ∂ρ +∇ · j =0 ∂t

(2.26)

as well, where ρ is the charge density and j is the current density (Jackson 1999, p. 553 ff.). This equation has the same shape as the invariant divergence of a 4-vector modulus a factor c in the time derivative, Eq. 2.18. Consequently, it is straight forward to define a current 4-vector J α = (cρ, j ). The continuity equation in covariant form is (Jackson 1999, p. 555) ∂α J β = 0.

(2.27)

To calculate the electromagnetic field emitted by a relativistic electron, the inhomogeneous Maxwell equations have to be employed. They can be expressed in a totally antisymmetric matrix equation in the 4-vector notation (Jackson 1999, p. 557):

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2 X-ray Generation by Laser-Electron Interaction

∂α F αβ = μ0 J β ,

(2.28)

where μ0 is the magnetic permeability of free space and ⎛

F αβ

⎞ E 0 − Ecx − cy − Ecz ⎜ E x 0 −Bz B y ⎟ c ⎟ =⎜ ⎝ E y Bz 0 −Bx ⎠ . c Ez −B y Bx 0 c

(2.29)

Recall from classical non-relativistic electrodynamics, that the magnetic field is defined in terms of a vector potential B = ∇ × A and the electric field E = −∇φ − ∂t∂ A in terms of a scalar potential φ and this particular vector potential A. Furthermore, these potentials are not uniquely defined which leaves the choice of a gauge function. Dealing with relativistic particles, the Lorentz-gauge ∇ · A+

1 ∂ φ = ∂α Aα = 0 c2 ∂t

(2.30)

is chosen due to its invariance under Lorentz-transformations of the ) ( potentials α (Jackson 1999, p. 555). Consequently, the 4-vector potential is A = φc , A . This potential can be used to replace the electromagnetic field in the inhomogeneous Maxwell expressions since F αβ = ∂ α Aβ − ∂ β Aα .

(2.31)

Inserting this expression into Eq. 2.28 yields ( ) ∂α ∂ α Aβ − ∂α ∂ β Aα = ∂α ∂ α Aβ − ∂ β (∂α Aα ) = μ0 J β .

(2.32)

The last term is brackets is the Lorentz condition ∂α Aα = 0. Therefore, this equation simplifies to ∂α ∂ α Aβ = μ0 J β ,

(2.33)

which is the wave equation of the 4-potential including a source term (Jackson 1999, p. 555). For example, a single electron or an electron bunch moving at relativistic velocities forms such a source. Therefore, this is the fundamental equation for the subsequent calculation of the radiation emitted by a relativistic charged particle.

2.2.2 Solution of the Wave-Equation The inhomogeneous wave equation ) for the 4-potential, Eq. 2.33, can be solved by ( finding a Green function D r, r ' which fulfils

2.2 From Covariant Electrodynamics to Liénard-Wiechert Fields

◻D(r, r ' ) = δ (4) (r − r ' ),

17

(2.34)

) ( ) ( where δ (4) (r − r ' ) = δ ct − ct ' δ r − r ' is a four-dimensional delta function (Jackson 1999, p. 612). The Green function can only depend on the 4-vector difference z = r − r ' in the absence of any boundaries. Both, D(z) and the δ-distribution, can be rephrased as the inverse Fourier transformation of their representations in frequency space: ◻

1 (2π)4

{

ik·z ˜ dk (4) D(k)e =

1 (2π)4

{

dk (4) eik·z ,

(2.35)

˜ denotes the Green function in frequency space and k is the frequency where D(k) 4-vector. Exchanging the order of integration and differentiation on the left hand side due to the Fourier derivative theorem and summarising both sides of the equation in one integral yields 1 (2π)4

{

] [ ˜ − 1 eik·z = 0, dk (4) (ik)2 D(k)

(2.36)

which means that the term inside the braces has to vanish. As a result, the solution of the Green function in frequency space is (Jackson 1999, p. 612) 1 1 ˜ =− 2 . D(k) =− k·k k 0 − κ2

(2.37)

Here, k0 is the time component of the 4-vector and κ = |k| the space component. Finally, inverse Fourier transformation of the result, Eq. 2.37, generates the shape of the potential in real space: D(z) = −

1 (2π)4

{

∞ −∞

{ dk0 eik0 z0

dk

e−i k·z . k02 − κ2

(2.38)

Obviously, the two singularities of the Green function in frequency space require special attention in this task. A convenient tool to solve this kind of integration is residuum calculus. First, the integral over the spacial frequencies is expressed in polar coordinates. The integration over the angles yields { ∞ { ∞ 2κ sin (κR) 1 ik0 z 0 dk0 e dκ 2 Dr (z) = (2π)3 R −∞ k 0 − κ2 0 { ∞ { ∞ 1 ∂ 2 cos (κR) dk0 eik0 z0 dκ , =− 3 ∂ R k02 − κ2 (2π) R −∞ 0

(2.39)

∂ −iκR cos θ where R = |z| and the identity e−iκR cos θ κ2 sin θ = iκR ∂θ e was used. The cosine can be expressed as a sum of two exponential functions once again, which

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can be summarised into exp [iκR] if the integration interval of κ is chosen from minus infinity to infinity instead. Afterwards, this integral can be solved by residuum calculus: { ∞ ( ) 1 (2.40) dk0 ei (z0 −R)k0 + ei (z0 +R)k0 . Dr (z) = 2 8π R −∞ For z 0 > 0, the latter is always zero since R > 0. Consequently, the Green function is Dr (r − r ' ) =

) Θ(ct − ct ' ) ( δ ct − ct ' − R . 4π R

(2.41)

This solution is called retarded or causal Green function because the source point time ct ' is always earlier than the observer point time. In other words, the delta function relates the potential observed at a time t and at position r to the time t ' when it was generated by the source at a position r ' . Analogously, for z 0 < 0 this results in the advanced Green function Da (r − r ' ) =

) Θ[−(ct − ct ' )] ( δ ct − ct ' + R . 4π R

(2.42)

Equations 2.41 and 2.42 can be put into a covariant form transforming the delta 1 [δ(ct − ct ' − R) + δ(ct − ct ' + R)]. functions with the identity δ[(r − r ' )2 ] = 2R One of the two delta functions in the sum is always zero and the covariant formulations of the Green functions are Θ(ct − ct ' ) δ[(r − r ' )2 ] 2π Θ(ct ' − ct) δ[(r − r ' )2 ]. Da (r − r ' ) = 2π Dr (r − r ' ) =

(2.43)

It should be noted that the retarded (advanced) Green functions are different from zero only in the forward (backward) light cone of the source point. Any potential can be constructed from either of the two Green functions by the addition of an arbitrary solution to the homogeneous wave equation. The complete solution of the wave equation in terms of the Green functions are (Jackson 1999, p. 614) α

A =

Aαin (r )

{ + μ0 {

or α

A =

Aαout (r )

+ μ0

d (4)r ' Dr (r − r ' ) J α (r ' )

(2.44)

d (4)r ' Da (r − r ' )J α (r ' ),

(2.45)

where Aαin (r ) and Aαout (r ) are such solutions of the homogeneous wave equations (Jackson 1999, p. 614). Equation 2.44 is called the retarded potential and Eq. 2.45

2.2 From Covariant Electrodynamics to Liénard-Wiechert Fields

19

the advanced potential, respectively. Usually, the retarded potential is used in calculations due to the human perception of cause and effect, but there is no compelling mathematical reason for choosing this solution (Zangwill 2012, p. 722 f.). Aαin (r ) has the interpretation of an incoming or incident potential, while Aαout (r ) is the asymptotically outgoing potential.

2.2.3 The Liénard-Wiechert Fields This section discusses the radiative effects of a point charge e moving on a trajectory r(t) in the inertial frame K in the absence of incoming fields Aαin (r ) = 0. The particle’s charge and current density are (Zangwill 2012, p. 871) ρ(x ' , t) = eδ[x ' − r(t)] j (x ' , t) = ev(t)δ[x ' − r(t)].

(2.46)

v(t) = ∂r(t) is the particle’s velocity. This transforms into the current 4-vector by ∂t considering the motion of the particle as a 4-vector coordinate of its proper time τ and integrating over proper time (Jackson 1999, p. 661): J α (x ' ) = ec

{

dτ V α (τ )δ (4) (x ' − r (τ )).

(2.47)

This current and the retarded Green function are plugged into Eq. 2.44: { { ecμ0 d (4) x ' dτ Θ(x0 − x0' )δ[(x − x ' )2 ]V α (τ )δ[x ' − r (τ )] 2π { (2.48) ] [ ecμ0 dτ V α (τ )Θ[x0 − r0 (τ )]δ (x − r (τ ))2 . = 2π

Aα (x) =

Integration yields only a contribution at τ = τ0 , defined by the light cone constraint [x − r (τ )]2 = 0 and fulfilling the retardation requirement defined by the thetafunction. Evaluation of the delta-distribution is simplified using the identity δ[ f (x)] =

∑ i

δ(x − xi ) |, |( | | ∂ f (x) ) | | | ∂x x=xi |

(2.49)

where xi are singular zeros of f (x) (Jackson 1999, p. 662). The time-derivative of the argument of the delta-distribution is −2[x − r (τ )]α V α . The result of Eq. 2.49 is inserted into the potential and the integral is evaluated. This generates the LiénardWiechert potential in its covariant form (Jackson 1999, p. 662):

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Aα =

| | Vα ecμ0 | . 4π V · [x − r (τ )] |τ =τ0

(2.50)

Most of the time, the non-covariant explicit form of the Liénard-Wiechert potentials are found in literature, which will be employed later on in the derivation of the undulator formulas. Therefore, these forms are derived as well. A simple transformation of the light-cone condition yields x0 − r0 (τ ) = |x − r(τ )| ≡ R. This expression can be used to evaluate the dot product in the denominator: ( ) ˆ = γc R 1 − β · nˆ , V · [x − r (τ )] = V0 [x0 − r0 (τ )] − v · [x − r(τ )] = γc R − γv · nR

(2.51) where nˆ denotes the unit vector in the direction of R. In combination with the definition of Aα = (φ/c, A) the more familiar forms of the Liénard-Wiechert potentials emerge (Jackson 1999, p. 663): [ φ=

e 1 ) ( 4πε0 R 1 − β · nˆ

[

] and A = ret.

ecβ μ0 ) ( 4π R 1 − β · nˆ

] .

(2.52)

ret.

β has to be evaluated at the retarded time cτ0 = ct − R calculated from the light-cone constraint. The electromagnetic field (tensor) is obtained from the potential with Eq. 2.31. However, the integral equation for the potential, Eq. 2.48, is advantageous in this respect. Integration and derivation can be exchanged. Derivation of the integrand with respect to the observer position x contributes to the integral only in the derivation of the delta function. The derivative of the theta function is the delta function δ[x0 − r0 (τ )] which constraints δ{[x − r (τ )]2 } = δ(−R 2 ). Since the only contribution at R = 0 is excluded, this differentiation does not contribute to the fields. Therefore, the derivative reads (Jackson 1999, p. 663) ∂ α Aβ =

ecμ0 2π

{

dτ V β (τ )Θ[x0 − r0 (τ )]∂ α δ{[x − r (τ )]2 }.

(2.53)

This can be further simplified by ∂ α δ[ f ] = ∂ α f

∂τ ∂ δ[ f ], ∂ f ∂τ

(2.54)

which results in ∂ α δ{[x − r (τ )]2 } =

[x − r (τ )]α ∂ δ{[x − r (τ )]2 }. V · [x − r (τ )] ∂τ

After integration by parts, the differential ∂ α Aβ is

(2.55)

2.3 The Undulator Equations

ecμ0 ∂ A = 2π α

β

{

∂ dτ ∂τ

21

[

] [x − r (τ )]α V β Θ[x0 − r0 (τ )]δ{[x − r (τ )]2 }. (2.56) V · [x − r (τ )]

Recognising that this equation is the same as (2.48) with V α (τ ) replaced by the derivative term, the solution of the integral is obtained similarly (Jackson 1999, p. 663): F αβ =

1 ∂ ecμ0 4π V · [x − r (τ )] ∂τ

[

]| [x − r (τ )]α V β − [x − r (τ )]β V α || . | V · [x − r (τ )] τ =τ0

(2.57) This is the covariant expression of electromagnetic field. The equation has to be evaluated at the retarded time after derivation as well. Starting from the general equation for the electromagnetic field tensor, the components of the electric and magnetic fields can be calculated explicitly. This work is quite tedious and therefore omitted here and only the final fields are presented. Jackson provides hints and partial results for the calculation of the individual contributions on page 663 f. in Jackson (1999).3 The equations [ ] c B = nˆ × E ret. e E= 4πε0 (

[

nˆ − β 2 γ (1 − nˆ · β)3 R 2 )(

velocity field Ev (static field)

]

{ }⎤ ⎡ ˙ e ⎣ nˆ × (nˆ − β) × β ⎦ (2.58) + (1 − nˆ · β)3 R 4πε0 c ret. ) ret. )( ) ( accelleration field Ea (radiation field)

describe the electromagnetic field of an electron moving on a certain trajectory and are called the Liénard-Wiechert fields. β˙ denotes the temporal derivative of β.

2.3 The Undulator Equations In this section, the general form of the spectrum emitted from a single particle is going to be calculated from the Liénard-Wiechert fields first. This result is then employed to derive the radiation formulas of an undulator in a second step.

2.3.1 The Liénard-Wiechert Spectrum This section is based on (Zangwill 2012, p. 880 ff.) Since the acceleration fields, in addition to falling off as 1/R, constitute an orthogonal triad with the retarded unit vector of the particle motion, they form a radiation 3

Alternatively, an explicit derivation of the Liénard-Wiechert field can be found in Zangwill (2012, p. 874 ff.).

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2 X-ray Generation by Laser-Electron Interaction

field. The directional instantaneous energy flux radiated by an accelerated particle is described by the Poynting vector (Zangwill 2012, p. 880) S=

1 [E a × B a ]ret. = ε0 c |E a |2 nˆ ret. , μ0

(2.59)

where ε0 denotes the dielectric permittivity of free space. To calculate the power emitted into a certain solid angle, the Poynting vector must be integrated over the surface of a sphere with the surface element R 2 dΩ. Accordingly, the power emitted into a unit solid angle is d P (t) = R 2 S (t) · nˆ r et . (2.60) dΩ From this result the total energy E can be calculated by integrating over time. dE = dΩ

{ dt

d P (t) = R2 dΩ

{

{ dt S (t) · nˆ ret. = ε0 c R 2

dt |E a (t)|2ret. .

(2.61)

|2 | |2 { | { 1 Applying Parseval’s theorem ( dt | F (t) | = 2π dω | F˜ (ω) | with F˜ (ω) the Fourier transform of F (t)) Eq. 2.61 takes the form (Zangwill 2012, p. 887) dE ε0 c R 2 = dΩ 2π

{

{ | | |2 |2 ε0 c R 2 ∞ |˜ |˜ | | dω | E a (ω)| = dω | E . a (ω)| ret. ret. π 0

(2.62)

In the last step, the reality condition E (t) = E ∗ (t) was applied, which implies ˜ ∗ (ω) = E ˜ (−ω). With (Zangwill 2012, p. 887) the relation E dE = dΩ

{

d 2E dω = dωdΩ

{ dω

d I (ω) dΩ

(2.63)

the equation for the angular intensity becomes4 |2 d I (ω) d 2E ε0 c R 2 || ˜ ε0 c R 2 | = = | E a (ω)| = ret. dωdΩ dΩ π π

|{ |2 | | | dt E a (t) e−iωt | . | |

(2.64)

ret.

The last step remaining is the evaluation of the Fourier transformation of the radiation part of the Liénard-Wiechert field (Eq. 2.58) μ0 ce2 d 2E = dωdΩ 16π 3

|{ |2 [( ) ] | nˆ × nˆ − β × β˙ −iω(tr +R(tr )/c) || | e | dtr | . )2 ( | | 1 − β · nˆ

(2.65)

Please keep in mind that the engineering convention for the phase term φ = ωt − k · r is used in this thesis.

4

2.3 The Undulator Equations

23

Fig. 2.2 Sketch of the electron trajectory through an undulator (a) and of the geometry used in the calculation of the undulator equations (b). b is inspired by Fig. 26.3 of Wiedemann (2015)

Here, the retarded time ctr = ct − R was inserted for t and the corresponding ˆ ret (Zangwill 2012, transformation of the integration constant ∂t/∂tr = [1 − β · n] p. 874) was performed. Since the radiation is usually observed sufficiently far from the source, the direction of R (tr ) is approximately constant. Thus, the first part of the fraction of the integrand is a perfect derivative [ ( )] [( ) ] nˆ × nˆ − β × β˙ d nˆ × nˆ × β ) , ( = )2 ( dtr 1 − β · nˆ 1 − β · nˆ

(2.66)

which enables integration by parts (Zangwill 2012, p. 889). In the realistic case that the particle velocity vanishes for tr → ±∞, the Liénard-Wiechert spectrum simplifies to |{ | ( ) −iω(tr +R(tr )/c) |2 μ0 ce2 ω 2 || d 2E | , ˆ ˆ = dt n × n × β e r | dωdΩ 16π 3 |

(2.67)

the form that is going to be employed in the following analysis of the undulator radiation (Zangwill 2012, p. 890).

2.3.2 Particle Trajectory and Wiggling Strength Parameter K From here on the derivation follows the one given by Wiedemann in his book Particle Accelerator Physics, Chap. 26 Wiedemann (2015) until Sect. 2.5, except stated differently explicitly. Before Eq. 2.67 can be evaluated, the particle trajectory inside the undulator R (τ ) has to be calculated. In the following, the z-axis is oriented parallel to the undulator axis. The undulator is of finite length and sufficiently large transverse extent that the vertical magnetic field can be assumed constant in the range containing the particle’s deflection in the horizontal direction. Both, a laser undulator and (to first order)

24

2 X-ray Generation by Laser-Electron Interaction

a permanent magnet undulator, consist of sinusoidally oscillating magnetic fields. For the analysis of the particle trajectory, the origin of the coordinate system is located at the symmetry centre of the undulator at the cusp of the magnetic field, i.e. B y = −B0 cos(ku z) and Bx = Bz = 0, where ku = 2π/λu and λu is the undulator period. For a permanent magnet undulator this position is located in the middle of a magnetic pole. At t = 0 the particle is located at this position and its transverse velocity is x˙0 = 0 while its longitudinal velocity is z˙ 0 = βc. It is deflected by the Lorentz force: d (2.68) γmcβ = −ec [E + (β × B)] . dt This results in two equations of motions along the x- and z-axis: d eB0 dz βx = − cos (ku z) dt γmc dt d eB0 d x βz = + cos (ku z) . dt γmc dt

(2.69)

The left hand side of the first equation is integrated over dz first, before time integration is performed on both sides, leading to βx = −

eB0 K sin (ku z) ≡ − sin (ku z) , γ γmcku

(2.70)

with K =

eB0 mcku

(2.71)

the wiggling strength or K -parameter of an undulator (Wiedemann 2015, p. 820). βz can be calculated by inserting the result for βx and integrating βx cos (ku z) by parts. Considering the initial condition z˙ 0 = βc, the integration constant C = β and the longitudinal velocity becomes ] [ ] K2 K2 K2 2 βz = β 1 − 2 2 sin (ku z) = β 1 − 2 2 + 2 2 cos (2ku z) . 2β γ 4β γ 4β γ [

(2.72)

The transverse motion, Eq. 2.70, exhibits the expected oscillatory behaviour. The associated increase in transverse momentum reduces the one along the longitudinal direction due to conservation of energy, thereby decreasing the particles velocity along the z-axis. Due to the sin2 -term in βz , two oscillations occur in the longitudinal direction within one oscillation in the transverse direction. A convenient Cartesian reference frame for such a system is the one moving along at the particles average velocity. The latter, defined by ⟨˙z ⟩ = β¯ c , can be calculated from Eq. 2.72 to (Wiedemann 2015, p. 898)

2.3 The Undulator Equations

25

( ¯β = β 1 −

K2 4γ 2 β 2

) .

(2.73)

Replacing z = β¯ ct and integrating Eqs. 2.70 and 2.72 respectively directly yields the deflections in the horizontal and longitudinal direction (Wiedemann 2015, p. 898) ( ) K ¯ cos ku βct βγku ( ) K2 z = β¯ ct + 2 2 sin 2ku β¯ ct , 8β γ ku

x=

(2.74)

which equals a figure-eight trajectory in the reference frame moving at the particle’s average longitudinal velocity. The maximum distance of the particle during its transverse oscillation is the amplitude of the transverse oscillation (Wiedemann 2015, p. 898) K . (2.75) a= βγku

2.3.3 The Effective Undulator Parameters of Electromagnetic Waves In the preceding discussion, a general magnetic field has been assumed, which can be easily determined for permanent magnet undulators. In contrast, the intensity, thus the electric field, is commonly measured for lasers. Recall in this respect that the electric and magnetic fields are interconnected via c B = n × E. Without loss of generality, the laser’s electric field is assumed to be polarised in the x-axis resultˆ where k L is the laser’s spacial frequency. If ing in E = −E 0 cos (ck L t − k · r) x, these expressions are inserted into the Lorentz-force equation (2.68), the particle’s transverse velocity becomes βx,laser und. = −

eE 0 sin (ck L t − k · r) . γmc2 k L

(2.76)

A comparison of this result with Eq. 2.70 yields the laser K-parameter K L , often denoted as the dimensionless normalised peak vector potential a0 of the laser: / √ e 2cμ0 I L 2re I L eE 0 a0 ≡ K L = λL , = = 2 2 mc k L mc k L πmc3

(2.77)

where λ L is the laser wavelength, I L the laser’s intensity and re the classical electron radius (Kim et al. 2017, p. 70).

26

2 X-ray Generation by Laser-Electron Interaction

Finally, the electromagnetic wave of a laser propagates at the speed of light contrary to the static magnetic field of a classical permanent magnet undulator. Consequently, a transformation into the rest frame of the electron Lorentz-contracts the latter only, while the former is undergoing a relativistic Doppler-shift. Since the electron emits dipole-radiation at the frequency of its oscillation in its rest frame, this frequency is different for a permanent magnet undulator and a laser one, even if their (hypothetical) period is identical. If the same formulas should remain valid for both types of undulators, an effective undulator period has to be defined for the laser undulator which takes into account the different results of their Lorentz-transformations. To determine the effective undulator period of a laser λu,L , Eqs. 2.70 and 2.76 are Lorentz-transformed into the average rest frame of the particle moving at a velocity β¯ parallel to the z-axis employing Eq. 2.5. For the laser, the equations for the relativistic Doppler-shift (Eq. 2.25) fulfil this purpose. Furthermore, the particle is located at the origin in its rest frame, i.e. z ' = 0. permanent magnet undulator x ∝ cos (ku z) ( ) x ∝ cos ku γβ¯ ct ' ( ) x ∝ cos ku γβ¯ ct '

laser undulator x ∝ cos (k L ct − k L · r) ( ( ) ) x ∝ cos k L γ 1 − β¯ · nˆ ct ' − k⊥,L · r⊥' ( ) x ∝ cos ku,L γβ¯ ct ' − k⊥,L · r⊥' ,

where in the last step

ku,L ≡

( ) k L 1 − β¯ · nˆ β¯

λ L β¯

λu,L ≡ (

1 − β¯ · nˆ

)

(2.78)

was used. Since β¯ = β¯ zˆ and nˆ contains a negative component in the z-axis for a counter-propagating electromagnetic wave, the sign is inverted leading to the expected frequency up-shift. Consequently, the laser undulator parameter is λu,L =

λ L β¯ 1 + β¯ cos (Ψ)

(2.79)

in the typical case of a laser counter-propagating to the electron at an angle Ψ (Kim et al. 2017, p. 70). In conclusion, the only differences between permanent magnet undulators and optical ones are that K and λu have to be exchanged for a0 and λu,L , respectively, but the physics remains the same. Therefore, only the general terms K and λu as well as quantities derived from them are used throughout the following derivations due to their general validity.

2.3 The Undulator Equations

27

2.3.4 Undulator Radiation A very intuitive access to the radiation process of an undulator is to perceive the generated radiation as a superposition of individual radiation fields generated from the Nu sources, where Nu is the number of undulator periods. Their interference results in the generation of quasi-monochromatic (harmonic) radiation (Wiedemann 2015, cf. p. 899 f.). The time T it takes the electron to travel over one undulator period on its trajectory is determined by the undulator period and the electron’s average velocity (Eq. 2.73) (Wiedemann 2015, p. 899): T =

λu λu [ ( )] . = cβ 1 − K 2 / 4γ 2 cβ¯

(2.80)

Accordingly, the wave front of the radiation travels a distance sr f = T c =

λu [ ( )] , β 1 − K 2 / 4γ 2

(2.81)

which is longer than a single undulator period λu . For constructive interference, the difference between the distance covered by the electron and the wave front has to be an integer multiple of the wavelength of the emitted radiation, i.e. mλm = sr f − λu cos (ϑ) =

( ) λu 1 ( )] − λu 1 − ϑ2 , 2 β 1 − K 2 / 4γ 2 [

(2.82)

where ϑ accounts for small observation angles. A schematic for the resonance condition is depicted in Fig. 2.3. Under the assumptions K ≪ γ 2 and β ≈ 1, this equation can be rearranged to the well known undulator equation (Wiedemann 2015, p. 900) mλm =

Fig. 2.3 Interference condition for the undulator radiation. Figure after Fig. 26.2 in Wiedemann (2015)

λu 2γ 2

( 1+

) K2 + γ 2 ϑ2 . 2

(2.83)

28

2 X-ray Generation by Laser-Electron Interaction

If the undulator would be infinitely long, the emitted radiation would be a comb of spectral lines determined by Eq. 2.83. In particular, the highest frequency of the radiation is emitted directly in the forward direction, while it is red-shifted at finite angles ϑ due to the Doppler effect. For K → 0, as it is usually the case for a laser undulator in the linear Thomson regime like at the MuCLS, the particle oscillation in the electron rest frame remains purely non-relativistic suppressing high harmonics as will be derived in the next section.

2.3.5 Spatial and Spectral Distribution of Undulator Radiation After the determination of the particle trajectory in the undulator and the introduction of generally valid parameters, like the K -parameter or λu , these results can be plugged into the Liénard-Wiechert spectrum (Eq. 2.67) yielding the spectral and spatial distribution of the undulator ( )radiation. As a first step, the nˆ × nˆ × βˆ term has to be evaluated. nˆ denotes the unit vector of the spatial frequency of the radiation, i.e. the unit vector pointing from the observer to the source. From the geometry displayed in Fig. 2.2b (Wiedemann 2015, p. 902) follows that nˆ = − cos (ϕ) sin (ϑ) xˆ − sin (ϕ) cos (ϑ) ˆy − cos (ϑ) zˆ .

(2.84)

In the next step, β is jointly determined with the vector R = nˆ · R which connects the observer and the source and is contained in the retarded time. This vector can be expressed by a constant vector r 0 connecting the origin of the coordinate system with the observer and by the varying vector r p connecting the origin of the coordinate system and the particle. Thus, R = −r 0 + r p , cf. Fig. 2.2b. Since the first term is constant, it just adds a constant phase which is cancelled by taking the absolute square. This justifies its direct omission. The components of r p have been determined previously and are displayed in Eq. 2.74. rp =

] [ 2 K ¯ r + K sin (2ωu t) zˆ , cos (ωu tr ) xˆ + βct γku 8γ 2 ku

where 1/β ≈ 1 was applied and

ωu = ku β¯ c

(2.85)

(2.86)

was introduced (Wiedemann 2015, p. 902). The derivative of r p is the desired solution for β (Wiedemann 2015, p. 902): [ ] K K2 ¯ ¯ β = −β sin (ωu t) xˆ + β 1 + 2 cos (2ωu t) zˆ . γ 4γ

(2.87)

2.3 The Undulator Equations

29

( ) With these expressions, the cross-product nˆ × nˆ × βˆ is evaluated. Its solution simplifies greatly, if the following approximations are applied (Wiedemann 2015, p. 903): • • • • •

Emission of radiation into a narrow cone with ϑ ≪ 1. Low wiggler strength parameter K ≪ γ. β¯ ≈ β. β ≈ 1 whenever β does not appear as a difference to unity. Disregard second order terms in ϑ and K /γ.

The final result is (Wiedemann 2015, p. 903) ) ) ( ( ) K ¯ ¯ ˆ nˆ × nˆ × β = β ϑ cos (ϕ) + β sin (ωu t) xˆ + β¯ ϑ sin (ϕ) ˆy. γ (

(2.88)

In the approximation ϑ ≪ 1, the retarded time in the exponential term of Eq. 2.67 becomes (Wiedemann 2015, p. 903) ) K β¯ ϑ ( 1 K 2 β¯ tr + nˆ · r p = tr 1 − β¯ cos (ϑ) − cos (ϕ) cos (ωu tr ) − 2 sin (2ωu tr ) 8γ ωu γωu c (2.89) if only linear terms are included. Please note, that for cos (ϑ) ≈ 1 − ϑ2 /2, the first term ( ) ) ( 1 K2 ωu + γ 2 ϑ2 = (2.90) 1 − β¯ cos (ϑ) = 2 1 + 2γ 2 ω1 reproduces the resonance condition derived in Eq. 2.83 (Wiedemann 2015, p. 904). ω1 denotes the fundamental frequency of the undulator. The final step is the integration of the Liénard-Wiechert spectrum of Eq. 2.67 with the expressions (2.88) and (2.89). This integral is not straight forward to solve. However, a detailed walk-through is provided in Chap. 26 (pp. 904–910) of the book “Particle Accelerator Physics” by Wiedeman (2015).5 After integration, the final Liénard-Wiechert spectrum is ] ∞ [ ω 2 ∑ sin (π Nu Δωm /ω1 ) 2 d 2E 2 = hαβ¯ Nu2 2 dωdΩ ωu m=1 π Nu Δωm /ω1 ( ∑ ) )2 K ∑ 2 2 ( ∑ sin (ϕ) ˆy2 , × ϑ cos (ϕ) − xˆ + ϑ 2 1 1 2γ

(2.91)

where α is the fine structure constant, Δωm ω = −m ω1 ω1 5

Note: There seem to be errors (typos?) in the equations of a few intermediate results.

(2.92)

30

and

2 X-ray Generation by Laser-Electron Interaction

∑ ∑ 1, 2 are sums of Bessel’s functions Jl (u): ∑ 1

∑ 2

= =

∞ ∑ l=−∞ ∞ ∑

J−l (u) Jm−2l (ν)

(2.93)

[ ] J−l (u) Jm−2l−1 (ν) + Jm−2l+1 (ν) ,

(2.94)

l=−∞

with the arguments u=

ω K 2 β¯ ) ( ω1 4 1 + K 2 /2 + γ 2 ϑ2

and

ν=

ω 2 K β¯ γϑ cos (ϕ) . ω1 1 + K 2 /2 + γ 2 ϑ2

(2.95)

The unit vectors in the x- and y-direction were kept to indicate the polarisation of the radiation.

2.4 Discussion of the Undulator Line Spectrum The width of the Liénard-Wiechert spectrum is given by the first zero-crossing of the sin(x)/x-term, which is typical for multi-beam interference, in this case of the radiation produced at the individual undulator periods. This occurs at π Nu Δωm /ω1 = ±π, leading to (Wiedemann 2015, p. 909) 1 Δωm =± . m Nu ωm

(2.96)

Since the relative bandwidth is inversely proportional to the number of undulator periods as well as the harmonic order, the relative bandwidth is typically very narrow. Thus, the term ω/ωu in Eq. 2.91 can be approximated well by ω ωm ω1 2γ 2 ≈ =m =m . ωu ωu ωu 1 + K 2 /2 + γ 2 ϑ2

(2.97)

Equation 2.90 was applied in the last step. If this is plugged into Eq. 2.91, the equation simplifies to: ] [ ∞ ∑ sin (π Nu Δωm /ω1 ) 2 d 2E 2 = hαβ¯ γ 2 Nu2 m2 dωdΩ π Nu Δωm /ω1 m=1 ( ( )2 ∑ ∑ )2 ∑ 2γϑ 1 cos (ϕ) − K 2 xˆ 2 + 2γϑ 1 sin (ϕ) ˆy2 . × ( )2 1 + K 2 /2 + γ 2 ϑ2

(2.98)

2.4 Discussion of the Undulator Line Spectrum

31

Another quantity that is very useful, especially for planning experiments, is the number of photons Nph that are produced at a certain energy or within a certain energy window. This can be easily derived from Eq. 2.98 by dividing the total energy of the radiation by the energy of the photon, i.e. E/ (hω) (Wiedemann 2015, p. 910, Eq. 26.54/Δω): [ ] ∞ d 2 Nph 1 ∑ 2 sin (π Nu Δωm /ω1 ) 2 2 = αβ¯ γ 2 Nu2 m ω m=1 dωdΩ π Nu Δωm /ω1 ( ( )2 ∑ ∑ )2 ∑ 2γϑ 1 cos (ϕ) − K 2 xˆ 2 + 2γϑ 1 sin (ϕ) ˆy2 . × ( )2 1 + K 2 /2 + γ 2 ϑ2

(2.99)

While the sin (x) /x-term defines the bandwidth of the emitted harmonic radiation only, the Bessel’s functions determine their intensity. Since K ≪ 1 for most laser undulators, e.g. at the MuCLS, the argument of the Bessel’s function u ∝ K 2 (cf. Eq. 2.95) is very small. Therefore, higher order Bessel’s functions barely contribute to the spectrum which in this case consists of the fundamental line only. For stronger undulators, such as typical permanent magnet ones installed at synchrotrons, K > 1 and the higher order content of the Bessel’s functions increases and higher order harmonic radiation emerges in the line spectrum of the undulator. In the forward direction (ϑ = 0), the intensity of the polarisation in the x-axis is strongly peaked, while the radiation in the y-axis vanishes completely. The same holds true for all even harmonics m in the x-axis. The latter follows from the fact that the second argument of the Bessel’s functions ν is zero at ϑ = 0, since it is proportional to ϑ. Since the Bessel’s functions Jm−2l±1 (ν) are uneven for all even harmonics m, they are zero at ν = 0 and in turn no radiation is produced at even harmonics in the forward direction. Moreover, only the ∑ lowest order Bessel’s function does not vanish for ν = 0. Therefore, the sum 2 simplifies greatly, since only the two elements of the summation remain for which the Bessel’s function’s index m − 2l ± 1 = 0 which results in (Wiedemann 2015, p. 915 ff.) ∑ 2

=

∞ ∑

[ ] J−l (u ϑ=0 ) Jm−2l−1 (0) + Jm−2l+1 (0)

l=−∞

(2.100)

= J− m−1 (u ϑ=0 ) + J− m+1 (u ϑ=0 ) . 2 2 In this case the photon number can be expressed by | [ ]2 ∞ ∑ d 2 Nph || K2 2 sin (π Nu Δωm /ω1 ) ¯ 2γ2 N 2 1 ( = αβ m JJ 2 , ) u ω 1 + K 2 /2 2 m=1 dωdΩ |ϑ=0 π Nu Δωm /ω1 (2.101)

32

2 X-ray Generation by Laser-Electron Interaction

(

where JJ = J− m−1 2

mK2 4 + 2 K2

(

) + J− m+1 2

mK2 4 + 2 K2

) .

(2.102)

The amplitude of the emitted intensity scales with the square of the number of undulator periods Nu . This result arises from the interference of multiple beams which concentrates the generated radiation ever more into a single frequency (and its harmonics) as the number of interfering beams, or in other words undulator periods, increases. An effect which is much more difficult to directly extract from Eq. 2.101 is that the radiated intensity increases with the wiggling strength parameter for weak undulators (K ≪ 1). Consequently, an optimal K -parameter for maximum photon flux exists for a given harmonic because at large K-parameters the flux decreases again. Finally, the opening angle of the forward radiation cone is calculated, which is determined by the sin(x)/x-term again. The opening angle is defined as the angle at which the first zero crossing of the aforementioned term occurs, i.e. π Nu Δωm /ω1 = 2γ 2 ±π. The fundamental wavelength ω1 = ωu 1+K 2 /2 , the one of the m-th harmonic N mγ 2 ϑ2

2

m u m ωm = ωu 1+K 2γ 2 /2+γ 2 ϑ2 and Eq. 2.92 are inserted which leads to 1+K 2 /2+γ 2 ϑ2 = 1. Solvm m ing the latter for ϑm yields (Wiedemann 2015, p. 916)

/ 1 ϑm = γ

1 + K 2 /2 . m Nu − 1

(2.103)

For a very long undulator (m Nu ≫ 1), the root-mean-square (rms) value of the opening angle is (Wiedemann 2015, p. 916) / 1 1 σr ≈ √ ϑm = γ 2

1 + K 2 /2 . 2 m Nu

(2.104)

The total amount of radiation emitted into this small solid angle ΔΩ = πσr2 . With this result, the total number of photons emitted into a narrow bandwidth Δω by a 2 single electron is calculated to be (Wiedemann 2015, p. 917, factor β¯ neglected therein) | π 2 Δω m K 2 JJ2 . (2.105) Nph (ωm )|ϑ=0 = αβ¯ Nu ωm 1 + K 2 /2 2 The photon flux is obtained by multiplying above equation with I /e, where I is the current flowing through the undulator. This assumes the ideal situation, in which the single particle flux is just multiplied by the number of electrons which traverse the undulator per second. In reality, the electron beam has a certain radius, duration and divergence, which influence the photon number and spectrum. As long as the angular distribution of the electron spread is much smaller than the one of

2.5 Comparison of a MuCLS-Type Laser Undulator with a Permanent Magnet One

33

the undulator radiation as well as the electron distribution is uncorrelated in energy and position, the single-particle energy density distribution in the forward direction results in (Kim et al. 2017, p. 61 (there only for the fundamental harmonic)) [ ( )) ]2 ( { ∞ ∑ ( ) sin π Nu 2η j − Δωm /ω1 d2E K2 2 2 ( ) dη = hαγ 2 Nu2 ( m JJ f η , j j )2 dωdΩ π Nu 2η j − Δωm /ω1 1 + K 2 /2 m=1

(2.106) ( ) where f η j describes the probability distribution function of the electron with particle index j.

2.5 Comparison of a MuCLS-Type Laser Undulator with a Permanent Magnet One Figure 2.4 displays properties of the X-ray radiation emitted from a single particle according to Eq. 2.99 for different undulator configurations. For this comparison, the highest electron energy achievable at the MuCLS, i.e. 45 MeV, was chosen for the electrons travelling through the laser undulator. In conjunction with the Nd:YAG laser operating at 1064 nm, an effective length of 3350 periods and a pulse energy of 3.9 mJ (250 kW power stored inside the cavity) this electron energy results in a peak X-ray energy of 36.15 keV. The two permanent magnet undulator configurations were chosen such that the same X-ray energy is generated as well, once at the fundamental of the undulator and once at its 11th harmonic. In the calculations, an undulator period of 16 mm, a length of 100 periods and a magnetic field strength of 1 T, i.e. typical undulator parameters, were used. This results in a K-parameter of 1.5 of the undulator. Accordingly, the electron energy has to be 11.4 GeV at this undulator to generate fundamental X-ray radiation at 36.15 keV, while only 3.4 GeV are required to produce this radiation at the 11th harmonic, i.e. an electron energy in the range of typical medium-sized electron storage rings. Figure 2.4a–c displays the angular intensity distribution of the radiation cones with the polarisation along the x-axis for the three undulator configurations–laser undulator, permanent magnet undulator operated at the fundamental wavelength and the same permanent magnet undulator at the 11th harmonic. The intensity distribution of the polarisation along the y-axis is shown in Fig. 2.4d–f. The most prominent difference between intensity distributions of the radiation emitted by a laser undulator and a permanent magnet one is the latter’s much narrower radiation cone. Since the opening angle of the radiation cone scales as 1/γ, the radiation emitted from a typical laser undulator is compressed into ≈160 µrad in (b) and ≈0.5 mrad in (c), cf. Fig. 2.4. Figure 2.4g displays a zoom-in of Fig. 2.4b which demonstrates that the shape of the intensity distributions in Fig. 2.4b, e look alike Fig. 2.4a, d because they all are radiation cones at the fundamental harmonic. However, the shape of the radiation emitted at the 11th harmonic is very different, as can be seen from the zoom-in

34

2 X-ray Generation by Laser-Electron Interaction

Fig. 2.4 Comparison of 36 keV-radiation emitted by a single electron for a laser undulator, the fundamental of a permanent magnet undulator, and a permanent magnet undulator operating at harmonic 11. The laser wavelength is 1064 nm and the electron energy 45 MeV. The permanent magnet undulator has a period of 16 mm and a magnetic field strength of 1 T resulting in a Kparameter of 1.5. The electron energies are 11.4 GeV at the fundamental and 3.4 GeV at the 11th harmonic, respectively. a–c display the normalised spatial intensity distribution of the resonant wavelength of the three different undulator configurations for X-ray radiation polarised along the xaxis, while d–f display the intensity distribution with polarisation along the y-axis. Since permanent undulators require a much higher γ-factor of the electrons, the radiation cone is narrowed down by the ratio of the gamma factors. However, the shapes of the radiation cones of b and e are the same as a and d as can be seen from the zoom-in of b depicted in g. The distribution of the radiation is very different for higher harmonics, thus h is a zoom-in on the cone for the horizontal polarisation of the 11th harmonic and i the one corresponding to the vertical polarisation

2.5 Comparison of a MuCLS-Type Laser Undulator with a Permanent Magnet One

35

Fig. 2.5 Spectral intensity comparison of a laser and permanent magnet undulator in the single particle picture. The normalised spectral intensity for a laser undulator, a permanent magnet one at the fundamental harmonic and the 11th one are displayed. As expected the spectral intensity bandwidth decreases for the laser undulator due to its higher number of periods and the higher harmonic of the permanent magnet undulator compared to its fundamental. Although the shape of the spectrum is similar, the spectral flux of the permanent magnet undulator is much higher due to its stronger K -parameter and operation at the fundamental at high electron energy. The modulations at the 11th harmonic arise from the complex shape of the radiation lobe depicted in Fig. 2.4h, i. The small rapid oscillations in the spectral flux of the laser undulator arise from the numerical integration

onto the horizontal polarisation depicted in Fig. 2.4h and the one onto the vertical polarisation displayed in Fig. 2.4i. The former exhibits a series of 11 peaks along the horizontal axis, while the latter incorporates a double comb of peaks, but no intensity along the horizontal axis, as would be expected from the sin (ϕ)-term in Eq. 2.99 and holds true for the fundamental as well. The change of the radiation cone’s shape at the higher harmonics arises from the sum of Bessel’s functions which contain the harmonic number m in the index sum, cf. Eqs. 2.93 and 2.94. The peaks with X-ray polarisation along the x-axis show an asymmetry between the non-deflecting plane and the deflecting plane, very well visible in the lobes of the fundamental harmonic (Fig. 2.4a, g) as well. This asymmetry originates from the particle’s motion, cf. the cos(ϕ)-term in Eq. 2.91. In summary, the distribution of the radiation arising from a laser undulator is the same as the one from a permanent magnet one, but emitted into a wider cone due to the reduced γ-factor of the electron. Figure 2.5 displays the integral spectral flux emitted into a cone angle of ± 4 mrad for the laser undulator The corresponding cone angles of the permanent magnet undulator were derived downscaling the ±4 mrad -cone angle by a γ-dependent √ spectral compaction factor 2γe,laser /γe,magnet . The square root of two originates from the relativistic Doppler-shift of the laser undulator compared to the magnetic one. The spectra have been normalised to the maximum of the spectral flux emitted from

36

2 X-ray Generation by Laser-Electron Interaction

the permanent magnet undulator operated at the fundamental. Its flux is about two orders of magnitude larger than the one achieved by operation at the 11th harmonic. Nevertheless, this is still three orders of magnitude higher than the one of a laser undulator. However, the spectrum at the 11th harmonic features prominent energydependent flux modulations. The latter originates from the intensity pattern of the 11th harmonic displayed in Fig. 2.4h, i which is heavily modulated by the terms containing the sum of Bessel’s functions compared to the radiation cone of the fundamental depicted in Fig. 2.4a, b and g. Consequently, both spectra obtained from undulators radiating at their fundamental exhibit a continuous decrease in flux with decreasing X-ray energy. All spectra possess a sharp low energy cut-off at about 32 keV originating from the chosen aperture size. Its steepness depends on the natural bandwidth of the undulator configuration which is given by Eq. 2.96. Consequently, the broadening of both, the on-axis harmonic energy- and the aperture cut-off, are most prominently affected for the short permanent magnet undulator operated at the fundamental. Here, the bandwidth is about 360 eV. This edge-spread is reduced to approximately 33 eV by switching to the 11th harmonic and lower electron energy while for the long laser undulator the energy spread basically vanishes. As a final remark, please be aware, that the discussion in this section was based on a single particle picture and does not include 3D-effects, such as the transverse variation of the K -parameter in a real laser with a 3D-Gaussian intensity profile or properties of an extended electron beam, such as its emittance. These effects could be included by statistical averaging over an ensemble of electrons which is distributed according to the laser and electron beam properties. The simplicity of the undulator model might be beneficial for the description of X-rays in electron-laser interaction where the beam size is much larger compared to the electron beam in all three directions of space, because the laser can be approximated by a one-dimensional undulator with constant K -parameter.

2.6 X-ray Generation by Inverse Compton Scattering The wave-particle duality of light allows an equivalent description of the interaction between the laser and the relativistic electron beam in terms of particle collisions in the rest frame of the electron. Therefore, the calculations in the scattering picture are presented in the following. Since momentum transfer in this process shifts the photon energy to the X-ray regime in the laboratory frame, this process is commonly called inverse Compton scattering.

2.6.1 The Scattered Photon’s Frequency The scattered photon’s frequency ωs can be derived from energy and momentum conservation using the 4-vectors of the momenta of the photon

2.6 X-ray Generation by Inverse Compton Scattering

37

k = (k0 , k) = (hω/c, hk)

(2.107)

p = ( p0 , p) = (γmc, γmcβ) = ( p0 , p0 β).

(2.108)

and electron

Momentum conservation requires that momenta before and after the collision are the same, i.e. (2.109) p + k = ps + k s , in which the subscript s denotes scattered quantities. Solving this equation for the scattered momentum and taking the square of the resulting equation yields: ps · p s = p · p + p · k − p · k s + k · p + k · k − k · k s − k s · p − k s · k + k s · k s . (2.110) The scalar products can be evaluated independently. In the following calculations the free space dispersion relation ω = kc is used and nˆ is a unit vector as usual. k · k = ks · ks = 0

(2.111)

ˆ p · k = p0 k0 (1 − β · n) p · ks = p0 k0,s (1 − β · nˆ s )

(2.112) (2.113)

p · p = p s · ps = m 2 c 2 k · ks = k0 k0,s (1 − nˆ · nˆ s )

(2.114) (2.115)

Using these results, Eq. 2.110 can be solved for the frequency of the scattered radiation when the explicit forms of Eqs. 2.107 and 2.108 are inserted. The final form of the scattered photons frequency is ωs =

ˆ ω(1 − β · n) . hω (1 − nˆ · nˆ s ) 1 − β · nˆ s + γmc 2

(2.116)

Consider a highly-relativistic electron colliding with a counter-propagating lowenergy laser photon which both move along the z-axis. In this case nˆ and β are antiparallel. For backscattered radiation, nˆ s and β are parallel. This simplifies Eq. 2.116 to ω(1 + β) , (2.117) ωs = 2hω 1 − β + γmc 2 which can be rephrased in terms of the electron energy Ee , the laser photon energy El and the X-ray photon energy EX-ray =

(1 + β)2 El Ee γ 2 (1 + β)2 El Ee = . Ee + 2γ 2 (1 + β)El (1 − β 2 )Ee + 2(1 + β)El

(2.118)

38

2 X-ray Generation by Laser-Electron Interaction

As long as γ 2 El ≪ Ee , the electron recoil can be neglected while β ≈ 1 for the highly relativistic electron. Therefore, Eq. 2.118 simplifies to EX-ray = 4γ 2 El

(2.119)

for backscattering a counter-propagating laser photon from a highly relativistic electron. Since the maximum momentum transfer takes place at backscattering, this equation describes the high energy cut-off of the X-ray radiation obtained via linear inverse Compton scattering.

2.6.2 The Klein-Nishina Differential Cross-Section This section is based on “Laser pulsing in linear Compton scattering” by Krafft et al. (2016). A prerequisite required to calculate the quantitative spectrum resulting from inverse Compton scattering is the differential cross section for this process. The detailed derivation of this cross-section is out of the scope of this thesis and was already performed by Klein and Nishina (1929). For linearly polarised incident and scattered photons, it can be expressed by r2 dσ = e 4 dΩ

(

ωb,s ωb

)2 [

]

ωb ωb,s + − 2 + 4(εb · εb,s ) ωb ωb,s

(2.120)

in the electron beam rest frame. The subscript b denotes quantities specified in this rest frame. re is the classical electron radius and ε = (0, ε) a polarisation 4-vector. The frequencies and wave vectors in the electron beam frame can be easily expressed with Eq. 2.24 in terms of the laboratory frame: ˆ ωb = γ(1 − β · n)ω

) ) ( ˆ ˆ (β · n)β (β · n)β ˆ − β + k n − β2 β2 γ−1 ˆ = −γkβ + k nˆ + k 2 (β · n)β β ωb,s = γ(1 − β · nˆ s )ωs γ−1 kb,s = −γks β + ks nˆ s + ks 2 (β · nˆ s )β. β (

kb = k∥,b + k⊥,b = γk

(2.121)

Again, the invariant scalar products of k · k and ks · ks vanish and accordingly ˆ 2 kb · kb = γ 2 (1 − β · n) kb,s · kb,s

( ω )2

c ( )2 2 2 ωs . = γ (1 − β · nˆ s ) c

(2.122) (2.123)

2.6 X-ray Generation by Inverse Compton Scattering

39

Using this, the unit vector in the electron beam rest frame can be expressed as ] [ 1 γ−1 ˆ (β · n)β −γβ + nˆ + nˆ b = ˆ γ(1 − β · n) β2 ] [ γ−1 1 (β · nˆ s )β . −γβ + nˆ s + nˆ b,s = β2 γ(1 − β · nˆ s )

(2.124) (2.125)

From these expressions for the transformation of the unit vector and performing a gauge transformation to eliminate the zeroth component of the polarisation 4-vector, the polarisation vectors εb = (0, εb ) and εb,s = (0, εb,s ) can be deduced from the ones in the laboratory frame ε = (0, ε) and εs = (0, εs ), respectively: γ−1 (β · ε)β β2 γ−1 + εs + (β · εs )β. β2

εb = γ(β · ε)nˆ b + ε + εb,s = γ(β · εs )nˆ b,s

(2.126)

These expressions at hand, the scalar product of the polarisations εb · εb,s in the Klein-Nishina differential cross section, Eq. 2.120, can be evaluated to

εb · εb,s = ε · εs −

( p · ε)(k · εs ) ( p · εs )(ks · ε) ( p · ε)( p · εs )(k · ks ) − + ≡ P(ε, εs ). p·k p · ks ( p · k)( p · ks )

(2.127)

p denotes the electron 4-momentum in this equation and k the photon 4momentum. Combining the rest frame expressions for the 4-vector scalar product of the polarisations, Eq. 2.127, and the frequencies, given in Eq. 2.121, in terms of the laboratory frame components, the Klein-Nishina differential cross section in the laboratory frame is ] ( ω )2 [ ω ((1 − β · nˆ ) ˆ dσ re2 ω((1 − β · n) s s s 2 . = + − 2 + 4 ε )] [P(ε, s ˆ dΩ ω((1 − β · n) ωs ((1 − β · nˆ s ) ˆ 2 ω 4γ 2 (1 − β · n)

(2.128) If the polarisation of the scattered photon is not observed, the total cross section can be separated into the ones for the two orthonormal final state polarisation vectors, evaluated individually and summed up. Details are provided in Krafft et al. (2016). This procedure yields the differential cross section ( ω )2 [ ω (1 − β · nˆ ) ˆ re2 ω(1 − β · n) dσ s s s = 2 + 2 ˆ ˆ ω(1 − β · n) ωs (1 − β · nˆ s ) 2γ (1 − β · n) ω dΩ )2 ] 2 2 ( 2m c p·ε (k · ks ) − ks · ε − , p·k ( p · k s )2 which may be beneficial in numerical calculations.

(2.129)

40

2 X-ray Generation by Laser-Electron Interaction

2.6.3 The Compton Scattered Spectrum This section is based on “Laser pulsing in linear Compton scattering” by Krafft et al. (2016). The electron beam propagates along the z-coordinate of the laboratory frame. Similar to the derivation of the Liénard-Wiechert fields, the derivation starts with the Poynting vector, Eq. 2.59. Using Parseval’s theorem once again, the time dependency of the Poynting vector can be replaced with one on the angular frequency, leading to 1 dEs = 2π dΩ

{

∞

−∞

dω R 2 S˜ s (ω) · nˆ ret ,

(2.130)

cf. derivation of Eq. 2.62. ω is the frequency of the incident photon. Within a “semiclassical” treatment, the number of scattered photons per solid angle can be expressed as (Krafft et al. 2016) 1 dEs = 2π dΩ

{

∞ −∞

dω| S˜ inc. (ω)|

dσ ωs . dΩ ω

(2.131)

dσ/dΩ is the Klein-Nishina differential cross section and the term ωs /ω accounts for the relativistic Lorentz-contraction in this semiclassical picture. Inserting the pointing vector of Eq. 2.59 and a change of variable yields the spectrum [ ] ε0 c ˜ ωs dω d 2 Es 2 dσ = | E inc. (ω)| . 2π dΩ ω dωs dωs dΩ

(2.132)

The last equation serves as the basis for calculation of the spectrum of a single interaction. Similarly to the discussion of the undulator spectrum, effects like electron beam emittance and energy spread can be included by an ensemble average over a set of particles having the same distributions as the real electron beam. Transverse effects of the extended laser beam can be included by appropriately varying the modulus ˜ inc. (ω). One such code based on the above description square of the electric field E is the Improved Code for Compton Simulation (ICCS) by Ranjan et al. (2018). Its three-dimensional version has been recently made available at the MuCLS by the authors Balsá Terzi´c and Geoffrey Krafft. If quantitative results are desired, the fact that inverse Compton scattered photons are the result of two extended focussed (Gaussian) beams colliding under an crossing angle has to be appropriately included. First, the transverse interaction area is not constant for focussed, thus diverging, beams. Since the beam’s diameter increases parabolically along its propagation direction, the luminosity reduction due to this effect is called “hourglass effect” inspired by its geometric similarity to sand flowing through an hourglass. The reduction factor accounting for the hourglass effect was calculated by Furman (1991) for colliders. Additionally, a crossing angle reduces the luminosity further because both beams start to overlap only partially. For very

2.6 X-ray Generation by Inverse Compton Scattering

41

Fig. 2.6 Effect of electron beam parameters on the inverse Compton scattered spectrum. In all simulations, the real laser parameters were employed while electron source size, energy spread and emittance are added one by one. The almost monochromatic black curve is obtained, if only the real source size for electron beam is considered. Adding electron energy spread to the electron beam broadens the spectrum symmetrically, cf. the light blue curve. Once electron beam emittance is added, i.e. an angular divergence of the electrons, the typical spectrum observed at the MuCLS emerges. Electron beam emittance also causes a slight red-shift of the radiation. Simulations were performed with ICCS3D (Krafft et al. 2016; Ranjan et al. 2018)

small angles and very short pulses, the latter effect becomes negligible. Miyahara (2008) calculated the effect of the crossing angle and also derived a formula for a combined reduction factor due to hourglass effect and crossing angle. With a reduction factor of around 0.78, i.e. a flux deprivation of 22%, this reduction is noticeable for the interaction geometry at the MuCLS and parameters measured at the 25 keV configuration under typical operation conditions.

2.6.4 Effects of Electron Beam Parameters on the Inverse Compton Spectrum The presentation of laser-electron interaction on the basis of inverse Compton scattering is closed with a brief discussion of the individual beam parameters that influence the spectral shape at the MuCLS. The laser can be considered as monochromatic due to a Fourier-transform-limited (or bandwidth-limited) laser pulse length of about 30 ps. Consequently, the photon energy spread is negligible. Furthermore, a photon beam’s emittance is constant and rather small compared to the one of the electron

42

2 X-ray Generation by Laser-Electron Interaction

beam which is on the order of 10 mm mrad. Additionally, the electron beam energy spread typically amounts to a couple of tenth of a percent outweighing the one of the laser significantly. As a result, the electron beam parameters will determine the inverse Compton scattered spectrum at the MuCLS. Figure 2.6 displays spectra, where the aforementioned quantities have been added one by one. The spectra are calculated with ICCS3D (Krafft et al. 2016; Ranjan et al. 2018) using the MuCLS laser beam parameters. The electron energy was chosen such that it corresponds to the one at the 25 keV setting at the CLS. The black curve was calculated only considering a finite source size of the electron beam in addition to the aforementioned parameters. It is almost purely monochromatic, demonstrating the negligible effect of laser beam emittance and energy spread. When the electron energy spread (0.3%) is added, the spectrum broadens symmetrically quite a bit, as expected for a symmetric energy spread. However, once the restriction that all the electrons travel along the identical trajectory is omitted, i.e. an angular divergence of the electron beam is introduced expressed by a finite emittance, the spectrum depicted in green emerges. It was calculated with a realistic emittance of 10 mm mrad. Two prominent changes compared to the preceding spectrum are visible. First, if emittance is considered, the peak occurs at a lower X-ray energy. While the total electron energy is conserved, its momentum is partially redistributed into components perpendicular to the laser beam’s propagation direction. Thus, the average momentum of the electrons in the direction counter-propagating the laser beam is reduced effectively red-shifting the spectrum. Second, an asymmetric wide long low energy tail emerges as a result of the multitude of scattering angles present. Accordingly, the shape of the low energy tail allows to draw conclusions on the quality of the stored electron beam in terms of emittance. If the low energy-tail of the X-ray spectrum suddenly changes significantly, this may be an indication that the electron beam of the MuCLS is misbehaving. Therefore, spectra should always be recorded for every experiment which relies on spectral information, since the electron beam could have changed slightly. This little analysis and reminder concludes the chapter about X-ray generation via laser-electron interaction. The following one treats the propagation of electromagnetic waves, such as the laser, in free space which is a prerequisite to comprehend the generation of high-power laser pulses at the CLS.

References Einstein A (1905) Zur Elektrodynamik bewegter Körper. Annalen der Physik 322:891–921 Furman MA (1991) Hourglass effect in asymmetric colliders in proceedings of PAC 1991, pp 422–424 Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley. ISBN: 0-471-30932-X Kim K-J, Huang Z, Lindberg R (2017) Synchrotron radiation and free-electron lasers. Cambridge University Press, Cambridge. ISBN: 978-1-107-16261-7

References

43

Klein O, Nishina Y (1929) Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamic von Dirac. Zeitschrift für Physik 52:853–868 Krafft GA et al (2016) Laser pulsing in linear Compton scattering. Phys Rev Accel Beams 19:121302 Miyahara Y (2008) Luminosity of angled collision of strongly focused beams with different Gaussian distributions. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 588:323–329 Ranjan N et al (2018) Simulation of inverse Compton scattering and its implications on the scattered linewidth. Phys Rev Accel Beams 21:30701 Wiedemann H (2105) Particle accelerator physics, 4th edn. Springer. ISBN: 978-3-319-18316-9 Zangwill A (2012) Modern electrodynamics. Cambridge University Press, Cambridge. ISBN: 9780-521-89697-9

Chapter 3

Scalar Wave Theory

After the generation of electromagnetic radiation by an undulator has been presented in the preceding chapter, this one focusses on the X-rays emitted in this process, their properties and their evolution in free space. Most derivations in this chapter follow either “Coherent X-ray Optics” by David M. Paganin (Paganin 2006), “Introduction to Fourier Optics” by Joseph W. Goodman (Goodman 2005) or “Lasers” by Anthony E. Siegmann (Siegman 1986). Nevertheless, the concepts presented here are covered by many textbooks, including (Zangwill 2012; Träger 2007; Jackson 1999) among others.

3.1 Wavefields in Free-Space 3.1.1 From the Maxwell- to the Helmholtz Equations This section loosely follows (Paganin 2006, pp. 1–6 and 77–83). In free space, the current four-vector J α = 0 resulting in ∂ α F αβ = 0, cf. Eq. 2.28. This property of free space in conjunction with the two homogeneous Maxwellequations ∇ · B (r, t) = 0 (3.1) ∂ ∇ × E (r, t) + B (r, t) = 0 ∂t can be used to formulate a wave-equation similar to Eq. 2.33, just for the fields this time. Taking the curl of the last equation, employing Gauss’ law and vector equalities, the d’Alembert wave equation (Paganin 2006, p. 3)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_3

45

46

3 Scalar Wave Theory

(

) 1 ∂ 2 − ∇ E (r, t) = 0 c2 ∂t 2

(3.2)

is derived. An identical equation for the magnetic field is obtained if Ampere’s law is used instead of Faraday’s and enforcing the absence of a source of the magnetic field, i.e. the absence of magnetic monopoles. Since no cross-terms are present, the components of the electric and magnetic field are decoupled. Thus their behaviour can be summarised into a single scalar function Ψ (r, t) (Paganin 2006, p. 4): (

) 1 ∂ 2 − ∇ Ψ (r, t) = 0. c2 ∂t 2

(3.3)

Decomposing the wave-field Ψ (r, t) into its spectral components via the Fourier transformation, ∫ ∞ 1 (3.4) ψ˜ (r) ei ωt dω, Ψ=√ 2π 0 provides the basis for the time-independent wave-equation. The negative angular frequencies are typically omitted as the full spectral information is contained in the positive angular frequencies already. ψ˜ (r) is the scalar function in terms of the angular frequencies ω. Plugging expression (3.4) into Eq. 3.3, exchanging the order of integration and differentiation, multiplication of the whole expression by −1 and employing the free-space dispersion relation ω = ck establishes the desired time-independent wave equation (

) ∇ 2 + k 2 ψ˜ (r) = 0,

(3.5)

the so-called Helmholtz equation (Paganin 2006, p. 6). In the presence of a point scatterer (or point source of the electro-magnetic field), the Helmholtz equation becomes (Paganin 2006, p. 77 ff.) (

) ( ) ( ) ∇ 2 + k 2 Dω r, r ' = −4π δ (3) r − r ' .

(3.6)

( ) Its right hand side was multiplied by 4π by convention and D r, r ' is the Green function. Since this expression is very similar to Eq. 2.34, just in three dimensions and multiplied by −1, it is solved analogously to the procedure described in Sect. 2.2.2. Therefore, the general solution of the Green function in this 3D-case is given by Eq. 2.40 times 4π when the final Fourier-transform in this equation is omitted and z 0 = 0: '

'

( ) ( ) ( ) e−ik0 ∥r−r ∥ eik0 ∥r−r ∥ ' ' + Dω r, r ' = ' = Dr r, r + Da r, r . ' ∥r − r ∥ ∥r − r ∥

(3.7)

3.1 Wavefields in Free-Space

47

( ) The first term D(r r, )r ' describes an outgoing (retarded) spherical wave, while the second one Da r, r ' characterises an incoming (advanced) spherical wave. The latter is unphysical for the process in question and thus disregarded. Accordingly, the final solution for the wave scattered by a point scatterer or point source, respectively, is governed by Paganin (2006, p. 83)1 (

Dr r, r

'

)

'

e−ik0 |r−r | = . |r − r ' |

(3.8)

However, one should keep in mind that any specific situation can be delineated by a certain linear combination of those solutions.

3.1.2 Rayleigh-Sommerfeld Diffraction Theory This section is based on (Goodman 2005, Sects. 3.3–3.6) and (Paganin 2006, Sect. 1.6). In this section, the spherical wave solution is employed in determining the diffracted field at a position P in the half-space behind an aperture. The goal is to determine a scalar field ψ˜ which fulfils the Helmholtz Eq. 3.5. Figure 3.1a displays the geometry of the problem. In the following, the spacial function variables are omitted in favour of better readability. Since both, the general Green’s function D and ψ˜ obey the Helmholtz equation, they comply with the requirements of Green’s theorem: ∫ ∫ ∂D ∂ ψ˜ (3) 2 2 ˜ ˜ −D , (3.9) d V ψ∇ D − D∇ ψ = d (2) S ψ˜ ∂n ∂n where ∂/∂n denotes a partial derivative in the outward normal direction at each point ˜ 2 to the integrand in on S (Goodman 2005, p. 39). Addition and subtraction of D ψk the volume integral reproduces the two Helmholtz equations for D and ψ˜ ∫

˜ 2 D − D∇ 2 ψ˜ = d (3) V ψ∇

∫ ∫

=

) ( ) ( d (3) V ψ˜ ∇ 2 + k 2 D − D ∇ 2 + k 2 ψ˜ ( ) ( ) d (3) V ψ˜ (−4π ) δ (3) r − r p − D · 0 = −4π ψ˜ r p .

(3.10) Consequently, the scalar field at the position P can be calculated via

1

Please note that the opposite sign compared to Paganin (2006) originates from the opposite definition of the phase as ωt − kr in this thesis.

48

3 Scalar Wave Theory

Fig. 3.1 Geometries in Kirchhoff and Sommerfeld diffraction. a is the general geometry for the problem of diffraction through an aperture, while b displays the same situation but including a mirror source Q considered by Sommerfeld and resulting in the Rayleigh-Sommerfeld diffraction integral

( ) 1 ψ˜ r p = 4π

∫∫ dSD S1

∂ ψ˜ ∂D 1 − ψ˜ + ∂n ∂n 4π

∫∫ dSD S2

∂ ψ˜ ∂D − ψ˜ ∂n ∂n

(3.11)

if an appropriate Green’s function D is chosen (Paganin 2006, p. 21). To this end, P is considered a “virtual point scatterer” emitting the retarded spherical wave of Eq. 3.8. The closed surface S has been split in two parts, the plane surface S1 at which the aperture is located and the curved one S2 of the hemisphere, cf. Fig. 3.1a. To determine the contribution of S2 the derivative of the Green’s function Dr ( ) ∂ Dr ∂ e−ik R p 1 = = Dr −ik − cos (θ ) ∂n R p ∂n Rp

(3.12)

) ( has to be calculated. cos (θ ) = cos n, R p is the angle between the surface normal and the vector pointing towards P and R p = |r − r p |. For R p → ∞, cos(θ ) R p →∞

approaches 1 and 1/R p converges to 0 and in turn ∂ Dr /∂n ≈ −ik Dr . The contribution of the curved surface becomes ) ∫ ) ( ( ∫∫ ˜ ∂ ψ ∂ ψ˜ 2 d S Dr + ik ψ˜ = dΩR p Dr + ik ψ˜ . (3.13) ∂n ∂n Ω S2

Since R p Dr is bounded on the surface S2 , this integral vanishes only if ( lim R p

R p →∞

∂ ψ˜ + ik ψ˜ ∂n

) = 0,

(3.14)

in other words, if the scalar field or disturbance ψ˜ is decreasing at least as fast as a retarded (outgoing) spherical wave. Under this Sommerfeld radiation condition,

3.1 Wavefields in Free-Space

49

ψ˜ =

1 4π

∫∫ dSD S1

∂ ψ˜ ∂D − ψ˜ . ∂n ∂n

(3.15)

If the Kirchhoff boundary conditions are applied to this equation, the Kirchhoff diffraction integral for an aperture in an opaque screen is obtained, for details see, e.g. (Goodman 2005, p. 44 ff). Although remarkably successful, Kirchhoff’s diffraction theory has inconsistencies which can be overcome if either Dr or ∂ Dr /∂n vanishes on the entire surface S1 . Such a Green’s function can be constructed with the concept of mirror images. Let Q be a second point source at the position of the mirror image of P on the opposite side of the screen. A sketch of the geometry is depicted in Fig. 3.1b. Moreover, Q emits radiation of the same wavelength, but phase shifted by 180◦ . The corresponding Green’s function D− (r) =

e−ik|r−r q | e−ik|r−r p | − |r − r q | |r − r p |

(3.16)

conforms with this requirement. This choice of the Green’s function strongly simplifies Eq. 3.15 to ∫∫ ∂ D− 1 ˜ d S ψ˜ , (3.17) ψ =− ∂n 4π S1

since D− is zero on the surface of integration and in turn its first integrand disappears. Moreover, the disturbance ψ˜ disappears outside the aperture due to the opaqueness of the screen, which reduces the integration area to the aperture ∑ itself. ψ˜ = −

1 4π

∫∫

d S ψ˜

∑

∂ D− 1 =− 2π ∂n

∫∫

d S ψ˜

∑

∂ Dr ∂n

(3.18)

is the first Rayleigh-Sommerfeld solution (Goodman 2005, p. 48 f.). The last equality arises from the symmetry of the source geometries. Due to the fact that Q is a mirror image of P, cos(θ p ) = − cos(θq ) in the derivative of the two spherical waves contained in D− , cf. Eq. 3.12 for the derivative of a spherical wave with respect to the surface normal. Consequently, ∂ D− /∂n = 2∂ Dr /∂n. Finally, the explicit form of the first Rayleigh-Sommerfeld solution for diffraction in the limit λ ≫ R p is2 ( ) i ψ˜ r p = λ

2

∫∫ ∑

d S ψ˜ ∑

e−ik R p cos (θ ) . Rp

(3.19)

Again, the sign difference compared to (Goodman 2005, p. 49) results from the opposite sign in the phase definition.

50

3 Scalar Wave Theory

Instead of interpreting P as a secondary source, every point on the surface of the aperture ∑ can be interpreted as a secondary source emitting a spherical wave with a complex amplitude ψ˜ (r) and an obliquity factor cos(θ ). The disturbance at the position P follows from the superposition of the fields of all secondary sources at r p . This interpretation connects the Rayleigh-Sommerfeld diffraction integral with the Huygens-Fresnel principle (Goodman 2005, p. 52). The last equality in Eq. 3.19 accounts for the effect that the source and observation point are interchanged in the Huygens-Fresnel principle compared to the derivation of the Rayleigh-Sommerfeld diffraction integral. Although the diffraction integral remains unchanged as it depends only on the modulus of the distance vector, the latter variant of the RayleighSommerfeld diffraction integral is employed from here on because it agrees better with human perception of the diffraction process.

3.1.3 The Paraxial Wave Equation This section loosely follows (Siegman 1986, Sects. 16.1 and 16.2). The Sommerfeld radiation condition, Eq. 3.14, requires the scalar field ψ˜ to fall like a spherical wave for the radius approaching infinity. Thus, a generic spherical wave is an appropriate choice for the wave field. Consider a point source located close to the optical axis at a position r 0 = (x0 , y0 , z 0 ) radiating a spherical wave e−ik0 |r−r0 | ˜ . ψ(r) = |r − r0 |

(3.20)

Without loss of generality, the z-axis of the coordinate system is aligned with the optical axis. In case of the region of interest being restricted to the vicinity of the optical axis and of a sufficiently large propagation distance, the radius ρ(r, r 0 ) = |r − r0 | can be expanded in a power series3 ( ) 1 (x − x0 )2 + (y − y0 )2 + . . . . ρ(r, r 0 ) = (z − z 0 ) · 1 + 2 (z − z 0 )2

(3.21)

Quadratic terms need to be kept in the exponent, since small phase changes can alter the value of the exponential function significantly, while they do not affect the radius in the denominator notably. In this limit, the scalar wave field is approximated by ˜ = ψ(r)

3

+ y−y x−x 1 −ik ( 02) z−z( 0 ) ( 0) = u˜ sph. (r ⊥ , z)e−ik(z−z0 ) , e−ik(z−z0 ) e z − z0

The power series of

2

√ 1 + x is used.

2

(3.22)

3.1 Wavefields in Free-Space

51

where u˜ sph (r ⊥ , z) is a slowly varying envelope function and r = (r ⊥ , z) with r ⊥ = (x, y). In many cases the scalar wave field can be decomposed into such a paraxial wave field consisting of a rapidly oscillating plane wave and a slowly varying envelope function. Accordingly, a more general formulation of Eq. 3.22 is ˜ = u(r ˜ ⊥ , z)e−ikz . ψ(r)

(3.23)

Inserting Eq. 3.23 into the homogeneous Helmholtz Equation 3.5 results in the paraxial Helmholtz equation, ( ∇⊥2 − 2ik where ∇ 2 = ∇⊥2 +

∂2 ∂ z2

∂ ∂z

) u(r ˜ ⊥ , z) = 0,

(3.24)

and the relation

∇ 2 ψ˜ = e−ikz ∇⊥2 u˜ + e−ikz

∂2 ∂ u˜ − 2ike−ikz u˜ − k 2 e−ikz u˜ ∂z ∂z 2

(3.25)

was used. Due to the fact that u(r ˜ ⊥ , z) describes a slowly varying envelope, | | 2 | | | | | | |2ik ∂ u˜ | ≫ | ∂ u˜ | , | ∂z 2 | | ∂z |

(3.26)

which allows to neglect the derivative of second order in z. Hereinafter, the paraxial ˜ ⊥ , z) form the basis of the description and wave Equation 3.24 and its solutions u(r analysis of the optical systems in this thesis. In a similar fashion, the paraxial approximation of the scalar spherical wave field, Eq. 3.22, can be employed for the Green’s function inside the Rayleigh-Sommerfeld diffraction integral, Eq. 3.19. The resulting paraxial Rayleigh-Sommerfeld diffraction integral is4 (Siegman 1986, p. 634 ff.) ∫∫ u˜ (r ⊥ , z) = ∑

+ y−y x−x i −ik ( 02) z−z( 0 ) ( 0) e = d S u˜ ∑ (z − z 0 )λ 2

2

∫∫

d S u˜ ∑ K˜ (x, x0 ).

(3.27)

∑

Later, the one-dimensional Rayleigh-Sommerfeld integral is needed as well, which has a very similar form (Siegman 1986, p. 777 f.): √

∫ u˜ (x, z) =

4

∑

d x u˜ ∑

x−x i −ik ( 0 ) e 2(z−z0 ) = (z − z 0 )λ 2

∫ ∑

d x u˜ ∑ K˜ 1D (x, x0 ).

(3.28)

A consequence of the small angles involved in the paraxial approximation is that cos(θ ) ≈ 1.

52

3 Scalar Wave Theory

3.1.4 Free-Space Propagation of Paraxial Waves The derivation in this section follows roughly (Paganin 2006, pp. 6–12). ˜ ˜ Let ψ(x, y, z = 0) be a paraxial wave field and ψ(x, y, z = z) be this wave field propagated by a distance z in free-space. The goal is to find an operator Oz that relates the initial wave with the propagated one via ˜ ˜ y, z = z) = Oz ψ(x, y, z = 0). ψ(x,

(3.29)

˜ y, z = 0) in the basis of the transverse plane waves yields Decomposition of ψ(x, ˜ ψ(x, y, z = 0) =

1 2π

∫∫

dk x dk y ψ(k x , k y , z = 0)e−i (kx x+k y y ) ,

(3.30)

where ψ(k x , k y , z = 0) is the conjugated field in frequency space. Each of these plane waves can be propagated by a distance z by multiplication with e−ikz z . Since plane waves fulfil the Helmholtz equation, Eq. 3.5, and if k x2 + k 2y + k z2 = k 2 , the wave √ vector k z can be expressed as k z = k 2 − (k x2 + k 2y ). This term can be expanded in a power series because k z ≫ k x , k y for paraxial waves leading to ( kz ≈ k 1 −

k x2 + k 2y

) .

2k 2

(3.31)

Multiply Eq. 3.30 with e−ikz z to obtain the propagated wave field 1 ˜ y, z = z) = ψ(x, 2π

∫∫ dk x dk y ψ(k x , k y , z = 0)e

−i (k x x+k y y )

e

) ( k 2 +k 2 −i k− x 2k y z

.

(3.32)

Consequently, the Fresnel propagation operator Oz can be expressed as Oz =

F⊥−1 e

) ( k 2 +k 2 −i k− x 2k y z

(

F⊥ = e

−ikz

F⊥−1 e

i

k x2 +k 2y 2k

) z

F⊥ = e−ikz OFresnel ,

(3.33)

where F⊥ denotes the Fourier transform in the directions transverse to the propagation direction and F⊥−1 its inverse. OFresnel summarises the terms which are most commonly referred to as the Fresnel operator. Assuming that the spacial distribution of the field is known at a certain position, e.g. at the exit surface of an optical element, the following steps are necessary to calculate the field at the entrance surface of the next element: ˜ ⊥ , z = 0) to obtain ψ(k x , k y , z = 0). i Fourier transform ψ(r ii Multiply the elements of the plane wave decomposition with the propagator 2 2 ei(kx +k y )Δz/(2k) .

3.2 Gaussian Beams

53

Fig. 3.2 Parameters of a Gaussian beam which are required to describe its evolution. The confocal parameter indicates the range over which the Gaussian beam can be considered as collimated

iii Fourier back-transform the resulting expression into real space. iv Multiply this field with the constant phase factor e−ikΔz . Alternatively, this step can be included into ii as well. In this thesis, the scalar fields obey the paraxial approximation and free-space propagation of these beams was simulated with the described procedure.

3.2 Gaussian Beams This section is based on Chaps. 15–17 and 20 of “Lasers” by Siegman (1986). The uniform paraxial spherical wave as stated in Eq. 3.22 and derived in Sect. 3.1.3 cannot represent the situation of a real physical beam for several reasons despite satisfying the paraxial Helmholtz equation. First, such a wave extends to infinity in the transverse direction which results in a large phase error far off the optical axis since the paraxial approximation is no longer fulfilled. Second, it carries infinite energy and power across the transverse plane as a consequence of its transverse extent (Siegman 1986, p. 638). The transition from the paraxial spherical wave to a beam-like wave, namely the Gaussian beam, is illustrated in the following. The meaning of its main parameters is depicted in Fig. 3.2.

3.2.1 The Fundamental-Mode Gaussian Beam A generalised form of the paraxial spherical wave is (Siegman 1986, p. 642) u(x, ˜ y, z) = A(z)e−ik

x 2 +y 2 2q(z) ˜

,

(3.34)

54

3 Scalar Wave Theory

where both amplitude A(z) and the radius of curvature q(z) are unknowns and allowed to take on complex values. q(z) ˜ can be interpreted as a complex radius of curvature. Inserted into the paraxial wave equation, this Ansatz for the paraxial spherical wave generates (Siegman 1986, p. 642 ff.) [( ) ( ) )] ( ) 2ik q˜ ∂ A ( 2 k 2 ∂ q˜ 2 −1 x +y − +1 A = 0, q˜ ∂z A ∂z 2

(3.35)

which is satisfied for all x and y values only if each of the two differential equations inside the round brackets are 0 independently. This leaves the following set of differential equations: ∂ q(z) ˜ ∂ A(z) A(z) = 1 and =− . ∂z ∂z q(z) ˜

(3.36)

While integration of the first equation is straightforward, solving the second one is a bit more complicated. The resulting solutions are q(z) ˜ = q˜0 + z and A(z) =

A0 q˜0 , q(z) ˜

(3.37)

where q˜0 denotes the complex integration constant, which can be interpreted as a complex source point. The case A0 = 1 and q˜0 real reproduces the paraxial spherical wave. Since q(z) ˜ can be interpreted as a complex radius of curvature, 1/q(z) ˜ can be separated into a real and imaginary part (Siegman 1986, p. 639) 1/q(z) ˜ = 1/q˜r (z) − i /q˜i (z),

(3.38)

which transforms the paraxial wave to u(x, ˜ y, z) =

+y 2 A0 q˜0 −ik x22q(z) A0 q˜0 −k x22q˜+y(z)2 −ik x22q˜+y(z)2 ˜ r i e . e e = q(z) ˜ q(z) ˜

(3.39)

The middle term on the right hand side of above equation looks very similar to a normal distribution with the variance σ 2 (z) = q˜i (z)/k. Equivalently, q˜i can be expressed in terms of the beam √ waist, which is commonly defined as the 1/e beam radius of the electric field w = 2σ in optics, via q˜i (z) = kσ 2 (z) =

k 2 π w (z) = w 2 (z). 2 λ

(3.40)

Similarly, q˜r (z) can be interpreted as the real radius of curvature of the parabolic phase term (x 2 + y 2 )/(2q˜r (z)). Thus, the real part of the complex radius of curvature is denominated R(z) from here on. In conclusion, the expression for the complex radius of curvature is (Siegman 1986, p. 640)

3.2 Gaussian Beams

55

1 1 λ ≡ −i . q(z) ˜ R(z) π w2 (z)

(3.41)

Inserting this expression into Eq. 3.39, a solution for the paraxial Gaussian beam u(x, ˜ y, z) =

2 +y 2 A0 q˜0 − xw22+y(z)2 −ik x2R(z) e e q(z) ˜

(3.42)

is obtained. Let the reference plane be located at z = 0 where q˜0 is purely imaginary. In this ˜ = −i /q˜i (0) at the reference plane (cf. Eqs. 3.37 and situation, q(0) ˜ = q˜0 and 1/q(0) 3.38), leading to ˜ = i q˜i (0) = i q˜0 = q(0)

π ω2 (0) π ω02 =i = i zR, λ λ

(3.43)

where z R denotes the Rayleigh length (Siegman 1986, p. 663 f.). Consequently, the complex radius of curvature can be expressed replacing q˜0 in Eq. 3.37 as q(z) ˜ = z + i zR.

(3.44)

˜ = 1/(1 − i z/z R ) with Combining Eqs. 3.43 and 3.44 and the quotient q˜0 /q(z) the paraxial Gaussian beam, Eq. 3.42, the latter becomes5 2 2 2 +y 2 x 2 +y 2 w0 i arctan (z/z R ) − xw22+y(z)2 −ik x2R(z) A0 − xw2+y (z) e −ik 2R(z) = A e e e . 0 z e 1 − i zR w(z) (3.45) The infinite two-dimensional integral of the square modulus of Eq. 3.45 deter√ mines the normalisation factor A0 = 2/π /w0 . Thus, the solution of the paraxial wave equation for a normalised Gaussian beam is (Siegman 1986, p. 664)

u(x, ˜ y, z) =

( ) 21 ( ) 2 +y 2 2 1 i arctan zz − xw22+y(z)2 −ik x2R(z) R e e e . u(x, ˜ y, z) = π w(z)

(3.46)

Comparing the inverse of Eq. 3.44 with Eq. 3.41 the relations for the real radius of curvature R(z) of a Gaussian beam, its waist ω(z) and Gouy-phase ϕG are found (Siegman 1986, p. 665):

5

1 1−i zzr

=

w0 w z w0 −i z0r

=

( ) w z w0 w0 +i z0r w2 z 2 w02 + 02 zR

arctan (z/z R ) the first factor

1 1−i zzr

=

=

w0 Bei ϕ . w2 (z)

With B = w(z) and ϕ = arctan

w0 i arctan (z/z R ) . w(z) e

( w0 z ) zR

w0

=

56

3 Scalar Wave Theory

z2 R(z) = z + R z √ ( )2 z w(z) = w0 1 + zR ( ) z ϕG = arctan zR

(3.47)

These three equations characterise the evolution of a Gaussian beam during freespace propagation. At z = 0, the waist of the Gaussian beam is smallest and its radius of curvature becomes infinite, in other words its curvature is zero. Consequently, this position corresponds to the focus of a Gaussian beam.

3.2.2 Higher-Order Gaussian Modes This section follows (Siegman 1986, pp. 643–648, 672 and 691). The Gaussian beam solution to the paraxial wave equation is not the only one, but just the fundamental one of a set of solutions which can be either expressed in Cartesian coordinates, called Hermite-Gaussian modes u˜ nm (x, y, z), or in cylindrical coordinates, called Laguerre-Gaussian modes u˜ pm (r, θ, z). In this section, the Hermite-Gaussian solutions are discussed in detail because these are the most commonly used ones. Nevertheless, the Laguerre-Gaussian modes are presented in brief as well to provide a complete picture.

3.2.2.1

The Hermite-Gaussian Modes

In Cartesian coordinates, the elementary solutions to the paraxial wave equation can be separated into a product of the solutions along the x- and y-directions as (Siegman 1986, p. 643) (3.48) u˜ nm (x, y, z) = u˜ n (x, z)u˜ m (y, z). Consequently, it is sufficient to find solutions for one coordinate and extend this solution to the second one by analogy. Accordingly, only solutions for the x-direction are investigated in the following. The trial function of Eq. 3.34 can be generalised by extending it with additional unknown functions of both coordinates: ( ˜ u˜ n (x, z) = A(q(z))h n

) x2 x ˜ , e−ik 2q(z) p(z) ˜

(3.49)

where p(z) ˜ is a distance-dependent scaling factor in the argument of h n , A(q) ˜ and ˜ are the additional unknown functions (Siegman 1986, p. 643). Under the h n (x/ p)

3.2 Gaussian Beams

57

assumption that ∂ q/∂z ˜ = 1 remains valid, this trial solution converts the paraxial ˜ (Siegman 1986, p. 643): wave equation into a differential relation for h n (x/ p) h ''n

[

] [ ] ik p˜ 2 p˜ 2q˜ ∂ A ' ' − 2ik − p˜ xh n − 1+ h n = 0. q˜ q˜ A ∂ q˜

(3.50)

In this equation h 'n and h ''n denote the first and second total derivative of h n with ˜ Equation 3.50 respect to their total argument, i.e. h 'n ≡ ∂h n (ξ )/∂ξ , and p˜ ' ≡ ∂ p/∂z. becomes equivalent to the Hermite differential equation, provided that the following two conditions are fulfilled simultaneously (Siegman 1986, p. 644): ] ∂ p˜ ∂ p˜ p˜ i p˜ 2 − or = + = 2ik ∂z ∂z q˜ k p˜ q˜ p˜

(3.51)

[ ] −ik p˜ 2 2ink p˜ 2 2q˜ ∂ A 2q˜ ∂ A = − 1. 1+ = 2n or q˜ q˜ A ∂ q˜ A ∂ q˜

(3.52)

[

and

Equation 3.51 can be solved by (Siegman 1986, p. 644) √ 2 1 = w(z) p(z) ˜

(3.53)

while the second condition (Eq. 3.52) is fulfilled for (Siegman 1986, p. 645) ( A(q) ˜ = A0

q˜0 q(z) ˜

) 21 (

q˜0 q˜ ∗ (z) ˜ q˜0∗ q(z)

) n2

.

(3.54)

After normalisation by a proper choice of A0 , the complete set of HermiteGaussian mode functions in one dimension and free space are obtained (Siegman 1986, p. 645): (√ ) ( ) 14 ( ) 21 ( )1 ( )n x2 2x 2 1 q˜0 2 q˜0 q˜ ∗ (z) 2 ˜ Hn . e−ik 2q(z) u˜ n (x, z) = ∗ n ˜ w(z) π q(z) ˜ q˜0 q(z) 2 n!w0 (3.55) An alternative, very detailed derivation of the Hermite-Gaussian mode functions is presented in (Träger 2007, p. 147 ff.). The intensity distribution of the first few Hermite-Gaussian transverse electro-magnetic mode functions is depicted in Fig. 3.3. Since the Hermite-Gaussian mode functions form an orthonormal basis characterised by the complex parameter q˜0 at any reference plane z 0 , each scalar wave ˜ ψ(x, y, z) can be decomposed into this basis via (Siegman 1986, p. 646) ∫∫ cnm =

˜ ψ(x, y, z)u ∗n (x, z)u ∗m (y, z)d xd y,

(3.56)

58

3 Scalar Wave Theory

Fig. 3.3 Intensity profile of Hermite-Gaussian transverse electro-magnetic modes (TEM) up to order three. The complexity of the intensity as well as the diameter of the Gaussian beam increases with increasing mode number. Except for rare special purposes, the desired mode is a fundamental Gaussian-shaped TEM00 -mode

where cnm are the coefficients of the decomposition. Accordingly, the scalar wave can be expressed in terms of the respective basis function and these coefficients by ˜ ψ(x, y, z) =

∑∑ n

m

cnm u˜ n (x, z)u˜ m (y, z)e−ikz .

(3.57)

3.2 Gaussian Beams

3.2.2.2

59

The Generalised Gouy Phase

In the derivation of the fundamental mode, it has been shown that the term q˜0 /q(z) ˜ is equivalent to w0 ei (ϕ(0)−ϕ(z)) /w(z) = w0 ei (ϕG (z)) /w(z). This term can be generalised to w0 ei (ϕG (z)−ϕG,0 ) /w(z), where ϕG,0 = ϕG (z 0 ) (Siegman 1986, p. 645). Higher-order modes contribute an additional phase factor (

( )∗ ) n2 w0 ei (ϕG (z)−ϕG,0 ) w(z) = = w(z) w0 ei (ϕG (z)−ϕG,0 ) ( ) n2 w0 ei (ϕG (z)−ϕG,0 ) w(z)ei (ϕG (z)−ϕG,0 ) = = ein (ϕG (z)−ϕG,0 ) . w0 w(z) (3.58) The general expression for the Hermite-Gaussian higher order modes in a form equivalent to the one of the fundamental mode given in Eq. 3.46 is6 (

q˜0 q˜ ∗ (z) ˜ q˜0∗ q(z)

) n2

(

q˜0 q(z) ˜

(

q(z) ˜ q˜0

)∗ ) n2

(√ ) ( ) 14 ( i (2n+1)(ϕG (z)−ϕG,0 ) ) 21 2 kx 2 2x 2 e − wx2 (z) −i 2R(z) H u˜ n (x, z) = . e e n 2n n!w(z) w(z) π

(3.59)

This variation of the Gouy phase shift for paraxial Gaussian beams of different order has some important implications, which are discussed later on.

3.2.2.3

The Laguerre-Gaussian Modes

An alternative family of solutions of the paraxial wave equation is obtained if cylindrical coordinates are employed. These solutions are called Laguerre-Gaussian modes, whose form are √ u˜ pm (r, θ, z) =

ei (2 p+m+1)(ϕG (z)−ϕG,0 ) 2 p! w(z) (1 + δ0m ) π (m + p)! ( √ )m ) ( 2 2r 2r 2 +imθ −i 2kr m q(z) ˜ × , Lp e w(z) w 2 (z)

(3.60) where L mp are the generalised Laguerre polynomials, the parameter p ≥ 0 is the radial index and m is the azimuthal mode index (Siegman 1986, p. 647). The LaguerreSiegman sometimes switches the sign of the Gouy phase from eiϕG to e−i ϕG , e.g. in the definition of the Hermite-Gaussian mode functions. Among others, positive definitions for the Gouy phase are used in Eqs. 16.58 (Siegman 1986, p. 646) and 17.41 (Siegman 1986, p. 686), while 16.60 (Siegman 1986, p. 646) has a negative sign. However, this affects only the direction of the relative phase shift. In this thesis, I tried to maintain a self-consistent definition of the Gouy phase as eiϕG = ei(ϕ(0)−ϕ(z)) .

6

60

3 Scalar Wave Theory

Gaussian modes are presented for completeness, although the Hermite-Gaussian basis is typically used in the framework of this thesis.

3.2.2.4

Higher-Order Transverse Mode Aperturing

A few important properties of higher-order Gaussian modes are discussed in the following. Figure 3.3 displays the first few Hermite-Gaussian higher order modes. Their radii grow with increasing mode-index number n. A reasonable measure for this radius is the position of the outermost peak. It can be shown that the higher-order mode (≥1) half-width rn can be approximated by mode half-width rn ≈

√ nw(z),

(3.61)

where w(z) denotes the waist of the fundamental mode (Siegman 1986, p. 691). In conjunction with this radius, the spatial period of the electromagnetic field modulation can be calculated.√ A higher-order mode with n intensity peaks has n/2 periods across a diameter of 2 nw(z). The resulting 4w(z) spatial period Ʌn ≈ √ . n

(3.62)

Equations 3.61 and 3.62 are good approximations to the exact mode half-width and spatial period, especially at high mode-index numbers n (Siegman 1986, p. 691). The correlation between mode-index number and beam diameter, Eq. 3.61, can be employed to constrain the Gaussian-beam to low mode numbers by adding an appropriate aperture. Only higher orders n with a radius rn ≤ ra pass an aperture with radius ra without significant losses, i.e. (Siegman 1986, p. 691) ( transverse mode aperturing n ≤ Nmax. ≈

ra w(z)

)2 .

(3.63)

Apertures can be mirrors, pinholes or simple blades. Such transverse mode control apertures are often employed in stable laser cavities to prevent higher-order mode resonances. A rule of thumb for the aperture size is 2ra ≈ (3.5 to 4.0)w(z), or slightly larger than the 99 % criterion 2ra, 99% = π w(z) (Siegman 1986, p. 691). This technique is used for higher-order mode suppression at the enhancement cavity of the CLS at the MuCLS. An implementation very similar to the one at the CLS was published recently by researchers involved in the ThomX-project Amoudry et al. (2020).

3.2 Gaussian Beams

3.2.2.5

61

Multi-mode Hermite-Gaussian Beams

Although real laser beams often cannot be represented by a single mode, they can be expanded in the basis of Hermite-Gaussian ones. Therefore, a measure regarding the quality of any optical beam can be expressed by the similarity of the multi-mode beam to a fundamental Gaussian one. The far-field beam angle for a fundamental Gaussian beam is (Siegman 1986, p. 672) θ1/e = lim

z→∞

w(z) λ = . z π w0

(3.64)

The solid angle covered by this Gaussian beam can be defined as a circular cone 2 Ω1/e = π θ1/e =

λ2 . π w02

(3.65)

Similarly, the waist area A1/e can be defined as A1/e = π w02 . Its product with the far-field beam angle is constant: 2 = λ2 , A1/e Ω1/e = π w02 π θ1/e

(3.66)

i.e. the radiation wavelength squared (Siegman 1986, p. 672). In the one-dimensional case, the beam parameter product becomes A1/e Ω1/e |1D =

λ π

(3.67)

as one can easily calculate. In case the laser√consists of multiple modes up to a mode = Nmax. ra as well as its beam divergence index Nmax. ,√its beam diameter ra, N√ max. θ1/e, Nmax. = Nmax. θ1/e increase by Nmax. compared to the one of the fundamental Gaussian beam. Consequently the beam parameter product of waist size at the focus and beam divergence is (Siegman 1986, p. 696) 2 2 2 A1/e, Nmax. Ω1/e, Nmax. = πra, Nmax. π θ1/e,, Nmax. = (Nmax. λ)

(3.68)

and in the one-dimensional case A1/e, Nmax. Ω1/e, Nmax. |1D =

Nmax. λ . π

(3.69)

Such a multi-mode beam is often called “N-times diffraction limited”. Accordingly, this value is very closely related to the M 2 -factor, which is the ratio between the beam parameter product of a multi-mode beam and the fundamental Gaussian one usually determined by the D4σ method Siegman (1998). The latter method was used in the characterisation of the cavity laser system, Sect. 7.2.

62

3 Scalar Wave Theory

3.2.3 Propagation of Gaussian Beams Through Optical Systems This section is based on (Siegman 1986, pp. 581–595 and Chap. 20). 3.2.3.1

The ABCD Ray Matrices

Any laser system consists of a sequence of optical elements like (curved) mirrors or lenses, even free-space propagation can be considered as a component of the optical system since it may result in changes of the wavefront curvature and beam diameter for divergent waves. A powerful, yet simple, technique has been developed which can be used for the analysis and design even of complicated paraxial systems. It is based on ray matrices and called the ABCD-matrix method. In this theory all optical elements are represented in a 2 × 2 matrix and the optical ray is represented by a 2 × 1 vector containing the ray’s displacement from the optical axis ri and reduced slope ri' = n∂r/∂ z, where n is the refractive index7 (Siegman 1986, p. 583) ( ) ( )( ) r2 A B r1 = . r2' r1' C D

(3.70)

This matrix equation is a set of linear equations which describe how the optical element affects the ray’s transverse position and propagation direction from the entrance plane of the element to its exit plane, explicitly r2 = Ar1 + Br1' r2' = Cr1 + Dr1' .

(3.71) (3.72)

An intuitive example is propagation in free space, where A = D = 1, B = L is the effective length of the propagation and C = 0. One important property of the ABCD-matrices is that their determinant is equal to unity (Siegman 1986, p. 584): AD − BC = 1.

(3.73)

The ray transformation through multiple elements can be performed simply by matrix multiplication of the individual optical element’s ABCD-matrices Mi (Siegman 1986, p. 593), r 2 = Ml Ml−1 . . . M2 M1 r 1 . (3.74) A very good step by step introduction into ABCD-matrices offers Chap. 15 in “Lasers” by Siegman (1986). 7

Since the waves propagate in free space only and all transmissive optical elements can be considered as thin elements in the framework of this thesis, n = 1 and the reduced slope equals the real slope.

3.2 Gaussian Beams

63

Fig. 3.4 This sketch displays the geometry employed for the generalisation of the ABCD-matrix formalism to Gaussian beams. Inspired by Fig. 20.3 in Siegman (1986)

3.2.3.2

Extension of the ABCD-Matrix Method to Gaussian Beams

The ABCD-matrix formalism can be extended to Gaussian beams as well. Its theoretical foundation is briefly motivated following the description by (Siegman 1986, p. 779 ff.) which is based on the Rayleigh-Sommerfeld diffraction integral and the paraxial approximation. The result is a generalised version of the Rayleigh-Sommerfeld diffraction integral for free space propagation, cf. Eq. 3.28. Figure 3.4 displays the geometry of the problem to be solved. The points P1 and P2 are conjugate points, which means that they are object-image points. Therefore, the optical path length between them has to be independent of the actual trajectory. Consequently, the on-axis path length (Siegman 1986, p. 781) P1 P2 = n 1 R1 + L 0 − n 2 R2 ,

(3.75)

∑ where n i is the refractive index and L 0 = k n k L k is the optical path length through the system. R1 , R2 are the distances from the object point P1 to the entrance plane of the optical system at z 1 and the distance from the image point P2 to the exit plane at z 2 , respectively. Due to this convention, R2 is negative. Equivalently, the path of the off-axis ray is (Siegman 1986, p. 781) P1 X 1 X 2 P2 = P1 X 1 + X 1 X 2 + X 2 P2 ( )1/2 ( )1/2 = n 1 R12 + x12 + ρ(x1 , x2 ) − n 2 R22 + x22 ) ) ( ( x12 x22 = n 1 R1 + + ρ(x1 , x2 ) − n 2 R2 + . 2R2 2R1

(3.76)

In the last step, the paraxial approximation was applied. Since P1 P2 has to be equal to P1 X 1 X 2 P2 , the optical path ρ (x1 , x2 ) inside the optical system is

64

3 Scalar Wave Theory

ρ (x1 , x2 ) = L 0 −

n 1 x12 n 2 x22 + . 2R1 2R2

(3.77)

The final step is to determine R1 and R2 . The distance Ri is simply the ratio n i xi /xi' , because xi' is the reduced slope. Solving Eq. 3.71 for the reduced slope, x1' in this case, n 1 Bx1 . (3.78) R1 = x2 − Ax1 Replacing x1' in Eq. 3.72 and exploiting the condition for the determinant, Eq. 3.73, the solution for R2 is n 2 Bx2 . (3.79) R2 = Dx2 − x1 Inserting Eqs. 3.78 and 3.79 into Eq. 3.77, the trajectory through the arbitrary ABCD-system is ρ (x1 , x2 ) = L 0 +

) 1 ( 2 Ax1 − 2x1 x2 + Dx22 . 2B

(3.80)

Using this expression for ρ (x1 , x2 ) in Eq. 3.19, the generalised form of Eq. 3.28 is obtained (Siegman 1986, p. 781)8 : √

∫ u˜ (x2 , z 2 ) =

∑

d x1 u˜ ∑

i −ik 1 ( Ax12 −2x1 x2 +Dx22 ) e 2B = Bλ

∫ ∑

d x1 u˜ ∑ K˜ 1D (x2 , x1 ).

(3.81) This generalised Kernel K˜ 1D (x2 , x1 ) allows propagation of a Gaussian beam through an arbitrary ABCD-matrix system. Lowest-Order Hermite-Gaussian Beams A simple Gaussian beam is chosen for simplicity in this demonstration: u(x ˜ 1, z1) = e

x2

−ik 2q˜1

1

.

(3.82)

If this beam is plugged into Eq. 3.81, the beam transformed by the optical ABCDsystem is √ ∫ x2 i 1 2 2 −ik 1 u(x ˜ 2 , z2 ) = d x1 e 1 2q˜1 e−ik 2B ( Ax1 −2x1 x2 +Dx2 ) Bλ ∑ √ (3.83) x2 1 −ik2 2q˜2 2 = e . A + n 1 B/q˜1

The formula in Siegman (1986) is for the wave field ψ, while here it is stated for the paraxial wave field u, ψ = ue−ik L 0 . 8

3.2 Gaussian Beams

65

In the last step, the identity complex radius of curvature

∫

e−ax

2

−2bx

=

√π a

eb

2

/a

has been used and the new

q˜2 A (q˜1 /n 1 ) + B = . n2 C (q˜1 /n 1 ) + D

(3.84)

has been introduced (Siegman 1986, p. 783). Finally, the refractive index can be ˜ incorporated into the q-parameter by defining the reduced q-parameter qˆ = q/n with n nλ 1 λ0 1 = −i = −i . (3.85) R π w2 π w2 qˆ Rˆ This simplifies Eq. 3.84 to (Siegman 1986, p. 784) qˆ2 =

Aqˆ1 + B , C qˆ1 + D

(3.86)

which describes the full paraxial wave transformation of a Gaussian beam with a q-parameter qˆ1 by an optical system using the latter’s ABCD-matrix elements. In addition to the transformation of the q-parameter, the Gaussian beam’s amplitude is also transformed by (Siegman 1986, p. 784) u˜ (x2 = 0, z 2 ) = u˜ (x1 = 0, z 1 )

√ 1 , A + B/qˆ1

(3.87)

cf. Eqs. 3.82 and 3.83. For purely real-valued ABCD-matrices, such as used in this thesis, algebraic manipulation converts the amplitude transformation into9 u˜ (x2 = 0, z 2 ) = u˜ (x1 = 0, z 1 )

√

w1 i ϕ G e 2 , w2

(3.88)

where ϕG = ϕG (z 2 ) − ϕG (z 1 ). Finally, the Gouy-phase can be calculated exploiting the equivalence of Eqs. 3.87 and 3.88 via (Siegman 1986, p. 785) A + B/qˆ1 = e−i ϕG . |A + B/qˆ1 |

(3.89)

9 The switch in sign in Eqs. 3.88 and 3.89 compared to 20.25 and 20.26 in Siegman (1986, p. 784 and 785) originates from the inconsistent definition of the Gouy phase mentioned earlier. This sign change here is necessary for the Gouy phase’s convergence to ϕG = arctan (B/zr ), the classic result defined in Eq. 3.47, for z 1 = 0 at the focus, i.e. qˆ1 = i z R , and free space propagation over a distance B = z2 .

66

3 Scalar Wave Theory

Higher-Order Hermite-Gaussian Beams The propagation of higher-order modes can be calculated equivalently by inserting the higher-order Hermite-Gaussian beam u˜ n (x, z) formula for u˜ ∑ in Eq. 3.81. While the propagation law for the q-parameter turns out to be identical to Eq. 3.86, as expected, the Hermite-Gaussian’s beam amplitude transforms according to (Siegman 1986, p. 799) u˜ n (x2 = 0, z 2 ) = u˜ n (x1 = 0, z 1 )

is

[

1 A + B/qˆ1

]n+1/2 .

(3.90)

Consequently, a one-dimensional Hermite-Gaussian beam’s Gouy phase shift ϕG,n ( ) 1 ϕG,n = n + (3.91) ϕG . 2

The factor 1/2 in the one-dimensional case accounts for the fact that both transverse dimensions contribute equally to the total Gouy-phase shift. For a nonastigmatic two-dimensional Hermite-Gaussian TEMmn -mode, the Gouy phase shift becomes (Siegman 1986, p. 685) ϕG,mn = (m + n + 1) ϕG .

(3.92)

Again, this result is a manifestation of the increased phase-shift that higherorder modes experience, which was derived earlier in the definition of the HermiteGaussian solution to the paraxial wave equation, cf. Eqs. 3.55 and 3.59.

3.2.3.3

ABCD-Matrices Used in this Thesis

The ABCD-matrix formalism is applied to model the enhancement cavity of the CLS for a fundamental Gaussian beam in this thesis. Consequently, three matrices are necessary, which are listed below: First, the one for free space propagation, second, the one for (curved) mirrors and finally, the ABCD-matrix for a thin lens. The latter is required to calculate properties of the mode-matched laser beam outside the cavity, since the curved entrance mirror at the enhancement cavity of the CLS acts as a lens. • Free-space propagation: Propagation over a “free-space” region with a refractive index n and length L: ) 1 Ln . 0 1

( P=

• Curved mirror, arbitrary angle of incidence: The radius of curvature R > 0 for a concave mirror. θ is the angle of incidence.

(3.93)

References

67

Re = R cos(θ ) in the plane of incidence, called “tangential” plane. Re = R/ cos(θ ) perpendicular to the plane of incidence, called “sagittal” plane: (

) 1 0 M= . − R2e 1

(3.94)

• Thin lens: Focal length f > 0 for a converging lens: (

) 1 0 L= . − 1f 1

(3.95)

These and further matrices can be found in Siegman (1986, p. 585 f.).

References Amoudry L et al (2020) Modal instability suppression in a high-average-power and high-finesse Fabry-Perot cavity. Appl Opt 59:116–121 Goodman J (2005) Introduction to Fourier optics, 3rd edn. McGraw-Hill Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley. ISBN: 0-471-30932-X Paganin DM (2006) Coherent X-ray optics. Oxford University Press, Oxford. ISBN: 978-0-19967386-5 Siegman AE (1998) How to (maybe) measure laser beam quality in DPSS (diode pumped solid state) lasers: applications and issues. Optical Society of America. MQ1 Siegman AE (1986) Lasers. University Science Books, Sausalito. ISBN: 0-935702-11-3 Träger F (2007) Handbook of lasers and optics, 1st edn. Springer, New York. ISBN: 978-0-38730420-5 Zangwill A (2012) Modern electrodynamics. Cambridge University Press, Cambridge. ISBN: 9780-521-89697-9

Chapter 4

Enhancement Cavities

The fundamental principles of Gaussian beam transformation discussed in the preceding section can be used to analyse fundamental properties of laser resonators, such as the eigenmode, generalised stability parameter or resonant longitudinal frequencies among others. In the following, the ABCD-matrices of the optical components forming the resonator are assumed to have only real components.

4.1 Stable Optical Resonators An arrangement of optical mirrors that resonates at certain eigenmodes thereby generating either standing light waves (linear resonator) or travelling light waves (ring resonator) is called optical resonator. The enhancement cavity of the CLS can be considered a real and geometrically stable resonator. Therefore, this section focusses on this type of resonators and the description follows (Siegman 1986).

4.1.1 Stability Criteria of Resonators This section follows (Siegman 1986, p. 815 ff.). A resonator is considered to be stable if it fulfils the following three criteria (Siegman 1986, p. 818 f.): 1. A self-consistent eigenmode must exist, i.e. the initial beam has to be reproduced after one round-trip in the resonator. 2. This mode has to be confined to a real and finite transverse waist size. 3. Finally, it must be stable against small perturbations, or in other words, initially small deviations from the transverse profile of the eigenmode must not grow over time or multiple round-trips, respectively. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_4

69

70

4 Enhancement Cavities

The first criteria requires that qˆ2 = qˆ1 . Employing Eq. 3.86 for qˆ2 , this condition can be transformed into a quadratic equation for 1/qˆ1 . Its solutions are the two self-consistent eigenmodes (Siegman 1986, p. 816) D−A 1 1 1 , = ∓ 2B B qˆa qˆb

/(

A+D 2

)2 − 1.

(4.1)

For a real geometrically stable resonator, the confined Gaussian beam solution depends on the sign of B. The confined self-consistent eigenmode must contain a negative imaginary part as described by Eq. 3.41. Otherwise, its field does not fall off at large radii which is required for a physical solution carrying finite power. Consequently, the value below the square root in Eq. 4.1 has to be negative. The remaining step is to prove stability against small perturbations. To this end, assume an input beam qˆ1 = qˆa,b + Δqˆ1 and calculate the resulting mode after one round-trip via Eq. 3.41 and a Taylor expansion about qˆa,b : ( ) [ ]2 A qˆa,b + Δqˆ1 + B Aqˆa,b + B 1 ) ≈ + Δqˆ1 = qˆa,b + Δqˆ2 . qˆ2 = ( C qˆa,b + D C qˆa,b + D C qˆa,b + Δqˆ1 + D (4.2) Consequently, the growth ratio of the perturbation is ⎤2 ⎡ / [ ( ]2 [ ]2 )2 Δqˆ2 D 1 1 A + D A + = = =⎣ ± − 1⎦ ≡ λa2 , λ2b , 2 Δqˆ1 C qˆa,b + D A + B/qˆa,b 2

(4.3) where λa , λb denote the perturbation eigenvalues (Siegman 1986, p. 818). In the second step, the condition on self-consistency was used and in the last step the solution for qˆa,b given in Eq. 4.1 was inserted. This formula can be simplified by introducing a generalised m-value m = (A + D)/2, which is half of the trace of the round-trip matrix of the resonator. As such it is independent of the reference plane unlike qˆa,b . Furthermore, the product λa λb = 1 for all m. In conclusion, a real geometrically stable resonator can be described by the single value (Siegman 1986, p. 820) A+D . (4.4) −1  m  1, m = 2 Since |m| √ < 1, the perturbation eigenvalues can be expressed in terms of m by λa,b = m ± i 1 − m 2 = cos(ϕ) ± i sin(ϕ) = e±iϕ . Comparing this equation with Eq. 4.3 and the defining equation of the Gouy phase, Eq. 3.89, the fundamentalmode’s roundtrip Gouy-phase inside a real and geometrically stable resonator can be expressed alternatively by Siegman (1986, p. 821, 836)

4.1 Stable Optical Resonators

71

( ϕG = sign(B) arccos(m) = sign(B) arccos

A+D 2

) .

(4.5)

Please keep in mind that two-dimensional higher-order TEMmn -beams experience a phase-shift that is m + n + 1 times as strong, cf. Eq. 3.92.

4.1.2 Axial Resonances of Passive Resonators This section follows (Siegman 1986, p. 761 f.), but generalised to multi-mirror resonators, such as ring resonators. The absolute phase shift of a Hermite-Gaussian TEMmn beam including the on-axis shift k L is given by −k L + ϕG,mn . Since a resonant axial mode’s phase shift must be a multiple of 2π, those axial resonances can be determined via k L − ϕG,mn = 2πl, l ∈ N,

(4.6)

where l denotes the axial mode number and L the resonator length for one roundtrip sometimes called perimeter as well (Siegman 1986, p. 762, generalised to multimirror resonators).1 As a matter of fact, the roundtrip ABCD-matrices in the tangential and sagittal plane are not the same in the case of an astigmatic resonator. In turn, the Gouy phase shift along the tangential plane is unequal to the one along the sagittal plane in this most general case, leading to ( ) ) ( 1 1 sagittal tangential − m+ = 2πl, l ∈ N. ϕG ϕG kL − n + 2 2 sagittal

(4.7)

tangential

and ϕG can be calculated using Eqs. 3.91 and 4.5 with the Both, ϕG matrix elements of the sagittal roundtrip matrix and tangential one, respectively. As a result, higher-order modes are resonant at frequencies different from the fundamental one. Moreover, even higher-order modes of the same mode index sum may not be degenerate in an astigmatic resonator, depending on the resonator’s astigmatism and bandwidth of the resonances. Consequently, resonators can be efficient spatial mode filters. Summarising the Gouy-phase effects of Eq. 4.7 into the single quantity ϕG,mn = sagittal tangential + (m + 0.5) ϕG , this equation can be rephrased for a (n + 0.5) ϕG dispersion-free resonator into ( ϕG,mn ) 2πc ( ϕG,mn ) = l+ ΔωFSR , ωlmn = l + 2π 2π L

(4.8)

√ arccos( g1 g2 ) on p. 762 in “Lasers” are the Gouy phase shift for half the roundtrip in the twomirror case, while arccos(m) is the one for the complete roundtrip.

1

72

4 Enhancement Cavities

where ΔωFSR is the spacing between two fundamental axial modes, often referred to as free spectral range (FSR). It is straight forward to see that the frequency difference between the higher-order modes and the fundamental one, the transverse mode range (TMR), depends on the roundtrip Gouy phase. Therefore, the roundtrip Gouy phase can be extracted from a measurement of the higher-order mode resonances. In the framework of this thesis, this technique has been employed to model thermal deformations of the mirror radii of curvature. Thus it evaluates changes of the waist size at the focus since the roundtrip Gouy phase is dependent on the resonators position in the stability range.

4.1.3 Misaligned Optical Elements Another important aspect of resonators is their stability against small misalignments of their optical components, because of the technically limited alignment precision. Each component may be displaced from the optical axis by a distance d at its front plane and tilted by a small angle α with respect to the optical axis. Such an analysis can be performed with ray-matrix methods as well, cf. Chap. 15.4 in “Lasers” by Siegman (1986, p. 607 ff.). In terms of the ray matrices, deviations from the optical axis can be expressed with a misalignment vector added to the usual ABCD-matrix equation: )( ) ( ) ( ) ( A B r1 Δx r2 = + . (4.9) r2' r1' C D Δα The components of the misalignment vector are linked to the element’s action and positional deviations via (Siegman 1986, p. 609) Δx = (1 − A)d + (T − n 1 B)α Δα = −Cd + (n 2 − n 1 D)α.

(4.10)

T is the thickness of the optical element represented by the ABCD-matrix and Δr = (Δx, Δα) is the misalignment vector. Expression 4.9 is inconvenient for cascading multiple elements. However, it can be transformed into the 3 × 3-matrix equation ⎛ ⎞ ⎛ ⎞⎛ ⎞ A B Δx r2 r1 ⎝r2' ⎠ = ⎝C D Δα⎠ ⎝r1' ⎠ , 1 0 0 1 1

(4.11)

which can be multiplied equivalently to the 2 × 2 ABCD-matrices in the preceding discussion (Siegman 1986, p. 609). This allows to model the effects of misalignment in complex optical systems, such as in an optical resonator. For such a system, an effective axis ray can be defined for which the axis ray is reproduced after one roundtrip, i.e. the condition

4.1 Stable Optical Resonators

73

Mr 0 + Δr = r 0

(4.12)

is fulfilled, where M denotes the roundtrip ABCD-matrix. The solutions to this matrix equation at this particular reference plane are the displacement r0 and the slope r0' with respect to the optical axis (1 − D)Δx + BΔα 2− A− D CΔx + (1 − A)Δα r0' = , 2− A− D r0 =

(4.13)

which constitute a self-consistent axis ray for the misaligned system r 0 (Siegman 1986, p. 612). A measure for the effect of misalignment on the operation of a passive resonator or enhancement cavity is the change in the spatial overlap between the incoming external beam and the mode circulating inside the cavity. This is a meaningful quantity because the overlap between the two beams determines the efficiency of the coupling (Carstens et al. 2013). Such perturbations are not limited to transverse ones, like lens displacements or mirror tilts expressed by the misalignment vector, but include longitudinal ones which affect the eigenmode of the resonator as well. Examples for the latter are thermal lensing, thermal mirror deformations or changes in the distances between the resonator elements. The overlap between the two beams can be calculated with the overlap integral U (Joyce and DeLoach 1984) |2 |  | | ∗ (x, y)|| , U = || d xd yψideal (x, y)ψpert

(4.14)

where ψideal (x, y) is the eigenmode of the unperturbed beam while ψpert (x, y) is the one in the equilibrium after the perturbation took place. An advantageous feature of this metric is that it is conserved upon propagation through lens-like elements, like mirrors (Carstens et al. 2013). In the special case of transverse misalignments, the perturbation can be expressed by (Joyce and DeLoach 1984) '

ψpert (x, y) = ψideal (x − r0 , y)eikr0 x ,

(4.15)

whose extension to perturbations along both transverse planes is straightforward. These ingredients at hand, the sensitivity of enhancement cavities against misalignments can be calculated now using the change of overlap as a metric.

74

4 Enhancement Cavities

4.2 Enhancement Cavities The most important parameter of an enhancement cavity employed in an ICS is the power stored inside the resonator since the latter is proportional to the laser pulse’s energy or in other words the number of laser photons available for collision in a single interaction of the two beams. Therefore, this section discusses the (steady state) enhancement of the external laser, the methods for keeping the cavity on resonance with the external laser, ways for experimentally determining the stored power and processes which can induce unwanted loss terms.

4.2.1 Mode Matching To efficiently store laser power inside a single transverse mode of a passive resonator, the mode of the external laser beam must agree with this particular transverse eigenmode of the resonator, Eq. 4.1, at the inside of the input coupler (Siegman 1986, p. 412). Typically, excitation of the fundamental transverse mode is desired since its field distribution is most confined and exhibits lowest leakage or diffraction losses (Siegman 1986, p. 412). To this end, properly designed lens- or mirror telescopes or other mode-matching optics are required which transform the external seeding laser’s shape and divergence such that it agrees with the ones of the resonator eigenmode. During the design process, the effect of the input coupler has to be considered, especially if its radius of curvature is not infinite because it acts like a lens on the transmitted laser beam. Moreover, the circulating beam inside the cavity may contain contributions of higher-order Zernike polynomials at the input coupler, e.g. because roundtrip astigmatism compensation is performed at a single deformable cavity mirror. In this case, the external beam shape should be manipulated by deformable mirrors such that the same higher-order Zernike perturbations are imprinted on the external laser beam to improve geometric overlap and in turn coupling efficiency. Although no heterodyne diagnostic for mode-match exists at the CLS, multiple different approaches have been proposed for cavities in general, e.g. (Mueller et al. 2000; Fulda et al. 2017; Maganã-Sandoval et al. 2019).

4.2.2 Phase Shift at a Multilayer Dielectric Mirror The subsequent derivation follows (Siegman 1986, p. 403 ff.). Fresnel’s equation for the reflection of electromagnetic waves at a dielectric interface states that the phase change is either 0 or π depending on whether the refractive index of the reflective medium is lower or higher than the one of the medium in which the wave is propagating (Zangwill 2012, p. 591 f.; Träger 2007, p. 251). Nevertheless, typically a slightly different convention has been adopted for the analysis of modern

4.2 Enhancement Cavities

75

Fig. 4.1 Schematic of a multilayer dielectric mirror. The general situation with two beams impinging onto the mirror (a1 , a2 ) and two reflected ones (b1 , b2 ) matches the situation at an input coupler of an enhancement cavity. The system of equations links the incoming beams to the reflected ones, ri j and ti j are complex field reflection- and transmission coefficients

complex dielectric optics (Bond et al. 2016). A typical multilayer mirror consists of a stack of alternating layers of high and low refractive index with typically a quarterwavelength optical thickness. A brief introduction into multilayer mirrors can be found in Träger (2007, p. 373 ff.). An input/output coupler additionally may contain an antireflective coating on the back of the substrate, cf. Fig. 4.1. The magnitude of the mirror’s reflection- and transmission coefficient are provided by the manufacturer, while the exact number of layers, thus the absolute phase shift across the mirror, is not. However, only the relative phase shift between the reflected and the transmitted beam is important to calculate the interference effects between the reflected and transmitted waves at such a mirror. The relative phase shift of the beam upon reflection and transmission at a lossless multilayer dielectric mirror can be calculated by introducing two reference planes equidistant on either side of the mirror and requiring energy conservation. The beams impinging on this two-port system, cf. Fig. 4.1, are interconnected by the following system of equations, where the relative phase shift between the two reference planes is incorporated into the respective field reflection (ri j )-/transmission (ti j )-coefficient: )( ) ( ) ( r11 t12 a1 b1 = = Sa. b= b2 t21 r22 a2

(4.16)

In a lossless system, the power flow into the mirror must agree with the power flow out of the mirror. Accordingly, b† b = (Sa)† Sa = a† S† Sa ≡ a† a.

(4.17)

The last equality arises from energy conservation in a lossless system which requires S† S = 1. Moreover, a dielectric multilayer mirror does not contain a DC-

76

4 Enhancement Cavities

magnetic field. Therefore, the system is reciprocal, i.e. |Si j | = |S ji | (Siegman 1986, p. 405). Applying those constraints to Eq. 4.16, the following set of conditions is obtained (Siegman 1986, p. 405)2 : |r11 | = |r22 |, |t12 | = |t21 |,

(4.18)

|r11 | + |t21 | =|r22 | + |t12 | = 1, ∗ ∗ +t21r22 = 0. r11 t12

(4.19) (4.20)

2

2

2

2

These conditions are met by the asymmetric matrix for reflection and transmission at a dielectric interface ( ) r t (4.21) Sinterface = t −r as well as a symmetric matrix ( Smirror =

r it it r

) (4.22)

with π/2 phase shift upon transmission among others, where i = eiπ/2 is the imaginary unit (Siegman 1986, p. 406). For a mirror embedded into air or vacuum (refractive index n = 1), the phase shift for reflection at both reference planes has to be identical. Contrary to the symmetric solution, Eq. 4.22, the asymmetric one, Eq. 4.21, cannot fulfil this last constraint. Accordingly, the symmetric solution is a proper choice for the description of reflection and transmission at a multilayer dielectric mirror. A less formal derivation of the relative phase shift upon transmission through a multilayer optic can be found in Bond et al. (2016, Sect. 2.4, p. 16 ff.), albeit in a beam-splitter application. Nevertheless, the same calculation can be performed for a two-port system as well, e.g. the input coupler of an enhancement cavity, yielding the same results.

4.2.3 Steady State Power Enhancement and Finesse of a Passive External Cavity The relative phase shifts for the reflection- and the transmission coefficient at a multilayer dielectric mirror were derived in the preceding section. With these results, the power enhancement of a resonator can be calculated in the steady state situation depicted in Fig. 4.2.

2

The condition (4.20) follows from the preceding ones.

4.2 Enhancement Cavities

77

Fig. 4.2 Scalar fields in the steady state situation at the input coupler of a enhancement cavity. Losses by absorption at all optical components and scattering can be summarised into a single roundtrip transmission A(ω) and a single roundtrip phase shift eiθ(ω)

4.2.3.1

Field Relationships

This section loosely follows (Siegman 1986, ch. 11.3). The external laser beam is described by the scalar field ψi (ω), the input coupler by its intensity reflectivity R(ω) = |r |2 and its intensity transmission T (ω) = |it|2 . In this section, the values for the intensity are used, because those are the ones usually provided by the manufacturer. Upon one roundtrip in the resonator, the beam acquires a round-trip phase shift θ(ω) and its intensity is attenuated by diffraction, scattering and non-unity reflectivity. The latter effects can be summarised into a single intensity attenuation value A(ω). After one roundtrip and before reflection at the input coupler, the circulating beam ψcirc., ret. (ω) is ψcirc., ret. (ω) =

√

A(ω)eiθ(ω) ψcirc. (ω).

(4.23)

Using Eq. 4.16 and the symmetric reflection matrix, Eq. 4.22, the reflected beams can be calculated from the incoming ones, ψi (ω) and ψcirc.,ret. . The circulating field ψcirc. itself in the steady state is given by √ √ √ ψcirc. = i T (ω)ψi (ω) + R(ω) A(ω)eiθ(ω) ψcirc. (ω).

(4.24)

By solving Eq. 4.24 the field inside the enhancement cavity can thus be related to the one of the external laser: ψcirc.

√ i T (ω) = ψi (ω) ≡ iψc (ω). √ √ 1 − R(ω) A(ω)eiθ(ω)

Accordingly, Eq. 4.24 can be transformed into

(4.25)

78

4 Enhancement Cavities

ψc (ω) =

√

T (ω)ψi (ω) +

√

√ R(ω) A(ω)eiθ(ω) ψc (ω),

(4.26)

an equation containing only real-valued transmission and reflection coefficients. Equation 4.25 can be converted into an equation for the field enhancement inside the resonator: √ ψc (ω) T (ω) = . (4.27) √ √ ψi (ω) 1 − R(ω) A(ω)eiθ(ω) Combining Eqs. 4.23 and 4.25, the field reflected off the input coupling mirror can be expressed as ψr (ω) =

√

R(ω)ψi (ω) −

√

√ T (ω) A(ω)eiθ(ω) ψc (ω).

(4.28)

Finally, plugging the expression for ψc (ω), Eq. 4.25, into Eq. 4.28, yields ψr (ω) = ψi (ω)

√

√ R(ω) − [R(ω) + T (ω)] A(ω)eiθ(ω) . √ 1 − R(ω)A(ω)eiθ(ω)

(4.29)

When the numerator of this equation is equal to zero, the seeding laser’s energy is coupled completely into the enhancement cavity. Such a resonator is called impedance matched. This condition is fulfilled for a certain frequency ω if the input coupler’s reflectivity and transmission are matched to the roundtrip losses in conjunction with the roundtrip phase shift accumulating to a multiple of 2π. In case of a lossless input coupler R(ω) + T (ω) = 1, the first requirement for impedance matching simplifies to R(ω) = A(ω).

4.2.3.2

Power Enhancement

The enhancement cavity’s power enhancement P and reflected power Refl are the absolute squares of Eqs. 4.27 and 4.29, respectively. In the following, the formulas for power enhancement P(ω) and reflectivity Refl(ω) are provided for the general situation as well as two useful special cases. • General case. √ | |2 | | T (ω) | | , P(ω) = | √ | iθ(ω) 1 − R(ω) A(ω)e √ | |√ | R(ω) − [R(ω) + T (ω)] A(ω)eiθ(ω) |2 | . Refl(ω) = || √ | 1 − R(ω) A(ω)eiθ(ω)

(4.30)

• Assumptions: The cavity is on resonance, i.e. θ(ω) = 2πm. The input coupler is lossless and the dispersion is negligible, i.e. no frequency-dependence.

4.2 Enhancement Cavities

79

P=

T √

, (1 − R A)2 (√ √ )2 R− A Refl = . √ 1 − RA

(4.31)

• Special case: The same as in the preceding case, but additionally the cavity is impedance matched, i.e. R(ω) = A(ω). 1 , T Refl = 0, P=

(4.32)

F = π P.

4.2.3.3

Finesse

The finesse F of a resonator is a measure for its resolving power if the cavity is employed as a transmission filter (Siegman 1986, p. 436). Frequency discrimination improves either if the longitudinal modes are spaced further apart from each other, i.e. the free spectral range increases, or if the resonances’ FWHM decreases. Consequently, the finesse is defined as (Siegman 1986, p. 436) √ π rtot ΔωFSR = F= ΔωFWHM 1 − rtot

(4.33)

√ with the roundtrip reflectivity rtot = R A. The free spectral range ΔωFSR was defined in Eq. 4.8. Therefore, the last parameter to be determined to arrive at the last expression is the resonance’s FWHM ΔωFWHM . The frequency ω ' at which the power enhancement of the passive resonator drops by a factor of two compared to the resonance condition at ωres , Eq. 4.31, defines the position of the half width of the FWHM: | |2 √ | | T T P(ωres ) | | ! ' P(ω ) = | = . (4.34) √ √ | = ' | 1 − R Aeiθ(ω ) | 2 2(1 − R A)2 For simplicity, a dispersionless and lossless resonator is considered here. This requires that | | √ √ √ √ ) ! ( ' |2 | |1 − R Aeiθ(ω ) | = (1 − R A)2 + 2 R A 1 − cos (θ(ω ' )) = 2(1 − R A)2 . (4.35) Since ) cos (θ(ωres )) = 1 on resonance, 1 − cos (θ(ω ' )) = 2 sin2 ((θ(ωres )− θ(ω ' ))/2 . Finally, θ(ωres ) − θ(ω ' ) = (ωres − ω ' )L/c = ΔωFWHM L/(2c). Inserting

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4 Enhancement Cavities

these transformations into Eq. 4.35, the FWHM of the resonance turns out to be (Siegman 1986, p. 436) ΔωFWHM

( ) 1 − rtot 4c 1 − rtot 2πc 1 − rtot = arcsin = ΔωFSR √ . ≈ √ √ L 2 rtot π rtot L π rtot

(4.36)

For a given resonator geometry or free spectral range, respectively, the higher its finesse, the smaller the resonance’s bandwidth and the higher the roundtrip reflectivity rtot . In turn, the latter corresponds to a stronger power enhancement. As a result, a high finesse resonator allows for high intra-cavity power with low seeding power of the external laser beam on the one hand. On the other, the narrowband resonances increase the resonator’s sensitivity against detuning between the longitudinal cavity resonances and the frequency comb, in other words the excited axial modes, of the external seeding laser. To ensure an overlap of the external laser’s frequency comb with the passive enhancement cavities’ axial modes, active feedback loops are required which are discussed in the next section.

4.2.4 Locking a Laser Oscillator to a Passive Resonator Naturally occurring relative shifts between the axial mode combs, e.g. driven by thermal fluctuations, inhibit a continuous resonance between a seeding laser and an enhancement cavity due to its narrowband resonances. Thus, an active feedback system is required which keeps the two systems on resonance. In this section, the frequency comb of a potentially dispersive resonator, like an oscillator, is presented first. Afterwards, different locking techniques are briefly discussed.

4.2.4.1

Frequency Combs of Dispersive Resonators

When calculating the axial mode spacing of a dispersion-free resonator, Eq. 4.8, an offset frequency ϕG,mn ΔωFSR /(2π) was obtained. Such an offset frequency is called carrier-envelope offset frequency ωCE . Dispersion in a resonator produces an additional contribution to the offset frequency since the dispersion relation becomes ω = k(ω)c which can be expanded in a Taylor series about the frequency ωc (Siegman 1986, p. 335 & 337) by k(ω) =

ω − ωc 1 ωc + + GVD(ω − ωc )2 + . . . , 2 vph vg

(4.37)

where vph denotes the phase velocity, vg the group velocity and GVD the group velocity dispersion. Phase-locking the axial modes of a laser oscillator, so called mode-locking, generates short laser pulses and balances non-linear dispersion terms (Pupeza 2011, p. 11). For simplicity, the dispersive medium is assumed to fill the

4.2 Enhancement Cavities

81

resonator completely and the Gouy phase is assumed to be identical in the sagittal and tangential planes. In this case the wave vector in Eq. 4.7 can be simply replaced by the terms of Eq. 4.37 up to the second order. The solution for the longitudinal resonances is ωlmn

( ) ( ϕG,mn ) 2πvg vg = l+ + ωc 1 − = 2π L vph ( ) ϕG,mn vg ωr + ωc 1 − = = lωr + 2π vph

(4.38)

= lωr + ωCE , where ωr is the repetition rate / free spectral range of the laser system. If the dispersive medium, e.g. the active medium in a laser oscillator, does not fill the whole cavity, but only a fraction of length d, the roundtrip phase shift k(ω)L → kair (L − d) + k(ω)d in Eq. 4.7. This introduces multiplicative terms “weighting” ωr and ωc in the second line of the equation by the contribution of this small extent compared to the cavity roundtrip distance. 4.2.4.2

Optimum Overlap Between the Seeding Laser and the Enhancement Cavity

The actual spectrum of the laser pulse emitted from a mode-locked laser oscillator is not an infinite comb of equally spaced as well as equally intense lines. Instead, the laser medium has a limited (non-uniform) gain bandwidth. As a result, the actual frequency comb of the mode-locked laser is multiplied with a frequency-dependent gain curve, for example (Udem et al. 2002). To maximise the power stored inside the enhancement cavity, the spectrum must overlap exactly with the peaks of the resonances of the enhancement cavity, cf. Fig. 4.3a. This requires adjustment of two parameters of the seeding laser’s spectrum, its carrier-envelope offset frequency as well as its free spectral range, or in other words, the oscillator’s repetition rate. Adjusting the total cavity length, for example by a motorised mirror, achieves the latter. Feedback systems that keep the free spectral range of the seeding laser on resonance with the enhancement cavity are presented later. Accordingly, matching the carrier envelope offset frequencies remains. The sole contribution to the carrierenvelope offset frequency in a dispersion-free enhancement cavity is the roundtrip Gouy phase shift. If residual dispersion exists in the enhancement cavity, it adds to the carrier-envelope offset frequency, of course. As an enhancement cavity for inverse Compton scattering is an empty, thus basically dispersion-free, resonator, its carrier-envelope offset frequency will not be equal to the one of the seeding oscillator in general. Additionally, the latter’s carrier-envelope offset frequency can change, e.g. due to thermal fluctuations. Consequently, the oscillator’s carrier-envelope offset frequency must be adjustable and ideally stabilised by a feedback loop.

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Fig. 4.3 Effect of carrier-envelope offset drift-compensation via free spectral range adjustment. a the situation of optimum overlap. The passive resonator’s normalised enhancement is coloured in blue. The orange lines indicate the intensity enhancement of the individual frequencies of the seeding laser’s frequency comb. In optimum overlap, the comb frequencies perfectly coincide with the maxima of the enhancement. If the frequency stabilisation loop compensates drifts of the carrierenvelope offset frequency by adjusting the free spectral range, the drift is only compensated perfectly at ωc . For all other frequencies of the seeding laser’s frequency comb, a mismatch to the optimum overlap exists that increases with the difference to ωc . b displays this situation. All frequencies apart from ωc experience a reduced enhancement decreasing the power stored in the enhancement cavity. This effect becomes more severe the larger the seeding laser’s bandwidth and the higher the enhancement cavity’s finesse

4.2.4.3

Effect of Locking Solely the Repetition Rate

The derivation in this section is similar to the presentation in (Pupeza et al. 2011). If only one active control loop exists that keeps the two combs on resonance by adjusting the free spectral range, this affects the power enhancement. Assume that an optimum overlap of the seeding laser’s modes and the enhancement cavity’s modes exists initially. Moreover, the central mode of the oscillator’s spectrum ωc = lc ωr + ωCE , l ∈ N,

(4.39)

is to be locked to the peak of the corresponding cavity resonance. If the mode spacing ωr changes, this can be perfectly corrected for by this feedback. But what happens if the carrier envelope frequency changes by ΔωCE ? This drift shifts the complete frequency comb laterally while the spacing between the modes remains constant, in other words it affects the combs offset from zero. The control loop tries to counteract the shift of ωc by adjusting the free spectral range by Δωr which recovers ωc = lc (ωr − Δωr ) + ωCE + ΔωCE .

(4.40)

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83

Fig. 4.4 a Magnitude and phase of the field enhancement, Eq. 4.27, and of the reflection coefficient (b), Eq. 4.29, for an impedance-matched resonator with a power enhancement of 100. The phase of the circulating field switches sign across the resonance which is true for the reflected field as well. This can be exploited to generate a bipolar error signal via Pound-Drever-Hall locking

This implies that lc Δωr = ΔωCE . Consequently, the drift of the carrier-envelope offset frequency is only compensated completely for the frequency ωc on which the active feedback is operating. Any other frequency with mode index lc + x deviates from its position at optimum overlap afterwards. Its new frequency ' = ωc+x − xΔωr ωc+x

(4.41)

has been calculated by replacing lc in Eq. 4.40 by lc + x, x ∈ Z. Figure 4.3b displays the situation in which a drift of the carrier-envelope offset frequency was compensated by adjustment of the free spectral range. The effect of this mismatch becomes ever more prominent for modes of the seeding comb with an increasing mode-index difference to the central mode as well as with an increasing finesse of the enhancement cavity as the latter is inversely proportional to the bandwidth of the cavity resonances. As a result, seeding lasers with a wide spectrum are more prone to this effect, or -for transform limited laser pulse durationsthe shorter the seeding laser’s pulse the stronger the power drop if only a single feedback system acting on the laser’s free spectral range is employed. The next paragraph discusses the feedback method used to lock the laser to the enhancement cavity at the MuCLS, before potential solutions to the issue discussed above are presented.

4.2.4.4

Pound-Drever-Hall Locking

Like any active feedback loop, the Pound-Drever-Hall locking requires a bipolar error signal indicating the direction and magnitude of the deviation from the target

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4 Enhancement Cavities

position, locking electronics which translate the error signal into an actuator signal for the control mechanism and the control mechanism itself. Since the Pound-DreverHall laser frequency stabilisation acts on the seeding laser’s free spectral range, usually, the control mechanism is a piezo-actuated mirror which enables seamless adjustments of the cavity perimeter. The locking electronics are standard tools and therefore not discussed here. Instead, the main ideas for generating the error signal are described in the following. For a detailed introduction, description and formulas, the interested reader is referred to the tutorial paper by Black (2001)3 which also includes tips and tricks for practically setting up such a system. The basic foundations of this technique date back to the 1940s when Pound introduced a similar technique for microwave applications (Pound 1946) which was translated to optics by Drever and Hall in the 1980s (Drever et al. 1983). The main idea of this technique is to translate the phase of the circulating field into a measurable intensity signal. The phase change across resonance is imprinted in the reflected wave as well, i.e. the argument of Eqs. 4.27 and 4.29 depicted in Fig. 4.4. To this end, the carrier frequency is periodically phase modulated which generates non-resonant sidebands to the carrier frequency if the modulation frequency is higher than the width of the resonance. In this case, the sidebands are completely reflected, while the carrier is transmitted into the enhancement cavity. Thus, the phase of the carrier’s field reflected from the input coupler carries information on the cavity’s frequency. As a result, oscillations in the reflected intensity occur at the modulation frequency due to interference between the reflected field of the carrier and the sidebands. Accordingly, the amplitude of the temporally varying signal, which is proportional to the imaginary part of the reflection coefficient, can be separated with a mixer. Sign and magnitude of the retrieved amplitude encode the absolute frequency shift because the reflection coefficient’s phase changes sign at the resonance, cf. Fig. 4.4b.

4.2.4.5

Other Common Frequency Stabilisation Techniques

Although the Pound-Drever-Hall locking scheme is very robust and widely applied, other techniques exist to keep an external laser resonant with a passive cavity as well. Since the Pound-Drever-Hall frequency stabilisation technique is implemented at the MuCLS, examples for other techniques are provided for interested readers. For polarisation discriminating resonators, Hänsch and Couillaud proposed a technique similar to Drever and Hall in which they exploit the relative phase shift between the resonant polarisation and the rejected one to generate the error signal (Hänsch and Couillaud 1980). While the authors inserted an intra-cavity polariser to generate a strong polarisation discrimination at the cost of the finesse in their proof-ofprinciple experiment (Hänsch and Couillaud 1980), the non-orthogonal incidence on 3

Black seems to use a different convention for reflection and transmission at a mirror compared to the one used in this thesis. If the convention employed in this thesis is used, F(ω) in the manuscript has to replaced with −F(ω). Moreover, the enhancement cavity is assumed to be impedance matched in the manuscript which is not mentioned explicitly.

4.2 Enhancement Cavities

85

the mirrors in a ring resonator can provide sufficient polarisation discrimination for this technique enabling the application at high-finesse resonators, e.g. Pupeza et al. (2010). This technique was later refined by controlling the amount of polarisation discrimination by specifically designed multilayer mirrors, which allow to operate the cavity at a design polarisation (Liu et al. 2016). Alternatively to sidebands or polarisation, the interference between non-resonant higher-order transverse modes and the fundamental provides the desired error signal. This family of techniques can be referred to as transverse mode mismatch locking, because the transverse mode of the seeding laser has to be slightly detuned from the cavity eigenmode to excite a non-resonant higher-order mode. This can be achieved, for example, by a non-mode matched seeding laser (Wieman and Gilbert 1982; Harvey and White 2003) or by a tilted input beam (Shaddock et al. 1999). Advantages of this technique are its straight forward implementation and low cost at the expense of its sensitivity to changes of the exciting laser’s or cavity’s mode shape and misalignment. The latter can be improved with the implementation by Miller and Evans (2014).

4.2.4.6

Feedback on the Carrier-Envelope Offset Frequency

Drifts of the carrier-envelope offset frequency can lead to a degraded performance of the enhancement cavity if only one feedback loop acting on the free spectral range is implement as discussed earlier. In case of an octave-spanning laser frequency spectrum, “f-to-2f” interferometry reveals the carrier-envelope offset frequency (Telle et al. 1999; Jones et al. 2000). A variable amount of dispersive material, e.g. a motorised pair of glass wedges, as well as the pump laser power or the laser head temperature adjust the carrier-envelope frequency by acting on the Kerr effect and the temperature dependence of the dispersion in the gain medium, respectively. A summary on carrier envelope phase stabilisation can be found in Yamanouchi and Midorikawa (2013, Chap. 6). The narrow laser spectrum in the picosecond regime, where typical laser systems designed for ICS-applications operate, prevent the application of “f-to-2f” interferometry, even if an octave-spanning super-continuum was generated from the picosecond pulse (Börzsönyi et al. 2013). However, Börzsönyi et al. (2013) demonstrated two techniques quantifying the carrier envelope offset frequency: (a) multi-beam interferometry in which consecutive pulses interfere and (b) an additional PoundDrever-Hall analysis at a second mode at the wings of the comb. Precise measurement and controlled change of the carrier envelope offset frequency of picosecond pulses have been demonstrated as well (Börzsönyi et al. 2013; Jójárt et al. 2014). Although, these methods would work at the enhancement cavity of the CLS as well, none of them have been implemented yet. In contrast to quite complex and space consuming multi-beam interferometry, implementation of the second Pound-Drever-Hall-based feedback is straight forward. Nevertheless, the latter option still requires isolating a mode of the wings of the comb, e.g. by spatially separating the individual modes of the comb with a grating. Although its sensitivity is limited to the bandwidth of the

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resonances like the normal Pound-Drever-Hall locking which results in a narrower dynamic range compared to the multi-beam interferometer, this should not be an issue if only stabilisation of the carrier-envelope offset frequency is desired. Implementation of a carrier envelope offset frequency feedback system would significantly improve the X-ray flux stability as drifts severely affect the cavity gain. Börzsönyi et al. (2013) observed a reduction of the intra-cavity gain from ≈ 7000 to ≈ 4000 for a carrier envelope phase drift of one radian at a cavity with a finesse of 28000, which is similar to the one at the CLS. As their laser pulse length of 2 ps was shorter than the one at the CLS, the effects at the CLS might be smaller. Nevertheless, significant flux drops during hands-off runs were observed at the MuCLS, especially in non-equilibrium thermal conditions, which could be recovered by manual adjustment of the carrier envelope phase. Therefore, a second feedback loop stabilising the carrier envelope offset frequency is highly desirable, not only at the MuCLS.

4.2.5 Determination of the Circulating Power Various approaches exists for the determination of the power stored inside a passive resonator. Most of them require precise knowledge of the cavity’s free spectral range, which can be determined easily, especially if the seeding laser is locked to the cavity. This ingredient at hand, calculation of the stored power via measurement of the resonance’s linewidth, e.g. (Bondu and Debieu 2007; Locke et al. 2009), Doppleror ringing effect, e.g. (Li et al. 1991; Poirson et al. 1997; Lawrence et al. 1999; Matone et al. 2000), and ring-down or decay-time method, e.g. (Rempe et al. 1992; An et al. 1995; The Virgo Collaboration 2007), can be discussed. A brief comparison of the three methods can be found in Isogai et al. (2013). This section is based on the literature mentioned above. All of these techniques work in both, reflection as well as transmission.

4.2.5.1

Measurement of the Resonance’s Linewidth

Determination of the stored power via measurement of the resonances’ FWHMband-width is straight forward. Scanning the seeding laser’s frequency across one longitudinal resonance while recording the power transmitted or reflected from the cavity yields the cavity’s resonance curve. Ideally one of the fundamental modes should be tested, since the roundtrip losses for higher-order ones may differ from those of the TEM00 one. The finesse, Eq. 4.33, connects the free spectral range and the resonance’s FWHM-bandwidth to the total losses of one roundtrip. Knowing the transmission of the input coupler, Eq. 4.31 calculates the power enhancement. The stored power is obtained multiplying the power enhancement with the power coupled into the resonator. In reality, only a fraction ccoupled of the total power incident on the input coupler Pin might be matched to the cavity eigenmode, for example, due to a residual mode

4.2 Enhancement Cavities

87

mismatch and/or non-resonant side-bands for Pound-Drever-Hall-locking. Therefore, the power reflected from the input coupler Pref when the laser is locked to the cavity is (4.42) Pref = Pin − Pcoupled + ReflPcoupled . Refl is the reflectivity of the cavity for the matched beam, cf. Eqs. 4.30 and 4.31, Pcoupled = ccoupled Pin is the power matched to the cavity mode, in other words the incoming seeding power that can be coupled into the cavity (Loewen 2003, p. 78). The coupling coefficient ccoupled can be calculated by ccoupled =

Pin − Pref , Pin (1 − Refl)

(4.43)

from which Pcoupled can be determined. Finally, a potential solution for the enhancement cavity of the CLS based on Locke et al. (2009) is proposed for non-invasive determination of the resonator finesse and in turn stored power. To this end, the seeding laser is locked to the enhancement cavity, e.g. by the Pound-Drever-Hall mechanism. A second electro-optical modulator creates another set of sidebands in the seeding laser which are scanned across another fundamental longitudinal mode of the enhancement cavity, for example the closest one located at a distance of one free spectral range in frequency space. Plotting the transmitted- or reflected power against the modulation frequency generates the resonance curve. Its shape is similar to the magnitudes displayed in Fig. 4.4, except for the amplitude of the modulation and the off-resonance boundary values. Another example is displayed in Fig. 4.5a. The reason for this is that only a small fraction of the total power is distributed into these sidebands while the dominant fraction remains on resonance at all times. The modulation frequency at the resonance’s peak corresponds to the free spectral range, however, its value is already precisely known at the CLS, since the enhancement cavity is locked to the CLS’s master frequency by a second feedback. Division of the resonance’s free spectral range by its FWHM-bandwidth provides the finesse which in turn allows for determination of the stored power again if the input coupler’s reflectivity is known. In their manuscript, Locke et al. (2009) also derive an analytic model, which allows to extract the roundtrip losses and input coupler transmission from the reflection geometry by fitting a spectral shape curve to the measured resonance curve. However, their model is only valid for high-finesse, low diffraction- and absorption loss cavities. Thus, a careful analysis is necessary, if all assumptions of their model are valid for the specific case.

4.2.5.2

Ring-Down or Decay-Time Method

A second method is based on measuring the decay time of the power stored inside the cavity. An examplary measurement is depicted in Fig. 4.6. Once the seeding laser

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Fig. 4.5 Linewidth-measurement and Ringing-effect. a is a typical measurement of a fundamental resonance’s linewidth in transmission. b are two measurements of the ringing-effect, once in transmission and once in reflection. The exponential decay of the envelope can be clearly observed. a & b are reprinted with permission from Isogai et al. (2013) ©The Optical Society

pulse is switched off rapidly, the power stored inside the enhancement cavity decays exponentially at a characteristic rate determined by the cavity’s roundtrip losses. The power stored inside the cavity Pcavity decays according to Pcavity = PPcoupled (R A)n ,

(4.44)

where P is the power enhancement, Pcoupled the seeding laser’s power coupled into the cavity, R A the roundtrip intensity reflectivity and n the number of roundtrips after the seeding laser was turned off. The number of roundtrips can be transformed into a function continuous in time via t

!

(R A)n = (R A) trt = e

−τ

t decay

,

(4.45)

where trt denotes the roundtrip time and τdecay a decay time constant. The solution for the decay time constant is τdecay = −

2π τD ≡ . 2ΔωFSR ln(rtot ) 2

(4.46)

2 and trt by 2π/ΔωFSR . The decay constant’s In this equation R A was replaced by rtot negative sign can be explained by the fact that rtot is less than unity. Accordingly, the logarithm is negative and the whole decay constant takes a positive value as expected. Combining Eqs. 4.44, 4.45 and 4.46, an equation for the exponentially decaying stored power is obtained from which the transmitted and reflected power after the laser was turned off can be derived to be

Ptrans = Ttrans PPcoupled e Prefl = T APPcoupled e

− τ2t

− τ2t

D

D

(4.47)

.

(4.48)

4.2 Enhancement Cavities

89

Fig. 4.6 Finesse determination via ringdown/decay-time method in transmission geometry. This measurement was performed at the enhancement cavity at the CLS. The retrieved losses were 171 ppm, resulting in a finesse of about 36700 or a power enhancement of about 16920

Ttrans is the power transmittance of the mirror at which the transmission signal is recorded which is typically much lower than the input coupler’s transmission T . Thus the total losses for one roundtrip are derived by fitting an exponential to the power decay. Knowing the total losses, the finesse can be determined and if the input coupler transmission was measured as well, the power enhancement and in turn the stored power can be calculated. The cavity decay time measurement is the standard tool at the CLS to determine the enhancement cavity’s stored power, cf. Fig. 4.6. However, this technique typically requires the seeding laser to be turned off or, at least, to be unlocked from the cavity in order for the intracavity power to decay. Accordingly, this is an invasive technique and does not allow for monitoring the resonator’s state during X-ray experiments. This limit can be overcome if the ringing effect is exploited, a term which is also present in the name ring-down. This ringing effect is discussed next.

4.2.5.3

Doppler-/Ringing Effect

Another technique for finesse determination which is very closely related to the first one is the so-called Doppler- or ringing effect. Instead of sweeping the laser frequency slowly across the cavity’s resonance so that the steady state condition always builds up, the scanning time in this method is shorter than the decay time of the cavity. The power that builds up inside the enhancement cavity as the frequency is scanned across the resonance decays exponentially with a decay time constant which depends on the photon life-time inside the resonator. As a result, the decay is superimposed with a beating note which originates from the interference between the decaying resonant cavity field and the seeding laser’s non-resonant field, cf. Fig. 4.5b. If the frequency is swept significantly faster than the cavity decay time,

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the response becomes impulse-like with very narrow low amplitude modulations for approximately the duration of the sweep followed by a purely exponential decay like in the classic ring-down discussed in the preceding paragraph (Gagliardi and Loock 2014, p. 34 f.). The decay time and subsequently the roundtrip losses can be extracted either from the envelope of the ringing function or directly from the exponential decay in the case of an impulse-like response. Due to this technique’s close similarity to the linewidth scan, it could be implemented into the CLS with an analogous design which would provide the additional advantage of shorter scanning times.

4.2.6 Thermally Induced Mode Coupling The last aspect of enhancement cavities that is going to be discussed briefly are thermal effects and transverse mode-coupling. The cavity-roundtrip Gouy phase ϕG,mn determines the transverse modes’ location in frequency space, cf. Eq. 4.7 and 4.8, which can be a multiple of 2π compared to the roundtrip Gouy phase of the fundamental mode ϕG,00 for certain mode(s) ϕG,ab . Consequently, these modes are degenerate and resonant at the same frequencies. Mismatch between the seeding mode and the eigenmode of the resonator directly excites these degenerate higher-order modes, while scattering, thermal deformations of the mirrors or mirror imperfections contribute to the coupling between the fundamental mode and degenerate higher-order modes, e.g. (Klaassen et al., 2005; Bullington et al., 2008). Degenerate transverse modes may be disadvantageous because higher-order modes either exhibit an intensity minimum or a weaker secondary intensity peak on-axis, cf. Fig. 3.3. This is detrimental for applications requiring strong on-axis fields at the focus, like inverse Compton scattering or high-harmonic generation. Choosing the operation point of the enhancement cavity in the design process such that the lowest-order transverse modes retain a significant frequency shift with respect to the fundamental one, avoids the appearance of mode-degeneracy. Almost degenerate modes with a high mode-index sum can be suppressed due to their much larger beam radius, e.g. by spatial filtering discussed earlier in Sect. 3.2.2.4. To maintain the resonator’s non-degenerate state, the mechanisms shifting the resonance frequencies of the higher-order modes must be considered. Absorption in the mirror coatings and/or substrates causes their thermal expansion which increases the mirrors’ effective radii of curvature, e.g. (Bullington et al. 2008). This reduces the cavity’s roundtrip Gouy phase as the resonator geometry remains fixed. In turn transverse modes with a low positive frequency shift compared to the fundamental one move closer to the latter in frequency space, while transverse modes with a low negative frequency shift compared to the fundamental one move further away from the latter in frequency space. Since this effect is proportional to the power stored inside the resonator, the frequency difference between the fundamental mode and the closest low-order mode of the empty resonator has to decrease (positive frequency shift) or increase (negative frequency shift) as well with stored power. Alternatively,

References

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compensation of the thermal deformation can be attempted. One potential device for thermal compensation is presented in Chap. 8. A much more extensive discussion of thermal effects in an enhancement cavity and potential compensation mechanisms can be found in the review “On Special Optical Modes and Thermal Issues in Advanced Gravitational Wave Interferometric Detectors” by Vinet (2009) and references therein. At the operation point of the CLS’s enhancement cavity, the closest low-order mode has a mode-index sum of 2 and possesses a frequency difference to the fundamental mode of about −2 MHz. The closest high-order mode located slightly above the fundamentals resonance is located at about 3 MHz and has a mode-index sum of 9 which can be suppressed by transverse mode aperturing.

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Liu X et al (2016) Laser frequency stabilization using folded cavity and mirror reflectivity tuning. Opt Commun 369:84–88 Locke CR et al (2009) A simple technique for accurate and complete characterisation of a FabryPerot cavity. Opt Express 17:21935–21943 Loewen RJ (2003) A compact light source: design and technical feasibility study of a laser-electron storage ring X-ray source. PhD thesis, Stanford University Maganã-Sandoval F et al (2019) Sensing optical cavity mismatch with a mode-converter and quadrant photodiode. Phys Rev D 100:102001 Matone L et al (2000) Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect. Phys Lett Sect A: Gener Atom Solid State Phys 271:314–318 Miller J, Evans M (2014) Length control of an optical resonator using second-order transverse modes. Opt Lett 39:2495–2498 Mueller G et al (2000) Determination and optimization of mode matching into optical cavities by heterodyne detection. Opt Lett 25:266–268 Poirson J et al (1997) Analytical and experimental study of ringing e ects in a Fabry–Perot cavity. Application to the measurement of high & finesses. J Opt Soc Am B 14:2811–2817 Pound RV (1946) Electronic frequency stabilization of microwave oscillators. RSI 17:490–505 Pupeza I (2011) Power scaling of enhancement cavities for nonlinear optics. PhD thesis, LudwigMaximilians- Universität, München Pupeza I et al (2010) Power scaling of a high-repetition rate enhancement cavity. Opt Lett 35:2052– 2054 Pupeza I et al (2011) Power scaling of femtosecond enhancement cavities and high-power applications. In: Proceedings of the SPIE, fiber lasers VIII: technology, systems, and applications, vol 7914, p 79141I Rempe G et al (1992) Measurement of ultralow losses in an optical interferometer. Opt Lett 17:363– 365 Shaddock DA, Gray MB, McClelland DE (1999) Frequency locking a laser to an optical cavity by use of spatial mode interference. Opt Lett 24:1499–1501 Siegman AE (1986) Lasers. University Science Books, Sausalito. ISBN: 0-935702-11-3 Telle HR et al (1999) Carrier-envelope offset phase control: a novel concept for absolute optical frequency measurement and ultrashort pulse generation. Appl Phys B: Lasers Opt 69:327–332. ISSN: 09462171 The Virgo Collaboration (2007) Measurement of the optical parameters of the Virgo interferometer. Appl Opt 46:3466–3484 Träger F (ed) (2007) Springer handbook of lasers and optics, 1st edn. Springer, New York. ISBN: 978-0-387-30420-5 Udem T, Holzwarth R, Hansch TW (2002) Optical frequency metrology. Nature 416:233–237 Vinet JY (2009) On special optical modes and thermal issues in advanced gravitational wave interferometric detectors. Living Rev Relativ 12:5 Wieman CE, Gilbert SL (1982) Laser-frequency stabilization using mode interference from a reflecting reference interferometer. Opt Lett 7:480–482 Yamanouchi K, Midorikawa K (eds) (2013) Progress in ultrafast intense laser science IX. Springer, Berlin Heidelberg. ISBN: 978-3-642- 35052-8 Zangwill A (2012) Modern electrodynamics. Cambridge University Press, Cambridge. ISBN: 9780-521-89697-9

Chapter 5

Fundamentals of X-ray Imaging and Spectroscopy

Up to now, the generation of X-rays via laser-electron interaction has been presented as well as the fundamentals of scalar wave theory including free-space propagation. However, in most scientific cases X-rays are employed as a diagnostic tool to gain insight into material properties which requires understanding the interactions of Xrays with matter. Accordingly, the goal of this chapter is to familiarise the reader with the main interaction mechanisms between X-rays and matter. Furthermore, applications and experimental techniques are introduced that exist at the MuCLS and have been employed within this thesis. Contrary to the preceding chapters, these topics are presented in a compendious fashion and the interested reader is directed to relevant PhD-theses or textbooks for details throughout this chapter. The textbooks “Elements of Modern X-ray physics” by Als-Nielsen and McMorrow (2011), “An Introduction to Synchrotron Radiation” by Willmott (2019), “Coherent X-ray Optics” by Paganin (2006), “Handbook of X-ray Imaging” edited by Russo (2018) and “Xrays and Extreme Ultraviolet Radiation” by Attwood and Sakdinawat (2017) build the main sources of this chapter.1

5.1 X-ray Interactions with Matter Although the boundary for the transition between ultraviolet radiation and the (soft-) X-ray regime is very smooth, the term X-ray commonly denominates electromagnetic radiation with photon energies surpassing about 100 eV. On the high energy side, the term γ -ray is frequently employed for X-ray energies in the MeV. Both, the γ -ray as well as the ultraviolet regime are not relevant for this thesis. Therefore, the discussion is restricted to the X-ray energy range well below 100 keV. Consequently, effects 1 The reader is reminded that the phase convention ωt − kr is used throughout this thesis, while these textbooks employ −(ωt − kr). Sign differences between formulas in those books and this thesis are a consequence thereof.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_5

93

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like pair-production which occurs in matter for X-rays exceeding twice the electron rest energy, cf. (Russo 2018, p. 15 f.), or photo-nuclear interactions, cf. (Russo 2018, p. 18 f.) are neglected.

5.1.1 Differential Cross Section and Complex Refractive Index X-rays mainly interact with bound electrons of the material’s atoms in the energy range considered here in which their photon energy approaches or surpasses the binding energy of the electrons. In the semi-classical model, those Z electrons in an atom with atomic number Z are bound at discrete energies to a positively charged nucleus (charge Z e). Scattering of an X-ray at an electron involves elastic and inelastic processes (Attwood and Sakdinawat 2017, p. 61). Accordingly, its interaction can be described by a damped driven harmonic oscillator in which the dissipative processes are responsible for the damping and the Lorentz-force is the driving mechanism (Attwood and Sakdinawat 2017, p. 46): m

∂x ∂2x + m[ + mωs2 x = −e (E i + v × B i ) ≈ −e E i , 2 ∂t ∂t

(5.1)

where m is the electron mass, x the trajectory, [ the damping, ωs the frequency of a resonance and E i = E i rˆ E = E 0 eiωt rˆ E and B i are the incident electric and magnetic fields. As long as the fields are sufficiently weak, the bound electron’s velocity is small compared to the speed of light and the magnetic part of the Lorentz-force can be neglected. Its solution is (Attwood and Sakdinawat 2017, p. 46) x=

5.1.1.1

ω2

e Ei 1 . 2 − ωs − i[ω m

(5.2)

The Differential Cross Section

Taking the time derivative of the trajectory and division by the speed of light yields β, which can be inserted into the radiation part of the electric field in Eq. 2.58.2 E rad

( ( )) ω2 nˆ × nˆ × rˆ E e−i ω R/c e−ik R e2 =− E = f Ei , i R 4π ε0 mc2 ω2 − ωs2 − i[ω R

(5.3)

where

2

Please recall that the symbol e in Eq. 2.58 is the general unsigned charge and has to be replaced by −e for the electron here.

5.1 X-ray Interactions with Matter

95

( ( )) ( ( )) ω2 nˆ × nˆ × rˆ E ω2 nˆ × nˆ × rˆ E e2 f =− = −re 2 4π ε0 mc2 ω2 − ωs2 − i[ω ω − ωs2 − i[ω

(5.4)

is the scattering amplitude. re is the classical electron radius. Accordingly, a measure for the efficiency of a scattering process can be defined by the absolute square of the scattering amplitude which is called the differential (scattering) cross section (Zangwill 2012, p. 777) dσscat. = | f |2 . dΩ

(5.5)

Since an atom contains multiple electrons of which a certain amount contributes to each of these resonances, each of them has a different oscillator strength gs , indicated by integer numbers in the semi-classical model. Consequently, the sum of all oscillator strengths has to be equal to the atomic number Z of the element, or, in other words the atom’s number of electrons (Attwood and Sakdinawat 2017, p. 55). Therefore, the scattering amplitude is modified to ( ( )) Σ gs ω2 nˆ × nˆ × rˆ E )) ( ( f = −re = − ( f 1 + i f 2 ) nˆ × nˆ × rˆ E . 2 2 ω − ωs − i[ω s

(5.6)

In the direction of forward scattering, the scattering amplitude f simplifies to f = ( f 1 + i f 2 ) rˆ E ,

(5.7)

( ) ( ) ) ( where the vector identity nˆ × nˆ × rˆ E = − rˆ E − nˆ · rˆ E nˆ and the fact that nˆ · rˆ E ≈ 0 in forward scattering were used.

5.1.1.2

The Complex Refractive Index

Similarly, the current density induced in a homogeneous material by the incoming electric field can be expressed in terms of the electron’s trajectory, Eq. 5.2, as (Attwood and Sakdinawat 2017, p. 62) j (r, t) = −eρa

Σ

gs x˙s ,

(5.8)

s

where ρa is the atomic number density. Since forward scattering is sufficient for describing refractive effects and absorption in the X-ray regime (Attwood and Sakdinawat 2017, p. 61), this approximation is used in the following. The current density of Eq. 5.8 can be inserted into the inhomogeneous vector wave equation for the electric field (Zangwill 2012, p. 715)

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(

) ) ( 1 ∂2 1 1 ∂ 2 − ∇ E(r, t) = − j (r, t) + ∇ρ , c2 ∂t 2 ε0 c2 ∂t

(5.9)

where ρ denotes the charge density, which can be related to the current using the continuity equation. Exploiting the approximation for forward scattering in a similar fashion as before generates the wave equation (Attwood and Sakdinawat 2017, p. 63) ) ( 2 2 n ∂ 2 − ∇ (5.10) E(r, t) = 0, c2 ∂t 2 which is very similar to the d’Alembert equation for free space, cf. equation 3.2, except for a multiplicative factor n(E ) in front of the time derivative. This factor is defined as the refractive index. The negative sign in front of β inside the refractive index n = 1 − δ − iβ originates from the phase convention ωt − kr. (Attwood and Sakdinawat 2017, p. 63 f, p. 66) n(E ) = 1 −

2πρa (hc)2 [ f 1 + i f 2 ] ≡ 1 − δ − iβ. E2

(5.11)

It is related to the scattering amplitude, Eq. 5.7, in the approximation to forward scattering. A comparison of the last equality provides a relation for the refractive index decrement δ (Attwood and Sakdinawat 2017, p. 66) δ=

2πρa (hc)2 f1. E2

(5.12)

The absorption index β in its most general form is (Attwood and Sakdinawat 2017, p. 66) 2πρa (hc)2 β= (5.13) f2 . E2 The leading contribution to f 2 arises from photoelectric absorption, discussed at a later stage, for the X-ray energy range considered in this thesis (Willmott 2019, p. 23, p. 38).

5.1.1.3

The Lambert-Beer Law of Attenuation

Since Eq. 5.10 neither couples components of the respective fields nor the magnetic and electric field, it can be transformed analogously to section 3.1 into a scalar inhomogeneous Helmholtz equation3

3

A detailed derivation based on the dielectric function can be found in (Paganin 2006, p. 69 ff.).

5.1 X-ray Interactions with Matter

97

(

) ˜ ∇ 2 + k 2 n 2 (E ) ψ(r) = 0,

(5.14)

with k the vacuum wave number. Consequently, the solution to this wave equation is a scalar field of the form (Attwood and Sakdinawat 2017, p. 67) kr ˜ ψ(r) = ψ˜ 0 e−in(E )kr = ψ˜ 0 e−i 

iδkr e

kr e−β  .

(5.15)

propagation phase shift attenuation

Unfortunately, the field is not a quantity that can be accessed directly as detectors are sensitive to the intensity I (r) only. Therefore, they perceive the signal 2 ˜ I (r) = |ψ(r)| = |ψ˜ 0 |2 e−2βkd ≡ |ψ˜ 0 |2 e−μd ,

(5.16)

where d is the distance traversed inside the medium with an attenuation coefficient μ along the propagation direction. This exponential decay of the X-ray intensity while traversing matter is known as the Lambert-Beer law (Willmott 2019, p. 39). The attenuation coefficient μ relates to the absorption index β and the absorption cross section σa via (Willmott 2019, p. 39 f.) μ = 2βk = ρa σa ,

(5.17)

while the phase shift ϕ of the scalar field relates to the refractive index decrement via (5.18) ϕ = δkd. d The transition to inhomogeneous materials requires just xd → z0 x dz with x ∈ {μ(r ⊥ , z), β(r ⊥ , z), δ(r ⊥ , z)}. Figure 5.1a visualises the described effects of attenuation and phase shift upon propagation through matter. Both effects, the reduction in amplitude as well as the phase shift, provide contrast mechanisms for several imaging techniques, sketched in Fig. 5.1b. While absorption allows for attenuation imaging, the gradient of the phase shift is proportional to the refractive angle and free-space propagation of the exit wave generates interference fringes proportional to the phase shift’s second derivative. The latter two mechanisms are responsible for the image contrast in grating-based phase-contrast and propagation-based phase-contrast imaging discussed in Sect. 5.4.

5.1.2 Interaction Mechanisms The preceding section treated the interaction mechanisms in a very general fashion by including them into an universal driven oscillator model. This section aims to briefly introduce the different microscopic interactions that make up the harmonic

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Fig. 5.1 Effects of a wave field’s interaction with matter. a Visualisation of an X-ray wave field’s ψ interaction with a medium of a refractive index decrement δ and an absorption index β. The incident wave field is depicted in blue, the interacting wave field propagating inside matter in green and the transmitted wave field in orange. The result of the interaction is a reduction in the wave field’s amplitude and a phase shift ϕ. b Effects of the varying phase shift upon traversing a homogeneous cylindrical object, which generates a curved wave front and accordingly a refraction angle α proportional to the derivative of the phase shift. Free-space propagation of the outgoing wave field generates interference fringes f related to the second derivative of the phase shift. These effects can be exploited for phase-contrast imaging. Panel a is adapted from Fig. 2.1 in Hehn (2019) with permission of the author. Panel b is a reprint of Fig. 3.2 (b) in Gradl (2019) with permission of the author

oscillator model relevant in the sub-MeV X-ray regime. Figure 5.2 presents those, namely photoelectric absorption (b), elastic (Thomson/Rayleigh-) scattering (c) and inelastic Comptom scattering (d). Those effects have in common that the X-ray photon interacts with the shell electrons of the atoms and not with the atomic core itself. Apart from these interactions, X-ray photons can penetrate matter unperturbed with a certain probability as well, Fig. 5.2a.

5.1.2.1

Photoelectric Absorption

This section is mainly based on Sects. 2.6.3 and 2.7 of Willmott (2019). Photoelectric absorption constitutes the main interaction mechanism responsible for absorption in the energy range between 15 keV and 35 keV in which the CLS operates. The X-ray photon is absorbed via the photo-effect by a single bound electron promoting the electron from its bound state into the vacuum. Its kinetic energy originates from the difference between the X-ray energy and the binding energy of this electron. Accordingly, the binding energy constitutes a lower bound on the required X-ray energy for which this particular transition can take place. The corresponding frequencies are the resonance frequencies ωs in the semi-classical oscillator model. Since these electrons are removed from the atom, this is a dissipative process contributing to the attenuation [ in this model. However, the exact energy (E −3 ) and Z -dependence is not captured in this simple oscillator model which predicts only a narrow-band peak in the attenuation instead of a step increase. Both effects can be accounted for by introducing a continuous superposition of oscillators with oscillator strength gs instead of the aforementioned integer numbers for each shell (Als-Nielsen and McMorrow 2011, p. 277). The calculation of the energy and Z -dependence

5.1 X-ray Interactions with Matter

99

Fig. 5.2 Main interaction mechanisms between X-ray photons and matter. In the X-ray energy range relevant to this thesis, Thomson-scattering, photoelectric absorption and Compton scattering are the main interaction mechanisms. Between 15 keV and 35 keV, photoelectric absorption dominates although the contribution by Compton scattering becomes increasingly significant at higher X-ray energies, cf. (Willmott 2019, p. 23). Image adapted from Fig. 3.1 in Gradl (2019) with permission of the author

requires a full quantum mechanical treatment based on Fermi’s golden rule for the transition matrix elements, details can be found, e.g. in Sect. 7.1 of Als-Nielsen and McMorrow (2011). The resulting proportionality of the absorption cross section for photoelectric absorption σph is (Als-Nielsen and McMorrow 2011, p. 247) σph,theoretical ∝

Z5 . E 3.5

(5.19)

The deviation from the experimental behaviour σph,exp. ∝

Z4 E3

(5.20)

arises from neglecting the Coulomb interaction between the photo-electron and the positively charged ion (Als-Nielsen and McMorrow 2011, p. 247). Additionally, Compton scattering (explained in Sect. 5.1.2.3) is a small, but nevertheless existing, contribution in the experimental data. The electron vacancy in the atom’s shell is filled by an outer shell electron which dissipates the excess energy either via the

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5 Fundamentals of X-ray Imaging and Spectroscopy

Auger-effect in which a secondary electron is emitted or via X-ray fluorescence. Since each element has distinct transition energies, X-ray fluorescence results in element-specific characteristic fluorescence spectra. Auger electron emission is the preferential decay mechanism for low atomic-number atoms, while for high atomic number atoms X-ray fluorescence dominates. For details on Auger-emission and X-ray fluorescence, the reader is referred to Sect. 2.7 of Willmott (2019).

5.1.2.2

Elastic Scattering

In elastic scattering, non-relativistic electrons are accelerated by the electric field of the incident photon and in turn re-irradiate dipole-radiation at the exact same frequency. Consequently, no net energy transfer to the electron takes place (Willmott 2019, p. 25). Thus the differential cross section of this process can be calculated from the driven oscillator model, Sect. 5.1.1, by plugging the scattering amplitude, Eq. 5.4, into the definition of the differential cross section, Eq. 5.5. Assuming unpolarised radiation and integrating over the full solid angle yields the cross section for elastic scattering at a single electron (Als-Nielsen and McMorrow 2011, p. 279): σelastic, e =

ω4 8π 2 re 2 . 2 3 (ω − ωs )2 + ([ω)2

(5.21)

For [ → 0 and ω ≪ ωs , this equation reduces to the classical formula for the Rayleigh cross section ( ) 8π 2 ω 4 σRayleigh, e = re , (5.22) 3 ωs while for [ → 0 and ω ≫ ωs , the classical Thomson cross section σThomson, e =

8π 2 r 3 e

(5.23)

is retrieved. Within the latter assumption, i.e. far from absorption edges, the imaginary part of the scattering amplitude vanishes and the scattering amplitude for real matter, Eq. 5.6, in forward scattering, Eq. 5.7, becomes f ω≫ωs,max , [→0 = f 1, ω≫ωs,max , [→0 = re

Σ

gs = Zre .

(5.24)

s

Accordingly, the Thomson cross section of an atom is proportional to the number of electrons squared σThomson = Z 2 σThomson, e ,

(5.25)

5.1 X-ray Interactions with Matter

101

and all electrons scatter coherently, except for very high atomic number atoms, where corrections to the free-electron approximation are required (Attwood and Sakdinawat 2017, p. 56). Equivalently, the real part of the scattering amplitude far from the absorption edges, Eq. 5.24, can be inserted into Eq. 5.12 to determine the refractive index decrement for forward scattering 2π(hc)2 re ρe . (5.26) δ= E2 Since ρe denotes the electron density, the refractive index decrement is proportional to the atomic number Z .

5.1.2.3

Inelastic (Compton) Scattering

This section is mainly based on Sect. 2.3 of Willmott (2019). Contrary to the elastic processes described above, Compton scattering is an inelastic incoherent process. The X-ray photon transfers a part of its momentum to the electron during this scattering process thereby liberating the latter from the atom to which it is bound. The energy of the scattered photon Escattered can be derived from the energy of the incident photon Eincident applying momentum and energy conservation via (Willmott 2019, p. 23) λC Eincident =1+ (1 − cos θ ). λincident Escattered

(5.27)

λincident is the wavelength of the incident radiation, while λC = h/m e c = 2.43 pm is the Compton wavelength and θ is the angle of the scattered radiation with respect to the incident one. The efficiency of the Compton scattering process can be described by the Klein-Nishina cross section for an atom of atomic number Z which approaches [ σCompton = 2πre2

1+𝜣 𝜣2

(

ln(1 + 2𝜣) 2(1 + 𝜣) − 𝜣 1 + 2𝜣

) +

] 1 + 3𝜣 ln(1 + 2𝜣) − Z, (1 + 2𝜣)2 2𝜣

(5.28) when energy transfer is sufficiently large that the electrons are able to escape from the atom (Willmott 2019, p. 24). 𝜣 = hω/(mc2 ) is the X-ray photon energy divided by the electron rest mass. The proportionality to the number of electrons would be expected for an incoherent process since the interaction takes place between a single photon and a single electron. Although Compton scattering is the dominating interaction only at very high X-ray energies well above 100 keV, its contribution to the total cross section becomes noticeable already at the higher end of the CLS’s energy range.

102

5.1.2.4

5 Fundamentals of X-ray Imaging and Spectroscopy

The Total Cross Section

All aforementioned contributions are additive and make up the total cross section σtotal = σph + σelastic + σCompton .

(5.29)

5.2 Coherence This section is based on Chap. 4 of Attwood and Sakdinawat (2017). Coherence is a property of the X-ray wave field originating from the X-ray source and characterising the ability of its X-rays to interfere one with another. In order to do so, the individual wave fields must be connected by a well-defined phase relation. Monochromatic electromagnetic fields radiated by a single point source are such fields since they are perfectly correlated in space and time. However, real X-ray sources have a finite spatial extent and emit radiation within a finite bandwidth. The former can be described by a spatial distribution of point sources. Their emission typically is incoherent, except for lasers and other coherent processes like high-harmonic generation. This reduces the correlation of the radiation fields perceived at a certain point in space since the field emitted by each individual point source travels a different distance to the observation point and in turn acquires a different phase shift. Accordingly, the correlation reduction increases with the transverse size of the X-ray source and the angle between the observation point and the optical axis. Assuming a Gaussian distribution of the X-ray source, the transverse coherence angle θcoh., trans , i.e. the angle to the optical axis below which the wave field’s correlation is still sufficiently high, can be expressed as θcoh., trans. =

λ , 2πd

(5.30)

where λ is the X-ray’s wavelength and d = 2σtrans. is twice the transverse rms-source size σtrans. . Since this angle is extremely small, a transverse coherence length lcoh., trans. =

λ z 2π d

(5.31)

can be defined via the small angle approximation, cf. (Attwood and Sakdinawat 2017, p. 117)4 . z is the distance to the X-ray source along the optical axis. In other words, the detrimental effect of an increased source size can be mitigated by a small observation angle, i.e. a large source-to-detector distance. This is the reason why many beamlines at synchrotrons which work with phase-sensitive techniques are very long. 4

Other authors use different criteria in their derivation. Thus their results may vary slightly.

5.3 K-Edge Subtraction Imaging

103

A further complication arises from the finite bandwidth. Dissimilar wavelengths de-phase upon propagation along the optical axis which constrains the longitudinal extent in which the radiation is correlated as well. This range shortens with an increasing spectral bandwidth of the radiation since a monochromatic wave’s phase is proportional to kz, where k denotes the wave number. The longitudinal coherence length λ2 , (5.32) lcoh., long. = 2Δλ defines the longitudinal range over which the spectrum still sufficiently interferes, cf. (Attwood and Sakdinawat 2017, p. 115). A detailed analysis of coherence based on the mutual coherence function is out of scope of this thesis and may be found in textbooks, e.g. by (Paganin, 2006) or (Born and Wolf, 2019).

5.3 K-Edge Subtraction Imaging Knowing the interaction mechanisms between X-rays and matter, this and the following sections describe techniques exploiting the interaction’s effects for imaging and spectroscopy. Figure 5.3a displays the mass attenuation coefficient of calcium and carbon, two materials common in bones and soft tissue, respectively. As carbon’s attenuation is much smaller than calcium’s, contrast agents such as iodine are employed to enhance soft tissue visibility. In contrast to the latter’s continuous E −3 dependence in the displayed energy range, the third material, iodine, exhibits a step increase in the attenuation at 33.17 keV. This step is a result of the K-edge which opens up a new channel for photoelectric absorption. Jacobson (1953) recognised that this step increase in the attenuation coefficient, whose location in energy is element-specific, can create an element-specific contrast image. To do so, an image taken with monochromatic radiation slightly below this element’s absorption edge is subtracted from a second one acquired right above this absorption edge. In the resulting subtraction image, the contributions to the overall attenuation signal by all the other elements basically vanish since their attenuation coefficients barely change. This leaves behind only the element exhibiting a step increase in absorption. As this technique is usually employed at the K-edge of the material under investigation, this technique is named K-edge subtraction (KES) imaging. The image contrast in the difference image can be approximated by ln

Ihigh I0, high

− ln

Ilow I0, low

) ( ≈ μc, low − μc,high dc ,

(5.33)

where Ihigh/low are the intensities in the high/low energy image, I0, high/low are the intensities in the respective reference images, μc, high/low the attenuation coefficients

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5 Fundamentals of X-ray Imaging and Spectroscopy

Fig. 5.3 a Mass attenuation coefficient and b δ/β ratio of relevant elements. Carbon is one of the main building blocks of soft matter, while calcium is contained in bones. Iodine on the other hand is an important contrast agent in medical imaging. The discontinuous step increase in the mass attenuation coefficient, in a displayed for the iodine K-edge, compared to the other materials is exploited in K-edge subtraction imaging as a contrast mechanism. This can be used to separate bones and iodine-contrasted vessels in angiography (Kulpe et al., 2018). The large δ/β-ratio for carbon indicates that phase-contrast imaging is beneficial if soft tissue is imaged. Since this ratio exhibits a peak around 25 keV for carbon, this X-ray energy is used in such experiments at the MuCLS. The displayed data is tabulated in Henke et al. (1993). Figure adapted from Fig. 3.4 in Gradl (2019) with permission of the author

of the element with the K-edge at the high/low X-ray energy and dc denotes the thickness of this element (Kulpe 2020, p. 14). At the MuCLS, the energy switching was first established by tuning the CLS’s quasi-monochromatic spectrum such that it partially surpasses the K-edge energy of the element. This spectrum was used to acquire the high energy images, while the low energy spectrum was generated by a filter made of the same element as the one under investigation. Accordingly, this filter removes the energies located above the K-edge of this element. Since the filter does not remove the high energies entirely and since the two spectra are quasi-monochromatic an additional weighting of the images is required. This procedure is described in detail in Kulpe et al. (2018) and Kulpe (2020, p. 41 ff.). Those images are subtracted considering the E −3 -dependence of the photoelectric effect as a finite difference in the mean X-ray energies of the high energy image and the low energy image exists: (

)

ImageKES = ln Imagehigh −

(

Emean, low Emean, high

)3 ln (Imagelow ) .

(5.34)

ImageKES is the final contrast image, Imagehigh/low are the weighted high and low energy images and Emean, high/low the mean energies of the weighted images (Kulpe et al., 2018). For further details on theory, a historic overview and various application examples, the reader is referred to the PhD-Thesis by Stephanie Kulpe (Kulpe, 2020).

5.4 X-ray Phase Contrast Imaging

105

5.4 X-ray Phase Contrast Imaging The goal of X-ray phase-contrast imaging is to translate the phase shift imprinted onto the incident wave field upon its passage through the specimen into a measurable intensity modulation. Such an approach becomes especially beneficial for substances which affect the wave field’s phase strongly while negligibly reducing its amplitude, i.e. which feature a high δ/β-ratio. Figure 5.3b displays this ratio for different elements. Steep drops in the ratios of calcium and iodine arise at their absorption edges. Among them, carbon, one of the main building blocks of soft tissue, exhibits a very high δ/β-ratio which indicates that image contrast of soft tissue should be improvable by phase-contrast imaging techniques. Accordingly, multiple approaches have been demonstrated transferring phase information into intensity modulations. In an approximate historic order, those are: Crystal interferometry (Bonse and Hart 1965), in-line phase-contrast or propagationbased imaging (PBI) (Snigirev et al. 1997; Cloetens et al. 1996; Wilkins et al. 1996), analyser-based or diffraction-enhanced imaging (Davis et al. 1995; Ingal and Beliaevskaya 1995; Chapman et al. 1997), edge-illumination (Olivo et al. 2001), gratingbased imaging (Momose et al. 2003; Weitkamp et al. 2005; Pfeiffer et al. 2008), single-grid (Wen et al. 2010; Morgan et al. 2011) and speckle-based imaging (Morgan et al. 2012; Berujon et al. 2012). Of those available techniques, propagation-based imaging and grating-based imaging have been implemented at the MuCLS to date. Therefore, these two techniques are presented in more detail in the following.

5.4.1 Propagation-Based Phase-Contrast Imaging This section is mainly based on Paganin et al. (2002). Good introductions into the topic are presented in the PhD-theses by Regine Gradl (2019) and Lorenz Hehn (2019) as well. Additionally, the latter contains explicit derivations of the steps presented below. Propagation-based phase-contrast imaging is an optic-free free-space propagation method, as its name already suggests. Consequently, its implementation into existing set-ups for attenuation imaging is straight forward given three requirements are fulfilled: 1. The X-ray source must provide a sufficiently coherent X-ray beam. 2. The set-up must allow for an adequate propagation distance. 3. The X-ray detector resolves the formed intensity modulations. One of the most popular implementations is the single-material phase-retrieval algorithm developed by Paganin et al. (2002). In order to derive the phase shift, it is assumed that 1. The specimen is homogeneous, i.e. it consists of one single material.

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2. The X-ray wave field is monochromatic and can be approximated by a plane wave. 3. The propagation distance between specimen and detector is constrained to the near-field region, i.e. a Fresnel number F = sr /(zλ) ≫ 1 (Willmott 2019, p. 378 f.). In other words, the propagation distance z is much smaller than the ratio of the structure size to be resolved sr and the wavelength of the radiation λ. Multiplication of the paraxial Helmholtz equation, Eq. 3.24 with the complex conjugate of the paraxial transverse wave field u˜ ∗ (r ⊥ , z) and solving for the imaginary part results in (Paganin 2006, p. 295)5 k

∂ I (r ⊥ , z) = ∇⊥ [I (r ⊥ , z)∇⊥ ϕ(r ⊥ , z)] , ∂z

(5.35)

where intensity I (r ⊥ , z) is defined as usual and the phase ϕ is defined according to Eq. 5.18. This is the so-called transport-of-intensity equation. Approximation of the differential term on the left hand side with the finite difference

results in

∂ I (r ⊥ , 0) I (r ⊥ , z) − I (r ⊥ , 0) = ∂z z

(5.36)

z I (r ⊥ , z) = I (r ⊥ , 0) + ∇⊥ [I (r ⊥ , 0)∇⊥ ϕ(r ⊥ , 0)] . k

(5.37)

With I (r ⊥ , 0) = I0 e−μd(r ⊥ ) , the solution to this equation is the projected thickness [ ( ]] {I , z)} /I 1 (r F ⊥ ⊥ 0 d(r⊥ ) = − ln F⊥−1 zδ 2 μ k +1 μ ⊥

(5.38)

of the specimen (Paganin et al., 2002). Multiplication of the last term with δk results in the phase shift [ ]] ( 1 −1 F⊥ {I (r ⊥ , z)} /I0 ϕ = − ln F⊥ . β 2 + 2kz k2⊥ δ

(5.39)

Consequently, this term depends solely on the sample’s δ/β-ratio which has to be included as a priory information. Although the homogeneity assumption poses a strong restriction, this algorithm remains widely applicable, as long as the δ/β-ratio remains approximately constant. The latter is the case either for pure phase-shifting objects or for media with a homogeneous elemental composition, but varying density 5 The opposite sign of the term on the left-hand side with respect to (Paganin 2006, p. 295) is a consequence of the definition of the phase term with an opposite sign in this thesis.

5.4 X-ray Phase Contrast Imaging

107

(Gradl 2019, p. 36). Slight variations of the δ/β-ratio usually still generate good images, although it impairs the quantitativeness of this algorithm. More details on considerations for a PBI-set-up can be found in Gradl (2019). Other approaches for phase retrieval as well as more sophisticated phase-retrieval algorithms, e.g. including the image formation process or iterative methods, are described in Hehn (2019).

5.4.2 Grating-Based Phase Contrast Imaging Since a lot of introductory literature exists already at the chair, this section summarises just the most important properties. Extensive introductions into gratingbased phase-contrast imaging can be found, e.g. in Sect. 2.5 of the PhD-Thesis by Fabio De Marco (2021) or Chap. 2 of the one by Lorenz Birnbacher (2018). This section is based on the aforementioned theses as well as Sect. 49.11 of “Handbook of X-ray imaging” (Russo, 2018) and “Elements of Modern X-ray Physics”, Sect. 9.3.2 (Als-Nielsen and McMorrow, 2011). Contrary to propagation-based imaging, grating-based phase-contrast imaging is an interferometric technique first developed in the early 2000s at synchrotrons (David et al., 2002; Momose et al., 2003; Weitkamp et al., 2005) and later transferred to incoherent X-ray sources Pfeiffer et al. 2006. Another difference to PBI is that grating-based imaging (GBI) generates three different image modalities: A conventional attenuation image, a phase-contrast image and a so-called dark-field image which originates from small-angle scattering Bech et al. (2012).

5.4.2.1

Talbot-Effect

Inserting a grating into the beam path of a (partially-)coherent wavefield produces self-images of this structure at certain distances downstream the grating. This effect was first observed by Henry Talbot (1836) with visible light and successfully described by Lord Rayleigh (1881). The grating modulates the incident wave field periodically. Interference due to Fresnel-propagation of this modulated wave field is responsible for the formation of the modulating structure’s self-images at certain (fractional) Talbot distances. For a periodic amplitude modulator, the Talbot distance is (Goodman 2005, Sect. 4.5.2) dTalbot = m

2 p2 , λ

(5.40)

m = 1, 2, 3, ... is the Talbot order and p the period of the grating. If purely phaseshifting gratings are employed, self-images occur at rational fractions of the Talbotdistances. The distances downstream the grating at which these self-images occur depend on the phase shift’s amplitude and profile (Suleski, 1997). For gratings with a duty cycle of 0.5, the fractional Talbot distances are given by

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dfrac.Talbot = m

p2 , 2η2 λ

(5.41)

where η = 1 for a π/2-phase shift and η = 2 for a π-phase shift.

5.4.2.2

Set-Up of a Talbot-Interferometer

Grating-based imaging exploits this Talbot self-imaging effect for interferometry. To this end, a first grating, typically called G1 , generates a Talbot-carpet downstream the grating. When a specimen is placed inside this Talbot carpet or in front of G1 , it attenuates, refracts and scatters the incident wave field. This alters the observed pattern at the Talbot-distance compared to the undisturbed reference. The disturbance introduced by the specimen can be resolved directly if a high-resolution detector is employed, as it is done in single grid imaging (Wen et al. 2010; Morgan et al. 2011). However, the grating periods typically are on the order of 10 µm for reasonable Talbot-distances, cf. Eqs. 5.40 and 5.41, and coherence lengths of the X-ray beam, cf. Eq. 5.31, which prevents the use of efficient large pixel detectors. To overcome this issue in a grating interferometer, a second grating (G2 ) with an appropriate period and typically made of a highly absorbing material is placed at a (fractional) Talbotdistance and laterally translated across one period in at least three steps. This transfers the distortion of the Talbot-carpet into a sinusoidal intensity modulation6 with step position. This modulation occurs in each detector pixel, thus making this technique compatible with large pixel detectors. Such a Talbot-interferometer is schematically depicted in Fig. 5.4. For details on the design of the interferometer at the MuCLS, the reader is referred to Sect. 11.5 of this thesis, Chap. 4 of “Biomedical X-ray Imaging at the Munich Compact Light Source” by Elena Eggl (2017) and Chap. 6 of “X-ray Vector Radiography for Biomedical Applications” by Christoph Jud (2019).

5.4.2.3

Information Retrieval in Grating Interferometry

The three modalities absorption, refraction and small-angle scattering affect the intensity modulation I (l, m) recorded at each pixel (index l, m) differently. Figure 5.5 qualitatively displays the effect of the respective modalities on the stepping curve for ideal hypothetical samples contributing only to a single modality each. A purely absorbing sample, Fig. 5.5a, reduces the transmitted intensity by a constant amount while a purely refracting sample, Fig. 5.5b, laterally shifts the Talbot-carpet selfimages and in turn the modulation curve. In contrast to the aforementioned modalities, scattering, Fig. 5.5c, redistributes the intensity incoherently in the lateral image plane effectively diminishing the amplitude of the modulation. Consequently, the 6

In an ideal case, the modulation would be triangular, but grating imperfections, detector blur, finite coherence length, etc. create a sinusoidal curve (Als-Nielsen and McMorrow 2011, p. 325 ff.).

5.4 X-ray Phase Contrast Imaging

109

Fig. 5.4 Set-up of a Talbot-interferometer for grating-based X-ray imaging. The grating G1 produces a self-image at the position of G2 , which is altered when a sample is placed inside the beam. The reference interference pattern is graphed in blue while the one modified by the sample is plotted in orange. Adapted from Fig. 3.6 in Gradl (2019) with permission of the author

Fig. 5.5 Effects of attenuation, refraction and scattering on the stepping curve. Three ideal objects which exhibit either only absorption (a), phase shift (b) or small-angle scattering (c) are displayed together with the resulting stepping curves. The reference scan is coloured in blue, the sample scan in orange. a attenuation reduces the intensity by an constant amount. b Refraction affects solely the phase offset φ. c Scattering only reduces the amplitude of the intensity modulation. Adapted from Fig. 3.7 in Gradl (2019) with permission of the author

intensity modulation can be expressed as (Pfeiffer et al., 2008) ( I (l, m) ≈ a0 (l, m) + a1 (l, m) cos

) 2π x G + φ(l, m) . p2

(5.42)

a0 is the average intensity, a1 the amplitude of the modulation, p2 the period of G2 , x G the step position and φ a phase offset. Normalised quantities can be extracted by comparison of the stepping curve acquired with a specimen with the reference one. In the following, the pixel indices are omitted for better readability.

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T =

a0sam. a0ref.

(5.43)

recovers the conventional transmission signal T (Pfeiffer et al., 2008). The superscripts sam. and ref. denote the sample scan and the reference one, respectively. The difference between the phase offset of the sample scan and the reference scan produces the differential phase-contrast image: DPC = φ sam. − φ ref. .

(5.44)

This phase offset itself is related to the differential phase via (Weitkamp et al., 2005) φ=

λdTalbot ∂ϕ , p2 ∂ x

(5.45)

where p2 is the period of the G2 . The differential phase-shift is related to the refraction angle α via, e.g. Pfeiffer et al. (2006), α=

λ ∂ϕ . 2π ∂ x

(5.46)

Consequently, integration of φ along the lateral dimension reproduces the phase shift ϕ induced by the sample. However, it is worth noting that whenever refraction shifts the pattern by more than a full grating period, i.e. a phase offset larger than 2π , the stepping curve can no longer be distinguished from the principal one. This effect is called phase wrapping (Bech 2009). The third and final sample property, the dark-field DF, is defined as normalised reduction of the visibility V , DF =

V sam. a1sam. a0ref. Imax. − Imin. a1 = , where V = = sam. ref. ref. Imax. + Imin. a0 V a0 a1

(5.47)

and Imax./min. denote the intensity maxima and minima of the stepping curve (Pfeiffer et al., 2008).

5.5 X-ray Absorption Spectroscopy

111

5.5 X-ray Absorption Spectroscopy This section is based on Sect. 7.2 of “Elements of Modern X-ray Physics” by AlsNielsen and McMorrow (2011) and Chap. 4 of “Neutron and X-ray spectroscopy” edited by Hippert et al. (2006). For details about the implementation at the MuCLS, the reader is referred to the PhD-thesis by Juanjuan Huang, TUM, currently in preparation. X-ray absorption spectroscopy (XAS) is a technique that enables the determination of the local geometry surrounding an atom of interest. In addition, it can also provide information on the specimen’s chemical species, oxidation state, site symmetry among others (Yano and Yachandra, 2009). Consequently, this technique is especially beneficial for materials that are difficult to crystallise or liquids, etc. XAS is based on the photoelectric absorption and therefore performed at Xray energies in the vicinity of the absorption edge, e.g. the K- or L-edge. A XAS spectrum can be subdivided into two regimes based on the X-ray energy range around the absorption edge. The X-ray Absorption Near-Edge Structure (XANES) extends only a couple of tens of electron volt above the absorption edge followed by the Extended X-ray Absorption Fine Structure (EXAFS) which extends several hundred electron volt to one kilo electron volt past the absorption edge. Generation of the XAS signal can be described by one electron that is liberated from the atomic core state by the photoelectric effect generating an outgoing particle wave which gets backscattered at neighbouring atoms. This backscattered wave interferes with the outgoing one which results in oscillations of the absorption cross section and thus the attenuation coefficient. Although the underlying physics is the same in both regimes, some approximations can be introduced in the EXAFS-region which simplify quantitative interpretation of the data using the so-called “EXAFS-equation” (Eq. 5.49) compared to the XANES-region (Bunker, 2010). The process to retrieve the next neighbours’ shell radii in EXAFS is sketched in the following. First, the oscillations of the absorption profile (μχ − μ0 )d are extracted from the measured data by a set of cubic splines and normalised. The resulting dimensionless parameter χ (k) that describes the oscillation is / ) μχ (E ) − μ0 (E ) d 2m ( ) , where k = χ (k) = (E − E0 ) 8h2 k 2 h2 J 1 − 6 mE 0 (

(5.48)

is the X-ray photon’s wave vector, m the electron’s mass, μ0 is the attenuation coefficient of the isolated atom, E0 is the exact edge energy in the material under investigation and J is the height of the absorption jump (Hippert et al. 2006, p. 145). The first inflection point at the absorption edge is typically selected for E0 which can be treated as a free parameter similarly to J in an iterative analysis. The rapidly oscillating function χ is multiplied with k n to amplify weak signals at high k. Fourier transformation of this quantity (χ k n ) yields a radial distribution function in

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5 Fundamentals of X-ray Imaging and Spectroscopy

real space. The individual peaks of the radial distribution function constitute a direct picture of the different coordination shells. To extract the radii of the closest shells and their occupation number more accurately, χ (k) is fitted with the following model, cf. (Als-Nielsen and McMorrow 2011, p. 255 ff.) (Hippert et al. 2006, p. 136 ff.): χ (k) =

Σ N j f j (k) 2 2 e−2k σ j e−2r j /Δ sin(2kr j + φ j (k)). 2 kr j j

(5.49)

In this formula, Δ is the photo-electron’s mean free path, f j is the scattering amplitude and φ j the phase. The actual numbers to be retrieved are the occupation number N j , the interatomic radii r j and their spread σ j . Since these multiple parameters and their k-dependence complicate analysis, this is usually solved by a combination of theory and reference measurements.

References Als-Nielsen J, McMorrow D (2011) Elements of modern X-ray physics, 2nd edn. Wiley, Chichester. 978-0-470-97394-3 Attwood D, Sakdinawat A (2017) X-rays and extreme ultraviolet radiation: principles and applications, 2nd edn. Cambridge University Press, Cambridge. 9781107477629 Bech M (2009) X-ray imaging with a grating interferometer. University of Kopenhagen, PhD-Thesis Bech M et al (2012) Experimental validation of image contrast correlation between ultra-smallangle X-ray scattering and grating-based dark-field imaging using a laser-driven compact X-ray source. Photon Lasers Med 1:47–50 Berujon S, Wang H, Sawhney K (2012) X-ray multimodal imaging using a random-phase object. Phys Rev A 86:063813 Birnbacher L (2018) High-sensitivity grating-based phase-contrast computed tomography with incoherent sources. PhD thesis, Technical University of Munich Bonse U, Hart M (1965) An X-ray interferometer. Appl Phys Lett 6:155–156 Born M, Wolf E (2019) Principles of optics, 7th edn. Cambridge University Press. ISBN: 978-1108-47743-7 Bunker G (2010) Introduction to XAFS. Cambridge University Press. ISBN: 9780511809194 Chapman D et al (1997) Diffraction enhanced X-ray imaging. PMB 42:2015–2025 Cloetens P et al (1996) Phase objects in synchrotron radiation hard X-ray imaging. J Phys D Appl Phys 29:133–146 David C, Nohammer B, Solak HH (2002) Differential X-ray phase contrast imaging using a shearing interferometer. Appl Phys Lett 81:3287–3289 Davis TJ et al (1995) Phase-contrast imaging of weakly absorbing materials using hard X-rays. Nature 373:595–598 Eggl E (2017) Biomedical X-ray imaging at the Munich Compact Light Source. PhD thesis, Technical University of Munich Goodman J (2005) Introduction to Fourier optics, 3rd edn. McGraw-Hill Gradl R (2019) Dynamic phase-contrast X-ray imaging at an inverse Compton source. PhD thesis, Technical University of Munich Hehn L (2019) Model-based iterative reconstruction for X-ray computed tomography using attenuation and propagation-based phase contrast. PhD thesis, Technical University of Munich

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Henke B, Gullikson E, Davis J (1993) X-ray interactions: photo absorption, scattering, transmission and reflection at E = 50–30000 eV, Z = 1–92. Atom Data Nucl Data Tables 181–342 Hippert F et al (eds) (2006) Neutron and X-ray spectroscopy. Springer. ISBN: 978-1-4020-3336-0 Ingal VN, Beliaevskaya EA (1995) X-ray plane-wave topography observation of the phase contrast from a non-crystalline object. J Phys D Appl Phys 28:2314–2317 Jacobson B (1953) Dichromatic absorption radiography. Dichromatography. Acta Radiol 39:437– 452 Jud C (2019) X-ray vector radiography for biomedical applications. PhD thesis, Technical University of Munich Kulpe S (2020) K-edge subtraction and X-ray fluorescence imaging at the Munich Compact Light Source. PhD thesis, Technical University of Munich Kulpe S et al (2018) K-edge subtraction imaging for coronary angiography with a compact synchrotron X-ray source. PLOS ONE 13:e0208446 Marco FD (2021) Image reconstruction, pre-clinical studies, and signal formation investigations at a dark-field chest radiography setup. PhD thesis, Technical University of Munich Momose A et al (2003) Demonstration of X-ray Talbot interferometry. JJAP 42:L866–L868 Morgan KS, Paganin DM, Siu KKW (2011) Quantitative X-ray phase-contrast imaging using a single grating of comparable pitch to sample feature size. Opt lett 36:55–57 Morgan KS, Paganin DM, Siu KKW (2012) X-ray phase imaging with a paper analyzer. Appl Phys Lett 100:124102 Olivo A et al (2001) An innovative digital imaging setup allowing a low-dose approach to phase contrast applications in the medical field. Med Phys 28:1610–1619 Paganin D et al (2002) Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J Microsc 206:33–40 Paganin DM (2006) Coherent X-ray optics. Oxford University Press, Oxford. 978-0-19-967386-5 Pfeiffer F et al (2008) Hard-X-ray dark-field imaging using a grating interferometer. Nat Mater 7:134–137 Pfeiffer F et al (2006) Phase retrieval and differential phase contrast imaging with low-brilliance X-ray sources. Nat Phys 2:258–261 Rayleigh L (1881) On copying diffraction gratings and on some phenomenon connected therewith. Philos Mag 11:196 Russo P (ed) (2018) Handbook of X-ray imaging: physics and technology. CRC Press, Boca Raton. 978-1-4987-4152-1 Snigirev A et al (1995) On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation. RSI 66:5486–5492 Suleski TJ (1997) Generation of Lohmann images from binary-phase Talbot array illuminators. Appl Opt 36:4686–4691 Talbot HF (1836) Facts relating to optical science. Philos Mag 9:401 Weitkamp T et al (2005) X-ray phase imaging with a grating interferometer. Opt Express 13:6296– 6304 Wen HH et al (2010) Single-shot X-ray differential phase contrast and diffraction imaging using two-dimensional transmission gratings. Opt Lett 35:1932–1934 Wilkins SW et al (1996) Phase-contrast imaging using polychromatic hard X-rays. Nature 384:335– 338 Willmott P (2019) An introduction to synchrotron radiation, 2nd edn. Wiley, Chichester. 978-1119-280378 Yano J, Yachandra VK (2009) X-ray absorption spectroscopy. Photosynth Res 102:241–254 Zangwill A (2012) Modern electrodynamics. Cambridge University Press, Cambridge. 978-0-52189697-9

Part II

R&D at the Inverse Compton X-ray Source of the MuCLS

Chapter 6

Overview on Inverse Compton X-ray Sources

Following the theoretical foundations of inverse Compton X-ray sources developed in the preceding part, this chapter reviews the experimental work focussing on the realisation of these sources. First, a brief historic summary on the path from the discovery of Compton scattering to the development of modern inverse Compton X-ray sources is presented. Subsequently, the different types of modern inverse Compton sources are presented including an extensive overview on various realisations.

6.1 From Compton Scattering to Modern Inverse Compton Sources Inelastic interaction of highly energetic electromagnetic radiation with free charged particles (at rest), typically electrons, was first observed by Compton (1923). In this process, momentum is transferred from the photon to the electron resulting in a red shift of the scattered radiation which depends on the angle between the incident photon and the scattered one. A quarter century after Compton published his work, Feenbert and Primakov proposed that collisions of highly relativistic electrons with thermal photons emitted from stars in which the photons are promoted in energy and momentum might explain the observed relative lack of high-energy electrons compared to ions in the cosmic radiation which can be observed close to the earth (Feenberg and Primako 1948). Their idea was heavily discussed during the 1960s in astrophysics while at the same time first considerations to exploit this effect on our earth emerged after the first brilliant—and in terms of energy “thermal”—photon sources emerged. This source, the maser (microwave amplification by stimulated emission of radiation), had been invented by Gordon et al. (1954), Basov and Prokhorov (1954, 1955), Karlov et al. (2010) (who also suggested optical pumping of an active medium

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_6

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for the first time). Shortly after the maser, the laser (light amplification by stimulated emission of radiation) was proposed by Bloembergen (1956), Schawlow and Townes (1958) and realised for the first time by Maiman (1960). Only three years after the laser was experimentally demonstrated, “inverse” Compton scattering was proposed as an elegant method to create brilliant quasi-monochromatic X/γ-rays by Arutyunian and Tumanian (1963) as well as Milburn (1963) independently in the former USSR and USA. It should be noted that the term inverse Compton source—although widely used for this kind of X/γ-ray generation—might not be the most suitable one to describe the physics behind it. The actual physical process taking place is either Thomson- or Compton scattering in the rest frame of the electron, depending on the recoil of the electron. Nevertheless, the term inverse Compton scattering will be used in the following to remain concordant with the convention. The term “inverse" probably refers to the laboratory frame, where the electron momentum is significantly larger than the photon momentum, which is the opposite of the situation described by Compton. While the aforementioned studies were of theoretical nature, Fiocco and Thompson (1963) observed Thomson scattering from a low energy electron beam in the same year and experimental validation of inverse Compton scattering of laser photons (Kulikov et al. 1964; Bemporand et al. 1965) followed soon. This early work sparked intense research in this field during the following years, ranging from refinement of the theory (Brown and Kibble 1964; Jones 1968) and their confirmation (Kulikov et al. 1969; Ballam et al. 1969) to new proposals for Compton-lasers by stimulated Compton scattering (Pantell et al. 1968). In this proposal, microwave photons were to collide with a relativistic electron bunch in order to generate (far-) infra-red photons which are stored in an optical cavity. However, a finite interaction length between the low energy photons in a cavity can significantly reduce the gain of the stimulated emission (Sukhatme and Wolff 1973). Madey (1971) proposed to replace the microwave photon beam by a periodic magnetic field, an undulator (Motz 1951; Motz et al. 1953), generating a higher density of virtual photons compared to the one achievable with microwave or laser technology at that time. This aspect rendered this technology favourable for research on free-electron lasers (FELs) operating in between the deep-IR and the UV, where resonators for the emitted radiation could be built, and resulted in the first demonstration of an FEL-amplifier in 1976 (Elias et al. 1976) as well as an FEL-oscillator (Deacon et al. 1977) the following year. An interesting feature of Madey’s theory is that, albeit being quantum mechanical in nature, its results are independent of Plank’s constant h (Madey 1971). This indicates that a classical formulation should be possible which was developed later on, some early works include Colson (1976, 1977), and Hopf et al. (1976). Around the same time, renewed interest in (spontaneous) inverse Compton scattering with optical photons arose when optical laser systems were able to deliver considerable intensities, e.g. by cavity dumping (Maydan 1970). Due to their higher momentum than the virtual undulator photons, in other words their shorter period of the optical wave compared to the permanent magnet undulator period, these sources reach higher X-ray energies for the same electron energy. As the brilliance of wigglers at synchrotrons drops significantly at high photon energies above ∼100 keV,

6.1 From Compton Scattering to Modern Inverse Compton Sources

119

the performance gap between inverse Compton sources and synchrotron radiation is mitigated constituting inverse Compton scattering as the sole mechanism to efficiently achieve quasi-monochromatic γ-rays in the MeV-range. Therefore, the first photon sources based on inverse Compton scattering were built mainly for nuclear physics research and operated in the γ-ray regime (Federici et al. 1980; Sandorfi et al. 1983; Yamazaki et al. 1985; Kezerashvili et al. 1993; D’Angelo et al. 2000). On the other hand, synchrotrons had been providing very brilliant photon beams in the X-ray regime outperforming inverse Compton scattering sources by orders of magnitude in flux, even more so when the 3rd generation storage rings came on-line at the beginning of the 1990s (Willmott 2019, p. 16). Consequently many techniques exploiting this kind of radiation were developed in various fields, like diagnostic imaging, non-destructive testing, material science, structure determination, lithography and many more. However, one unresolved issue of these facilities has been remaining: Their size and more importantly their cost for both construction and operation resulted in a few large synchrotron research centres which provide synchrotron radiation services to researchers or industry through a proposal-based system or for a compensation, respectively. As a result, beam time is scarce and expensive inhibiting the transfer of synchrotron techniques into local research centres, medical settings, industrial research complexes or production lines. Advances in laser technology, such as chirped-pulse amplification (CPA) and developments in solid-state lasers, resulted in short laser pulses at high (peak) intensities and rendered tunable brilliant compact synchrotron X-ray sources based on inverse Compton scattering feasible, which can overcome the aforementioned issue. The prospect of compact local synchrotron radiation sources has been re-initiating the development of inverse Compton X-ray sources at the end of the 1980s and beginning of the 1990s (Sprangle et al. 1989; Carroll et al. 1990; Sprangle et al. 1992; Blum 1993; Tompkins et al. 1993; Ting et al. 1995; Schoenlein et al. 1996; Huang and Ruth 1998; Pogorelsky 1998; Bulyak et al. 1999). As compact inverse Compton X-ray sources have been continuously improving and the first sources reached a sufficient performance to demonstrate their potential in real world applications (Bech et al. 2009; Ikeura-Sekiguchi et al. 2008; Oliva et al. 2010; Abendroth et al. 2010) around a decade ago, interest for user facilities based on compact ICSs in the X-ray regime has taken off. Since then, the number of ICSs has been growing significantly with many X-ray sources currently under construction or commissioning and several proposals for ICSs currently seeking funding. An overview of the ICSs proposed within the last decade (2009–2019) and their distribution around the earth including both X-ray and γ-ray sources is depicted in Fig. 6.1. All sources are on the northern hemisphere, which is where most industrialised countries are located that also have a history in accelerator and laser physics.

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Fig. 6.1 Overview of the light sources generating their photons by inverse Compton scattering of laser photons with RF-accelerated relativistic electrons. Light sources which are intended to serve as user facilities or technology demonstrators are depicted. Although most user facilities which exist are γ-ray sources based at synchrotrons at the moment, a significant number of X-ray sources are planned or already under construction and commissioning. This indicates a widespread conviction that this technology might play a key role for providing brilliant X-ray radiation in local laboratories, industrial screening or pre-clinical settings

6.2 Modern Inverse Compton Sources

121

6.2 Modern Inverse Compton Sources Most of the ICS which are currently under development rely on RF-driven particle acceleration due to its maturity compared to laser-driven plasma-based particle acceleration. Therefore, this thesis focusses on ICSs based on the former technology. A comprehensive overview of laser-driven particle acceleration can be found, e.g. in Esarey et al. (2009). Two different concepts for the technical realisation of RFdriven particle acceleration are possible. Either the electron beam can be collided with the photon bunch directly after acceleration in the linear accelerator, in which case every electron bunch is used only once. Or it can be transferred into a small electron storage ring, where recirculation of the electron bunch permits to reuse it multiple times. Albeit their fundamental difference in design, both concepts aim to maximise the same quantity, the luminosity of the X-ray generation. Nevertheless, different parameters contributing to the luminosity are optimised in the various ICSdesign approaches, which in turn influence the brilliance of the generated X-rays. Therefore, benefits and drawbacks of those concepts are going to be discussed and several realisations are going to be presented, starting with the linear acceleratorbased systems first which are followed by storage ring-based ones. Pulse lengths, focii and source sizes are rms-values in the following if not stated differently.

6.2.1 Linear Accelerator-Based ICSs Linear accelerator-based ICSs have one obvious simplification of the accelerator system: They lack a electron storage ring. This feature reduces costs of construction (as long as room temperature accelerators are employed) as well as the complexity of the accelerator resulting in a more compact footprint. Although these benefits are desirable on their own, the main reason for the development of linear acceleratorbased ICSs is the lower emittance achievable with linear accelerators compared to storage rings (Graves et al. 2014).

6.2.1.1

ICSs Based on Room Temperature Linear Accelerators

For room temperature accelerators, these advantages come at the cost of a repetition rate significantly reduced to typically less than 100 Hz, where water cooling can be employed to compensate the microwave thermal dissipation in the accelerator (Ovodenko et al. 2016). This is four to five orders of magnitude lower than repetition rates at storage ring-based ICSs in the X-ray regime, which are tens of MHz, e.g. (Rui and Huang 2018; Eggl et al. 2016; Variola et al. 2014a). Consequently, this significant reduction in luminosity has to be overcome by other means. Since storage ring-based ICSs require a linear accelerator as well, the achievable electron-bunch charge for linear accelerator-based ICSs is of the same order of magnitude as for storage ring-

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based ones, for example 400 pC at the Tsinghua Thomson X-ray source (TTX) (Chi et al. 2018) and 250 pC at the Munich Compact Light Source (Eggl et al. 2016). Therefore, increasing the photon number or laser pulse energy is most straightforward in order to enhance the luminosity of linear accelerator-based ICS-designs. This requirement is fulfilled by modern ultra-short high-power laser-systems, which provide Joule-level (sub-)picosecond laser pulses at repetition rates of several Hz. Such ICS-variants were chosen at the Lawrence Livermore National Laboratory to produce X-rays (Gibson et al. 2004; Brown et al. 2004) and γ-rays (Gibson et al. 2010; Albert et al. 2010), at TTX (Chi et al. 2018; Du et al. 2013; Tang et al. 2009), the National Institute of Advanced Industrial Science and Technology (AIST, Japan) (Kuroda et al. 2010), SPARC_LAB (Vaccarezza et al. 2016) or STAR (Bacci et al. 2016, 2014) among others. • PLEIADES (LLNL) PLEIADES (Picosecond Laser-Electron Inter-Action for the Dynamical Evaluation of Structures) operates at a peak X-ray energy of 70 keV to 80 keV (Brown et al. 2004; Gibson et al. 2004). The electron pulse of ∼300 pC charge and duration of a few picoseconds is accelerated to an energy of 50 MeV to 60 MeV in an S-Band linear accelerator while preserving a low energy spread of 0.2 % and focused to ∼50 µm at the interaction point. Even sub-picosecond pulse durations can be achieved if active bunch compression is applied. This bunch collides with 54 fs-long laser pulse produced by a CPA-Ti:Sapphire laser system (Gibson et al. 2004). Its pulse energy is 180 mJ and its central wavelength 820 nm. Compton scattering generates a total X-ray flux of up to 4.4 × 106 ph/shot with a divergence of 6 mrad × 3 mrad and an X-ray pulse duration of ∼3 ps. Its bandwidth is 12.5 keV. The system repetition rate is 10 Hz (Brown et al. 2004). • T-REX (LLNL) In contrast to PLEIADES, the T-REX (Thompson-Radiated Extreme X-ray) source relies on Nd:YAG as the active medium for the main laser amplifier, which increases the laser wavelength to 1064 nm and the laser pulse length to 16 ps (FWHM). Electrons can be accelerated to up to 120 MeV in an S-band travelling wave accelerator, an upgraded system of PLEIADES, preserving the same energy spread at a much higher electron bunch charge of 800 pC which was reached by switching the copper cathode for a magnesium one. Combined with frequency doubling (532 nm, 150 mJ) and frequency trippling (351 nm), a wide X-ray window is opened up ranging from 75 keV to 0.9 MeV. The flux at 478 keV was measured to be 1.6 × 105 ph/shot at a repetition rate of 10 Hz (Albert et al. 2010). The Xray source size has been determined to be ∼36 µm and X-ray beam divergence is 10 mrad × 6 mrad (Gibson et al. 2010). This system was used for nuclear resonance fluorescence, e.g. of Lithium-7 (Albert et al. 2010). Based on this successful demonstration, a new γ-ray source is under construction at LLNL which relies on X-band technology in order to decrease its size (Albert et al. 2012; Barty and Albert 2011). A layout of this new facility is depicted in Fig. 6.2c.

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• TTX The Tsinghua Thomson X-ray (TTX) source has been the first of its kind in China operating in the hard X-ray regime (20–70 keV). Its X-ray beam of a few percent bandwidth created in an X-ray source of 36 µm is emitted into a cone of ∼6 mrad (full divergence angle). The flux generated by TTX is 2 × 107 ph/shot, which translates into 2 × 108 ph/s due to the repetition rate of laser and electron system of 10 Hz. The electron beam can be tuned between 30 and 54 MeV and has an energy spread of 0.3 %, its bunch charge is 400 pC (Chi et al. 2018), and its bunch duration ∼10 ps (FWHM) (Chi et al. 2017). Electrons are generated in the gun displayed on the left-hand side in Fig. 6.2d, accelerated in a 3 m-long travelling wave S-band cavity, followed by a bunch compressor (not installed when the picture was taken) and on the right-hand side the interaction chamber for inverse Compton scattering (Du et al. 2013; Tang et al. 2009). At this charge level, the electron beam can be focussed to 60 µm, while at a charge level of 200 pC ∼40 µm rms could be achieved. The interaction laser is a terawatt-class Ti:Sa laser operating at a central wavelength of 800 nm. Its pulse duration is ∼70 fs (FWHM), its pulse energy can reach up to 300 mJ (Du et al. 2013) and it can be focussed down to ∼3 µm (Chi et al. 2018). • AIST compact ICS At the National Institute of Advanced Industrial Science and Technology (AIST, Japan), a slightly different design for a linear accelerator-based ICS was realised. Instead of a completely linear design, an achromatic 90◦ arc was inserted in between the electron accelerator and the interaction chamber to reduce dark-current background (Kuroda et al. 2014). A schematic of this system is depicted in Fig. 6.2a and a top-view of the ICS in Fig. 6.2b. The electrons are accelerated by two 1.5 m long S-band standing wave structures (Kuroda et al. 2010). Moreover, the X-ray energy can be adjusted not only changing the electron energy (up to 42 MeV), but also by adjusting the angle between the laser and electron beam between 90◦ and 165◦ , which in addition affects the X-ray pulse length and flux. At 90◦ it is limited to ∼150 fs by the laser pulse duration of ∼100 fs (FWHM) while in the other case it is limited by the electron bunch duration of ∼3 ps. This enables an X-ray energy range from 10 to 40 keV with a maximum flux of 1 × 107 ph/s at 165◦ , i.e. close to backscattering geometry, and at the system repetition rate of 10 Hz (Kuroda et al. 2010; Yamada et al. 2009). By colliding short macro-pulses containing six individual bunches, the flux could be enhanced from originally 1 × 106 ph/s to the aforementioned 1 × 107 ph/s at 165◦ (Kuroda et al. 2009). The X-ray source size is ∼30 µm created by a laser focal spot of this dimension and a slightly larger electron beam (43 µm × 30 µm). The electron bunch charge is ∼1 nC and its energy spread is 0.2 %. The laser system is based on titanium sapphire, centred at 800 nm, and delivers pulses of 140 mJ (Kuroda et al. 2010; Yamada et al. 2009). • SPARC_LAB The SPARC_LAB (Sources for Plasma Accelerators and Radiation Compton with Lasers and Beams) inverse Compton X-ray source is located at Frascati, Italy. A femtosecond Ti:Sapphire laser system with a central wavelength of 800 nm is

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employed as interaction laser. This system delivers laser pulses with an energy ranging from 1 J to 5 J within 25 fs to 10 ps at a rate of 10 Hz. At the interaction point, laser pulses can be focussed to a diameter of 10 µm (FWHM) (Vaccarezza et al. 2014). The electron beam can be accelerated to energies between 30 MeV and 150 MeV. Each bunch contains a charge of 200 pC and electrons within this bunch have an energy spread of 0.1 % (Vaccarezza et al. 2016). Compared to the achievable laser focus, the electron beam focus of 160 µm to 240 µm is rather large, but was required due to an offset of the laser and electron beam at the interaction point resulting in a poor overlap (Vaccarezza et al. 2014). Later, this was corrected improving the electron beam focus to about 60 µm to 80 µm (Vaccarezza et al. 2016). However, a significantly increased X-ray background on the X-ray detector, arising from the strongly diverging electron beam in the beam dump which is located nearby, limited the electron beam size to 110 µm (Vaccarezza et al. 2016). At 22.5 keV the measured X-ray flux was ∼1 × 104 ph/shot, i.e. ∼1 × 105 ph/s, at a X-ray bandwidth of 20 % (Vaccarezza et al. 2016). For a high Xray energy experiment conducted with 50 MeV electrons, a flux of 6.7 × 104 ph/s was estimated for the X-ray beam with a mean energy of ∼60 keV (Vaccarezza et al. 2014). • STAR The Southern Europe Thomson Backscattering Source for Applied Research (STAR), which is also located in Italy, is based on a design similar to the one of SPARC_LAB. However, one notable difference in the technical design is the increased system repetition rate of 100 Hz (Bacci et al. 2014). In contrast to the sources described before, this one was still under construction at the time of writing (January 2021). The electron accelerator should deliver 1 × 1010 ph/s at around 10 MeV) and the sole inverse Compton source based on a free-electron laser (Wu 2013). • HIγS The High Intensity γ-ray Source (HIγS) produces γ-rays in the range between 1 MeV and 100 MeV (Wu 2013) with a flux above 1 × 108 ph/s across all energies (Weller et al. 2009) peaking at >1 × 1010 ph/s at around 10 MeV (Wu 2013). Such a high flux at very high photon energies is obtained by utilising a kilowattclass free electron-laser as interaction laser (Wu 2009; Litvinenko et al. 2001). This technology enables a high-power laser system even in the UV-range, at the moment down to 190 nm. The FEL comprises either of the two different kinds of wigglers and an optical resonator. Currently, either two planar wigglers (OK-4) or four helical wigglers (OK-5) are available in order to create linearly or circularly polarised light (Wu 2013). A drawing of the whole system is presented in Fig. 6.3a. The interaction laser wavelength is selected by the wavelength-dependent reflectivity of the resonator’s mirrors. As there is only a fixed selection of mirrors, the desired photon energy is tuned adjusting the electron energy. Electron acceleration at HIγS is identical to the one at synchrotrons: A short linear accelerator speeds up the electrons to relativistic energies (160 MeV to 270 MeV), followed by a booster synchrotron, which ramps the electron energy up to the desired one (0.24 GeV to 1.20 GeV). Finally, the electrons get injected into the storage ring at their final energy, which operates in the steady state condition. Lost electrons are replenished by top-up injection. The circumference of the storage ring is 107.46 m and the length of the Fabry-Perrot resonator is 53.73 m, which illustrates the large size of this facility compared to the ones discussed before. The current in the storage ring is about 80 mA. Both beams are mildly focussed, i.e. to a couple of hundred micrometer, except for the vertical extent of the electron beam of 20 µm to 70 µm (Weller et al. 2009). Figure 6.4 summarises the performance of the various types of ICSs in terms of their energy range and usable photon flux. Due to different collimation of the beams, the bandwidth for the flux may vary for the different sources. Although only ICSs which are in operation or under construction have been discussed in this section, funded as well as planed ICSs have been included into the Fig. 6.4 to provide an overview of potential future developments. Interestingly, almost all ICSs are constructed to either work in a low energy regime between 10 keV and 100 keV or in the

6.3 The CLS Installed at the MuCLS

137

high energy regime between 1 MeV and 100 MeV. The latter is very interesting for nuclear science as well as security applications, while the former is mainly interesting for biomedical and material science due to the ability of ICSs to provide synchrotron radiation in a laboratory environment. Hence, many new sources are emerging in this low energy region. Among ICSs, the MuCLS is the highest performing one at the moment. However, other sources which are proposed or currently under construction might surpass the current performance of the MuCLS if they reach their design goals. In order to keep the MuCLS competitive, several improvements for the inverse Compton source of the MuCLS have been developed and implemented in the framework of this thesis. They are going to be discussed after an introduction of the MuCLS.

6.3 The CLS Installed at the MuCLS This section is a slightly extended and modified version of the section “Design of the Compact Light Source at the MuCLS” which was previously published in Günther et al. (2020), licensed under CC BY 4.0. In the preceding section, it was already mentioned that the fundamental design concept of the CLS dates back to 1998 when Huang and Ruth proposed a “LaserElectron Storage Ring” (Huang and Ruth 1998). Initially, the main goal was to develop a device for radiative laser cooling of electron beams. This design was further refined by Loewen (2003) and Lyncean Technologies Inc. (Fremont, USA) over the years with the goal to convert it into a commercial inverse Compton X-ray source. An annotated CAD-rendering of the CLS is depicted in Fig. 6.5 and impressions of the CLS installed at the MuCLS are shown in Fig. 6.6. On the lower right side, the gun is visible, where electrons are produced at a copper photo-cathode. The photon energy of the drive laser has to be higher than the work function of copper (4.6 eV), which is achieved with a frequency-quadrupled Nd:YLF laser system operating at 266 nm wavelength and a pulse energy >100µJ. The oscillator of the photo-cathode laser system is operating at 64.91 MHz repetition rate, which corresponds to the revolution frequency of the electrons in the electron storage ring. Its frequency is locked to the 44th sub-harmonic of the CLS’s master frequency, the S-band at 2856 MHz. The laser pulses are transferred with an optical fibre to the regenerative amplifier, which picks pulses at a rate of 25 Hz from this pulse train and amplifies them. The pulse-picking rate of 25 Hz is derived from the European grid frequency of 50 Hz. Furthermore, a regenerative amplifier preserves a highquality mode shape during amplification due to its design as an optical cavity. After the pulses are extracted from the regenerative amplifier, frequency-quadrupling is performed by two nonlinear crystals. The UV-pulse is steered to the electron gun, where it produces electron bunches of typically about 250 pC. Although the photocathode can operate stably at much higher bunch charges, this value is chosen because charge injection efficiency into the storage ring starts to decrease notably at higher charges. This significantly reduces the expected life time of the kicker magnets’ electronics since they are located in very close proximity to the ring pipe. The charge

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Fig. 6.4 Summary of the performance of inverse Compton scattering experiments and facilities which have been proposed or conducted within the last decade (2009–2019). The flux values in this graph correspond to the flux that has been measured in the respective experiments. In most of them, this flux is contained in a small bandwidth of a few percent, but in some cases, denoted by f. Bw (full bandwidth), the bandwidth is larger than 10%. It should be noted that most sources that are under construction aim for a flux orders of magnitudes higher than what has been demonstrated in most experiments so far. The data for this graph can be found in the following references: (Utsunomiya et al. 2015; Vaccarezza et al. 2016; Chaleil et al. 2016; Chaikovska et al. 2016; Chi et al. 2017; Hwang 2018; Luo et al. 2010; LEPS Experiment 2020; Weller et al. 2009; Wu 2013; Niknejadi et al. 2019; Laundy et al. 2012; Ovodenko et al. 2016; Akagi et al. 2016; Carroll et al. 2003; Zen et al. 2016; Albert et al. 2010; Kezerashvili et al. 1998; An et al. 2018; Boscherini et al. 2019; Bacci et al. 2016; Drebot et al. 2019; Variola et al. 2014a; Graves et al. 2014; Deitrick et al. 2018, 2019; Luiten 2016; Adriani et al. 2014; Kärtner et al. 2016; Chen et al. 2019; MA et al. 2017; Shimizu et al. 2015; Graves et al. 2017a)

is kept constant during operation by a feedback adjusting the current through the diode pumping the laser crystal in the regenerative amplifier, i.e. by controlling the cathode laser’s pulse energy. Apart from the copper cathode, the electron gun is comprised of a half-cell accelerating structure (radio-frequency (RF)-cavity) and solenoids for electron collimation. Subsequently, the electron bunches are accelerated in up to five S-band (2856 MHz) standing wave accelerator modules to the desired electron energy between 29 MeV and 45 MeV. This energy can be freely adjusted choosing the number of accelerating structures and the gradient inside

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Fig. 6.5 Annotated schematic of the Lyncean Compact Light Source as installed at the MuCLS. This figure was previously published in Günther et al. (2020), licensed under CC BY 4.0

of them adequately. Three compact medical klystrons, high-power RF-amplifier systems, supply the RF-cavities with the accelerating electric fields. Between the individual accelerator modules, corrector magnets and beam position monitors confine the electron beam path to a straight line through the accelerator. To this end a feedback system adjusting the location of the laser beam position at the cathode is employed as well. Since the electron energy after acceleration determines the energy of the generated X-ray photons, variation of the X-ray energy between 15 keV and 35 keV is achieved with aforementioned adjustments to the electron accelerator. Directly after the acceleration section, a set of quadrupole magnets produces a secondary focus and re-collimates the electron beam. The transport line, the semi-circle arc visible below the storage ring on the left of Figs. 6.5 and 6.6a, images this focus onto the interaction point and guides the electron bunches to the fast kicker. This device is installed in one of the long straight sections of the storage ring and injects the electrons into the latter. In the opposite long straight section, which is shared with the optical cavity, the installed quadrupole magnets focus the electron bunch to a small size at the interaction point of electrons and laser photons in order to create a high luminescence and recollimate the electrons afterwards. In addition, a short L-band (1428 MHz) RF-cavity is installed right after the quadrupole magnets replenishing electron energy lost by synchrotron radiation and inverse Compton scattering. Exchanging the electron bunch in the storage ring every 40 ms, i.e. at the aforementioned rate of 25 Hz, preserves a circular electron beam shape. The current version of the interaction laser amplifier system is presented briefly in the following, since many aspects of it are going to be discussed in detail in the next chapter. In contrast to the photo-cathode laser, the laser used for inverse Compton scattering is based on Nd:YAG technology operating at a wavelength of 1064 nm. This system delivers up to 30 W at a repetition rate of 64.91 MHz. Pulse duration of

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Fig. 6.6 Impressions of the Lyncean CLS installed in Munich. a The transport line transferring the accelerated electrons to the storage ring. b The cavity laser system enclosed by the black box and the enhancement cavity. c The Lyncean Compact Light Source inside its radiation safety enclosure at the Technical University of Munich. View along the electron accelerator. a & c are photos taken by Andreas Heddergott, TUM

the sech-shaped laser pulse has been determined with an autocorrelator to be 26 ps (FWHM) and its M2 -value to be 1.3 in the horizontal as well as vertical direction. Laser pulses delivered by the system are mode-matched to the eigenmode of a highfinesse (>37,000) enhancement cavity, a four-mirror bow-tie ring resonator acting as a passive laser amplifier with a gain of >15,000, depicted in Fig. 6.6b. The coupling of the external seeding laser to the eigenmode of the enhancement cavity is >80 %. This increases the laser power from ∼25 W at the entrance window of the enhancement cavity to ∼350 kW. In order to increase luminescence, the cavity also focuses the laser beam in the interaction point, where it counter-propagates to the electron beam, albeit at a small angle of a couple of milliradians. This interaction angle avoids placing the laser beam directly onto the exit aperture for the X-rays. Since this aperture is a back-thinned area of the exit mirror, it is very sensitive to thermal deformations and contains higher surface errors, which strongly deteriorate the stored laser power if the laser beam is placed onto the aperture. The oscillator of the laser system is kept on resonance with the enhancement cavity via the Pound-Drever-Hall locking scheme (Pound 1946; Drever et al. 1983; Black 2001). Temporal coincidence of laser pulse and electron bunch in the interaction point is ensured by locking the roundtrip frequency of the cavity to the low-level radio frequency driving electron acceleration. Spatial overlap of the laser pulse and electron bunch is optimised by

References

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Table 6.1 Comparison of the Compact Light Source’s parameters after delivery in 2015 with the ones available in 2020 after several upgrades 2015 2020 Repetition rate Electron energy Storage ring current Electron pulse length

64.91 MHz 29–45 MeV 16 mA 50 ps (Schleede 2013)

Laser wavelength Laser power (seeding laser) Laser power (optical cavity) Laser pulse length X-ray energy X-ray flux

1064 nm/1.17 eV (Nd:YAG) ∼14 W

∼25 W

∼120 kW

∼350 kW

X-ray pulse length X-ray divergence X-ray source size X-ray bandwidth X-ray brilliance [ph/s/0.1%BW/mm2 /mrad2 ]

Unknown 26 ps (FWHM) 15–35 keV Up to 0.5 × 1010 ph/s (15 keV) Up to 1.5 × 1010 ph/s (15 keV) – – Up to 1.8 × 1010 ph/s (35 keV) Up to 4.5 × 1010 ph/s (35 keV) (Eggl et al. 2016) 60 ps (Feser et al. 2018) 4 mrad ∼50 µm (σ-value) 3 % (15 keV)–5 % (35 keV) (FWHM) 4.8 × 109 (Eggl et al. 2016) 1.2 × 1010

moving the electron beam orbit in the vertical and horizontal directions. In case that the ranges are not sufficient, the laser beam’s orbit inside the cavity can be adjusted as well. Table 6.1 compares CLS system parameters after installation with the ones available to date. Over the course of these five years, the available X-ray flux has been enhanced significantly. This short overview already reveals that mainly upgrades of the interaction laser system were responsible for this. The upgrades leading to these parameters and research on how the interaction laser system may be improved in the future are going to be the main topics of this part of my dissertation.

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Gibson DJ et al (2004) PLEIADES: a picosecond Compton scattering X-ray source for advanced backlighting and time-resolved material studies. Phys Plasma 11:2857–2864 Gladkikh P (2005) Lattice and beam parameters of compact intense X-ray sources based on Compton scattering. Phys Rev Spec Top Accel Beams 8:050702 Gordon JP, Zeiger HJ, Townes CH (1954) Molecular microwave oscillator and new hyperfine structure in the microwave spectrum of NH3 [7]. Phys Rev 95:282–284 Graves WS et al (2017) ASU compact XFEL. In: Proceedings of FEL 2017. TUB03 Graves WS et al (2017) Design of an X-band photo injection operating at 1 kHz. In: Proceedings of IPAC 2017. TUPAB139 Graves WS (2016) Compact X-ray light source R&D at Arizona State University in high-brightness sources and light-driven interactions, OSA Technical Digest (online). Optical Society of America. EM1A2 Graves WS et al (2014) Compact X-ray source based on burst-mode inverse Compton scattering at 100 kHz. Phys Rev Spec Top Accel Beams 17:120701 Graves WS et al (2009) MIT inverse Compton source concept. Nucl Instrum Methods Phys Res Sect A Accel, Spectrom Detect Assoc Equip 608:S103–S105 Günther B et al (2020) The versatile X-ray beamline of the Munich Compact Light Source: design, instrumentation and applications. J Synchrotron Radiat 27:1395–1414 Guo W et al (2008) Shanghai laser electron gamma source and its applications. Chin Phys C 32:190–193 Hong K-H et al (2018) Highly-stable, high-power picosecond laser optically synchronized to a UV photocathode laser for an ICS hard X-ray generation. In: Proceedings of IPAC 2018. TUPMK008 Hopf FA et al (1976) Classical theory of a free-electron laser. Opt Commun 18:413–416 Hornberger B et al (2019) A compact light source providing high-flux, quasi-monochromatic, tunable X-rays in the laboratory. In: Proceedings of the SPIE, advances in laboratory based X-ray sources, optics and applications VII, vol 11110, p 1111003 Huang Z, Ruth RD (1998) Laser-electron storage ring. Phys Rev Lett 80:976–979 Hwang Y et al (2017) Study of medical applications of compact laser-Compton X-ray source. In: Proceedings of IPAC2017, THOAB1 Hwang Y (2018) Characterization and applications of laser-Compton X-ray source. PhD thesis, University of California Irvine Ikeura-Sekiguchi H et al (2008) In-line phase-contrast imaging of a biological specimen using a compact laser-Compton scattering-based X-ray source. Appl Phys Lett 92:131107 Jones FC (1968) Calculated spectrum of inverse-Compton scattered photons. Phys Rev 167:1159– 1169 Jovanovic I et al (2007) High-power laser pulse recirculation for inverse Compton scatteringproduced γ-rays. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 578:160–171 Karlov NV, Krokhin ON, Lukishova SG (2010) History of quantum electronics at the Moscow Lebedev and General Physics Institutes: Nikolaj Basov and Alexander Prokhorov. Appl Opt 49:F32–F46 Kärtner FX et al (2016) AXSIS: exploring the Frontiers in attosecond X-ray science, imaging and spectroscopy. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 829:24–29 Kezerashvili GY, Milov AM, Wojtsekhowski BB (1993) The gamma ray energy tagging spectrometer of the ROKK-2 facility at the VEPP-3 storage ring. Nucl Instrum Methods Phys Res A 328:506–511 Kezerashvili GY et al (1998) A Compton source of high energy polarized tagged γ-ray beams. The ROKK-1M facility. Nucl Instrum Methods Phys Res Sect B: Beam Interact Mater Atoms 145:40–48 Kulikov OF et al (1964) Compton effect on moving electrons. Sov J Exp Theor Phys 47:1591–1594 Kulikov O et al (1969) Angular and energy distributions of gamma quanta in the Compton effect on relativistic electrons. Sov J Exp Theor Phys 56:115–121

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Oliva P et al (2010) Quantitative evaluation of single-shot inline phase contrast imaging using an inverse Compton X-ray source. Appl Phys Lett 97:134104 Ovodenko A et al (2016) High duty cycle inverse Compton scattering X-ray source. Appl Phys Lett 109:253504 Padamsee H (2001) The science and technology of superconducting cavities for accelerators. Supercond Sci Technol 14:R28–R51 Pantell RH, Soncini G, Harold E (1968) Puthoff. S-9—stimulated photon-electron scattering. IEEE J Quant Electron QE-4:905–907 Pogorelsky I et al (2014) High-repetition intra-cavity source of Compton radiation. J Phys B: Atom Mol Opt Phys 47:234014 Pogorelsky IV (1998) Ultra-bright X-ray and gamma sources by Compton backscattering of CO2 laser beams. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 411:172–187 Pound RV (1946) Electronic frequency stabilization of microwave oscillators. Rev Sci Instrum 17:490–505 Prantil MA et al (2013) Widely tunable 11 GHz femtosecond fiber laser based on a nonmode-locked source. Opt Lett 38:3216–3218 Priebe G et al (2008) Inverse Compton backscattering source driven by the multi-10 TW laser installed at Daresbury. Laser Part Beams 26:649–660 Rui T, Huang W (2018) Lattice design and beam dynamics of a storage ring for a Thomson scattering X-ray source. Phys Rev Accel Beams 21:100101 Sakanaka S et al (2015) Recent progress and operational status of the Compact ERL at KEK. In: TUBC1 Sandorfi A et al (1983) High energy gamma ray beams from Compton backscattered laser light. IEEE Trans Nucl Sci 30:3083–3087 Sannibale F (2016) Overview of electron source development for high repetition rate FEL facilities. In: Proceedings of NAPAC2016, TUB3IO02 Schawlow AL, Townes CH (1958) Infrared and optical masers. Phys Rev 112:1940–1949 Schleede S (2013) X-ray phase-contrast imaging at a compact laser-driven synchrotron source. PhD thesis, Technische Universität München Schoenlein RW et al (1996) Femtosecond X-ray pulses at 0.4 Å, generated by 90◦ , Thomson scattering: a tool for probing the structural dynamics of materials. Science 274:236–238 Shimizu H et al (2014) Development of a 4-mirror optical cavity for an inverse Compton scattering experiment in the STF. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 745:63–72 Shimizu H et al (2015) X-ray generation by inverse Compton scattering at the superconducting RF test facility. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 772:26–33 Sprangle P, Hafizi B, Mako F (1989) New X-ray source for lithography. Appl Phys Lett 55:2559– 2560 Sprangle P et al (1992) Tunable, short pulse hard X-rays from a compact laser synchrotron source. J Appl Phys 72:5032–5038 Sukhatme VP, Wolff PA (1973) Stimulated Compton scattering as a radiation source - theoretical limitations. J Appl Phys 44:2331–2334 Tang C et al (2009) Tsinghua Thomson scattering X-ray source. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 608:S70–S74 Ting A et al (1995) Observation of 20 eV X-ray generation in a proof-of-principle laser synchrotron source experiment. J Appl Phys 78:575–577 Tompkins PA et al (1993) Vanderbilt university Compton scattering X-ray experiment. Proceedings of PAC 1993:1448–1450 Toyokawa H et al (2007) A short-pulse hard X-ray source with compact electron linac via laserCompton scattering for medical and industrial radiography. In: Proceedings of PAC 2007, MOOAC02

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Urakawa J (2011) Nuclear instruments and methods in physics research a development of a compact X-ray source based on Compton scattering using a 1.3 GHz superconducting RF accelerating linac and a new laser storage cavity. Nucl Instrum Methods Phys Res A 637:S47–S50 Utsunomiya H, Hashimoto S, Miyamoto S (2015) The γ-ray beam-line at NewSUBARU. Nucl Phys News 25:25–29 Vaccarezza C et al (2014) The SPARC-LAB Thomson source commissioning. In: Proceedings of IPAC 2014, MOPRO078 Vaccarezza C et al (2016) The SPARC LAB Thomson source. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 829:237–242 Variola A et al (2014a) ThomX technical design report. HAL, Technical report, p 164. http://hal. in2p3.fr/in2p3-00971281 Variola A et al (2014b) The ThomX project status. In: Proceedings of IPAC 2014, WEPRO052 Weller HR et al (2009) Research opportunities at the upgraded HIγ S facility. Progr Part Nucl Phys 62:257–303 Williams O et al (2009) Characterization results of the BNL ATF Compton X-ray source using K-edge absorbing foils. Nucl Instrum Methods Phys Res Sect A Accel Spectrom Detect Assoc Equip 608:S18–S22 Willmott P (2019) An introduction to synchrotron radiation, 2nd edn. Wiley, Chichester. ISBN: 978-1-119-280378 Wu YK (2009) A kW intracavity power storage ring FEL. In: Proceedings of FEL 2009, MOOD01 Wu YK (2013) Light source and accelerator physics research program at Duke university. In: MOPEA077 Xu HS et al (2014) Design of a 4.8-m ring for inverse Compton scattering X-ray source. Phys Rev Spec Top Accel Beams 17:070101 Yamada K et al (2009) A trial for fine and low-dose imaging of biological specimens using quasimonochromatic laser-Compton X-rays. Nucl Instrum Methods Phys Res Sect A: Accel Spectrom Detect Assoc Equip 608:S7–S10 Yamazaki T et al (1985) Generation of quasi-monochromatic photon beams from Compton backscattered laser light at ETL electron storage ring. IEEE Trans Nucl Sci 32:3406–3408 Yu P, Wang Y, Huang W (2009) Lattice and beam dynamics for the pulse mode of the laser-electron storage ring for a Compton X-ray source. Phys Rev Spec Top Accel Beams 12:061301 Zelinsky A et al (2004) Kharkov X-ray generator based on Compton scattering. AIP Conf Proc 705:125–128 Zen H et al (2016) Generation of high energy Gamma-ray by laser Compton scattering of 1.94-Pm fiber laser in UVSOR-III electron storage ring. Energy Procedia 89:335–345

Chapter 7

The CLS Laser Upgrade

In the preceding chapter of this thesis, general aspects of ICSs were presented, in particular design considerations and general aspects of the ICS of the MuCLS were discussed. This chapter focusses on the interaction laser system of the Lyncean Compact Light Source. Its detailed description is accompanied by its characterisation, mainly in terms of stability. The first section contains a general description of the interaction laser system, the second one characterises its Nd:YAG laser, while the third one covers the enhancement cavity.

7.1 The Interaction Laser System of the MuCLS The CLS interaction laser system is based on Nd:YAG technology as already mentioned in the preamble of this chapter and in the overview about the Lyncean Compact Light Source. Figure 7.1 is a schematic of the complete interaction laser system including all components after its upgrade in 2017. Figure 7.2 shows pictures of the laser beam transport line in (a) and (b) as well as of the laser oscillator and its power amplification system in (c). The parts of the laser system on display in the pictures are highlighted with coloured boxes in the schematic of the laser system. Laser pulses are generated in a customised Lumentum Xcyte Nd:YAG oscillator, black box on the top right of Fig. 7.2c. It contains a direct-coupled pump laser head which is beneficial for side-pumping of crystals (Spühler et al. 1999). One end mirror of the cavity is a saturable Bragg-reflector, which is responsible for pulse generation via passive mode-locking. The repetition rate of the cavity has been adjusted to coincide with the revolution frequency of the electron beam of 64.91 MHz. However, thermal changes affect the cavity length and in turn the repetition rate of the laser. In order to compensate this elongation, the second end mirror is movable. A stepper motor is available for coarse adjustment of the laser’s repetition rate and a piezoelectric actuator stack, mounted on-top of the latter, for smooth in situ corrections © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_7

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Fig. 7.1 Schematic of the CLS laser system in its current status. The laser system can be divided into three parts: the laser amplification system on lower right side, the enhancement cavity on the upper right side and the transport line on the left. The latter contains measures to fit the laser exiting the amplifier system to the orbit inside the enhancement cavity and its diagnostics

of small drifts. Moreover, a fast feedback is required to keep the laser itself on resonance with the cavity. To this end, one tiny mirror inside the oscillator is glued onto a small rapidly-moving piezoelectric actuator controlled by the error-signal of the Pound-Drever-Hall (PDH) feedback (Pound 1946; Drever et al. 1983). The principle of keeping the laser both on resonance with the enhancement cavity and matched to the electron revolution frequency at the same time was discussed in Sect. 4.2.4.4. The oscillator delivers a laser power of about 4.5 W. After the laser beam exits the oscillator, it is collimated by a lens and steered to a straight section containing a Faraday-isolator and two electro-optic modulators (EOMs). The first device exploits time reversal symmetry breaking of the Faradayeffect to extinguish back-reflections originating downstream of the Faraday-isolator, e.g. caused by lenses or polarisers. The first EOM adds side-bands to the laserfrequency, which are offset to the centre frequency by the constant modulation frequency applied to the EOM and act as a reference in the PDH-locking. Operation of the second EOM is optional. Its main application is sweeping frequency side-bands over a certain frequency range, e.g. the one covering the first higher-order resonance of the enhancement cavity which provides information on the Rayleigh-range. After two turn mirrors, two RBAT 35 modules (Cutting Edge Optronics (CEO), St. Charles, USA) increase the laser power to ∼10 W. Thermal expansion of the rod crystal inside the amplifier results in a pronounced thermal lens focussing the laser beam. This effect is counteracted by several lenses re-collimating the beam to its original

7.1 The Interaction Laser System of the MuCLS

151

Fig. 7.2 CLS laser system in its current status. a is the complete laser transport and diagnostic assembly, while b only depicts the diagnostics on the right and the input steering mirror of the laser beam. c is the laser power amplifier assembly

transverse diameter. Two polarisers reject depolarised light exiting the pre-amplifiers before a Lumentum Argos multi-pass amplifier raises the laser energy to the desired value of ∼30 W. At the end of the laser amplifier chassis, a water-cooled beam block is installed. After the shutter a motorised half-wave plate and a quarter wave-plate (abbreviated as “WP”s in Fig. 7.1) are installed to adjust laser polarisation. Downstream, a telescope is implemented matching the transverse laser parameters to the ones of the circulating laser beam on OC50. The incoming beam’s diameter and its radius of curvature at OC50 are adjusted with a motorised delay stage. Its position on OC50 is determined by adjusting tip-tilt on the offset mirror (Fig. 7.1). Another

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mirror, the steering mirror, is used to overlap the incoming laser’s direction of propagation with the cavity orbit. Since only the cavity mirrors OC70 and OC80 are used to compensate astigmatism and skew astigmatism, the mode inside the enhancement cavity is astigmatic at OC50, cf. Fig. 7.8. Thus, the incoming laser beam needs to be astigmatic as well to be matched to the circulating mode, which is achieved by deliberately introducing astigmatism by deforming a mirror (abbreviated “ast.” in Fig. 7.1) in a controlled way.

7.2 Characterisation of the CLS Laser The laser beam arriving in front of OC50 was analysed in terms of its M2 -value, pulse duration and overall stability, i.e. parameters which are important for the generated X-rays in inverse Compton scattering.

7.2.1 The M2 Beam Propagation Factor The M2 -method is a measure for the transverse beam quality of arbitrary real lasers, which is introduced, e.g. in “How to (Maybe) Measure Laser Beam Quality” by Siegman (1998). A fundamental difficulty for assessing a laser beam’s transverse quality is defining the laser beam waist properly. Among the many possible definitions, like the 1/e or 1/e2 intensity criteria, the width at first nulls or a Gaussian fitted to the intensity distribution, the definition of the beam width based on the variance of the intensity distribution might be closest to a universal and mathematical rigorous formulation (Siegman 1998). This variance has to be calculated as the second moment of the intensity distribution in order to fulfil the equation for free-space propagation for arbitrary real beam profiles (Siegman 1998). As a result, the generalised beam waist w(z) propagates like an ideal Gaussian beam, except for the multiplication of the far-field divergence with the M2 beam propagation factor: ( w (z) = 2

w02

+M

4

λ πw0

)2 (z − z 0 )2 .

(7.1)

For the nomenclature, the reader is reminded of Fig. 3.2. Consequently, this method reveals the laser beam’s similarity to a diffraction-limited Gaussian TEM00 one. Therefore, an arbitrary beam can be propagated through an optical system like a fundamental Gaussian one. The only difference is that the beam width has to be scaled with the beam propagation factor of the beam in use. The M2 -factor has been determined with the beam reflected off of the OC50 incoupling mirror. It was refocussed with an auxiliary lens with a focal length of 606 mm. The laser was operated in standard conditions and delivered 33 W at the optical cavity (reflected power). When measuring the beam propagation factor, ISO

7.2 Characterisation of the CLS Laser

153

Fig. 7.3 Measurement of the M2 beam propagation factor of the new power amplifier system. a and b display the laser beam at the position of the horizontal and vertical focus. c shows the line profiles along the horizontal and vertical axis for both foci. d displays the beam waists calculated from the second moments. Solid lines are the fit functions from which the M2 -factor has been extracted

11146 should be followed, which requires at least five measurement points in the near-field, i.e. within a distance of one Rayleigh range to the focus. At least the same number of measurements has to be acquired at larger distances to accurately determine the beam’s far-field divergence. Figure 7.3a, b display the astigmatic laser beam at the position of the horizontal and vertical focus. The resulting line profiles along the principal axes of the beam for both foci are presented in Fig. 7.3c. At the respective focus, the horizontal and vertical profiles are very similar, however, the beam width in the vertical direction is slightly smaller. The latter can be seen very well in Fig. 7.3d displaying the beam width for the scan through the focus. Beam propagation factors Mx2 and My2 are extracted fitting Eq. 7.1 to the beam waist determined from the second moments. The beam diameter determined from this technique is also called D4σ . The resulting Mx2 - and My2 -factors of 1.31 and 1.27 are almost perfectly symmetric. Since both, beam propagation factor and waist size, agree quite well, the laser has an astigmatic transverse profile. The main contributions to this increased M2 -factor compared to an ideal TEM00 -mode are attributed to incoherent spontaneous emission during amplification and imperfect optical components (e.g. lenses, polarisers, etc.). Nevertheless, these values are still sufficiently small to allow for good mode-matching with the circulating laser beam.

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7.2.2 Laser Pulse Duration Direct measurement of the shape of short pulses is not straight forward and becomes more complex the shorter the pulses get. Therefore, indirect detection schemes are usually employed from which the pulse duration can be inferred. Since a detailed discussion of these techniques is out of the scope of this thesis, the interested reader is referred to Chap. 12.3 for femtosecond (also valid for picosecond pulses to a large extent) and Chap. 4.12 for attosecond pulses of the “Springer Handbook of Lasers and Optics” (Träger 2007). Pulse autocorrelation is a widely used technique to estimate pulse durations in the pico- and femtosecond regime. Its symmetric trace and absence of spectral phase information are characteristic. As a result, the pulse length can only be retrieved with prior knowledge of the pulse shape itself. Usually, Gaussian or sech2 -pulse shapes are assumed and fitted to the autocorrelation profile. The CLS’s laser pulse length was measured right after the laser beam exits the oscillator and after amplification to the desired power of 30 W with an FR-103XL (Femtocrome Research Inc., Berkeley, USA) autocorrelator. Both traces are depicted in Fig. 7.4a, b together with a Gaussian and a sech2 fit. Contradicting theory, the measured trace is slightly asymmetric. A typical reason for such an effect is a small misalignment of the autocorrelator. This is consistent with the fact that the autocorrelator was repaired by Lyncean Technologies

Fig. 7.4 Determination of the laser’s pulse duration from an autocorrelation measurement of the pulse. a is an autocorrelation trace that has been acquired with the laser pulses exiting the Xcyte oscillator (Lumentum Operations LLC, San Jose, USA) and b is the one acquired after amplification. Gaussian as well as sech2 pulses have been fitted to the measured data. Pulse duration is less than 30 ps for both fits (units of the retrieved FWHM pulse length are ps). Its elongation due to dispersion during amplification of ∼0.5 ps is unproblematic for the intended use of the laser in inverse Compton scattering with an electron beam of 50 ps duration (Schleede 2013)

7.2 Characterisation of the CLS Laser

155

Inc. at some point before the measurement took place.1 While the overall uncertainties on the retrieved values are very similar for both fits, a sech2 -shaped pulse describes the peak better than a Gaussian one, but exhibits larger deviations on one side of the tails of the pulse. The latter could be dominated by the imperfect autocorrelator. Thus, a sech2 pulse appears to be the most likely one. Its retrieved FWHM-pulse duration of 25.93 ps behind the oscillator and 26.37 ps after amplification are around 3.3 ps shorter than the ones of the respective Gaussian-fits. This can be understood from the different deconvolution factors for the Gaussian beam and the sech2 one. Moreover, the pulse elongation of ∼0.5 ps during amplification arises from dispersion inside the laser crystals, lenses, etc. However, such a small amount is negligible for the intended application in inverse Compton scattering, especially since the electron beam has a longer pulse duration of 50 ps (Schleede 2013).

7.2.3 Laser Stability Stability is a key aspect for any laser system in continuous operation. Above all, this is true for a laser system which is contained inside a radiation-shielding enclosure that cannot be accessed during operation of the instrument, here the CLS. Therefore, such a laser system should have a very rigid beam path and frequently installed diagnostics in order to detect misalignments early on. Static reference positions on the diagnostics should exist for repeatable realignment. Ideally, motorised mirror mounts would allow to remotely adjust the laser beam path while operating the CLS and therefore significantly increase uninterrupted X-ray beam time with optimal laser settings. A combination of diagnostics and motorised mirror mounts would finally enable a laser alignment feedback system. In contrast to this ideal setting, the laser system of CLS neither contained alignment markers nor any diagnostics for beam position or pointing when it was delivered with the compact light source. Consequently, there was no simple and fast way to optimise the laser alignment or identify creeping mirror mounts if power dropped noticeably, which typically happens in between two service visits. Often, the drop in laser power over the course of several months was so significant that the stored power in the enhancement cavity required to operate the CLS within specifications could not be reached any more, even with a constant performance of the enhancement cavity. As a first step to improve this situation, quadrant diode detectors were implemented into the laser amplifier system at certain locations in collaboration with Lyncean Technologies Inc. In an ideal world, beam position and pointing would be measured, but the spatially very constrained laser amplifier system did not allow for it. The quadrant detectors record the transmitted beam behind a mirror. Their positions are directly behind the first turn mirror after the oscillator, in front of the entrance to the Lumentum Argos power amplifier, after the laser beam exits it and 1

Information by Martin Gifford, Lyncean Technologies Inc.

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Fig. 7.5 24 h laser stability measurement. a is the signal reflected off the OC50 entrance window. b are quadrant detector signals of the laser power (sum signal) and positioned directly after the laser exits the oscillator. c The corresponding signals after the pre-amplifiers right in front of the main power amplifier and d after amplification in the main amplifier

on the diagnostic line where it detects the light reflected off of OC50, cf. Fig. 7.1. These detectors monitor the laser position and power at a frequency of 1 Hz. A typical 24 h period is depicted in Fig. 7.5. a depicts the laser power measured with a standard pyrolytic power sensor in dark blue and the horizontal as well as vertical signal of the quadrant detector located on the diagnostic assembly (QD4 in Fig. 7.1). b, c and d are the quadrant detector signals at locations QD1, QD2 and QD3. This data was acquired starting on September 19th, 2019 19:00h after the laser amplifier system reached thermal equilibrium. After a first drop in power, the reflected power (dark blue line in Fig. 7.5a) continues to decrease slowly, which is consistent with the vertical drift observed in Fig. 7.5c. As a consequence, the laser enters the Argos amplifier slightly misaligned, which seems to reduce the gain in the amplifier. One explanation for this behaviour would be a small aperture inside the amplifier. A more severe version of this effect is observed during the last ∼3.5 h of the stability scan, which correlates to an increase of the laser head temperature of 0.15 ◦C. Depending on the exact position of the thermistor, this might not be the actual temperature change of the laser head. Although the position of the laser recorded with QD1 are stable over the course of this measurement,2 small changes of the beam’s pointing might be only detected at QD2 which is located further 2

The sum signal of QD1 is 0 because archiving of this data was interrupted.

7.2 Characterisation of the CLS Laser

157

Table 7.1 Summary of the analysis of the 24 h stability measurement. Two periods have been analysed: the complete duration of the scan and a 12 h period indicated by vertical grey lines in Fig. 7.5a. In the 12 h, period the laser head temperature of the oscillator has been stable. Therefore this period resembles the fundamental limit of the laser stability, while the full 24 h period reflects the real situation in daily operation better Acquisition period Full 24 h 12 h stable Mean Std. Drift Mean Std. Drift Laser power [W] Hor. pos. QD 4 [V] Ver. pos. QD 4 [V] Sum sig. QD 3 [V] Hor. pos. QD 3 [V] Ver. pos. QD 3 [V] Sum sig. QD 2 [V] Hor. pos. QD 2 [V] Ver. pos. QD 2 [V] Hor. pos. QD 1 [V] Ver. pos. QD 1 [V]

27.60 −0.421 0.489 4.641 0.882 −0.969 6.260 0.478 0.031 −0.976 −0.621

0.12 0.014 0.010 0.045 0.010 0.009 0.005 0.013 0.020 0.001 0.002

0.53 0.031 0.035 0.184 0.011 0.007 0.020 0.011 0.059 0.002 0.005

27.62 −0.418 0.486 4.640 0.882 −0.969 6.261 0.477 0.033 −0.976 −0.620

0.07 0.013 0.007 0.040 0.010 0.009 0.004 0.013 0.016 0.001 0.001

0.22 0.011 0.005 0.051 0.012 0.006 0.008 0.012 0.006 0.002 0.002

than one meter downstream the beam’s path instead of only about 10 cm behind the oscillator’s end mirror. The position of the laser on QD3 is very stable throughout the complete period. In order to judge the laser system’s intrinsic stability, two periods were considered for the analysis. The first one is the complete 24 h range and the second one the 12 h range within the two grey vertical lines. In the latter period, the room- as well as laser head temperature of the oscillator are very stable. Therefore the 12 h range can be considered as the fundamental limit on the intrinsic stability of the laser system. Results are displayed in Table 7.1. In general, the standard deviation within the 12 h period is less than or equal to the one of the 24 h period, as expected. Comparing the results of the 12 h period with the full 24 h one reveals larger drifts of the laser power as well as sum signal on QD2, QD3 while the drifts of the horizontal position are the same for QD1, QD2 and QD3. However QD4 exhibits stonger drifts, especially in the very beginning of the measurement, cf. Fig. 7.5a, which might be related to cooling down of the entrance optic (assembly) after the laser was unlocked from the cavity. Nevertheless, the overall drifts as well as standard deviation of the mean values are sufficiently small that they do not impair daily operation of the CLS under normal circumstances. Information on the laser performance in a single run is only half the picture, as it does not contain evidence on the long term stability of the laser system. Figure 7.6 is an approach to estimate this stability. It displays the peak laser power recorded at the power sensor every week and the values that were recorded by the quadrant detectors closest to this time for half a year. Archiving of the reflected beam’s sum signal was accidentally stopped during the service visit in September of 2019. Additionally, QD1’s sum signal has not been recorded properly since. Unfortunately, this has not been discovered until the date of writing of these

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Fig. 7.6 Laser stability over the course of a 6 month period. The peak power measured with the laser power meter of the beam reflected off of the OC50-mirror for every week and the corresponding quadrant detector signals are depicted. These parameters indicate the long-term stability of the laser system. a are the signals of QD4 located on the diagnostic table. b are the signals of QD1 at the oscillator exit. Unfortunately, an incident during the September service visit resulted in incorrectly archived sum signals and jumps of the positional channels. At the same time, archiving of the QD4 sum signal was interrupted. This was only discovered in June 2020. c are the quadrant detector channels at the entrance of the Argos power amplifier (QD2) and d the corresponding signals at its exit (QD3)

lines in late June 2020. Therefore, QD4’s sum signal is only recorded until September while QD1’s sum signal remains at zero, even when the oscillator has been operated. At the same time, a big step change of the beam position on QD1 is observed which could be either related to the same issue that interrupted QD1’s archiving or oscillator alignment. Over time, the positional channels of QD1 still reflect changes of the laser beam position, albeit with the aforementioned offset. Further service visit work is reflected in the QD3-signals in December, where step changes of all three signals are observed as well, cf. Fig. 7.6d. The positions on QD4 shift in the opposite direction. Similar behaviour is visible on the sum signal and vertical position on QD2 after the pre-amplifiers, cf. Fig. 7.6c. Such actions during a service visit partially contradict the original intention of their implementation. While drifts remain detectable, long-term reproducibility is impaired and should be enforced in the future.

7.3 The CLS Enhancement Cavity

159

That said, the long-term stability is analysed under the constraints mentioned above. The overall laser beam position of the oscillator is very stable, while the sum signal contains a small peak after two weeks, which is not reflected in the sum signal of the other diodes. There, this signal is rather stable over the first three months. Afterwards, the trend of the QD2 sum signal, Fig. 7.6c, follows the one of the reflected laser power closely, Fig. 7.6a. During this period, the reduction in the QD2 sum signal, especially in October 2019, seems to be correlated with the drift in beam position. The trend of the sum signal at QD2 is imprinted on the one at QD3 as well, albeit in an enhanced fashion. This indicates that the Argos power amplifier is sensitive to slight misalignment and reduction in seeding power. In contrast, the QD4 diode signals after the power amplifier for the beam positions appear to be very stable and uncorrelated to changes in laser power. Changes in the laser power might be correlated to changes in the laser position for the October-December 2019 period as well. However, for the first three month period a clear correlation cannot be recognised. Furthermore, the effect of the service visit in between the two periods remains unknown. In order to observe long term trends, laser alignment has to be reproducible and should not change from service visit to service visit. A relation between the vertical beam position at the input to the Argos amplifier and output laser power was observed, which is indicated in Fig. 7.5. Its realignment to a reference value improved the situation significantly after the laser power had been dropping continuously for weeks, e.g. most recently in February and June 2020. Concurrent to the drift of the vertical position, the shape of the signal reflected from the polariser after the pre-amplifier changes. Figure 7.7a exhibits a quite symmetric reflection, which corresponds to the laser beam aligned vertically to the reference position. After the beam had drifted vertically, the reflected light is vertically asymmetric as displayed in Fig. 7.7b. Adjusting the vertical beam position on QD2 back to a reference value and optimising the symmetry of the reflected light by just tilting the first turn mirror after the amplifier increased the output of the Argos amplifier by several watts. The sensitivity of the quadrant detectors could be improved by focussing the beam onto the center of each detector, because small drifts result in stronger signal shifts on the quadrant detector. However, such interventions have to be performed by Lyncean Technologies Inc. for legal reasons and thus have not been implemented yet. Consequently, there is still a lot of room for improvement in both laser system diagnostic and its implementation into a motorised-mirror position feedback system. This would reduce the necessity of laser system realignment every three months, which is non-competitive with typical commercial systems.

7.3 The CLS Enhancement Cavity At the CLS, the laser system discussed above feeds an enhancement cavity, which basically is an “empty” high-Q resonator. Such a device is necessary for generating

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Fig. 7.7 Laser beam profiles. a is the shape of the reflected polarisation after the pre-amplifiers when the laser is aligned to the Argos amplifier entrance. b is the same reflection when the laser beam is misaligned vertically on QD2. c displays the reflected laser beam of OC50 when the laser is locked to a good mode circulating inside the cavity. d is the same laser beam when it is not on resonance with the enhancement cavity

very intense laser pulses at a repetition rate matching the electron revolution frequency in their storage ring resulting in an average optical power far above 100 keV. This section discusses the geometry of the enhancement cavity employed at the MuCLS and their implication on its performance as well as its feedback systems. A brief characterisation of the cavity’s stored power is provided and the resulting thermal effects are discussed.

7.3.1 The CLS Cavity System The resonator at the MuCLS is an all-curved mirror (ACM)-resonator operating at the inner stability edge. A schematic of the resonator geometry is depicted in Fig. 7.8a. Its length is constrained to a (sub-) harmonic of the electron beam’s revolution frequency in the storage ring in order to ensure Compton scattering at every roundtrip. If the intrinsic repetition rate of the enhancement cavity is a sub-harmonic of the electron beam’s revolution frequency, multiple pulses need to be stored inside the resonator to compensate the lower revolution frequency. Moreover, the mirror-to-mirror distance

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161

Fig. 7.8 Properties of the CLS enhancement cavity. a displays the geometry of the CLS enhancement cavity. This geometry results in a Rayleigh length of 9.5 cm at the interaction point with the electron beam in the strongly focused upper arm. b Gouy phase and laser beam waist for the cavity geometry in a. Three cases are shown, the ideal non-astigmatic case in blue as well as the sagittal and tangential waists with roundtrip astigmatism correction at OC70 in orange and green, respectively

is required to be longer than the storage ring’s extent, because such a small storage ring does not provide sufficient space for implementing a mirror assembly inside of it, e.g. mirrors with a small aperture for the electron beam. This limits their minimum spacing to about 2.5 m. Shorter inter-mirror distances could be realised by significantly increasing the angle between the laser and electron beam. As a result, a lower scattering efficiency as well as a reduced X-ray peak energy would have to be accepted. On the one hand, the latter could be overcome with higher electron energies, but would require a larger storage ring on the other hand. This increases system cost and footprint, which is undesirable for compact systems affordable to universities or laboratories. The realisation of short inter-mirror distances is further complicated by the fact that the electron beam has to be focussed tightly at the interaction point as well. This requires sets of quadrupole magnets close to the interaction point, whose positions are sketched in Fig. 7.13 and the real situation is displayed in Fig. 7.14a. Their presence inhibits wide interaction angles between the laser and electron beam in the current CLS electron storage ring design as well as the implementation of laser focus diagnostics. Therefore, at least one secondary focus would be beneficial since

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it facilitates implementation of such diagnostics. A straight-forward solution would be a ring resonator in standard bow-tie (SBT) geometry consisting of two curved mirrors of identical radius of curvature and two flat ones. Such resonators are stable in the range √ L − L 2 − 4d L , (7.2) R25 W. The two ringdown measurements displayed in Fig. 7.12 provide an overview of the overall improvement. At the enhancement cavity of the CLS, the photo-diode signal for the ringdown measurement samples intensity from the laser beam’s halo which is extracted at the position of the resonator’s secondary foci. Accordingly, the signal’s shape is identical to the one obtained from a ringdown in transmission geometry. This diagnostic assembly consists of the cylindrical aluminium vacuum can, which

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contains also the pellicle whose location in the laser system is shown in Fig. 7.1, and the black diagnostic box, depicted in Fig. 7.14a. The ringdown measurement is described in Sect. 4.2.5 in detail. At a coupling above 80 %, the laser power matched to the cavity mode is 20 W. Compared to the preceding laser system, the coupling was improved by a couple of percent. Both effects, higher coupling and overall increased laser power, approximately doubled the laser power matched to the cavity mode from ∼10.4 W to ∼20.5 W. Consequently, the achievable stored power should be about twice the one obtained with the old laser system. Instead of the expected stored power of about 280 kW, the enhancement cavity holds about 350 kW stored power under normal conditions. The residual gain of the enhancement cavity arises from improvements of the optical cavity itself. Losses were minimised during the laser system upgrade from close to 200 ppm to less than 175 ppm, i.e. by more than 12.5 %, by cleaning the cavity optics and optimisation of the laser beam’s orbit inside the resonator. In turn, the resonator’s finesse increased from ∼31,700 to ∼ 36,700 and the resonators performance as a passive amplifier improved from a gain of ∼13,600 to ∼16,900. Additionally, after installation of the laser upgrade by Lyncean Technologies Inc., the company told us that the input coupler’s transmission is now 124 ppm instead of the 133 ppm it used to be. Only mutual improvements of both, the laser cavity and the seeding laser system enables a stored power of 350 kW to be reached during standard conditions. One frequently observed issue at these higher laser powers was a rapid reduction of the X-ray photon flux after optimisation of the overlap between the laser and electron beam. A correlation between the flux drop and a vertical drift of the X-ray source position could be observed. This effect diminished the net X-ray flux enhancement due to the laser upgrade. Therefore, its origin and the measures to mitigate its impact on the X-ray flux are discussed in the following two subsections.

7.3.3.2

Origin of Source Position Drift

Although the overall losses of the resonator have been reduced by more than ten percent, the overall power deposited in the resonator increases significantly at 350 kW of stored laser power. Accordingly, the thermal effects caused by the laser have been much more pronounced after the upgrade. To understand where their influence on the system is strongest, the primary sources and directions of heat flow need to be identified. Orange arrows in Fig. 7.13a depict the main directions of heat flow and thus indicating the regions of the enhancement cavity that are subject to highest heat load. The positions where the enhancement cavity is connected to the optical table with flexible mountings are indicated in blue. Apart from the laser itself, the bending magnets are the second main heat source. They are located in close proximity to the mirror assemblies of the shared vacuum pipe of electrons and laser photons. The bending magnets’ influence is considered first. As a consequence of the shared vacuum tube, two of the electron storage ring’s bending magnets are located close to OC50 and OC90. Large currents well above 100 A running through these big magnets impose the dominant heat flow onto the two mirror assemblies nearby. The

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Fig. 7.13 a Schematic of the enhancement cavity of the CLS. Blue dots show the positions where the optical cavity is mounted to the optical table. Orange arrows indicate the direction of heat flow. The main heat load on the common leg of electron and laser beam is produced by the bending magnets of the electron storage ring. Laser losses in the back-leg are the main source of heat leading to an extension of the stainless steel tube between the optic containment mounted to the optical table, schematically depicted in b. This causes a tilt of the optic containment and therefore the optic shifting the laser orbit which corresponds to a shift of the X-ray source position

laser cavity orbit drifts until the magnets reach their thermal equilibrium and the elongation of the beam pipe between the two mirror assemblies settles, which may take up to five hours after the magnets are turned on. Since the laser upgrade does not alter the heat load provided by the magnets, their contribution cannot be the origin of the observed vertical shifts of the X-ray source position after the laser upgrade. Therefore, the thermal load introduced by the laser itself into the resonator’s vacuum assembly has to be responsible for it. Mainly absorption in the mirrors themselves and scattering of laser photons contribute to the laser-induced heat load. The latter primarily dissipates heat into the vacuum assembly of the enhancement cavity. This effect should be strongest, where the laser beam diameter itself is large. Since the transverse mode of the laser is collimated only in the back-leg of the cavity, scattering losses are mainly dissipated there. Heat deposition elongates the stainless steel tube connecting the two back-leg mirror assemblies. This tilts the optic assembly and thereby the optic shifting the orbit as well, schematically shown in Fig. 7.13b. As a result, a vertically shifted orbit at the opposite leg is observed as a shift of the X-ray source position which degrades the overlap between laser- and electron beam thereby reducing the X-ray flux. Since laser losses may change with the orbit of the laser in the resonator, a shift of the laser

7.3 The CLS Enhancement Cavity

171

orbit due to a thermally induced mirror tilt may alter the stored power inside the cavity. In turn, this acts on the tilt of the mirrors, which can increase the misalignment between photon and electron beam. Consequently, the resulting thermal drift needs to be counteracted by stabilising the mirror’s vertical orientation. This keeps the orbit of the laser beam constant and in turn source position and flux. The system developed for this task is the so-called align thermal compensation (ATC) system.

7.3.3.3

Thermal Compensation System

In order to compensate a deflection of the two mirror assemblies in the back-leg due to an elongation of the vacuum pipe connecting them, the tilt introduced by the thermal load has to be determined. To this end, a laser diode was mounted to each mirror assembly, OC70 as well as OC80, whose position is recorded with a quadrant photo-diode located close to the opposite mirror assembly, but mounted to the optical table. This way the deflection of the assembly relative to the optical table can be determined once each laser diode is aligned at the centre of their quadrant photo-diode. To facilitate this alignment, the laser diode has been fitted into a standard tip-/tilt mirror mount with an adapter. Figure 7.14b, c display the set-up at the OC70 assembly, consisting of the laser diode mounted to OC70 and the quadrant photodiode fixated onto the granite optical table and detecting the laser attached to the OC80 assembly. Figure 7.15a displays the reflected laser power as well as the horizontal and vertical signals of the quadrant photo-diode which detects deflection of the OC80 assembly. It tilts exclusively in the vertical direction if the seeding laser is locked to the enhancement cavity. This finding supports the hypothesis presented in the preceding section that the heat load due to losses extends the back-leg’s beam pipe. Consequently, the goal needs to be to keep the back-leg’s length constant, or, in other words, the heat load on the back-leg. This can be achieved by actively pre-heating the back-leg with a certain electric power that leaves enough overhead for power adjustments in both directions, depending on the direction of the recorded tilt. Such a system has been realised by attaching a commercial heating wire to the back-leg, cf. Fig. 7.14a. A closed-loop feedback adjusts the heating power. The closed-loop feedback’s readback is the signal from the quadrant photo-diode detecting the laser mounted onto the OC80 assembly. The initial reference position represents the state with the seeding laser unlocked. The deflection from this zero position multiplied with an empirically determined constant constitutes the error signal for the heater. Once the ATC-feedback is turned on, the position of the OC80 assembly remains much more stable when the seeding laser is locked to the cavity, cf. Table 7.2 and Fig. 7.15a, b. Nevertheless, small drifts can be still observed sometimes, e.g. at about 2 h in Fig. 7.15b, directly after the laser is locked to the enhancement cavity. The reason for this may be that the ATC feedback via the heating wire is layout for rather slow drifts that cannot cope with very rapid changes of the overall heat load, which can occur in this case. At all other instances the position of the OC80-assembly is kept very constant during both, the 24 h period and the stability run over 10 consecutive days, Fig. 7.15c.

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Fig. 7.14 Thermal compensation system, often called align thermal compensation (ATC) system. a depicts the actuator of the feedback system, a commercial heating wire. b displays the mirror assembly OC70 and the quadrant photo-diode detecting deflections of the OC80 assembly. The laser-diode mounted to OC70, analogously to the diode at OC80, is visible as well. c View along the backleg from OC70 to OC80. Red lines indicate the laser beams. The laser-diodes are mounted with mirror mounts in order to align their beam onto the centre of the corresponding quadrant photo-diode

7.4 Contributions

173

Table 7.2 Comparison of the enhancement cavity stability with and without the align thermal compensation system operating. The table depicts the mean value (mean), standard deviation (std) and the maximum drift (drift) for the three situations depicted in Fig. 7.15a–c. Stability of the horizontal position is not affected by the ATC system, while the stability of the vertical position is significantly improved, both in terms of absolute drifts as well as the standard deviation On (24 h) Heater feedback Off (24 h) On (10 d) Mean Std. Drift Mean Std. Drift Mean Std. Drift Hor. pos. QD OC80 [V] Ver. pos. QD OC80 [V]

0.133 0.011 0.048

0.045 0.011 0.028

0.108 0.016 0.055

−0.330 0.060 0.224 −0.198 0.019 0.136 −0.085 0.011 0.089

Although the ATC system maintains the tilt of the OC80 assembly constant by adjustment of the resonator’s back-leg length, this does not have to be automatically true for the OC70 assembly as well. Small thermal effects inside the OC80 assembly could be compensated via the ATC system or the other way round, effects inside the OC70 assembly could remain uncorrected. Consequently, a second degree of freedom is needed to keep the tilt of the OC70 assembly constant. One way to do so could be to actively adjust the mirror’s tip and tilt by piezoelectric actuators. The laser-diode mounted to the OC70 assembly and its corresponding quadrant photo-diode could serve as the readback signals. Although, such a solution, could keep the orientations of both, OC80- and OC70 assembly, constant, it may not be the optimum one because the lasers position at the interaction point has to remain constant as well to ensure a stationary X-ray source position. Changes on the two other mirrors of the bowtie cavity, OC50 and/or OC90, can affect the orbit as well. Accordingly, the better solution would be to employ the piezoelectric actuators attached to OC70 in order to keep the X-ray source position stationary by compensating the effective residual orbit shift of all mirrors via OC70. This feedback system and its instrumentation is presented in Chap. 10.

7.4 Contributions This chapter of my thesis has been realised with the following contributions from Lyncean Technologies Inc. (Fremont, USA). The upgrade of the seeding laser and the hardware for the transverse mode range measurement were installed by Lyncean Technologies Inc. The improved diagnostic inside the upgraded amplifier system and the ATC system were developed and installed in collaboration with Lyncean Technologies Inc. Data acquisition of the M2 -factor and the laser pulse duration was performed in collaboration with Martin Gifford, Lyncean Technologies Inc.

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Fig. 7.15 Characterisation of the align thermal compensation system. a displays a 24 h period during which the ATC system was turned off. Drifts of the vertical position are well visible, when the seeding laser is locked to the enhancement cavity. b Another 24 h period in which the ATC feedback system was turned on. Significant improvement of the stability of the vertical tilt of OC80 is achieved, although very rapid changes of the thermal load directly after turning on the laser and locking it to the enhancement cavity cannot be compensated completely by the rather slow feedback. c Long-term stability over the course of 10 days under various conditions of the enhancement cavity. Data acquired at 1 Hz

References Carstens H et al (2013) Large-mode enhancement cavities. Opt Exp 21:11606–11617 Djevahirdjian L, Méjean G, Romanini D (2020) Gouy phase shift measurement in a high-finesse cavity by optical feedback frequency locking. Meas Sci Technol 31:035013 Drever RWP et al (1983) Laser phase and frequency stabilization using an optical resonator. Appl Phys B Photophys Laser Chem 31:97–105 Hello P, Vinet JV (1990) Analytical models of thermal aberrations in massive mirrors heated by high power beams. J Phys 51:1267–1282 Hello P, Vinet JV (1990) Analytical models of transient thermoelastic deformations of mirrors heated by high power CW laser beams. J Phys 51:2243–2261 Mueller CL et al (2015) In situ characterization of the thermal state of resonant optical interferometers via tracking of their higher-order mode resonances. Class Quant Gravity 32:135018

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Pound RV (1946) Electronic frequency stabilization of microwave oscillators. Rev Sci Inst 17:490– 505 Schleede S (2013) X-ray phase-contrast imaging at a compact laser-driven synchrotron source. PhD thesis, Technische Universität München Siegman AE (1998) How to (maybe) measure laser beam quality in DPSS (diode pumped solid state) lasers: applications and issues. Optical Society of America, MQ1 Spühler GJ et al (1999) Diode-pumped passively mode locked Nd:YAG laser with 10-W average power in a diffraction-limited beam. Opt Lett 24:528–530 Stochino A, Arai K, Adhikari RX (2012) Technique for in situ measurement of free spectral range and transverse mode spacing of optical cavities. Appl Opt 51:6571–6577 Träger F (ed) Springer handbook of lasers and optics, 1st edn. Springer, New York. ISBN: 978-0387-30420-5 Uehara N, Ueda K (1995) Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry–Perot interferometer. Appl Opt 34:5611–5619 Uehara N, Ueda K (1995) Accurate measurement of ultralow loss in a high-finesse Fabry-Perot interferometer using the frequency response functions. Appl Phys B 61:9–15 Vinet JY (2007) Reducing thermal effects in mirrors of advanced gravitational wave interferometric detectors. Class Quant Gravity 24:3897–3910

Chapter 8

Development of a Deformable Exit Optic

In this chapter, an approach to overcome several technical constraints is investigated which limit the performance of the Compact Light Source currently. This work was performed at the site of Lyncean Technologies Inc., Fremont, USA. Research focussed on improvements of the optical cavity system only, since changes to this subsystem can be implemented much more easily and therefore faster. First, current limitations of the enhancement cavity are discussed before the envisioned solution is presented and evaluated.

8.1 Implications of the Current Interaction Geometry and Pressure Correction Although, the CLS provides synchrotron-like X-ray radiation, its similar X-ray source size at a wider X-ray emission angle compared to synchrotrons in combination with its lower X-ray flux inhibits applications which require a high spatial coherence of the X-rays. Reducing the X-ray source size benefits the beam’s coherence as well as luminosity which are both desirable. To do so the laser focus diameter at the CLS could be decreased. Consequently, the laser’s diameter on the optics increases. On the one hand, this reduces the power density on the optics, but on the other hand the finite extent of the reflecting surface limits the achievable minimal focal waist under high-finesse operation. The fundamental limit for a high-finesse cavity consisting of two 2 inch mirrors is well below the current focal spot size or Rayleigh range of ∼9 cm, cf. Sect. 8.6. This fundamental limit is further increased by the constraint that the laser spot must not be centred on the exit optic, the OC90, cf. Fig. 7.1. The central area of OC90 with a diameter of 6 mm is only 200 µm thin. This is much less than the bulk of the optics and constitutes the aperture for the X-ray beam. Residual absorption at this © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_8

177

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Fig. 8.1 Approach for compensation of mirror deformation. a is a schematic of the desired interaction geometry at the CLS. b displays the mirror surface deviations of an exit mirror of the CLS. The blue lines display the locations of the two lineplots below the mirror surface map. The mirror surface exhibits a dip in the central region where the mirror was back-thinned in order to act as an X-ray exit window. c is a schematic of a piece of hardware employed to correct the dip by introducing a background pressure

part results in a stronger heat deformation of the thinned area increasing diffraction losses. Moreover, this thinned area is subject to stress deformation from the coating process of the optic. Figure 8.1b displays a deformation map of one OC90-optic after coating, which reveals a dip in the region of its back-thinned area. Therefore, instead of the ideal geometry for inverse Compton scattering with a counter-propagating laser (sketched in Fig. 8.6a), the laser is angled at ∼5 mrad. This way the laser spot is located outside the region containing the dip. In order to overcome the drawbacks arising from this off-axis geometry, a solution compensating mirror deformations in the central region of the mirror has to be developed. Since the silicon crystal is thin like a membrane, the dip could be counterbalanced by increasing the background pressure behind the mirror by a small amount compared to the ultra-high vacuum inside the enhancement cavity. Controlling the background pressure behind the mirrors requires adjustments to the current mirror mount. The mirror mount has to be soldered to the optic (ideally at temperatures as low as possible) in order to create an air-tight connection to a “pressure chamber” which contains the correct background pressure to cancel the deformation. Once the laser is placed onto the thinned area, absorption will create an additional deformation in the positive direction, i.e. towards the inside of the resonator. Consequently, the background pressure has to be adjusted, i.e. reduced, because otherwise the dip

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would be overcompensated. Such a dynamic pressure control system could consist of two regulators connected to the pressure chamber, one for pressure reduction and one for pressure increase which are controlled by a feedback system. A basic schematic for such a piece of hardware is depicted in Fig. 8.6c. Although this system enables the reduction of the Rayleigh length, other factors than the laser beam might limit the Rayleigh range for optimal inverse Compton scattering. If the laser waist becomes smaller than the one of the electrons, the X-ray source size is laser beam size dominated counteracting the luminosity gain due to a smaller X-ray source size as not all electrons take part in the interaction any more. If this situation occurs the storage ring has to be upgraded to support smaller foci as well. Moreover, a shorter Rayleigh length reduces the acceptable timing jitter between the laser and the electron beam, since interaction should take place within the Rayleigh range for efficient X-ray generation. These additional constraints have to be kept in mind for the implementation of a pressure correction system for Rayleigh range reduction.

8.2 Pressure Correction of a Mirror One necessary ingredient for the analysis of the effects of the pressure deformation on stored power and laser beam waist is a measurement of the mirror surface at different background pressures. These measurements were conducted by Bruce Borchers, Lyncean Technologies Inc., at Coastline Optics (Camarillo, CA, USA) with a Dynafiz laser interferometer (Zygo Cooperation, Middleton, CT, USA). A transmission sphere was implemented in the beam path to correct for most of the mirror’s radius of curvature. The resulting data was further corrected for measurement artefacts as described in Fig. 8.2, shown for a 1 in mirror here. First, a circle was fitted to the raw mirror map, Fig. 8.2a. Its center coordinates together with its radius were used to crop the original measurement to the actual size of the mirror, displayed in Fig. 8.2b. Additionally, this procedure centred the optic in the map, which guarantees that the laser beam is centred on the deformation dip in the simulations. In the next step, the Cartesian height map was transformed into polar coordinates, Fig. 8.2d. Afterwards, a Zernike polynomial containing, piston, tip/tilt, astigmatism and skew astigmatism was fitted to the region ranging from 1.3 × rthinned to rclear aperture , which was manually chosen, e.g. 23 mm for a 2 in (diameter) optic and 10.5 mm for a 1 in (diameter) one. The polynomial was chosen such that it corrects for the uncompensated radius of curvature and parameters that can be compensated at the cavity of the CLS. Including the dip of the thinned region in the fitting procedure would heavily affect the fit’s accuracy for overall surface features. This is also true for the region outside the clear aperture, where the coating of the mirror is not uniform anymore. The obtained Zernike polynom is displayed in Fig. 8.2e. Figure 8.2c demonstrates the accurate representation of the overall surface parameters by the Zernike polynomial. Therefore, subtracting the Zernike polynomial, Fig. 8.2e, from the cropped raw mirror map, Fig. 8.2d, generates a map containing only higher order

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8 Development of a Deformable Exit Optic

Fig. 8.2 Pre-processing of mirror phase maps measured with a laser interferometer. a is the measured phase map. b is a cropped to the mirror size determined by fitting a circle (blue in a) to the mirrors size. d represents b in polar coordinates. e displays the Zernike polynom of order two (piston, tilt, curvature, astigmatism, skew astigmatism) fitted to the surface of d. c compares the Zernike surface with the measured one for the radial line at 180◦ . A map of the mirror’s residual surface error is displayed in f, which is the difference of d and e. In g values of f outside the mirror’s clear aperture are set to zero. h displays the data of g after a low pass filtering with a Savitzky-Golay moving average filter in both, the radial as well as azimuthal, direction. k and l are the Cartesian representations of g and h, respectively. This eliminates diffraction artefacts arising from particles like dust on optics, e.g. the compensation lens, or in the optical path of the interferometer as displayed in m. i is a comparison of the mirror profile corrected applying the Zernike polynomial only with the one which has undergone low-pass frequency filtering additionally

8.2 Pressure Correction of a Mirror

181

surface deviations (Zernike order 3), Fig. 8.2f, which cannot be compensated at the moment. The distorted area outside the clear aperture of the mirror is set to zero, which results in Fig. 8.2g. Since some diffraction rings are visible in the map of the mirror deviation, cf. Fig. 8.2f, which arise from diffraction of dust particles, the mirror map of Fig. 8.2g has been low pass filtered with a Savitzky-Golay moving average filter of 4th order and 51 points in radial direction and 31 points in the angular direction. The boundary conditions for the radial fit were the radial profiles shifted by π at r = 0 and at the radius of the clear aperture, the profile was mirrored. For the angular filtering, the boundary conditions were chosen to fulfil the 0/2π continuity. The result is displayed in Fig. 8.2h. The final mirror deviation map is obtained by transformation of Fig. 8.2h back into Cartesian coordinates, shown in Fig. 8.2l. Figure 8.2k is the Cartesian representation of Fig. 8.2g. The difference map, Fig. 8.2m, of Fig. 8.2g, l demonstrates that the low pass filter removes mainly interference fringes, which validates the earlier statement. The effect of the SavitzkyGolay filter is nicely visible in Fig. 8.2i, where cuts along the horizontal and vertical axis of Fig. 8.2k are compared to the corresponding ones of Fig. 8.2l. The mirrors installed at the enhancement cavity of the CLS have a diameter of 2 in. Therefore, the subsequent simulations were performed with a 2 in OC90-optic, whose intrinsic surface deformation is depicted in Fig. 8.3. The remaining surface exhibits a slight coma and a central dip of ∼35 nm depth, whose width is slightly larger in diameter than the nominal diameter of the thinned area which is 200 µm thick. A deformation of ∼35 nm for exit windows is rather low, typically the deformation is around 200 nm. Consequently, this optic was chosen for the construction

Fig. 8.3 Mirror deformation without background pressure. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

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8 Development of a Deformable Exit Optic

Fig. 8.4 Mirror deformation at a background pressure of 0.50 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

of the prototype assembly, discussed in Sect. 8.8.3, and in turn also for the simulation. For pressure correction of this mirror, an assembly like the one depicted schematically in Fig. 8.1c was constructed. Increasing the background pressure to 0.50 psi significantly reduces the peak-to-valley deformation to well below 10 nm both in the horizontal and vertical direction, cf. Fig. 8.4, which corresponds to less than λ/100 at a laser wavelength of 1064 nm. If the background pressure is further raised to 0.55 psi, only marginal changes are visible in Fig. 8.5, mainly a small bulge is emerging in the thinned region, observed best by comparing Figs. 8.5b, c to 8.4b, c. Stronger inflation to 0.60 psi barely affects the peak-to-valley deformation at first, shown in Figs. 8.6 and 8.7. At 1.0 psi, a notable bulge emerges with the same amplitude as the dip, cf. Fig. 8.8. At a background pressure of 2.0 psi, the thinned area is arching upward by more than 100 nm, cf. Fig. 8.9. Consequently, the region with lowest losses should be expected around 0.55 psi, where the initial surface error is compensated best.

8.3 Steady State Model of the Enhancement Cavity The main goal of the simulation is to determine diffraction losses arising from the deformations of the OC90-optic. This requires only knowledge about the steadystate optical field circulating inside a resonator. Consequently, the physics can be modelled using Fresnel-propagation of the coherent laser light. Figure 8.10 pictures

8.3 Steady State Model of the Enhancement Cavity

183

Fig. 8.5 Mirror deformation at a background pressure of 0.55 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

Fig. 8.6 Mirror deformation at a background pressure of 0.57 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

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8 Development of a Deformable Exit Optic

Fig. 8.7 Mirror deformation at a background pressure of 0.60 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

Fig. 8.8 Mirror deformation at a background pressure of 1.0 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

8.3 Steady State Model of the Enhancement Cavity

185

Fig. 8.9 Mirror deformation at a background pressure of 2.0 psi. a displays the residual surface deformation after the processing described in Fig. 8.2. b is a line profile through the center of the mirror along the vertical direction and c is one along the horizontal direction

Fig. 8.10 Steady-state model of the enhancement cavity. The length of the cavity leg containing the interaction point results in a Rayleigh range of 9 cm, identical to the current Rayleigh range at the enhancement cavity at the MuCLS

the individual steps in the calculation of one roundtrip of the laser beam inside the cavity. The general model of the steady-state calculation is described first, before the determination of the initial cavity eigenmode is described. To begin with, a laser beam is defined inside the cavity at position 1 (Fig. 8.10). This beam is expanded as a sum of basis functions, which corresponds to a Fourier transformation of the electric field into a basis formed by plane waves aei(ωt−kr) . Since steady-state results are desired, the time-dependent part can be neglected. As a reminder, the expansion in terms of the plane wave basis functions is briefly demonstrated. The general form of the electric field decomposed in this basis is

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8 Development of a Deformable Exit Optic

∫ E in (r) =

Eˆin (k)e−ikr dk,

(8.1)

which takes form of the inverse Fourier-transform of the electric field. For the twodimensional electric field considered here, the electric field at the first mirror is given by: ∫∫ E in (x, y, z = 0) = Eˆin (k x , k y )e−i(kx x+k y y) dk x dk y ∫∫ ˆ E in (k x , k y ) = E in (x, y, z = 0)ei(kx x+k y y) d xd y.

(8.2) (8.3)

From Sect. 3.1.4 it is known, that free space propagation in real space is equivalent to phase multiplication in frequency domain. Moreover, propagation of the laser beam in the enhancement cavity fulfils the paraxial approximation, since the transverse dimension is ∼10−3 times the longitudinal one. Therefore, the propagated beam is Eˆ prop (k x , k y , k z ) = Eˆin (k x , k y )e−ikz d = Eˆin (kx , ky )e−i(|k|−

kx2 +ky2 2|k|

)d

,

(8.4)

where d is the distance to the next mirror. Reflection at a curved surface introduces an optical path length difference which changes the laser beam’s wavefront curvature. Even real flat optics introduce wavefront distortions to some extent owing to their surface figure error. As this process is modelled easiest in real space, the electric field has to be transformed into real space again applying the inverse Fourier transformation, before the laser beam is reflected at a mirror surface. The optical path length difference introduced by a curved mirror OPM(x, y) is   √   √ OPM = 2s = 2 RofC − RofC2 − r 2 = 2 RofC − RofC2 − (x + y)2 .

(8.5) s is the mirror’s sagitta which depends on the radius of curvature RofC and the radial distance r to the central axis of the mirror, cf. Fig. 8.11. Additionally, the surface can be subject to distortions, like surface roughness, astigmatism, coma or other fabrication errors. In this case, a measured map of the surface errors, discussed in Sect. 8.2 contains all the relevant surface errors for the OC90-optic. This additional path length difference caused by distortions OPD(x, y) can be simply added. The total optical path length difference OPL(x, y) introduced by an optical element is OPL = OPM + OPD.

(8.6)

Considering the reflectivity r of the mirror as well as its finite clear aperture CA, the reflected electric field calculates as

8.3 Steady State Model of the Enhancement Cavity

187

Fig. 8.11 Definition of the sagitta s of a curved surface. RofC is the mirror’s radius of curvature and r is the transverse distance to the optical axis

E ref (x, y, d) = r E prop (x, y, d)e−i|k|OPL(x,y) CA.

(8.7)

All the intensity that is scattered outside the clear aperture amounts to the diffraction losses in a real cavity. After reflection at one mirror surface, the electric field is Fourier-transformed and propagated to the next surface, inverse Fourier-transformed and reflected. These steps are repeated until one round-trip of the cavity is completed. After the last reflection at the input coupler, step 9 in Fig. 8.10, the electric field transmitted through the input coupling mirror is added and the next roundtrip is computed. The simulation converges once the power stored in the optical cavity does not change any more from round-trip to round-trip, i.e. numerical changes fall below a threshold. Ideally, the electric field E in should be matched to the eigenmode of the optical cavity. A desired eigenmode is the Hermite-Gaussian HG00 -mode, i.e. a Gaussianshaped laser beam profile. This eigenmode can be calculated by Gaussian beam propagation with ABCD-matrices, cf. Sect. 3.2.3. Remember that reflection by a mirror Mi with a radius of curvature R, propagation of the beam Pi over a distance d and refraction of the beam by a lens Li with a focal length f is described by the following three matrices: Mi =

1 0 −2/R 1



Pi =

1d 01



Li =

1 0 . −1/ f 1

(8.8)

With these matrices, the ABCD-matrix of the enhancement cavity EC is EC = M1 PIP M4 PD M3 PBL M2 PD .

(8.9)

Indices refer to the ones in Fig. 8.10. The parameters of the Gaussian eigenmode of the cavity can be extracted from the ABCD-matrix EC (Siegman 1986, p. 820)

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8 Development of a Deformable Exit Optic

Rwavefront =

2B D−A

w=√

1 √ π 1 − ( A+D )2 λ|B| 2

1 λ 1 = −i Rwavefront πw 2 qˆ

(8.10) (8.11)

(8.12)

Rwavefront is the wavefront’s radius of curvature at the position of M1 , w is the beam waist, and qˆ is the complex q-parameter of a Gaussian beam (see Sect. 3.2.3) and λ is the laser beam’s wavelength. Note that in the special case considered here qˆ = q˜ because the refractive index is one inside as well as outside of the cavity. In order to get the mode-matched beam parameters of the laser beam if it is defined outside the optical cavity, the effect of the input coupler mirror M1 has to be considered, which acts as a concave lens on the transmitted beam. In front of the lens, the q-parameter takes the form: qˆ ' =

Aqˆ + B , C qˆ + D

(8.13)

where ABCD are the elements of Li . This beam is transmitted into the cavity and forms the mode-matched beam inside the cavity in the beginning as well as the beam that is added after each roundtrip in the simulation. The simulation package and some considerations on the simulation parameters are discussed in the next section.

8.4 Simulation Tool: Capabilities and Limitation The core of the simulation package responsible for the simulation of the optical cavity is a Matlab-based optical fast Fourier transform (FFT) code, called Oscar (Degallaix 2010). For the simulations performed in the framework of this thesis, version 3.16 was employed. This package was adapted for the pre-processing of the Zygo measurements described in Sect. 8.2. Beam preparation, e.g. the calculation of a matched input mode, was performed in Python using the standard libraries numpy and scipy (Jones et al. 2001). These parameters are accepted by a Matlab function which is called from Python via the Matlab-Python API built into Matlab for all versions since Matlab 2015. This Matlab function performs the cavity calculation and returns the results back to Python which is responsible for post-processing of the results. It is designed to be very flexible in order to allow the variation of a lot of different parameters and observe their effect on the cavity. Figure 8.12 provides an overview of the scan parameters and the returned results. Before the simulation tool can be applied extensively for analysis of the effects of the pressure compensation of a mirror, artefacts arising from improper parameter choice have to be investigated first, especially grid-dependent effects should be mit-

8.4 Simulation Tool: Capabilities and Limitation

189

Fig. 8.12 Capabilities of the optical cavity simulation package

igated. If the grid is small, the high frequency cut-off might be too low to support higher order modes propagating in the cavity thereby creating artificial losses. In case of the enhancement cavity of the CLS, the geometry of the cavity requires 2 in mirrors to support the laser beam size at the position of the mirrors for the desired short Rayleigh length. Consequently, the transverse size for the simulation is fixed at 51 mm. In order to explore grid-dependent effects, simulations were performed with a grid size of 512 × 512 points, 1024 × 1024 points and 2048 × 2048 points. Figure 8.13a displays the dependency of the grid size on the diffraction losses for different mirror background pressures. Doubling of the initial grid size of 512 × 512 points significantly reduces the observed diffraction losses. If the grid size is further increased to 2048 × 2048 points, diffraction losses barely change for a pressure compensated OC90-optic, i.e. around 0.5 psi and only at high pressure, where the deformation is already significantly overcompensated a noticeable change of diffraction losses can be detected. The main reason for the grid size dependent reduction of diffraction losses by larger grids is the increase in the high frequency cut-off, as demonstrated in Fig. 8.14, since the laser field at the focus of the cavity is the optical Fourier-transform of the circulating laser. Although a grid of 2048 × 2048 points is not sufficient to support all frequencies in extreme cases, this is acceptable because the laser should ideally maintain its Gaussian shape and consequently follow a Gaussian high-frequency drop off in the relevant cases. Moreover, simulating larger grids than 2048 × 2048 points is not feasible with the employed processing system. In addition to diffraction losses, the coupling of the external laser beam to the one circulating inside the enhancement cavity crucially influences the achievable stored power. Imperfections of the mirror’s surface as well as degenerate higher-order mode resonances detrimentally affect the circulating mode. Deviations from the HG00 mode grow with increasing finesse, because the stronger build-up inside the cavity

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8 Development of a Deformable Exit Optic

Fig. 8.13 Limitations of the simulation. a displays the dependence of diffraction losses on the grid size. b shows the dependence of the calculated stored power on the finesse of the cavity. The latter data is from simulations with a grid size of 1024 × 1024 points due to the long duration of the high-finesse simulation. The diffraction losses in both, low and high finesse simulations, are very similar. Therefore, the stored power of a mode-matched laser beam calculated from the losses agrees very well. However, if the incoming laser beam is mode-matched to the HG00 mode, while the stable one is a higher-order mode, the power stored inside the cavity deviates significantly from the one expected due to a mode mismatch between the external laser beam and the one circulating in the cavity. This effect increases with higher finesse because the cavity’s eigenmode prevails over the mode transmitted into the cavity more and more. c is the mode inside the cavity at high finesse and a background pressure of 0.60 psi and d is the one at low finesse at the same background pressure. A stronger high-order mode content is observed at high finesse compared to the one at low finesse. Moreover, the vertical stripes in both images are an indication of undersampling at a grid size of 1024 × 1024 points

enhances their impact. Figure 8.13b, displays diffraction losses, stored power, and the stored power calculated from the diffraction losses for a mode-matched beam for two simulations performed at a Rayleigh-range of 12.5 cm where mode degeneracy occurs, cf. Fig. 7.10, one performed at a high finesse of ∼44000 and one at a low finesse of ∼410. The grid size has been chosen to be 1024 × 1024 points in order to contain the duration of the high-finesse simulation to a reasonable time. Diffraction losses obtained from both simulations are very similar, except for the resonance with

8.4 Simulation Tool: Capabilities and Limitation

191

Fig. 8.14 Grid dependence of the electric field at the focus position for a background pressure of 2.0 psi. a is obtained for a grid size of 1024 × 1024 points and b for one of 2048 × 2048 points. a is clearly undersampled resulting in a early high frequency cut-off, which does not support sufficiently high frequencies. Although b still does not support all frequencies, the situation is much improved

Fig. 8.15 High-order mode-content visibility. a is the normalised circulating intensity on OC50. The perception of airy diagonal stripes could indicate the presence of some higher-order modecontent. In the normalised logarithmic intensity image, b, this contribution is well recognised

the degenerated higher-order mode at 0.60 psi where the losses double. Thus this is true for the stored power in the ideal case calculated from these values. However, the actual stored power deviates strongly from the ideal case. The reason for this behaviour is the aforementioned coupling. Figure 8.13c shows the circulating mode obtained by the high-finesse simulation at a background pressure of 0.60 psi, where the stored power drops by hundreds of kW, and Fig. 8.13d the mode obtained by the low-finesse simulation at the same background pressure. Normalised logarithmic intensities are displayed, because the onset of the higher-order modes becomes visible also in the low finesse case, which would otherwise remain unnoticeable. As a demonstration, Fig. 8.15 displays the normalised intensity for the background pressure of 0.60 psi in direct comparison to the normalised logarithmic one. Even in the high finesse case, the higher-order mode-content is barely visible in the normalised intensities, while it can be recognised clearly in the logarithmic illustration. The mode in the high-finesse case exhibits much stronger higher-order modes com-

192

8 Development of a Deformable Exit Optic

pared to the simulation at the low finesse case, although the higher-order modes are already emerging there, too. At high finesse, the circulating mode is dominated by the cavity eigenmode. Coupling between the fundamental mode and high-order modes increases with the finesse for frequency degenerate modes (Bullington et al. 2008). This explains the stronger higher-order mode content in the high-finesse case. Furthermore, the coupling of the external laser to this skewed mode reduces in the high-finesse case which in turn decreases the power stored inside the cavity. Simulations at low finesse take about eight hours (at a grid size of 2048 × 2048 points), while at high finesse more than four days are required (at a grid size of 1024 × 1024 points). Therefore, most simulations are performed for the low-finesse case at the highest gird resolution in the following. As a consequence, the circulating modes in combination with the map of the high-order-mode resonances, depicted in Fig. 7.10 for the ideal resonator, deserve special attention in the following discussions. Otherwise it cannot be reliably judged whether the stored power estimated from diffraction losses and mirror transmissions will be achievable with a mode-matched external HG00 -mode in reality.

8.5 Expected Effect of Pressure Correction on Diffraction Losses Following the discussion of the simulation and its limitations, this section focusses on the effect of the pressure correction on diffraction losses and mode shape. The aim is to evaluate whether the diffraction losses for a compensated dip remain similar to the ones achieved by placing the laser beam on the side of the optic, i.e. below 25 ppm for the OC90-optic.

8.5.1 Pressure Correction of an Existing Optic A detailed explanation of general aspects of the simulation have been provided in the previous two sections. Therefore, only the exact parameters employed in the simulation are provided in the following. The laser is defined outside of the optical cavity and assumed to provide a power of 25 W. Its waist is 180 µm and the laser is defined at a position of −77 cm in front of its focus, just on the outside of OC50. OC50 and OC90 have a radius of curvature of 1.172 m, while the one of OC70 and OC80 is 2.514 m. Astigmatism is corrected on OC70 over a diameter of 1 cm with a Zernike-coefficient of 1.4 × 10−7 . Skew astigmatism is accounted for on OC80 over a diameter of 1 cm with a Zernike-coefficient of 1.4 × 10−14 . The mirror transmission for OC50 was set to 124 × 10−4 and all the other mirrors to 1 × 10−4 . The clear aperture was set to 46 mm and the total transverse dimension to 51 mm with a grid of 2048 × 2048 points. In order to evaluate solely the effect of the OC90-optic, all

8.5 Expected Effect of Pressure Correction on Diffraction Losses

193

Fig. 8.16 Effects of pressure correction on an optic with a thinned diameter of 6 mm. The simulation was performed for the low finesse case. a displays the diffraction losses as well as the stored power inside the optical cavity for the low finesse case as well as the one expected in an ideal high finesse case. b is the power enhancement corresponding to a for ideal matching. The waist and its corresponding Rayleigh length are depicted in c for the parameters obtained from the simulation, i.e. including surface figure errors, and for the ideal Gaussian beam calculated with ABCD-matrices. d indicates the simulated effect on the power transmitted through OC90 (if it would be transparent) and the power reflected off of OC50

other optics were assumed to be ideal and only real surface figure error maps are included for OC90. The straight section containing the interaction point had a length of 2.52 m, the straight back-leg section one of 2.05 m, while the diagonals had a length of 2.33 m, which results in an angle of incidence of 6.0◦ . The result of the simulation is summarised in Fig. 8.16. Diffraction losses start to rise above 0.6 psi and become significant around 1.0 psi heavily diminishing the stored power (Fig. 8.16a) and power enhancement (Fig. 8.16b), respectively. Nevertheless, results of the high finesse simulation depicted in Fig. 8.13b indicate a drop of the stored power already at 0.6 psi background pressure due to a miss-match of the incoming beam with a degenerate cavity-mode resonant at this Rayleigh length, cf. Figs. 8.16c and 7.10. Small angle scattering couples the co-resonant modes (Klaassen

194

8 Development of a Deformable Exit Optic

Fig. 8.17 Effect of pressure correction on the circulating modes of an optic with a thinned diameter of 6 mm. The simulation was performed for the low finesse case. Subfigures are explained by their titles

et al. 2005), which is stronger at higher finesse (Bullington et al. 2008). This mode, is also visible in this simulation in Fig. 8.17d, although containing less intensity due to the lower finesse of the cavity in this simulation. For a background pressure of 0.6 psi and 2.0 psi, the amount of light scattered into lossier co-resonant high-order modes increases, while the decrease in stored power at 1.0 psi can be related to the mode-mismatch mainly (and due to increased scattering reflected in the diffraction losses), since the smooth curvature of the dip can be approximated well by a local reduction of the radius of curvature of the mirror. This approximation holds true for longer Rayleigh ranges, such as the 12.3 cm used in this simulation, where a good portion of the intensity is confined to the center of the mirror containing the dip. Figure 8.16c supports this conclusion, as the Rayleigh range increases continuously with the background pressure which agrees with the evolution of the dip profile depicted in Figs. 8.3, 8.4, 8.5, 8.6, 8.7, 8.8 and 8.9. Additionally, the simulated waist is very close to the nominal ideal one calculated from ABCD-matrices at 0.50 psi, where the mirror profile is basically flat. As a matter of fact, the external laser beam was matched to a Rayleigh length shorter than the compensated optic, alleviating the match between the circulating beam and the external one for higher pressures. The latter is underpinned by Fig. 8.16d, since the reflected power increases with increasing background pressure resulting in a lower stored power in the simulation (Fig. 8.16a, blue line) which is reflected in the reduced transmitted power in Fig. 8.16d, respectively. The sum of these effects explain the good performance without any background pressure as well. Albeit its slightly higher diffraction losses of 7 ppm, the background pressure at which the deformation seems to be compensated best is 0.50 psi due to its flat mirror surface map and decent circulating mode. Moreover, at this background pressure, the

8.5 Expected Effect of Pressure Correction on Diffraction Losses

195

Fig. 8.18 Simulated laser fields for the compensated exit optic at a background pressure of 0.50 psi. a is the intensity of the laser field defined outside the cavity. b The intensity profile of the circulating laser pulse on the input coupler (OC50). c is the intensity distribution at the focus at the interaction point. d The intensity reflected off of OC50, which is the sum of the transmitted field of b and the reflected one of a. e shows the intensity transmitted through OC90. f the map of the mirror deformation at 0.50 psi

aforementioned mode-degeneracy occurring at 0.60 psi is lifted. Figure 8.18 displays the laser profiles of the incoming laser beam, the mode on OC50 and at the interaction point as well as the mode reflected off of OC50 which is the sum of the reflected incoming beam and the circulating one transmitted through OC50 and the intensity transmitted through OC90. Both, the reflected as well as transmitted mode exhibit little content from higher-order modes. In conclusion, the diffraction losses arising from a compensated OC90-optic are below 10 ppm for long Rayleigh lengths, fulfilling the initial criterion of 2.0 cm, corresponding to a length of the IP-leg of 2.567 m.

8.6.2 Limitation on the Rayleigh Length Imposed by a Infra-red Laser Since the enhancement cavity geometry is intrinsically limited to Rayleigh lengths >2.0 cm, the simulation has been constrained to this region.

8.6.2.1

Rayleigh Length Reduction with an Existing Exit Optic Solution

Figure 8.23 summarises the results for an exit optic containing a thinned area of 6 mm in diameter and the current Nd:YAG infra-red laser operating at a wavelength of 1064 nm. Diffraction losses are relatively constant above a Rayleigh length of 7.5 cm and gradually increase down to 5 cm, cf. Fig. 8.23a. At shorter Rayleigh length, the diffraction loss rises steeply from 23 ppm at a Rayleigh length of 5.1 cm to 240 ppm at one of 3.2 cm. This is consistent with the large extent and/or strong high-order modecontent of the corresponding cavity modes, which are displayed in Fig. 8.24m–p. Furthermore, intense non-Gaussian cavity modes are apparent at Rayleigh lengths of 6.0 cm, 6.7 cm and 7.8 cm, depicted in Fig. 8.24i, h and f, respectively, which closely agree with the degenerate ones predicted in Fig. 7.10 for an ideal resonator. The stored power at these Rayleigh lengths drops notably in the low finesse simulation, depicted as the blue curve in Fig. 8.23a. This effect is not represented in the stored power for the high finesse case, which is shown as the green curve in Fig. 8.23a, as this simple scaling is based solely on the mirror- and diffraction losses. Therefore,

8.6 Effect of a Rayleigh-Range Reduction

201

Fig. 8.22 Effects of Rayleigh length reduction for an ideal enhancement cavity. The simulation was performed for the low finesse case. a displays the diffraction losses as well as the stored power inside the optical cavity for the low finesse case as well as the one expected in an ideal high finesse case. b is the power enhancement corresponding to a for ideal matching. The waist and its corresponding Rayleigh length are depicted in c for the parameters obtained from the simulation and for the ideal Gaussian beam calculated with ABCD-matrices. d indicates the simulated effect on the power transmitted through OC90 (if it would be transparent) and the power reflected off of OC50

it cannot account for low coupling of the external laser beam to the one circulating inside the cavity in these cases. This is well visible in Fig. 8.23d as the increase of the power reflected off of OC50 and corresponding decrease of the power transmitted through OC90. Appearance of these effects already at low finesse indicates that high power will not be achievable with a high finesse cavity at these particular Rayleigh lengths. In conclusion, shorter Rayleigh length than 5 cm seem to be infeasible for this type of optic. Moreover, frequency degenerate high-order modes appear at certain Rayleigh length only. This degeneracy can be lifted adjusting the IP-leg length by a couple of millimetres, as demonstrated in Fig. 8.23a, c and the modes corresponding to slightly shifted Rayleigh length in Fig. 8.24. Since a change of the mirror distance

202

8 Development of a Deformable Exit Optic

Fig. 8.23 Effects of Rayleigh length reduction on an optic with a thinned diameter of 6 mm and infra-red laser. The simulation was performed for the low finesse case. a displays the diffraction losses as well as the stored power inside the optical cavity for the low finesse case as well as the one expected in an ideal high finesse case. b is the power enhancement corresponding to a for ideal matching. The waist and its corresponding Rayleigh length are depicted in c for the parameters obtained from the simulation, i.e. including surface figure errors, and for the ideal Gaussian beam calculated with ABCD-matrices. d indicates the simulated effect on the power transmitted through OC90 (if it would be transparent) and the power reflected off of OC50

affects the round trip Gouy phase, this effect can be achieved by adjusting either the background pressure of the optic slightly or the length of the IP-leg in practise. Thus, this is not expected to pose an issue for daily operation. A Rayleigh length of 5 cm is longer than the one required for matching the laser waist at the interaction point to the one of the electron beam for the CLS installed at the MuCLS, see Sect. 8.5.3. Below 6 cm Rayleigh length, the 4σ-laser beam diameter at the exit optic also grows larger than the diameter of the thinned area of the mirror. Taking into account the preceding discussion, a pressure-compensated exit optic with 6 mm thinned area in diameter should fulfil the demands of diffraction losses 300 kW, the enhancement cavity is more sensitive to subtle changes, which may inhibit jumps in X-ray energy by 2.5 keV for long imaging experiments, like computed tomography. Accordingly, the 26.6 keV-configuration appears to be the most promising candidate for successful implementation of electron beam-based KES at the silver K-edge at the CLS at the time of writing.

1

The Covid-19 pandemic prevented a necessary service by Lyncean Technologies Inc. at the time these experiments were conducted.

238

9 Fast X-ray Energy Switching

Fig. 9.4 Different spectra tested for KES-subtraction imaging employing fast energy switching at the silver K-edge. The spectrum with a peak energy of 25 keV generates the low energy image, while an increasing fraction of the spectrum lies above the silver K-edge for the other three spectra. Spectra acquired with a pinhole

9.3.2.2

Demonstration of Electron Beam-Based Rapid Energy Switching

Figure 9.5 displays X-ray parameters when cycling between this 26.6 keV- configuration and the 25 keV-configuration. Each configuration was kept for 15 s. A single switch between the two X-ray energies takes about 2 s. The X-ray flux for a period of four minutes is displayed in Fig. 9.5a together with the RF-power that was delivered by the klystron K2 in order to demonstrate the switch between the energies and indicate the X-ray energy configurations. The X-ray flux was recorded with the beam position monitor presented in the next Chap. 10. It is slightly higher at the 26.6 keVconfiguration compared to the 25.0 keV-one. Moreover, shifts of the horizontal X-ray position and vertical X-ray source size can be observed in Fig. 9.5b which occur at each switch of the X-ray energy. Since the laser beam orbit is not affected by changes to the electron beam, those shifts have to be correlated with slightly different electron beam positions, -focussing and -direction of motion across the interaction region with the laser beam. In turn, this changes the spatial overlap of the beams and thereby the luminosity. As a consequence the flux for both configurations may vary depending on the initial optimisation of the overlap. Another contribution arises from the fact that the slightly increased γ -factor at the high energy X-ray configuration reduces the 1/γ -cone angle of the X-rays. Consequently, a slightly increased fraction of the total flux is extracted through the aperture with a fixed opening angle. The latter is a pure physical effect and cannot be avoided, while changes to the source size, which may be undesired, e.g. for techniques that rely on spatial coherence, can be avoided by properly optimising the electron beam orbit inside the storage ring. However, slight changes to the source size and position are irrelevant for pure absorptionbased imaging, such as the main application of fast X-ray energy switching at the

9.3 Rapid Electron Energy Switching

239

Fig. 9.5 Demonstration of fast energy switching. a Photon flux for the two different X-ray configurations with peak energies of 25.0 keV and 26.6 keV, displayed in Fig. 9.4. The power delivered by the second klystron, K2, is adjusted to switch between these two X-ray settings. One energy change takes about 2 s and each X-ray configuration was kept for 15 s. b X-ray source parameters during the period depicted in a. The X-ray flux varies slightly from configuration to configuration. The vertical source size and horizontal source position of the electron beam vary slightly. Since the laser beam position is constant, this has to be correlated with different electron beam orbits. As this affects the overlap between the photon and electron beam, the X-ray flux varies slightly from configuration to configuration as well, cf. a. The irregular spacing of data points in the RF-power, e.g. around 100 s, is due to the archiving procedure which saves data for this particular parameter only upon changes of its value

MuCLS, namely KES-imaging. Therefore, changes of the source size and source position were not obliterated for the following KES-demonstration experiment.

9.3.2.3

KES-Imaging with an Angiography Phantom

The feasibility of this approach for KES-imaging is going to be demonstrated with a home-made angiography phantom. It consists of three hoses attached to a 3.2 cm thick aluminium sheet. This construction was further supported by an acrylic glass in order improve the stability of the system. Consequently, three different background regions exist in Fig. 9.6: The one on the left is made up of the aluminium and the acrylic glass, the central one by acrylic glass only and the right one by air only. The three hoses (1 mm in diameter) were filled with different silver-nitride solutions with concentrations of 400 mg/ml, 150 mg/ml and 100 mg/ml, top to bottom in Fig. 9.6. The high- and low energy images were recorded with 0.3 s acquisition time using electron beam-based rapid X-ray energy switching. Additionally, the corresponding dark-current- and reference images were collected. 10 images were taken at each energy and averaged. Dark-current and reference correction was performed as well. Figure 9.6a displays the low energy image, Fig. 9.6b the high-energy one and Fig. 9.6c is the KES-image. An improved energy-scaling that incorporates Compton scat-

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Fig. 9.6 Demonstration of KES employing fast energy switching at the silver K-edge with an angiography phantom. It consists of three hoses containing a silver-nitride solution of 400 mg/ml, 150 mg/ml and 100 mg/ml top to bottom. The three background regions are air, air combined with an acrylic glass substrate and the acrylic glass substrate combined with a 3.2 cm thick aluminium sheet right to left. The aluminium plate simulates the presence of bone. a Image acquired at the low X-ray energy configuration with a peak energy of 25.0 keV. b The high energy image which was recorded with a peak X-ray energy of 26.6 keV. Those spectra are displayed in Fig. 9.4. c K-edge subtraction image. The factor for energy-scaling in the subtraction is 2.6. The optimisation of this value is described in Kokhanovskyi (2021)

tering in addition to photoelectric absorption was employed for the calculation of the KES-image. Its derivation can be found in Kokhanovskyi (2021). For the two X-ray energies used in this experiment, the scaling factor that minimises the contribution from aluminium is 2.6 (Kokhanovskyi 2021), i.e. slightly reduced from the approximate E −3 -scaling of the pure photoelectric effect (Als-Nielsen and McMorrow 2011, p. 23). Indeed, Δμd vanishes for aluminium in the KES-image, Fig. 9.6c. However, acrylic glass as well as air contain a slightly negative Δμd-value since their ratio of photoelectric absorption to Compton-scattering is different to (in these cases higher than) the one of aluminium for the same X-ray energy. Nevertheless, this allows for straight forward threshold based segmentation of the contrasted regions or separation of bones from contrasted areas, respectively. Consequently, this proof-of-principle experiment demonstrates that electron beambased rapid X-ray energy switching delivers a great discrimination of contrasted regions from heavily absorbing structures, such as aluminium mimicking bone in this experiment. Typically, the KES-image contrast using this technique surpasses the one of filter-based KES-imaging (Kokhanovskyi 2021). Since the feasibility of electron beam-based rapid energy switching is the scope of this section, the interested

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241

reader is referred to Kokhanovskyi (2021) for details on this comparative study on image contrast. Moreover, further applications of electron beam-based rapid X-ray energy switching and its extension to computed tomography can be found therein.

9.3.2.4

Benefits of Electron Beam Based Rapid X-ray Energy Switching for X-ray Absorption Spectroscopy

Finally, this method for rapidly switching the electron energy is very well suited for thickness correction in the energy-dispersive X-ray absorption spectroscopy (XAS) measurements performed at the MuCLS, cf. Huang et al. (2020). If the electron beam energy can be rapidly switched to generate X-rays with a peak energy in the post K-edge region, the spectrum for the XAS-measurement can be tuned such that the region containing the highest spectral flux density is covered. This was not possible in earlier work at the MuCLS by Huang et al. (2020) because the post-edge region had to be covered by the same spectrum that was used to record the XAS-data since rapid electron beam-based X-ray switching was not available back then. Accordingly, electron beam-based rapid X-ray energy switching paves the way to notably reduce the acquisition time for XAS-spectra once this technique is configured for the element under investigation as it allows both, the XAS-measurement as well as the thickness correction, to be performed at the highest spectral densities provided by the CLS.

9.4 Contributions The following colleagues contributed to Sect. 9.2 “Fast Spectral Filtration”: Madleen Busse developed the protocol to fabricate the solid-state iodine filters, fabricated the filter foils at an external company as well as the BSM 39 prototype. Johannes Brantl designed the adapter to the Owis FRM 40 filter wheel that contains the actual iodine filters, including the holder for the optical fork sensor and the filter wheel itself. Martin Dierolf operated the CLS during the spectrum measurements. Stephanie Kulpe fabricated the filters implemented in the filter wheel depicted in Fig. 9.3. The following colleagues contributed to Sect. 9.3.2 “Demonstration of Electron Beam-Based Rapid X-ray Energy Switching”: Martin Dierolf, Benedikt Günther and Ivan Kokhanovskyi performed the experiments, designed the new energy configurations and energy switching protocols. They operated the CLS as well. Ivan Kokhanovskyi derived the new energy-scaling and performed the data analysis of the KES image.

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References Als-Nielsen J, McMorrow D (2011) Elements of modern X-ray physics, 2nd edn. Wiley, Chichester. 978-0-470-97394-3 Huang J et al (2020) Energy dispersive X-ray absorption spectroscopy with an inverse Compton source. Sci Rep 10:8772 Jiles DC (1994) Frequency dependence of hysteresis curves in conducting magnetic materials. J Appl Phys 76:58495855 Kokhanovskyi I (2021) Rapid keV-switching at the Munich light compact source for K-edge subtraction imaging. Master thesis, Technical University of Munich Kulpe S (2017) Advanced angiography with K-edge subtraction imaging at the Munich compact light source. Master thesis, Technical University of Munich Kulpe S (2020) K-edge subtraction and X-ray fluorescence imaging at the Munich compact light source. PhD thesis, Technical University of Munich Kulpe S et al (2018) K-edge subtraction imaging for coronary angiography with a compact synchrotron X-ray source. PLOS ONE 13:e0208446 Li W et al (2016) Study of magnetic hysteresis effects in a storage ring using precision tune measurement. Chin Phys C 40:127002 NIST (2020) X-ray transition energies database 2020. https://www.nist.gov/pml/x-ray-transitionenergies-database

Chapter 10

X-ray Beam Position Monitoring and Stabilisation

Sections 10.2–10.4 of this chapter are published as Sects. 2–4 in “Device for source position stabilisation and beam parameter monitoring at inverse Compton X-ray sources”, Journal of Synchrotron Radiation, 26, 1546–1553 (2019) (Günther et al. 2019), licensed under CC BY 4.0. If high optical power is stored inside an optical cavity, slight thermal changes can have a detrimental effect on the laser orbit, stored power and in turn available X-ray flux. Therefore, a system to monitor and stabilise the X-ray beam parameters of the inverse Compton source in parallel to experiments is presented in this chapter.

10.1 Necessity of Source Position Stabilisation The laser upgrade presented in Chap. 7 has increased the optical power stored inside the optical cavity by more than 200 kW. This causes significantly higher thermal loads generating thermally induced stresses and deformations of the enhancement cavity, as discussed already in Sect. 7.3.3. As a result, a notably increased interplay between thermal fluctuations and the laser orbit was observed. These effects can be significantly mitigated by the align thermal compensation system, Sect. 7.3.3.3. Nevertheless, one degree of freedom cannot stabilise all four mirrors of the resonator at once. Especially, if the steering of the laser beam could be compromised by thermallyinduced stress in optics which are located close-by strong conventional magnets. The last effect is especially pronounced during the magnets’ heat-up. Although the electron beam in general is more rigid against small thermal transients, its position may slightly drift over time as well. All these effects can degrade the overlap between the electron beam and laser which reduces the X-ray flux and shifts the X-ray source position. In order to support the align thermal compensation system and to suppress © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_10

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this behaviour, a closed-loop X-ray beam position monitoring and stabilization system was developed, which has the capability to operate in parallel to experiments. Its design and performance is presented in this chapter.

10.2 An X-ray Beam Monitor (XBM) for Inverse Compton Sources 10.2.1 Design Constraints In order to monitor and compensate such shifts of the X-ray source position, especially during experiments, a closed-loop feedback system was developed. The main constraints for the design of such a system arise from the feedback frequency on the order of 1 Hz required to compensate aforementioned drifts, the X-ray flux density provided at state-of-the-art inverse Compton X-ray sources, like the MuCLS, and the respective beamline geometry. Knowing the geometry, two beam position monitors would be necessary to calculate the source position through extrapolation of the beam trajectory. The one closest to the X-ray source would have to be placed directly downstream of the X-ray exit window of the vacuum chamber, because it is not possible to install any monitor inside the vacuum vessel without severely deteriorating the laser pulses circulating in the enhancement cavity. This restricts the shortest distance to 1.4 m, where the X-ray beam size is already ∼6 mm in diameter. The second beam position monitor could be placed only outside the radiation shielding enclosure of the MuCLS just in front of the first experimental set-up at a distance of ∼3.0 m. At this distance, the X-ray diameter is 12 mm, which, on the one hand, is too large for commercially-available synchrotron beam position monitor systems. On the other hand, the typical X-ray flux density of at most 2.7 × 108 ph/s/mm2 is orders of magnitude lower than the ones available at synchrotrons, e.g. >1012 ph/s/mm2 at P05 at Petra III at DESY (DESY 2021) or beamline X25 at NSLS (Muller et al. 2012). Therefore, common solutions using the X-ray fluorescence of thin diamond screens or poly-crystalline quadrant-detectors do not deliver sufficient signal levels for online beam position measurements at flux levels available at inverse Compton sources (Bloomer et al. 2016). Single-crystalline diamond screens offer sufficient signal for beam position measurement, as they do not suffer from charge trapping in grain boundaries deteriorating detector efficiency (Tromson et al. 2000). Currently, their size is limited to a diameter of a few millimetres (Bloomer et al. 2016; Muller et al. 2012; Morse et al. 2010). This is smaller then the minimum accessible X-ray beam diameter at the MuCLS of about 6 mm discussed above. Consequently these systems cannot be employed as beam position monitors for large beam diameters like the ones provided at the MuCLS. Longitudinal segmented ion chambers or split ionization chambers provide information on X-ray flux as well as X-ray beam position, but like the diamond beam position monitors, two of these monitors are necessary to determine the X-ray source position and they lack information on the source size of

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the X-ray beam as well (Menk et al. 2007). The latter is the most important parameter for ensuring the correct relative timing between the electron and the laser pulse such that inverse Compton scattering takes place at their common focal position.

10.2.2 X-ray Beam Monitor Design Considering these constraints, our goal was to develop a more efficient system able to cope with this lower flux density. This can be achieved by recording a geometricallymagnified image of a knife edge with a photon-counting hybrid-pixel array detector (Eggl et al. 2016): The extended source leads to a (penumbral) blurring of the knife edge in the recorded image. Fitting an error function to the respective edges, the horizontal and vertical source sizes and the source position can be extracted, if the imaging geometry and detector point-spread function (PSF) are known. The flux of the source can be calculated from the number of photons absorbed in the detector. While this scheme is very robust and efficient, the implementation used in Eggl et al. (2016) was neither compact nor suited to work in parallel to experiments. Our new setup employing the same principle therefore replaces the previously rather big knife edge, which blocked almost 75% of the X-ray beam, with a tiny version that is inserted only at the very bottom of the beam. The second component of the new implementation is a detector that intercepts only that very small part of the X-ray beam and is placed at 2.98 m. This way, online X-ray source monitoring becomes possible in parallel to and without negatively affecting experiments. A schematic of the resulting set-up is depicted in Fig. 10.1a. As the tungsten knife edge intercepting the X-ray beam from the bottom is located at a distance of 1.45 m from source point, there is no geometrical magnification in this case. Thus, the X-ray camera has to provide a half-period resolution of at least 10 µm as the X-ray source-size is about 50 µm. We chose a commercial Basler Ava 1600–50 gm CCD-camera (Basler AG, Ahrensburg, Germany) for its low dark noise and small pixel size of 5.5 µm in a very compact housing. A good compromise of high spatial resolution and sufficient X-ray conversion for online measurements in the energy range between 15 and 35 keV is a 10 µm thick LuAG:Ce single crystal scintillator. It is covered with a thin reflective aluminum coating in order to enhance light collection on the one hand and inhibit visible light transmission from the environment into the detector on the other hand. Two fiber-optic plates consisting of fibers of 3 µm core diameter guide the visible light photons from the scintillator to the CCD-chip. Manufacturing constraints required a thin second fibre-optic plate glued to the CCD-chip, while the long fibre-optic plate, to which the scintillator is coupled, is interchangeable. This optic guarantees very efficient transmission while maintaining the resolution of the CCD camera. The long fiber-optic plate of 25 mm length is cut at an angle of 45◦ at the top in order to allow the rest of the X-ray beam to pass above the housing of the camera when the scintillator is adjusted to intercept only a very small part at the beam’s bottom, cf. Figure 10.1c. As a result, the detector system, depicted in Fig. 10.1a at position 7, is tilted by 45◦ with respect to the optical

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Fig. 10.1 a Schematic of the MuCLS front-end. The geometry constrains the minimal distance between X-ray source and knife edge to 1.45 m and the farthest distance for the X-ray beam monitor to 2.98 m in order to be able to stabilize the source position in parallel to experiments. b displays the unobstructed X-ray beam. c is a technical drawing of the customized X-ray camera system which is minimally intercepting the X-ray beam at the bottom. d The resulting X-ray beam usable for experiments when the X-ray beam monitor is inserted. This figure has been published in Günther et al. (2019), licensed under CC BY 4.0

axis resulting in an effective vertical pixel size of 3.9 µm. A comparison of Fig. 10.1b displaying the unobstructed X-ray beam with Fig. 10.1d where the X-ray beam monitor (XBM) is placed into its operating position, demonstrates the feasibility of our approach. Both images are taken with our Andor Zyla 5.5 sCMOS camera (Andor, Belfast, UK) equipped with an 2:1 fibre-optic taper and a 20 µm Gadox scintillator, which is part of our experimental end-station. Customization of the camera according to our design was performed by Crytur (Turnov, Czech Republic). The frames are directly read out from the camera via the EtherLink interface using the pypylon library, the official Python wrapper around the pylon 5 library of Basler. Gain map correction and fits are performed using standard SciPy-functions (Jones et al. 2001). The calculated source parameters are sent to the EPICS control system of the CLS and used in a software feedback loop to adjust the laser orbit in the cavity.

10.2.3 Characterization of the X-ray Beam Monitor The result of our analysis regarding the camera resolution is displayed in Fig. 10.2. For this analysis, the camera and the resolution pattern were oriented parallel to each other, i.e. both, horizontal and vertical effective pixel sizes, are 5.5 µm. Figure 10.2a

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Fig. 10.2 Analysis of the resolution of the X-ray beam monitor. a displays the line pattern recorded for determination of the X-ray camera resolution. The Siemens star b demonstrates isotropic resolution of the camera. c displays the intensity modulation of the line pattern indicated with a green box in a averaged along the lines. d The MTF-values calculated from the raw data depicted in c. The orange line is a moving average (Savitzky-Golay) of the MTF-values. At 65 lp/mm the MTF is 0.1 which fulfills the requirement of resolving the blur caused by a 50 µm X-ray source. This figure has been published in Günther et al. (2019), licensed under CC BY 4.0

shows a line pattern (Xradia, Pleasanton, USA), which was used for analysis of the modulation transfer function (MTF). The Siemens star in Fig. 10.2b demonstrates isotropic resolution of the camera system. Therefore, MTF-analysis of the line pattern enframed with a green box is representative for the whole camera. Figure 10.2c depicts the intensity modulation of the line pattern averaged in the vertical direction, i.e. along the direction of the lines. The MTF was calculated for the region marked by a blue box in Fig. 10.2a and is plotted in Fig. 10.2d. In order to account for substrate absorption, the reference contrast amplitude has been determined as the contrast between the large square structures and the background assuming a constant structure height as well as substrate thickness. The orange line is a moving average (Savitzky-Golay) of the MTF. 65 lp/mm are resolved at a MTF of 0.1. This corresponds to a resolution limit of ∼8 µm lines which fulfils the 10 µm requirement originally specified for the camera system. Consequently, online monitoring at a frequency of 1 Hz is possible. A typical image is depicted in Fig. 10.3a where the individual pixel response is accounted for with a gain map. The total X-ray flux is determined from the average value of counts within the region-of-interest depicted in black with an empirical flux conversion factor. The red and green boxes indicate the regions-of-interest for the error function fits, which are depicted in Fig. 10.3b– c in the corresponding colors. The raw data is averaged along the direction of the

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Fig. 10.3 Online operation of the X-ray beam monitoring system. a The image of the small knife edge inserted into the X-ray beam recorded with the X-ray beam monitor during online X-ray beam stabilization. The individual pixel response of the camera is corrected for with a gain map. The black box indicates the region of interest for X-ray flux determination while the green and red boxes indicate the regions of interests for the error-function fit along the vertical and horizontal knife edges. b and c depict the measured data as circles and fits as solid lines with the color corresponding to the respective regions of interest. This figure has been published in Günther et al. (2019), licensed under CC BY 4.0

edge before the error function fit. In contrast to photon counting detectors which exhibit a box-like PSF, any optic in the camera system—in our case a scintillator and a fiber-optic plate—increases the PSF. We determined both, the PSF and the flux conversion factor, by calibrating the source sizes and the flux measured with the integrating CCD-detector to the ones obtained with a photon-counting hybrid-pixel array detector (Pilatus 200k, Detris AG, Baden, Switzerland). Its photon counts can be converted into absolute photon flux of the source incorporating knowledge about the spectrum and spectral efficiency of the silicon sensor as well as the setup geometry (Eggl et al. 2016). Both measurements were recorded in parallel immediately demonstrating the desired capability of the XBM to work in parallel to experiments. Assuming a Gaussian shape of the PSF, a PSF of 5.73 px has been calculated in the vertical direction and a PSF of 6.56 px in the horizontal one. Source sizes are

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determined from the measured and PSF-deconvoluted error-function fits by multiplying the resulting pixel number with the corresponding effective pixel sizes for the vertical and horizontal direction. The source position can be tracked via the position of the edge in the image.

10.3 Closed-Loop X-ray Source Stabilization 10.3.1 Correction of Source Position Drift First, the offset from the optimum overlap position of laser and electron beam is calculated comparing the positions recorded with the X-ray beam monitor to the reference value after X-ray tuning. The shift of the source position is then corrected for by adjusting the steering of the laser beam with a motorized mirror mount. In the special case of the MuCLS, we use one of the mirrors of the enhancement cavity to perform this move. This procedure is not limited to correction of laser-driven source position movements in enhancement cavities, but is compatible with single-shot laser or laser recirculator-based inverse Compton sources. The calculated deviation of the source position from the desired one is multiplied with an empirical gain factor which translates the distance into an angular tip/tilt correction for the respective piezoelectric actuators attached to the motorized mirror. As a result, the position of the laser beam is slightly shifted back to its original position correcting the drift of the source position. The lower action limit of the actuator is set to 0.001 V in the software feedback and the determined conversion factor relating the voltage and the source position shift is 0.02 V/µm. Therefore, the minimum motion of the source position by the feedback is 0.05 µm. This is smaller than the sub-pixel accuracy of the source position determination of ±(0.3−0.4) µm given as the standard deviation of the fit parameters of the error function fits to the respective edges. Consequently, the sensitivity of the system is limited by the determination of the source position. The feedback usually runs with an update frequency of 1 Hz, which is more than sufficient to correct the substantially slower thermal drifts and allows to operate the XBM with an exposure time of 0.5 s. Image readout and calculation of the source parameters take another ∼0.07 s, online visualization of the acquired data and the knife-edge fits inside the analysis loop takes ∼0.25 s (but could simply be turned off). The design frequency of 1 Hz is achieved by implementing an additional wait time. In particular, the beam position monitor is also used for the initial optimization of the overlap between the laser beam and electron beam. Also in this case, the acquisition time of 0.5 s is sufficient to still obtain reliable results at a flux around 109 ph/s which is one order of magnitude lower than the typical case. Consequently, the acquisition time could be significantly reduced for the purpose of beam stabilization, but ultimately the feedback frequency is limited by the flux density provided by the X-ray source at the position of the beam monitor.

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Fig. 10.4 Comparison of the performance of the Munich Compact Light Source with active source stabilization a and without stabilization b over the course of three hours. Source sizes (blue and orange) are relatively stable in both cases. Contrary to this behaviour, the source positions (green and red) drift significantly if no active stabilization is performed, while they remain perfectly stable if the closed-loop feedback system is running. Actively pinning the source position improves X-ray flux stability in addition, as the optimum overlap between laser and electron beam is maintained. The source positions are shifted with an artificial offset of 35 µm for better display. Quantitative values are shown in Table 10.1. This figure has been published in Günther et al. (2019), licensed under CC BY 4.0

10.3.2 Evaluation of the XBM and Its Performance in Beam Stabilization The benefit of the feedback on source position is well visible comparing the two three-hour runs displayed in Fig. 10.4 where the closed-loop feedback was active in Fig. 10.4a and turned off in Fig. 10.4b. This duration was chosen because it reflects typical time scales for experiments frequently performed at the MuCLS, e.g., grating-

10.4 Significance of X-ray Source Position Stabilization

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Table 10.1 Effect of X-ray beam monitor on source stability. The origin of the source position is defined at the optimum overlap between laser and electron beam. Relative motion of the source position has physical consequences as it deteriorates source position sensitive X-ray experiments. This table has been published in Günther et al. (2019), licensed under CC BY 4.0 XBM-feedback

Active Mean

Horizontal position [µm] Vertical position [µm]

Off Variance Std.

3h-drift

Mean 5.31

4.79

2.19

8.94 −5.75

1.08

0.36

0.60

0.67

−0.40

Variance Std.

3h-drift

0.13

0.35

−0.46

2.94

2.70

1.64

Horizontal size [µm]

48.80

0.32

0.57

−0.61

56.38

0.76

0.87

1.08

Vertical size [µm]

48.99

0.09

0.29

0.36

45.97

0.12

0.35

−0.71

Flux [1010 ph/s]

1.64

0.0003

0.02

−0.03

1.41

0.001

0.03

−0.12

based phase-contrast tomography (Eggl et al. 2015). Quantitative numbers of the MuCLS performance during these runs are given in Table 10.1. A general observation is, that variances of source position, source size as well as flux are substantially lower, if the feedback is running. However, even more important for experiments are the maximum drifts. A drop in the X-ray flux of 8% is observed over the course of three hours when the feedback is turned off compared to only 2% when the beam stabilization is active. Instead of position drifts of 8.94 µm over the period of three hours in the horizontal direction and −5.75 µm in the vertical direction, the X-ray source position remains almost perfectly stable when the feedback is active resulting in drifts of 0.6 µm in the horizontal direction and −0.46 µm in the vertical one. Absolute shifts well below one micrometre over the course of three hours correspond to a relative shift of ∼1% of the source size when the feedback is running compared to shifts of up to ∼16% when the feedback is turned off. The observed improvement amounts to a factor of 14.9 in the horizontal source position stability and 12.5 in the vertical one, which is a reduction by more than one order of magnitude.

10.4 Significance of X-ray Source Position Stabilization Drifts of the source position presently limit the applicability of inverse Compton sources to certain types of experiments, e.g. scanning microscopy or phase-contrast imaging. It has been demonstrated here that a beam position monitor consisting of a knife edge and a customized commercial CCD-camera is sufficient to determine X-ray source parameters, such as X-ray flux, source size and position online. It therefore overcomes the limited applicability of commercial synchrotron beam position monitors at this class of sources due to the moderate flux densities and large beam diameters. In addition, it allows to stabilize the X-ray source in combination with a

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feedback system. This is achieved by acting upon changes in X-ray source position and adjusting the steering of the laser pulse within the resonator such that the shift of the source position is compensated. Most importantly, this is possible in parallel to and without impairing experiments. Instead of drifts of the source position of up to 9 µm within three hours, the feedback reduces the amplitude of the drift by more than one order of magnitude. This work enables very position sensitive experiments to be carried out at compact inverse Compton X-ray sources in the future. Moreover, this X-ray beam stabilization scheme is not limited to inverse Compton sources. Any kind of X-ray source providing a sufficient flux density and incorporating elements which can be used as actuators to actively influence the source position, like electron beam optics and/or motorized laser beam optics, can exploit our proposed scheme for source position stabilization. Instead of adjusting a mirror of the laser beam to keep the source position constant, an electron optic would have to be adjusted to steer the electron beam back to its original position.

10.5 Contributions This chapter of my thesis has been realised with contributions by Martin Dierolf, TUM. He helped with implementation of the X-ray beam position monitor into the operating system of the CLS. Additionally, all experiments presented in this chapter were performed together with Martin Dierolf.

References Bloomer C, Rehm G, Dolbnya IP (2016) An experimental evaluation of monochromatic X-ray beam position monitors at diamond light source. AIP Conf Proc 1741:030022 DESY P05 (2021) DESY Petra III, P05, Unified Data Sheet 2021. http://www.photon-science. desy.de/facilities/petra_iii/beamlines/p05_imaging_beamline/unified_data_sheet_p05/index_ eng.html Eggl E et al (2016) The Munich compact light source: initial performance measures. J Synchrotron Radiat 23:1137–1142 Eggl E et al (2015) X-ray phase-contrast tomography with a compact laser-driven synchrotron source. Proc Natl Acad Sci 112:5567–5572 Günther B et al (2019) Device for source position stabilization and beam parameter monitoring at inverse Compton X-ray sources. J Synchrotron Radiat 26:1546–1553. https://doi.org/10.1107/ S1600577519006453 Jones E, Oliphant T, Peterson P et al (2001) SciPy: open source scientific tools for Python. http:// www.scipy.org/ Menk RH et al (2007) Hiresmon: a fast high resolution beam position monitor for medium hard and hard X-rays. AIP Conf Proc 879:1109–1112 Morse J, Solar B, Graafsma H (2010) Diamond X-ray beam-position monitoring using signal readout at the synchrotron radiofrequency. J Synchrotron Radiat 17:456–464

References

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Muller EM et al (2012) Transmission-mode diamond white-beam position monitor at NSLS. J Synchrotron Radiat 19:381–387 Tromson D et al (2000) Geometrical non-uniformities in the sensitivity of polycrystalline diamond radiation detectors. Diam Relat Mater 9:1850–1855

Part III

X-ray Imaging and Spectroscopy at the MuCLS

Chapter 11

The MuCLS Beamline

This chapter is published as Sects. 3, 4 and Appendix A in “The versatile X-ray beamline of the Munich Compact Light Souce: design, instrumentation and applications”, Journal of Synchrotron Radiation, 27, 1395–1414 (2020) (Günther et al. 2020). The introduction to Sect. 4 and Appendix A of the publication are joined into the section “The MuCLS laboratory” in this thesis. Content licensed under CC BY 4.0. After various developments on the inverse Compton X-ray source of the Munich Compact Light Source have been discussed in preceding chapters, here its performance is evaluated and compared to synchrotrons and advanced X-ray tube sources. Based on these results, the versatile beamline of the Munich Compact Light Source is described in detail.

11.1 Evaluation of the MuCLS in Comparison to Synchrotrons and Advanced X-ray Tube Sources This section compares the MuCLS facility to synchrotrons and other laboratory setups based on advanced X-ray tube sources. To this end, the first part of this chapter provides a detailed analysis of these X-ray sources. The parameters of interest are evaluated in the following order: flux, source size, X-ray generation and its effect on X-ray beam divergence and bandwidth. Finally, these parameters are combined into the figure-of-merit for X-ray beams, the brilliance. Subsequently, the three facility types are compared in terms of costs for both installation and operation, support infrastructure and ease of access for users in the second part.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_11

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11.1.1 Comparison of the X-ray Sources’ Beam Parameters Flux Insertion devices at state-of-the-art synchrotrons provide several kilowatts of X-ray power, e.g. 9.5 kW at P11 at Petra III (DESY P11 2019). This flux enables such sources to deliver up to 1 × 1013 ph/s on the sample (after a double-crystal monochromator). In contrast to this narrow bandwidth synchrotron beam, the ICS at the MuCLS provides orders of magnitude lower flux within its full bandwidth of (3−5)%. From a value of 1 × 1010 ph/s (Eggl et al. 2016) at the time of installation, an upgrade of the laser system increased the flux of the ICS to up to 3 × 1010 ph/s (Günther et al. 2018) and further optimisation of our optical cavity system recently resulted in up to 5 × 1010 ph/s at 35 keV X-ray energy under optimal conditions. A measurement of this peak flux is depicted in the inset of Fig. 11.1. However, this peak flux has been reached only briefly during X-ray tuning so far. In such optimal circumstances a stable X-ray flux of 4.0 × 1010 – 4.3 × 1010 ph/s is achieved on time scales required for experiments, cf. the main graph of Fig. 11.1. Although micro-focus tubes typically do not reach this X-ray flux level (integrated flux: ∼1 × 109 ph/s (Procop and Hodoroaba 2008)), the integrated full-spectrum flux of high power rotating anodes as well as liquid-metal jet based X-ray sources can surpass the one available at the MuCLS. The integral flux of high-power rotating anodes is >1 × 1013 ph/s at the position of a clinical CT-detector (Schlomka et al. 2008). Considering the conversion efficiency from electrical power to X-ray power of ∼10−4 , the generated X-ray flux is as high as 1 × 1015 ph/s (Behling 2016), while liquid-metal jet sources reach

Fig. 11.1 Performance of the MuCLS operated at the 35 keV X-ray configuration. The inset depicts the maximum X-ray flux of 5 × 1010 ph/s achieved under optimal conditions. However, this flux has been reached only briefly during X-ray optimisation so far. Typically, small movements of the electron beam within the stability range of the storage ring in conjunction with adjustments of the temporal delay between laser and electron beam are sufficient for daily X-ray tuning. On timescales required for experiments, this X-ray flux stabilises at a level of 4.0 × 1010 – 4.3 × 1010 ph/s under such circumstances, as shown in the main figure. Data was recorded with the beam monitoring system described in Günther et al. (2019). This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

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>1 × 1012 ph/s to ∼1 × 1013 ph/s, estimated from the 160 kVp spectrum (Excillum 2019). Source size X-ray beams at current third-generation synchrotron facilities have an elliptical source size with their aspect ratio depending on the electron beam βfunction at the location of the insertion device. For state-of-the-art synchrotrons, like Petra III, typical source sizes for a high β-section are 140 µm × 5.6 µm and 36 µm × 6.1 µm for a low β-one (Barthelmess et al. 2008). The latter source size is comparable to the ones of emerging fourth-generation synchrotrons, such as Max IV (48.7 µm × 6.2 µm (Max IV Collaboration 2010)) and ESRF-EBS (30.2 µm × 5.1 µm (ESRF 2019)). The source size of the MuCLS of ∅ < 50 µm is comparable to synchrotons, at least in the horizontal direction and larger than the projected X-ray source size of the liquid-metal jet sources of ∅20 µm (Excillum 2019). Microfocus X-ray tubes provide source sizes ranging from a couple of micrometres to a couple of tens of micrometres, with micro-focus rotating anode sources (70 µm for the Rigaku FR-X) on the large spot size end of the spectrum. The smallest X-ray source size currently is provided by nano-focus transmission tubes, such as the Excillum NanoTube N1 60 kV which enables spot sizes of 200–300 nm (Müller et al. 2017), but their flux is orders of magnitude lower due to the use of thin transmission targets with low heat dissipation capability. Divergence and bandwidth Despite flux and source parameters varying strongly for the different X-ray tube concepts, there is one characteristic, which they all have in common: X-ray generation by bremsstrahlung. This results in a wide natural divergence as well as a broad spectrum of X-rays (basically from a few keV to eU , where e is the electron charge and U the acceleration voltage) containing additionally intense, narrow and material-specific emission lines. Therefore, high peak X-ray energies can be reached but high average X-ray energies are feasible only by filtering the spectrum appropriately, however, at the cost of a strongly reduced integral flux. In contrast to X-ray tubes, ICSs and synchrotron sources use a different mechanism to generate X-rays. X-ray generation at ICSs can be described equivalently to an undulator considering the electro-magnetic field of the laser as a “laserundulator”. Consequently, the divergence of their X-ray beam is inversely proportional to the γ-factor, i.e. electron energy, which is typically in the range of a few tens of MeV. The resulting divergence is a few milliradians to a few tens of milliradians, 4 mrad at the MuCLS defined by an fixed aperture. In contrast, the beam divergence is in the range of a few microradians to a few tens of microradians for synchrotrons due to their electron energies of a few GeV. Contrary to this disparity in divergence between ICSs and synchrotron sources, their spectra can be very much alike, i.e. with a natural bandwidth of a few percent. However, due to the significantly reduced flux at ICSs compared to synchrotrons, the former seldom use monochromators to further filter the spectrum, which could reduce their bandwidth down to E/E ≈ 10−4 with a double crystal monochromator. Brilliance A figure-of-merit combining the aforementioned X-ray source parameters is the brilliance. For synchrotrons, the corresponding brilliance is in the

260

11 The MuCLS Beamline

range of 1 × 1020 to 1 × 1021 ph (s mrad2 mm2 0.1%BW)−1 (Petra III, Hamburg) (DESY 2019), which is many orders of magnitude higher than for rotating anodes (up to 1.2 × 1010 ph (s mrad2 mm2 )−1 (Skarzynski 2013)). For 70 kVp and 200 W electron beam energy, liquid-metal-jet sources reach a value of 5 × 1010 ph(s mrad2 mm2 0.1%BW)−1 at their Ga-Kα-line, where these sources are most brilliant, while their brilliance is ∼2 × 109 ph (s mrad2 mm2 0.1%BW)−1 for the In-Kα-line (Wansleben et al. 2019). It has to be considered that the natural line-width of 2.6 eV for gallium and 10.6 eV for indium was scaled to the 0.1% BW-level of 9.25 and 24.4 eV in this calculation, respectively. Furthermore, Wansleben et al. (2019) collimated the X-ray beam to 2.5 × 10−5 sr. This might lead to a slight overestimation of the spatially averaged brilliance, as selfabsorption for off-axis angles of the beam with a full opening angle of 30◦ (0.054 sr) is not taken into account. However, considering the radiant flux at the Ga-Kα line of 6 × 1012 ph (s sr)−1 (4.6 × 106 ph (s mrad2 )−1 ) reported in their work, the total X-ray flux emitted from this line into the maximum full angle of 30◦ is ∼3.1 × 1011 ph/s. The ICS of the MuCLS is similarly brilliant to liquid-metal jets (1.2 × 1010 ph (s mrad2 mm2 0.1%BW)−1 ). Yet, it has the advantage that its brilliance is almost constant over the whole energy range. In contrast, the brilliance of bremsstrahlung-based sources drops by several orders of magnitude at energies other than the characteristic emission lines. Responsible for this behaviour at the MuCLS is the interaction cross-section of inverse Compton scattering. For our Xray energy range, it can be approximated by the Thomson cross-section and therefore as being energy independent. Note that the increase in total flux observed at the MuCLS when increasing X-ray energy is due to the decrease of the natural emission angle (∼1/γ) when increasing the electron energy. As a consequence, a larger portion of the total flux is extracted through the fixed X-ray exit aperture and the spectrum is broadened slightly, while the intensity at the peak stays constant.

11.1.2 Discussion of Typical Facility Implementations of Aforementioned X-ray Sources In the discussion so far, only machine parameters have been discussed, although these parameters do not provide an exhaustive evaluation of the X-ray source most suitable for specific tasks/a specific research environment. First of all, the instrument itself has to be acquired, maintained and operated. X-ray tube-based systems are inexpensive compared to other types of X-ray sources, turn-key, fit into a small laboratory and require little maintenance. Thus such an instrument may also be afforded and operated by a small research group, with the option of daily use. Contrary to that, ICS facilities like the MuCLS require a multi-million Euro commitment, a larger laboratory only for the X-ray source itself, elaborate radiation shielding due to the higher electron energies and trained operators for operation and maintenance. All of this significantly increases the costs for acquisition and operation compared to X-ray

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tube-based systems. Consequently, such X-ray sources will be mainly incorporated into central research or service facilities of bigger universities or larger non-university research institutes. In contrast to X-ray tube-based set-ups which are usually optimised for a specific task (e.g. crystallography or microtomography), this also means that experimental end-stations in such an environment have to be more flexible and have to provide complementary techniques in order to exploit the full potential of these sources. Furthermore, integration into a central facility also means that a better support infrastructure is going to be available at an ICS to satisfy requirements of the members of such an institution, like various sample preparation laboratories or complementary instruments. At the moment, the support infrastructure for the MuCLS available at CALA and the Munich School of BioEngineering include dedicated chemistry and electronic laboratories, different micro-CT systems, a clinical CT scanner as well as different light and electron microscopes, thereby surpassing the usual instrumentation of an average research group. While such support infrastructure might cover the main applications of the regular (internal) facility users, infrastructure for exotic experiments or external users might be missing. Such support infrastructure might be available at synchrotrons which provide a wide range of central laboratories for sample preparation or additional characterisation, which becomes feasible since they serve many beamlines at the same time. Moreover, synchrotron facilities employ a lot of staff for facility operation and user support at the beamline during experiments. Although synchrotrons provide the highest brilliance and probably best infrastructure, which might be optimal for the experiment, they have some drawbacks. First of all, getting beamtime at synchrotrons is often difficult since most beamlines are heavily oversubscribed. As a consequence, if granted, time for experiments is very limited. Furthermore, samples have to be transported to the facility, which can be complicated for certain classes of samples due to transport or import restrictions. Moreover, transportation could lead to degradation or damage of sensitive specimens, or requires them to be prepared on-site in an unfamiliar laboratory. In contrast, a central facility can offer easy regular on-site access to all members of the host institution. This also makes it easier to handle samples which are tricky to transport, either because they are too delicate or due to restrictions. In particular, it allows the on-site users to prepare the samples in their own laboratories and even take them back there for further treatment steps between parts of the X-ray experiments. The MuCLS facility was established as part of CALA with these advantages in mind. Access to CALA facilities is open to members of the two universities, TUM and LMU, and their collaborators. At the moment, there is thus no proposal system, but access to external users is granted through collaborations with local groups instead. External users who are interested in establishing a collaboration to perform experiments at the MuCLS are asked to use the contact information on https://www.bioengineering.tum. de/en/central-building/munich-compact-light-source/ for inquiries. The preceding discussion is summarised in Fig. 11.2, which provides a qualitative overview of the aforementioned parameters for the different X-ray sources. One technique that is well suited to the MuCLS’s source properties is dynamic (high-resolution) phase-contrast X-ray imaging. The larger (and symmetric) cone

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Fig. 11.2 Typical X-ray source parameters, available support infrastructure and system costs (operation as well as acquisition/development) are depicted qualitatively for synchrotrons, ICS sources and X-ray tubes (subdivided into micro-focus tubes and rotating anodes where appropriate). The combination of synchrotron-like radiation, especially in terms of energy range, spectral bandwidth and X-ray beam divergence, with easy access is unique to inverse Compton sources. Therefore, applications at ICS should rely on them ideally. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

angle compared to synchrotrons enables to capture larger objects, like the lungs of mice, in a single exposure with shorter acquisition times than other laboratory Xray sources (except for liquid-metal jet ones at much lower resolution, cf. Preissner et al. (2018)), which in turn enables in vivo small animal phase-contrast imaging. Consequently, X-ray imaging so far seemed to be the most suited candidate for transferring techniques limited to synchrotrons into a laboratory. Thus we designed flexible experimental end-stations keeping this goal in mind, which enables us to perform (propagation-based phase-contrast) microtomography, dynamic K-edge subtraction imaging, grating-based phase-contrast imaging and propagation-based imaging.

11.2 The MuCLS Laboratory The versatile X-ray application beamline of the MuCLS has been designed to accommodate a variety of imaging techniques as well as enabling the implementation of other techniques, if desired. X-rays exit the ICS of the MuCLS at a distance

11.3 The MuCLS Front-End

263

of 1.35 m to the X-ray source point at a height of 1.67 m above ground with a divergence of 4 mrad. This exit window defines the starting point of the MuCLS beamline, which is schematically depicted in Fig. 11.4 (bottom row) together with photos of selected parts (Fig. 11.4, top row). It can be divided into three sections, a front-end section and the two experimental end-stations. The beamline control system is SPEC (Certified Scientific Software, Cambridge, USA) which was chosen because of its built-in hardware support, its flexibility to add own code through user macros and its widespread use at many synchrotrons. This makes it easy for users to integrate their own equipment, if required, or existing SPEC routines which have been already developed for measurements at other facilities. At the MuCLS, the built-in hardware support directly interfaces with the motor controllers. Custom macros control the detectors through TCP/IP connections in a client-server model. Specifically developed data acquisition macros are optimised to automate common measurement tasks (e.g., computed tomography scans) while providing the necessary metadata for the in-house data processing pipeline. A to-scale sketch of the MuCLSlaboratory is depicted in Fig. 11.3. Only the experimental hall housing the MuCLS is depicted, other laboratories and offices are omitted. The size of the whole laboratory is 26.70 m × 7.05 m × 4.6 m of which the enclosure of the ICS itself occupies a volume of 10.27 m × 7.05 m × 3.34 m. Its walls are ∼80 cm thick and realised as a sandwich structure of heavy concrete filled with electro-furnace slag. The door of the enclosure is 0.9 m wide, which is sufficient to move a modulator out of the cave, e.g. in order to replace its klystron. The small area to the right of the enclosure in Fig. 11.3 houses the three chillers of the ICS, while most of the machine instrumentation, especially magnet power supplies and radio-frequency control systems, are located in a separate rack room which contains an additional forced-air cooling unit removing the heat dissipated mainly by the magnet power supplies. The ICS control as well as beamline control are located next to each other in the laboratory. Both end-stations can be controlled independently from each other enabling preparation of experiments in the second end-station while measurements are running in the first one. Moreover, workbenches for sample preparation are located between both end-stations with a chemical laboratory next door for advanced preparation protocols. The area that is coloured in ivory indicates the approximate extend of the new second end-station which is scheduled to replace the small one in 2021.

11.3 The MuCLS Front-End The front-end (Fig. 11.4a) contains a fast X-ray shutter (model SH-10-PI-B-L-24, Electro-Optical Products Corp., Ridgewood, USA), a knife edge combined with a special camera system for X-ray beam diagnostic and closed-loop X-ray beam stabilisation (Günther et al. 2019), a polycapillary optic (IFG, Berlin, Germany) with a long working distance and thus mildly focusing the beam to a diameter of ∼3 mm at the sample position in the first end-station, the X-ray safety shutter (inhouse design, motorised translation from Festo, Esslingen, Germany) and a slit-system (Huber

Fig. 11.3 To-scale schematic of the MuCLS laboratory in the Munich School of BioEngineering (MSB) of the Technical University of Munich. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

264 11 The MuCLS Beamline

11.4 MuCLS End-Station 1

265

Diffraktionstechnik, Rimsting, Germany) at the very end for beam shaping. The fast X-ray shutter can be controlled by SPEC and can be used to reduce the dose applied to the specimen under investigation, when synchronised with data acquisition. The polycapillary optic located in the front-end has originally been designed for microbeam radiation therapy studies enabling the implementation of this technique in the first experimental end-station. This is much more convenient than performing experiments inside the ICS enclosure like in the first trials (Burger et al. 2017). The recently developed custom-built X-ray beam monitor records the X-ray source parameters (flux, source size and -position) continuously. This enables manual reoptimisation of the collision of the laser and electron beams even during experiments. Moreover, data provided by this device can be used for a closed-loop feedback system stabilising the X-ray position significantly. Details on the system and its performance can be found in the publication by Günther et al. (2019). In addition, drops in X-ray flux, typically related to a loss of lock of the laser system to the enhancement cavity, can be detected in situ and used to automatically pause experiments thereby reducing data loss.

11.4 MuCLS End-Station 1 The first experimental end-station houses two optical tables for experiments. The first one is equipped with a very flexible multi-purpose set-up depicted in Fig. 11.4b, which has been employed for microtomography, propagation-based phase-contrast imaging and X-ray absorption spectroscopy so far. The sample manipulation system, as well as the detector assembly, are mounted on two different motorised stages (Limes 170, Owis, Staufen, Germany) with 1.5 m travel range along the X-ray beam axis each. Both systems can be completely moved out of the beam sideways by two additional motorised linear stages, which is necessary to perform experiments in the second end-station. The shortest source-to-sample distance achievable is 3.5 m. Therefore the maximum source-to-sample distance is 5 m, whereas the maximum sample-to-detector distance is ∼1.8 m. Accordingly, the X-ray beam diameter ranges from 14 to 20 mm. The sample is mounted on a rotation stage (either a PI Micos PRS-200 or a LAB RT-150U) placed on top of an elevator stage (Huber Diffraktionstechnik, Rimsting, Germany) with 90 mm range. Specimens can be centred on the rotation axis with two Newport MFA-PPD stages on which the actual sample holder (e.g. a goniometer head) can be mounted. To further enhance the set-up’s flexibility, the rotation stage is located inside a custom-built housing which is covered by a small breadboard (22.5 cm × 33 cm) with a threaded hole pattern (M6 on 25 mm square grid). It also contains an opening with 100 mm in diameter around the centre of the rotation axis. This breadboard can thus be used to mount additional equipment, e.g. a focusing optic with a short working distance, closely to the specimen while the latter can be rotated freely, nevertheless. A multi-axis overhead sample manipulation system can be mounted onto this small table if two axes of rotation or similar are required. The X-ray detectors are mounted on a horizontal breadboard

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Table 11.1 Overview of the camera systems available at end-station 1. The abbreviation PSF denotes point-spread-function, FOV denotes field-of-view and fps frames per second Model Hamamatsu Andor Zyla Ximea X-ray Photonic Andor Neo sCMOSsCMOS 5.5 CCD—xiRay science 4.2MP sCMOS 5.5 C12849-101U sCMOS (with 4x/10x optics) Scintillator Pixel size PSF (σ) FOV [pixel] [mm]

Max. frame rate Dynamic range Digitisation

Gd2 O2 S : Tb 10 µm 6.5 µm

Gd2 O2 S : Tb 20 µm 6.5 µm [13 µm]a 8 µm 11 µm [60 µm]a 2048 × 2048 2560 × 2160 13.3 × 13.3 16.6 × 14.0 [33.3 × 28.0]a 30 fps 100 fps

Gd2 O2 S : Tb 22 µm 9 µm

Gd2 O2 S : Tb 15 µm 11 µm

2.1 fps

18 fps

100 fps

18000:1 15 bit 16 bit

6000:1 13 bit 14 bit

38800:1 16 bit 16 bit

13000:1 14 bit 16 bit

12500:1 14 bit 16 bit

LSO 10 µm 1.63 µm/ 0.65 µm 12 µm 11 µm 1.5µm to 2µm (10x optic) 4008 × 2672 2048 × 2048 2560 × 2160 37.3 × 25.7 22.5 × 22.5 4.17 × 3.52/ 1.66 × 1.40

a

Indicates the values for the optional 2:1 fibre-optic taper available for the Andor Zyla sCMOS 5.5. This table has been published in Günther et al. (2020), licensed under CC BY 4.0

on top of an elevator stage (Humes 200, Owis, Staufen, Germany) to centre each detector in the X-ray beam. At the moment, mainly cameras with pixel sizes around 10 µm and fibre-optically coupled scintillators are installed, but a system with 4x or 10x optical magnification is available, too. Details on the detectors are summarised in Table 11.1. Furthermore, a Ketek AXAS-D silicon-drift detector (Ketek GmbH, Munich, Germany) is available, e.g. for measurements of the source spectrum or fluorescence. For the latter, a second polycapillary optic (IFG, Berlin, Germany) can be installed, which focuses the X-rays to a ∼50 µm spot at a working distance of 17 mm. The second optical table currently is reserved for temporary set-ups, e.g. of users who want to use their own equipment.

11.5 MuCLS End-Station 2 The second end-station is installed at a distance of 14.8 m to the source point and contains two permanently installed sample manipulation systems, an optional twograting interferometer for differential phase-contrast imaging (DPC) and detectors with pixel sizes in the range from 74.8 µm to 172 µm, which cover the full field-ofview of ∼60 mm in diameter. Details about the available detector systems (sourceto-detector distance ∼16.4 m) are presented in Table 11.2.

Fig. 11.4 The MuCLS X-ray beamline. e depicts a schematic drawing (not to scale) of the MuCLS consisting of the ICS housed in a radiation shielding enclosure, the beamline front-end located right after the exit window of the ICS at a distance of 1.35 m to the interaction point and the two end-stations designed for complementary X-ray imaging methods. The shortest source-to-sample distance is 3.0 m and the longest one is 15.6 m. a is a photograph of the front-end inside the radiation protection enclosure with the X-ray exit window on the left side, b is a possible experimental set-up in hutch 1 including a polycapillary optic and silicon drift detector. c shows the whole set-up of end-station 2 while d is a picture of the grating interferometer and the installed detectors. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

11.5 MuCLS End-Station 2 267

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Table 11.2 Overview of the camera systems available at end-station 2. The abbreviation PSF denotes point-spread-function and FOV denotes field-of-view Model Perkin Elmer Dectris Santis Direct Varian Dectris Pilatus Dexela 1512 (evaluation Conversion PaxScan 200K prototype) XC-Thor 2520D CsI

Max. frame rate Dynamic range

Gd2 O2 S : Tb, 150 µm 74.8 µm 71 µm (σ) 1536 × 1944 114.9 × 145.4 26 fps

GaAs, 500 µm 75 µm 75 µm (box) 1030 × 1064 77.25 × 79.80 25 fps

2000:1 12 bit

Digitisation

14 bit

65536:1 4096:1 16 bit for each 12 bit counter 32 bit 12 bit

Scintillator/ Sensor Pixel size PSF FOV [pixel] [mm]

CdTe, 750 µm 100 µm 100 µm (box) 1024 × 512 102.4 × 51.2

CsI, 600 µm 127 µm n.a. 1536 × 1920 195 × 243.8

Si, 1000 µm 172 µm 172 µm (box) 487 × 407 83.7 × 70.0

300 fps

10 fps

20 fps

n.a.

1048573:1 20 bit

16 bit

20 bit

This table has been published in Günther et al. (2020), licensed under CC BY 4.0

Three of these detectors can be installed at the same time. They are centred in the beam with two stages for horizontal and vertical movement. Detectors and interferometer are installed on an optical table with passive air damping. This avoids that external vibrations, also the ones arising from specimen movement with the sample manipulation stages, are coupled into the interferometer, which would strongly deteriorate its performance. The interferometer (Fig. 11.4d) consists of two gratings with a diameter of ∼65 mm covering the full beam. Currently, two different nickel phase gratings (G1) are available for design energies of 25 and 35 keV. In order to facilitate the lithographic production process, the same periodicity p (4.92 µm) was chosen for both phase gratings and a single gold absorption grating (periodicity 5.0 µm) was manufactured by the Karlsruhe Nano Micro Facility (KNMF, Eggenstein-Leopoldshafen, Germany). Thus, the phase grating for 35 keV has to be tilted by 6.3◦ to generate the design periodicity of 4.89 µm. Grating properties are summarised in Table 11.3. In grating-based phase-contrast imaging, one of the two gratings is stepped over one period. This phase stepping generates a sinusoidal intensity curve for each pixel, whose modulation strength, the visibility, is a crucial factor for the sensitivity of the interferometer. At 25 keV a visibility of >45 % can be achieved while the one at 35 keV is >30 %. DPC at X-ray energies in between is possible by choosing the G1 whose design energy is closer to the desired energy and tilting it until the desired periodicity is reached. The latter can be calculated by (Jud, 2019)

pG1 =

−Lη 2 λ +

/

2 Lη 2 λ(2npG2 + Lη 2 λ)

npG2

.

(11.1)

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Table 11.3 Parameters of the gratings installed in the interferometer in end-station 2 G1 (25 keV) G1 (35 keV) G2 Grating Period

4.92 µm

Height Material Type Duty cycle Geometry Substrate

4.39 µm Ni π/2 phase-shift 0.5 ∅70 mm 525 µm Si

4.92 µm 4.89 µm at 6.3◦ 6.15 µm Ni π/2 phase-shift 0.5 ∅70 mm 525 µm Si

5 µm >70 µm Au absorption 0.5 ∅70 mm 525 µm Si

This table has been published in Günther et al. (2020), licensed under CC BY 4.0

For the available G1s (phase-shift of π/2) η = 1 and the source-to-interferometer distance L ≈ 16 m. n denotes the Talbot order (the current design is n = 1), and λ the wavelength of the X-rays. The corresponding inter-grating distance d can be approximated by (Jud, 2019) d = 9.7

mm E + 5.5 mm. keV

(11.2)

Consequently, the grating holders are mounted on a sturdy rail running along the beam axis as the inter-grating distance has to be adjusted for different energies. The rail itself is mounted on a second one running perpendicular to the beam axis which allows for sliding the whole interferometer out of the beam without impairing its alignment. The end-station has two sample manipulation systems. They are mounted on an extra support frame. This decoupling prevents negative effects of sample stage movements on the stability of the grating interferometer. One system is built for projection imaging of large specimens, e.g. pig hearts for angiography, or mastectomy specimens. Therefore this system has a breadboard as sample support which can be translated by two linear stages (LTM 120, Owis, Staufen, Germany) in the plane perpendicular to the X-ray beam axis. The key component of the second system, depicted in Fig. 11.4c, is a large stage for rotation of the sample around the Xray beam propagation axis (Goniometer 440, Huber Diffraktionstechnik, Rimsting, Germany). In combination with the grating interferometer, it enables directional dark-field imaging, so-called X-ray vector radiography (XVR) (Jensen et al. 2010). Two linear stages (LTM 80, Owis, Staufen, Germany) are attached to this rotation stage for translation of the sample perpendicular to the beam axis. The specimen itself is mounted onto a second rotation stage (DMT 65, Owis, Staufen, Germany) used for tomography and placed on top of the translation stages. Consequently, this end-station is chosen if DPC or XVR measurements are performed or large samples are imaged.

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11.6 Contributions The beamline of the MuCLS was developed and built by the whole MuCLS team, consisting—in alphabetic order—of Klaus Achterhold, Martin Dierolf, Elena Eggl, Regine Gradl, Benedikt Günther, Christoph Jud and Franz Pfeiffer. The MuCLS front-end: Martin Dierolf, Benedikt Günther and Christoph Jud. Hutch 1: Martin Dierolf, Regine Gradl and Benedikt Günther. Hutch 2: Martin Dierolf, Benedikt Günther and Christoph Jud. Design of the grating interferometer: Elena Eggl. Radiation safety: Klaus Achterhold and Martin Dierolf. Supervision: Franz Pfeiffer.

References Barthelmess M et al (2008) Status of the PETRA III insertion devices. In Proceedings of EPAC08, WEPC133 Behling R (2016) Performance and pitfalls of diagnostic X-ray sources: an overview. Med Phys Int J 4:107–114 Burger K et al (2017) Increased cell survival and cytogenetic integrity by spatial dose redistribution at a compact synchrotron X-ray source. PLoS ONE 12:e0186005 DESY (2019) PETRA III facility information 2019. https://photon-science.desy.de/facilities/petra_ iii/facility_information/index_eng.html DESY P11 (2019) Desy Petra III, P11, unified data sheet 2019. http://photon-science.desy. de/facilities/petra_iii/beamlines/p11_bio_imaging_and_diffraction/unified_data_sheet_p11/ index_eng.html Eggl E et al (2016) The Munich compact light source: initial performance measures. J Synchrotron Radiat 23:1137–1142 ESRF (2019) EBS storage ring technical report. https://www.esrf.eu/files/live/sites/www/files/ about/upgrade/documentation/Designreduced-jan19.pdf Excillum (2019) Excillum liquid metal-jet 2019. https://www.excillum.com/products/metaljetsources/metaljet-d2-160-kv/ Günther B et al (2019) Device for source position stabilization and beam parameter monitoring at inverse Compton X-ray sources. J Synchrotron Radiat 26:1546–1553. https://doi.org/10.1107/ S1600577519006453 Günther B et al (2018) The Munich compact light source: flux doubling and source position stabilization at a compact inverse-compton synchrotron X-ray source. Microsc Microanal 24:316–317 Günther B et al (2020) The versatile X-ray beamline of the Munich compact light source: design, instrumentation and applications. J Synchrotron Radiat 27:1395–1414. https://doi.org/10.1107/ S1600577520008309 Jensen TH et al (2010) Directional X-ray dark-field imaging of strongly ordered systems. Phys Rev B 82:214103 Jud C (2019) X-ray vector radiography for biomedical applications. PhD thesis, Technical University of Munich Max IV Collaboration (2010) Max IV facility: detailed design report. https://www.maxiv.lu.se/ accelerators-beamlines/accelerators/accelerator-documentation/max-iv-ddr/ Müller M et al (2017) Myoanatomy of the velvet worm leg revealed by laboratory-based nanofocus X-ray source tomography. Proc Natl Acad Sci 114:12378–12383

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Preissner M et al (2018) High resolution propagation-based imaging system for in vivo dynamic computed tomography of lungs in small animals. Phys Med Biol 63:08NT03 Procop M, Hodoroaba VD (2008) X-ray fluorescence as an additional analytical method for a scanning electron microscope. Microchim Acta 161:413–419. ISSN: 00263672 Schlomka JP et al (2008) Experimental feasibility of multi-energy photon-counting K-edge imaging in preclinical computed tomography. Phys Med Biol 53:4031–4047 Skarzynski T (2013) Collecting data in the home laboratory: evolution of X-ray sources, detectors and working practices. Acta Crystallogr Sect D: Biol Crystallogr 69:1283–1288 Wansleben M et al (2019) Photon flux determination of a liquid-metal jet X-ray source by means of photon scattering. J Anal Atom Spectrom 34:1497–1502

Chapter 12

Full-Field Structured-Illumination Super-Resolution X-ray Transmission Microscopy

Sections 12.1 and 12.2 of this chapter are based on the publication “Full-field structured-illumination super-resolution X-ray transmission microscopy”, Nature Communications 10, 2494 (2019) (Günther et al. 2019). Content licensed under CC BY 4.0. Large parts of the aforementioned sections draw text and figures directly from the publication or its supplementary material. The text of the publication has been edited in this thesis such that it incorporates the publication’s supplementary material directly. Accordingly, summaries of the supplementary material that used to be in the publication’s main text are omitted to avoid repetitions. The introduction, Sect. 12.1, has been slightly extended from the published version. The transfer of synchrotron techniques back into laboratory or industrial environments drives the research on applications of inverse Compton X-ray sources. This chapter presents the development of a new rapid microscopy technique beneficial for both, classical synchrotrons as well as inverse Compton sources. First, a proofof-principle demonstration is performed at a synchrotron before the potential and challenges of this technique are investigated at inverse Compton sources.

12.1 A New Super-Resolution X-ray Transmission Microscopy Technique X-ray transmission microscopy is a standard method for non-destructive testing at low and medium X-ray energies providing valuable insights into a specimen’s microscopic structure due to the high resolution achievable. This technique can be classified into three categories: scanning transmission X-ray microscopy (STXM), full-field transmission X-ray microscopes (TXM) and microscopes based on geometric magnification in a cone beam geometry (Cosslett et al. 1951; Bleuet et al. 2009). In © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_12

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Fig. 12.1 Comparison of a “classical” STXM (a) with the proposed multi-beamlet STXM technique (b). Simultaneous illumination of large parts of the sample becomes possible in a STXM by employing a periodic array of beams. It speeds-up image acquisition by reducing the required translational movement. Mouse drawn by Regine Gradl, used with permission

STXM, the X-ray beam is focused onto the specimen and the transmitted intensity at that particular point is recorded. Subsequently, the object is scanned through the focus whose size determines the resolution of this technique (Rarback et al. 1988). In contrast to this procedure, in full-field TXM a condenser optic creates an extended illumination on the object, which in turn is re-imaged onto a two-dimensional detector by magnifying optics (Niemann et al. 1976). If sub-micrometre resolution is to be achieved, the field-of-view of all these techniques is typically limited to less than one millimetre (Sakdinawat et al. 2010). While the field-of-view increases when combining multiple images from adjacent regions of the specimen, so does the required data acquisition time. In the following, a new technique is introduced that is conceptually different to above-mentioned state-of-the-art approaches which rely on an isolated single Xray beam: Instead of using a single isolated pencil beam, creating a whole periodic array of such beams allows for simultaneous illumination of larger parts of the specimen also in a STXM, similar in concept to multi-beam scanning electron microscopy (Mohammadi-Gheidari et al. 2010; Eberle et al. 2015). The signals from the multiple beams have to be recorded with a two-dimensional detector on which the contributions of the individual beams have to be separable. A complete image is obtained by scanning the specimen over one period of the array of illuminating beamlets. Therefore, acquisition time can be drastically reduced if these beamlets are spaced closely, i.e. on the order of micrometres. A schematic of this idea is presented in Fig. 12.1. Such a periodic X-ray intensity modulation is created by the Talbot effect (Talbot 1836; Rayleigh 1881). It is the occurrence of self-images of a periodic structure at distinct distances along the optical axis when illuminated with a spatially coherent wavefield. In the X-ray regime, this effect has been mainly exploited for interferometric phase-contrast and dark-field imaging (Momose et al. 2003; Weitkamp et al. 2005; Pfeiffer et al. 2006, 2008). Self-images of typically employed binary gratings with periods on the order of a couple of micrometres exhibit roughly the same shape as the original grating structure. While this is not sufficient for sub-micrometre resolution,

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non-binary gratings change the longitudinal as well as the transverse intensity profile of the self-images (Yaroshenko et al. 2014). The desired structured-illumination with an array of narrow peaks is generated selecting an appropriate non-binary grating structure. In general, tilting a binary grating results in an effective trapezoidal height profile (Fig. 12.5). At certain angles, this becomes triangular which produces the sharpest intensity peaks. Numerical free-space propagation of the wave-front exiting a simulated triangular grating yields the Talbot-carpet in Fig. 12.2. Five grating periods have been included in the simulation and the lateral step size along the triangular height profile is 0.01 µm. In addition, only the phase shift of the nickel grating is included into the calculation, absorption within the grating is neglected. The wave field impinging on the grating is assumed to be a coherent monochromatic plane wave with an energy of 35 keV and with an amplitude set to unity. This is a good approximation to the experimental situation because the source-to-grating distance in the experiment is more than 86 m and the X-ray source is an undulator providing a very narrow divergence angle rendering the paraxial approximation valid. The propagated wave-field is calculated employing the Fresnel approximation for near-field diffraction (Suleski 1997; Goodman 2005). The interference pattern was simulated up to a propagation distance of one Talbot-distance, which is 1.411 m at an X-ray energy of 35 keV with 2823 equidistant propagation steps. This results in a resolution of 0.5 mm in propagation direction. Additional simulations including the effects of absorption, source-blur and detector resolution are displayed in Figs. 12.7 and 12.8. The width of the peaks is calculated as the square-root of the second moment of the transverse intensity distribution. For highest resolution, the specimen should be placed in the transverse plane with minimum peak width. This plane can be extracted from Fig. 12.2b. The intensity profile in this plane is depicted in Fig. 12.2c and features main peaks with a FWHM-width of 0.41 µm. The corresponding secondmoment width of the total intensity distribution is 0.73 µm (r.m.s., here and in the following used as the sigma-value of a corresponding Gaussian distribution), even at X-ray energies as high as 35 keV. This corresponds to a full width of the central peak where the intensity dropped down to ∼10% of the peak’s intensity. Thereby, triangular gratings provide a means to create a sub-micrometre structured illumination whose resolution is entirely independent of detector pixel size as long as the latter is smaller than the grating period. This separation of resolution and detector pixel size enables super-resolution imaging.

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Fig. 12.2 Simulated Talbot-carpet of a triangular grating at an X-ray energy of 35 keV. The grating’s period is set to 5 µm and its height to 32 µm. a One full Talbot-distance of the Talbot-carpet is depicted for two simulated grating periods. b Variance perpendicular to the propagation direction. The distances for structured illumination of the object are extracted from this graph. c The transverse intensity profile at the resulting specimen position. d–f show transverse intensity profiles at certain locations within the Talbot-carpet, which are indicated by lines in the corresponding color and Roman numerals in a. This figure has been published in Günther et al. (2019), licensed under CC BY 4.0

12.2 Proof-of-Principle Study at a Synchrotron 12.2.1 The Experimental Endstation P05 at Petra III at DESY The proof-of-principle experiments were performed at an X-ray energy of 35 keV at the micro-tomography end-station at the HZG beamline P05 at PETRA III at DESY (Hamburg, Germany) in May 2017. This energy was chosen in order to demonstrate that our proposed method is compatible with hard X-rays. The experimental set-up used for all experiments is depicted in Fig. 12.3. The X-ray source size is 36 µm × 6 µm with a divergence of 28 µrad × 4 µrad at 10 keV (Wilde et al. 2016). As the X-ray source parameters are barely varying with X-ray energy, these num-

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bers are also valid at 35 keV. X-rays were monochromatised by a double crystal monochromator. The full width at half maximum horizontal extension of the X-ray beam is 5.6 mm at the sample position of the micro-tomography end-station (sourceto-sample distance 86.5 m) (Wilde et al. 2016). The sample-to-detector distance can be adjusted up to 1.4 m. The detector was a CMOS-camera developed by the Karlsruhe Institute of Technology and based on the CMOSIS CMV20000 chip. A 10x optical magnification of the scintillator, lutetium-aluminum garnet (LuAG) with a thickness of 50 µm, onto the detector resulted in an effective pixel size of 0.68 µm. The scintillator is located at the very bottom of the nose of the detector system, cf. Fig. 12.3d (on the right) and Fig. 12.3e. All experiments were performed using the same magnification, but 2 × 2 software binning of the raw data was performed for the microscopy experiment increasing the pixel size above the width of the individual illuminations produced by the Talbot-carpet. The grating holder, depicted in Fig. 12.3b, c, has been slightly modified from the standard version build by Alexander Hipp to enable its placement in front of the sample. Figure 12.3d displays the set-up of the super-resolution experiment. Figure 12.4 shows a reference image of the resolution pattern acquired with a standard parallel beam imaging geometry. The X-ray energy was set to 35 keV, the effective pixel size in this measurement was 0.64 µm and a 50 µm LuAG-scintillator was used. 100 of each, dark frames, reference frames and projections of the resolution pattern have been taken at an acquisition time of 1000 ms. 2 × 2 binning has been performed for all images. Dark frames are subtracted from all reference images as well as projections in a first step. In a second step, a standard reference correction is performed dividing the projection by the reference frame. As the X-ray beam was unstable during data acquisition, this step can result in images containing strong artefacts if the intensity fringes of the X-ray beam shifted in the projection relative to the reference. In order to find the best combination of projections and references with lowest artefacts in the region of the edge-measurements, the following procedure was used: First, reference correction for each projection was performed separately with each of the references. The corrected images with the lowest global image variance were selected in a second step. Among those, the edge fit was performed on the one with the best reference correction in the area of the edge scans, which was selected by the following criteria: First, minimal intensity difference between the background area of the region of interest for the horizontal and vertical edge measurement. Second, lowest intensity variance in vertical direction, which was determined in a background area to the left of the square and parallel to the region of interest for the vertical edge measurement. The regions of interest for the two scans are displayed by red and blue boxes in Fig. 12.4a. Detector lines perpendicular to the edge have been averaged prior to the error-function fit to the respective edge. In the vertical direction, i.e. the horizontal edge, the detector resolution is 1.45 µm (r.m.s.), whereas in the horizontal one, it is 1.30 µm (r.m.s.). The corresponding line plots and error-function fits are depicted in Fig. 12.4b.

Fig. 12.3 The experimental set-up employed in the proof-of-principle experiment. a is a schematic of the set-up. X-rays emerging from an undulator are monochromatised and propagating to the triangularly-shaped grating, which can be moved in the horizontal direction with a piezo-electric actuator. The grating holder including the actuator is presented in b and c. At a distance of 65 mm downstream the grating, the specimen (resolution test chart) is placed on a sample positioning stage. Another 5 mm away, the detector is placed. This experimental setting is depicted in d. For the Talbot-carpet measurement, the grating to detector distance is slightly reduced first before the detector is moved in steps of 1 mm away from the grating, up to a maximum grating to detector distance of 1.43 m. The complete detector tower is shown in e. Panel a of this figure has been published in Günther et al. (2019), licensed under CC BY 4.0

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Fig. 12.4 Determination of the detector resolution. a Reference corrected image of the test pattern acquired with a standard parallel beam. b Edge profiles along the vertical (red) and horizontal (blue) detector direction together with the corresponding fits. Resolution in the horizontal direction (1.30 µm) is slightly better than in the vertical one (1.45 µm). This figure has been published in the supplementary material of Günther et al. (2019), licensed under CC BY 4.0

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12.2.2 Demonstration of Structured Full-Field Illumination As a first experimental step, the Talbot-carpet of the grating with an effective triangular profile was measured. This nickel grating with a period of 5 µm, a duty cycle of 0.5 and a height of 32 µm was produced by the Karlsruhe Institute of Technology (KIT, Karlsruhe, Germany). In order to produce an effective triangular height profile, lithographic exposure was performed at an angle of 4.5◦ with respect the surface normal resulting in tilted grating lines. These grating lines are interconnected by periodically arranged small structures, so called bridges, in order to stabilise the tilted grating lines. Two grating lines are connected with bridges spaced by a constant distance, while the bridges in the respective neighbouring gaps are offset by half this distance. This generates another periodicity with twice the grating period at the position of the bridges. A sketch of the grating structure is depicted in Fig. 12.5. Geometrical constraints of the set-up allowed a minimum sample-to-detector distance of 61 mm. The Talbot-carpet was measured in steps of 1 mm up to a propagation distance of 1.425 m. Acquisition time for each frame was 250 ms and five images were acquired at each distance. Only the last three images were averaged afterwards as the first two images were severely distorted by residual vibrations of the detector stage after movement. Two scans were performed, one reference scan without the grating in the beam, and one with the grating. Dark frames were acquired, averaged and subtracted from the individual averaged frames of both scans. The final image was obtained by dividing the corresponding frames of the grating scan by the ones of the reference scan. Although this usually eliminates intensity variations in the X-ray beam very well if the delivered X-ray beam is stable, vibrations of the monochromator resulted in some remaining background inhomogeneities in our case. In addition, limited positioning accuracy of the detector resulted in slight horizontal and vertical drifts of the Talbot-carpet over more than one grating period and bridge period respectively. If the same detector line is tracked over the whole distance, such vertical drifts imply that the Talbot-image of the bridge-like structure is visible at some point instead of the one of the gratings. Bridges, as mentioned before exist between all grating lines, but are shifted in their position between two adjacent ones. Averaging over all detector rows therefore eliminates these undesired contributions at the cost of a slightly increased background reducing the depth of the intensity modulation. The resulting Talbot-carpet is shown in Fig. 12.6. The measured intensity modulation in propagation direction agrees very well with the simulated one (cf. Fig. 12.2). The slight drift of the periodic illumination along the vertical axis of the image is attributed to the limited positioning accuracy of the detector. The modulation strength of the transverse intensity profiles is characterised through their variance (Fig. 12.6b): The higher the variance, the stronger the intensity changes. The longitudinal intensity modulation agrees well with the predicted one. The measured modulation depths of the transverse profiles in Fig. 12.6d–f are much smaller than the ones simulated for an ideal situation, cf. Fig. 12.2d–f. This is the result of mainly two contributions: First, elements stabilising the grating lines, so-called bridges (Vora et al. 2007; Mohr et al. 2012), discussed in detail above, introduce a homogeneous background signal which

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Fig. 12.5 Sketch of the grating structure used in the experiment. a Sideview of the real structure of the grating. b The resulting effective trigonal structure generated by such a grating. c Top view of the grating structure visualizing the bridges. This figure has been published in the supplementary material of Günther et al. (2019), licensed under CC BY 4.0

reduces the modulation depth when averaging over the detector lines. Second, source blur and, much more importantly, the intrinsic detector point-spread-function (PSF) of about 1.3–1.5 µm (r.m.s.) (Fig. 12.4), significantly reduce the detectable intensity modulation. The effect of the source blur and bridges on the Talbot-carpet is shown in Fig. 12.7. At the position of our structured illumination (Fig. 12.7c), the source blur of ∼27 nm is very small compared to the size of the individual illuminations of ∼0.7 µm and thus barely affects the intensity modulation. If the detector resolution is included into the simulation additionally, the modulation depth apparent on the detector is reduced to 30% and the intensity enhancement to 1 tSI our approach is faster. The number of required images with our method is 100.

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Abeam = 100. Afocus

Accordingly, our method would also outperform such a hypothetical TXM covering the same field-of-view with the same resolution. Parallel-beam microscope In a conventional parallel beam geometry, the detector pixel size has to be at least 500 nm × 500 nm, which can be reached with an optical microscope magnifying the scintillator plane onto the detector. In contrast, in our approach the flux of an area of 5 µm × 5 µm, so 100 times larger than in the parallel beam case, is focused with an efficiency of 52.8 %. Compared to the unfocused beam in the parallel beam microscope, the X-ray flux in the focus is thus enhanced by a factor 52.8. In an optimised setup for our multi-beam approach, the signal from one focus is then recorded in one detector pixel. Data acquisition with our proposed scanning approach requires 10 × 10 steps to cover the whole field-of-view. Considering the flux enhancement factor of 52.8, the exposure time per image can be reduced accordingly, which results in roughly twice the acquisition time as in the conventional parallel beam geometry for a full image. However, in one such 10 × 10 scan, 100 times the field-of-view of a single parallel beam image is covered by the multi-beam approach. To reach the same field-of-view in the parallel beam geometry, 100 images have to be taken, too. Therefore our proposed method is more than 50 times faster than stitching frames from conventional parallel beam imaging with the same resolution, which requires the same number of acquisitions and thus motor movements generating the same amount of data and a similar scanning-overhead. Furthermore, the required optical microscope imaging the scintillator plane onto the detector has typically a low efficiency. Another factor of 2 in efficiency thus could be gained using a fibre-coupled scintillator for the 5 µm pixels in the structured illumination case, which is not possible for the 0.5 µm effective pixel size provided by the 10x optical magnification. As large X-ray gratings with a diameter of around 10 cm are readily available without stitching (Schröter et al. 2017), the available field-of-view and thus the gain in acquisition time or resolution elements, respectively, is mostly limited by the extension of the X-ray beam and the number of pixels of the detector. Hard X-ray microscopy Moreover, focusing of hard X-rays has been demonstrated—even well above 30 keV—down to focal spot sizes of a few hun-

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dred nanometres with our method in a very simple and compact set-up. This is quite remarkable for a sub-micron scanning approach as other focusing devices like lenses or Fresnel zone plates suffer from a low efficiency in this energy range. For compound refractive lenses, e.g. less than 10% of the incident flux is concentrated into the focus at 25 keV (Schroer et al. 2003). In contrast, the phase-shifting grating employed here makes 52.8% available within the focus width of 0.7 µm. This efficiency is calculated from the simulation considering the absorption of the nickel structures with an average height of 16 µm. Although Kirkpatrick-Baez mirrors (Kirkpatrick and Baez 1948) provide a high reflectivity in the hard X-ray energy range and sub-micron resolution (Borca et al. 2009; Martínez-Criado et al. 2012), our approach is advantageous due to the high-number of parallel illuminations, as discussed above. In conclusion, (hard) X-ray nano-tomography of specimens with an extent of several millimetres at a resolution of several hundred nanometres becomes feasible. Accordingly, for now our full-field structured-illumination super-resolution X-ray transmission microscopy approach can be seen as a complementary technique for larger specimens to the sub-50 nm STXM and TXM microscopes, which are limited to very small specimen sizes on the order of a few hundred micrometres. Combining this method with recently developed brilliant compact inverse Compton sources, whose coherence has been demonstrated to be sufficient for Talbot interferometry (Bech et al. 2009; Eggl et al. 2015; Jud et al. 2017) is straightforward. Implementation of a source grating, known from Talbot-Lau interferometry (Pfeiffer et al. 2006), will make this technique feasible also at high-power (rotating-anode) X-ray tubes significantly decreasing data acquisition time while enlarging the field-of-view compared to current systems usually based on microfocus tubes. Accordingly, this technique paves the way to high-speed sub-micrometre imaging and computed tomography of large specimen in a laboratory environment.

12.3 Transfer to Inverse Compton X-ray Sources The evaluation of the proof-of-principle experiment performed at the beamline P05 of Petra III at DESY highlighted already the possibility to employ this technique at laboratory X-ray sources. Among them, inverse Compton X-rays sources are the ones most closely related to classical synchrotrons in terms of their intrinsic spectral bandwidth and spatial coherence. Accordingly, the MuCLS is an obvious candidate for exploring the potential application of full-field structured-illumination superresolution X-ray transmission microscopy in a laboratory setting. The most important difference between an inverse Compton source and a synchrotron is the former’s increased X-ray divergence angle. This originates from the electron’s lower γ-factor required at ICSs compared to synchrotrons to generate the same X-ray energy. This has two important implications: First, the X-ray beam size is typically much larger at the experimental endstations of ICSs compared to synchrotrons. For, example, the horizontal X-ray FWHM-beam size at P05 at Petra III is about 5.6 mm at a source-to-sample distance of 86.5 m

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Fig. 12.10 a Comparison of the coherence length at P05 with the one at the MuCLS for different monochromatic X-ray energies. The much longer coherence length at P05 can be explained by the much higher source-to-sample distance and a slightly smaller source size (36 µm rms (horizontal) at P05 (Wilde et al. 2016) versus 50 µm rms at the MuCLS). b Zoom-in to the coherence length range for the two configurations at the MuCLS

(Wilde et al. 2016). In contrast, the X-ray beam diameter at the MuCLS is already ∼14 mm at the shortest source-to-sample distance of ∼3.5 m. Using the parameters of the unified data sheet of P05 at Petra III (DESY 2021), the flux density at the sample position at P05 is about three orders of magnitude higher than the one at the MuCLS. Second, reducing the source-to-sample distance at an approximately constant source size (36 µm rms (horizontal) at P05 (Wilde et al. 2016) versus 50 µm rms at the MuCLS) by a factor of about 25 significantly impairs the X-ray’s transverse coherence length. The latter is important for the interference effect producing the structured illumination. Figure 12.10a compares the rms-coherence lengths, Eq. 5.31, available at P05 (green line) to the ones achievable at the MuCLS for different monochromatic energies. The sample position at the MuCLS for which the highest flux density is reached is displayed in blue, while the orange line indicates the longest source-to-sample distance and in turn highest coherence that can be realised at the MuCLS. Evidently, the rms-coherence length at P05 at the X-ray energy of 35 keV employed in the proof-of-principle experiment is about 7 µm. At the MuCLS, it amounts to below 1 µm, cf. Fig. 12.10b, even at the furthest sample position inside hutch 2. Since the coherence length at the very front of hutch 1 is only ∼0.5 µm at 15 keV, it was decided to perform the experiment in hutch 2, although the flux density is further reduced by one order of magnitude there.

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12.3.1 Simulation of the Expected Structured-Illumination Produced by 1D- and 2D-Gratings at the MuCLS The short coherence length available at the MuCLS requires a grating with a compatible period. Accordingly, the existing triangular grating which was used in the proof-of-principle experiment at P05 naturally fulfils this constraint. However, superresolution can only be achieved in one direction as long as a line grating is employed. This could be overcome using two-dimensional gratings. Currently, several small 2Dgratings exist at the Chair for Biomedical Physics. Among them, only one grating exists with a period of 5 µm. Thus and in contrast to the others which posses larger periods, this one might be suited for generation of a structured-illumination at the MuCLS. Accordingly, the expected performance of this grating and the triangular one is evaluated in the following two subsections.

12.3.1.1

Simulation of the Structured-Illumination Produced by a Line Grating

The triangular line grating that has been presented in Sect. 12.2.2 already is evaluated for generating a structured illumination under the constraints existing at the MuCLS and discussed above. The simulations are performed analogously to the ones presented in the proof-of-principle experiment at the synchrotron. Details on the simulation procedure are provided in Sect. 12.1. Figure 12.11 displays the expected Talbot-carpet including the penumbral blur originating from an X-ray source of 50 µm rms-size at a source-to-grating distance of 15.5 m. Compared to the corresponding Talbot-carpet expected at P05, cf. Fig. 12.7, a strong reduction of the Talbotcarpet’s visibility is observed. This is unsurprising considering the much shorter spatial coherence length at the MuCLS. Nevertheless, at the position of the structured illumination, Fig. 12.11c, the illumination pattern is still preserved, albeit slightly broadened by the penumbral blur of ∼210 nm. At distances above ∼20 cm downstream the grating, the penumbral blur eliminates the intensity modulations expected for a fully coherent beam (cf. Fig. 12.2) almost completely. In the proof-of-principle experiment, the dominant factor blurring the Talbot-carped at short distances downstream the grating was the detector point-spread function of 1.5 µm. Since the PSF of the camera system at the CLS is expected to posses a similar performance, this statement should remain valid at the MuCLS as the source blur at the focus should be about 210 nm. This is validated in Fig. 12.12. Although the amplitude of the intensity modulation is reduced to ∼0.18, cf. Fig. 12.12c, compared to ∼0.3 at P05, cf. Fig. 12.8c, the modulation at the sample position at ∼65 mm is clearly perceivable in Fig. 12.12a, b. Consequently, a structured-illumination should be obtainable at an X-ray energy of 35 keV at the MuCLS. Nevertheless, from an experimental point of view, reducing the X-ray energy might be beneficial for a couple of reasons. First, the scintillator thickness of the camera system at the MuCLS is only 10 µm instead of 50 µm at P05 which significantly reduces the scintillator’s absorption. Second, the

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absorption of LSO, the scintillator material, strongly decreases at 35 keV compared to 25 keV. Combined with a flux density at the designated sample position at the MuCLS that is about four orders of magnitude lower than at P05, sticking to an X-ray energy of 35 keV may not be the most promising strategy. Therefore, another simulation of the expected illumination pattern was carried out at 25 keV including the X-ray source blur as well as the detector PSF. Its result is depicted in Fig. 12.13. Apart from aforementioned practical considerations, reducing the X-ray energy also reduces the Talbot-distance which is inversely proportional to the light’s wavelength. Thus, the first focus of the triangular line grating moves closer to the grating. In turn, the penumbral blurring of the structured illumination at

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Fig. 12.12 Simulation of the Talbot-carpet with penumbral blurring, grating bridges and, in contrast to Fig. 12.11, the assumed detector PSF at the MuCLS. The Talbot-carpet of a triangular grating with a period of 5 µm and a height of 32 µm is simulated at an X-ray energy of 35 keV including X-ray absorption, the penumbral blur, the effect of the grating bridges as well as the detector PSF. For the latter, the same PSF of ∼1.5 µm (r.m.s.), which was measured for the detector at P05, was assumed. a depicts one full Talbot-distance of the Talbot-carpet for two grating periods. b Variance perpendicular to the propagation direction. c–f display transverse intensity profiles at the same location within the Talbot-carpet as Fig. 12.2. Even for a fully coherent beam the amplitude of the oscillation is reduced to ∼0.18 and the observable intensity enhancement basically vanishes

the designated sample position is reduced. Consequently, the amplitude of the intensity modulation increases compared to the 35 keV-case from ∼0.18 to ∼0.25, cf. Fig. 12.13c. Accordingly, the modulation depth obtained for the proof-of-principle experiment at P05 should be almost recovered. However, no net intensity enhancement can be achieved due to increased X-ray absorption by the grating at 25 keV. At 25 keV, the highest visibility should be reached at a distance of ∼35 mm downstream the grating, which is indicated by the position of the first peak in the variance, Fig. 12.13b. Figure 12.13d is at a position with an ideal intensity structure similar to the one at 0.35 m in the 35 keV-case displayed in Fig. 12.2a. As a result, this modulation vanishes almost completely when penumbral blur and detector PSF are

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normalised intensity

c

transverse profile [μm]

transverse profile [μm]

Fig. 12.13 Simulation of the Talbot-carpet with penumbral blurring, grating bridges and assumed detector PSF at the MuCLS at an X-ray energy of 25 keV. The Talbot-carpet of a triangular grating with a period of 5 µm and a height of 32 µm is calculated including X-ray absorption, the penumbral blur, the effect of the grating bridges as well as the detector point-spread function. For the latter, the same PSF of ∼1.5 µm (r.m.s.), which was measured for the detector at P05, was assumed. a depicts one full Talbot-distance of the Talbot-carpet for two grating periods. b Variance perpendicular to the propagation direction. c–f display transverse intensity profiles at various exemplary locations within the Talbot-carpet. Compared to the corresponding 35 keV-case, the amplitude of the oscillation increases to ∼0.25 at the position of the hypothetical sample, although increased absorption reduces the observable peak intensity further

considered. Figures 12.13e, f are at locations corresponding to the ones displayed in Fig. 12.2e, f. The improved visibility of the Talbot-carpet at 25 keV compared to 35 keV in conjunction with higher scintillator performance indicate that an X-ray energy of 25 keV should be chosen for generation of a structured illumination at the MuCLS. Lower X-ray energies, especially below 20 keV are not ideal since the performance of the CLS drops strongly compared to 25 keV. Thus, an X-ray energy of 25 keV appears to be the best compromise in terms of coherence and X-ray flux that the CLS can deliver for this particular application.

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Simulation of the Structured-Illumination Produced by a 2D-Grating

Section 12.2.4 already discussed that a two-dimensional grating enables superresolution imaging in two dimensions. Currently, the Chair for Biomedical Physics owns a few binary 2D-gratings designed for 10 keV, 15 keV and 20 keV with periods of 5.0 µm, 6.8 µm, 10 µm and 13.6 µm. All gratings are fabricated from silicon. Their duty cycle is 0.33 and their phase shift is 2π/3. Such gratings compress the X-rays into a binary focus with an extent of one third of the grating’s period at a propagation distance of one sixth of the Talbot-distance (Gustschin et al. 2021). More details on the gratings are presented in Gustschin et al. (2021) and its supplementary material. Among them, the ones fabricated for X-ray energies of 20 keV would be the most suited for structured-illumination super-resolution microscopy at the MuCLS albeit their design energy below the optimal working point of the CLS. Furthermore, small periods should be chosen due to the MuCLS’s much lower spacial coherence compared to synchrotrons. Similar to the one-dimensional line grating, its expected performance was assessed using the same Fresnel-propagation scheme and the same geometry, i.e. a sourceto-grating distance of 15.5 m. The penumbral blur as well as an assumed detector PSF of 1.5 µm were considered, while absorption was neglected in the simulation of the 2D-grating’s Talbot-carpet. Figure 12.14a displays the Talbot-carpet over one Talbot-distance. A clearly modulated intensity pattern is preserved albeit penumbral blur and detector PSF. At propagation distances larger than 0.2 m, the increasing penumbral blur extinguishes any intensity modulations expected from the Talboteffect, cf. Fig. 12.14b. The variance transverse to the X-ray propagation direction was normalised to the respective maximum. Therefore, no decrease in the peak of the variance is visible in Fig. 12.14b. The latter effect is clearly visible in Fig. 12.15, which displays cuts through the focal plane located at a distance of 0.134 m downstream the grating, which corresponds to the first peak of the variance in Fig. 12.14b. Figure 12.15a is a cut through the focal plane of the ideal fully-coherent self-image, Fig. 12.15b includes the penumbral blur and Fig. 12.15c both, penumbral blur and detector PSF. Figure 12.15d, e are line plots along the horizontal and vertical directions for the three cases of Fig. 12.15a–c. Although the visibility is strongly impaired mainly by the detector PSF, the amplitude of the intensity modulation is about 0.4 if both effects are included. Accordingly, the modulation strength expected for the 2D-grating is slightly larger than the one of a triangular line grating at 25 keV and 35 keV. The second-moment r.m.s-width of the triangular grating’s ideal intensity distribution at the focus is 0.73 µm which corresponds to a FWHM of 1.7 µm (more precisely, the D4σ-diameter should be used for the full beam diameter). The 2Dgrating’s binary intensity distribution’s hypothetical “FWHM” is 1.7 µm as well, but its intensity fall-off is different. While the triangular line grating contains multiple side-peaks and thus a long-range intensity fall-off, cf. Fig. 12.2c, the 2D-grating exhibits a very localised intensity distribution with almost binary edges. Therefore, the modulation depth is reduced in the case of the triangular grating.

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Fig. 12.14 Simulation of a 2D-grating’s Talbot-carpet with penumbral blurring and assumed detector PSF at the MuCLS. The Talbot-carpet of a 2D-grating with a period of 5 µm, a duty cycle of 0.33 and a phase-shift of 2π/3 is simulated at an X-ray energy of 20 keV including the penumbral blur and the detector point-spread function. For the latter, the same PSF of ∼1.5 µm (r.m.s.), which was measured for the detector at P05, was assumed. a depicts one full Talbot-distance of the Talbot-carpet for five grating periods. The intensity is normalised to the one at the grating plane. b Variance perpendicular to the propagation direction for the ideal coherent situation (blue line), the situation in which only penumbral blur is present (orange line) and the realistic situation including penumbral blur and detector PSF (green line). Each of the variances is normalised to its respective maximum

Although such a two-dimensional grating may be a good generator of a structuredillumination for super-resolution microscopy, experiments with the 2D-grating had been postponed for two reasons for the time being: First, results obtained with the line grating can be compared to the ones of the proof-of-principle experiment at the synchrotron directly. Second, this microscopy approach is very flux hungry. Since the X-ray flux scales approximately linearly with X-ray energy at the CLS, the gain in flux outweighs the small increase in visibility that may be expected for a twodimensional grating. If demonstration of a structured illumination with the linear grating is successful, a dedicated 2D-grating with 2π/3 phase-shift at an X-ray energy of 25 keV could be fabricated since simulations indicate a similar performance at this energy.

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Fig. 12.15 Simulation of a 2D-grating’s Talbot-carpet at the MuCLS. The Talbot-carpet of a 2Dgrating with a period of 5 µm, a duty cycle of 0.33 and a phase-shift of 2π/3 is calculated at an X-ray energy of 20 keV. a is a transverse slice through the Talbot-carpet at the focus position (one sixth of the Talbot distance) for the ideal coherent case. b is the same slice, but including the penumbral blur and the slice depicted in c includes both, penumbral blur and detector PSF. The intensity is normalised to the one at the grating plane. d–e display line plots along the horizontal and vertical intensity modulations in a–c. Their positions are outlined in a–c. While they are symmetric in the horizontal and vertical direction, the main contribution to the amplitude reduction is the detector PSF

12.3.2 Demonstration of Structured-Illumination at the MuCLS The triangular line grating employed in the proof-of-principle demonstration, Sect. 12.2.3, generates the structured-illumination at the MuCLS as well. Figure 12.16 displays the set-up designed and built for this purpose. A schematic of the set-up including all relevant distances is displayed in Fig. 12.16a. The set-up itself was built inside the second endstation of the MuCLS.1 Moreover, due to the high beam height above the optical table, the whole set-up resides on two bridges connected rigidly by a breadboard. The latter bears the linear stage required to move the camera system along the beam propagation direction (Fig. 12.16c). The grating is sustained by an overhead construction. Its position along the three spatial directions can be adjusted 1

To facilitate its implementation, the grating interferometer had to be removed.

Fig. 12.16 Experimental set-up for the demonstration of structured-illumination at the MuCLS. a Schematic of the set-up at the MuCLS. The X-rays produced by the CLS propagate to the triangular grating located 15.5 m downstream the X-ray source. The detector can be moved up to 30 cm downstream the grating. Optionally, a sample holder mounted onto a 3D-piezoscanner (P-562.3CD, Physik Instrumente GmbH, Karlsruhe, Germany) can be installed onto the breadboard visible in b. b Grating holder and X-ray detector. c displays the set-up as implemented into the second endstation at the MuCLS. d is an image of the set-up including a sample-holder

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manually with micrometer screws over a range of 25 mm, cf. Fig. 12.16b. The grating lines were oriented horizontally. The camera system visible on the right hand side of this image consists of a 10 µm thick LSO single-crystal scintillator, a 10× magnifying visible light optic, a turn mirror and an Andor Neo SCMOS camera. The first three components are contained in the light-tight black box on which the camera resides. Figure 12.16d displays an envisioned realisation of structured-illumination super-resolution X-ray transmission microscopy at the MuCLS. For the acquisition of the Talbot-carpet, the camera system was moved close to the grating such that the front of the camera system was located about 5 mm downstream the grating. Starting from this position, the Talbot-carpet was scanned over 15 cm in steps of 1 mm. Although the length of the stage would allow gratingto-detector distances up to 30 cm, the full range was not used. On the one hand, the position of the narrowest foci is expected 32 mm downstream the grating, on the other hand, the acquisition times required for one image rendered longer distances infeasible. The absolute minimum acquisition time at which image statistics may be still sufficient was 300 s. Consequently, one Talbot-carpet scan over this range requires 12.5 h of acquisition time alone, without overhead due to motor movement. A slice through the resulting Talbot-carpet is shown in Fig. 12.17a. All projections were dark-current and flat-field-corrected. The resulting images were rotated by 2.3◦ to correct for the remaining slight rotation of the grating lines. Furthermore, the mean along the grating lines was taken to improve statistics. A strong downward drift of the Talbot-carpet’s intensity modulation pattern in the vertical direction is present. This behaviour is attributed to the fact that the grating lines were aligned horizontally and the height of the two bridges was set manually. Thus a slight mismatch in their

Fig. 12.17 Talbot-carpet scan around the focal region at MuCLS. a Talbot-carpet over a range of 15 cm relative to the starting position with a grating-to-detector distance of ∼5 mm. The strong shift of the Talbot-carpet in the vertical image direction (about 3.3 pixel per step or this value multiplied with the grating period) is attributed to a vertical shift of the detector over its movement along the beam propagation direction. b Variance of the intensity modulation. Its peak occurs at a relative distance of ∼16 mm. Considering the initial grating-to-detector distance of ∼5 mm and the expected peak position at 32 mm, the scintillator must be located about 1 cm behind the detector’s entrance window

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Fig. 12.18 Transverse slice of the Talbot-carpet at its focal position at the MuCLS. a Image of the region of the Talbot-carpet at the focus indicated by the green square in Fig. 12.20a. b Line plot of the transverse intensity modulation of the first 10 periods in a. The mean along the direction of the grating lines was taken in order to improve statistics

height results in a slight linear shift of the illumination on the detector over the longitudinal scan direction. Moreover, the grating holder may move slightly as well during this long time span. Nevertheless, the variance of the intensity modulation, Fig. 12.17b, exhibits the expected shape since it is not affected by transverse shifts of each longitudinal slice. Its first peak occurs at a relative distance of ∼16 mm from the starting position. Considering the initially estimated grating-to-detector distance of ∼5 mm and the expected peak position at 32 mm, the scintillator must be located about 1 cm behind the detector’s entrance window. The visibility of an intensity modulation indicates already the existence of a structured-illumination pattern. Figure 12.18 proves this statement since the one-dimensional structuredillumination can be clearly recognised in Fig. 12.18a which is further supported by the strong peaks of the vertical frequency at 0.2 MHz, Fig. 12.20b, corresponding to a 5 µm-period of the intensity modulation. The region displayed in Fig. 12.18a is marked with a green rectangle in Fig. 12.20a. The mean along the grating lines of Fig. 12.18a is displayed for ten periods in Fig. 12.18b. The retrieved amplitude of the modulation is about 0.1 instead of the expected modulation depth of approximately 0.25. This reduction is understood if the actual detector resolution, analysed in Fig. 12.19, is considered. Instead of the assumed PSF of 1.5 µm identical to the one of the camera system at P05, the camera system at the MuCLS exhibits a resolution of 3.9 µm transverse to the grating lines. This deteriorates the attainable amplitude modulation significantly and explains the reduction of the measured amplitude of the intensity modulation compared to the simulation. Moreover, the resolution is not homogeneous across the detector, cf. Fig. 12.20. Already in the full image of the detector, Fig. 12.20a, a strong blur of the intensity modulation is visible, which is even more prominent if a line plot transverse to the illumination pattern is considered. The one displayed in Fig. 12.20c corresponds to the region marked in bright blue, while the one in Fig. 12.20d corresponds to the one marked in dark blue in

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Fig. 12.19 Determination of the Andor Neo detector’s resolution. a Dark-current and flat-field corrected image of the big square deposited on the X-radia resolution pattern (Xradia, Pleasanton, USA). The regions of interest for the error-function fit were located above the orange line in Fig. 12.20a, i.e. where the visibility of the intensity modulation is still good. b Mean of the regions of interest along their respective edge and the corresponding error-function fits

Fig. 12.20a. For good image quality of structured-illumination super-resolution Xray transmission microscopy, the intensity modulation should be homogeneous and as strong as possible. Only the top third of the image fulfils this requirement, which would restrict the effective field-of-view to about 400 µm in the vertical direction indicated by the orange line in Fig. 12.20a, c.

12.3.3 Conclusion for Structured-Illumination Super-Resolution X-ray Transmission Microscopy at the MuCLS Generation of a structured-illumination pattern could be demonstrated at the MuCLS at an X-ray energy of 25 keV with a triangular line grating. This—in principle— enables structured-illumination super-resolution X-ray transmission microscopy at inverse Compton sources if a suitable detector is employed. However, several aspects currently render this technique impractical. First and most importantly, the flux density provided by inverse Compton sources is orders of magnitude to low for efficient image acquisition. All measurements required 300 s acquisition time and averaging over multiple detector rows, which is possible if only information on the onedimensional structure of the intensity modulation is desired. In contrast, the statistic within one pixel needs to be sufficient for imaging applications which will significantly increase acquisition time to probably above 1000 s per image. Although the CLS was operating at about half of the typical X-ray flux at the time this experiment was carried out due to cancelled service visits during the COVID-19 pandemic, this renders microscopy infeasible even under normal conditions. Moreover, the detector system that is currently available has a very broad PSF and exhibits a very

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Fig. 12.20 Analysis of the homogeneity of the structured-illumination. a The complete transverse slice of the Talbot-carpet at the position of the structured-illumination. It was rotated by 2.3◦ to align the grating lines horizontally. b Frequency spectrum of a. Peaks at the grating period of 5 µm, i.e. a frequency 0.2 MHz, are prominent in the vertical direction indicating the generation of the expected structured-illumination. c and d Line plots transverse to the intensity modulations. The regions of interests are colour-coded in a. For better statistics, the mean in the direction along the grating lines inside the respective region of interest was taken. A comparison of c and d yields that only the first third of the field-of-view resolves the structured-illumination homogeneously. Especially the lower left side suffers from an extreme reduction of detector resolution

inhomogeneous resolution narrowing down the field-of-view significantly. Since the detector system contains an optical microscope, it is quite inefficient compared to fiber-coupled detector systems, as discussed in Sect. 12.2.4. Obviously, the overall efficiency of this microscopy technique could be significantly raised if a detector with a suited pixel size matched to the period of the used grating was acquired at the MuCLS or the other way round as discussed in Günther et al. (2019). The feasibility of this approach was demonstrated by Mamyrbayev et al. using refractive lens arrays (Mamyrbayev et al. 2019, 2021). This would be an important step towards a successful implementation of this technique at inverse Compton sources, since it will significantly boost the image statistics, even at current X-ray flux densities. At least, it lowers the bar for flux improvements on the X-ray source side. It could be tried to implement this technique in the first endstation which increases the flux density

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by an order of magnitude. However, it remains questionable if the available coherence length at these short source-to-grating distances is sufficient to create a usable structured illumination. In conclusion, given the current circumstances, i.e. the low X-ray flux density in combination with a very inefficient non-ideal detector system, it does not make sense to further implement structured-illumination super-resolution X-ray transmission microscopy at the MuCLS.

12.4 A Brief Outlook on Data Analysis in Frequency-Space This short section closes the chapter about structured-illumination super-resolution X-ray transmission microscopy with a brief outlook on a potential reconstruction technique which could be an alternative to the scanning transmission X-ray microscopy-type reconstruction. It is based on an analysis of different images acquired with different known phase shifts of the structured-illumination. This kind of analysis has been employed in visible light fluorescence microscopy first in the late 1990s (Heintzmann and Cremer 1999; Gustafsson 2000). Gustafsson (2005) also demonstrated that a non-linear structured-illumination can theoretically provide unlimited resolution. He extended the resolution improvement to three dimensions as well using the Talbot-effect for creating the required three-dimensional structured illumination (Gustafsson et al. 2008). A very up-to-date overview about the developments of structured-illumination fluorescence microscopy is provided by Masters (2020, “Superresolution Optical Mircroscopy”, Chap. 13). Here, the method employed by Heintzmann and Cremer (1999) is used for image reconstruction since it is well suited for the data-set acquired in the proof-of-principle experiment performed at DESY. The same data-set that was analysed in Sect. 12.2.3 is shown in the following. The illumination was phase shifted over 2π in 20 intervals. Remember that only every second image was included in the analysis of Sect. 12.2.3. In contrast, two sets of conjugated images are required for the analysis proposed by Heintzmann and Cremer (1999). One set, namely one image with a reference phase of 0 of the structured illumination and the one with a relative phase of π are contained in the sub-set for the STXM-analysis. The other conjugated set which is phaseshifted by π/2, i.e. acquisitions with relative phases of π/2 and 3π/2, is included in the set discarded in the STXM-analysis. Those four images were selected from the projections as well as from the reference images, corrected for dark current and binned by a factor of 2 to generate the same effective pixel size that was employed in the STXM-analysis. The projections and references were analysed individually first, before reference correction was performed. In the following, the individual processes of the analysis according to Heintzmann and Cremer (1999) are explained. More details on the process itself are available in the original publication. Each step of the

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Fig. 12.21 Steps 1 to 2 of the data analysis in frequency space. The left column contains the realspace images and the right one the logarithm of the magnitude of the Fourier-transformation. a, b Projection image at the reference phase. c, d Subtraction of the π-image from the one at reference phase. e, f Subtraction of the 3π/2-image from the π/2-image

reconstruction process is depicted in Figs. 12.21, 12.22 and 12.23 step by step for the projection image which contains the resolution pattern. In addition to the real space image, the magnitude of the Fourier-transformation is displayed for each step as well.

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Fig. 12.22 Steps 3 to 5 of the data analysis in frequency space. The left column contains the realspace images (if complex-valued, the real part is displayed) and the right one the logarithm of the magnitude of the Fourier-transformation. a, b Fig. 12.21e multiplied with the complex unit i and subtracted from Fig. 12.21c. c, d The sideband was shifted back to zero-frequency by multiplying the real-space image a with ei(kr+ϕ0 ) . e, f The complex conjugate of c added to itself

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Fig. 12.23 Steps 6 and 7 of the data analysis in frequency space. The left column contains the real space images (if complex-valued, the real part is displayed) and the right one the logarithm of the magnitude of the Fourier-transformation. a, b A weighted sum of all four raw real-space images was added to Fig. 12.22e to increase the suppressed low frequency content. c, d The final reconstruction after correction with the reference image

12.4.1 Reconstruction Process 1. Subtracting the π-image from the reference image eliminates the zero-order frequency peak, cf. Fig. 12.21c, d. Accordingly, the two frequency peaks left are those associated with the structured illumination. 2. Subtracting the 3π/2-image from the π/2-image eliminates the zero-order frequency peak in the shifted set, cf. Fig. 12.21e, f. 3. Subtraction of the result of step 2 multiplied with the complex number i from the result of step 1 leaves only one peak of the structured illumination, cf. Fig. 12.22a, b. 4. Shift of the image in frequency space by multiplication of the real space image with the phase-gradient ei (kr+ϕ0 ) , cf. Fig. 12.22c, d. k is the wave vector of the structured illumination and the global phase ϕ0 was chosen such that the zerofrequency peak in the shifted image is real and positive. 5. Addition of the complex conjugate of Fig. 12.22c to itself in real space, cf. Fig. 12.22e, f.

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6. The sum of all four raw real-space images weighted with an empirically determined factor of 1/26 is added to Fig. 12.22e. This ensures that only small negative values occur in the final reconstructed image, Fig. 12.23a, b. 7. After steps 1–6 are performed for the projections and the references, the final projection image is reference corrected. Figure 12.23c, d depicts the final result. Compared to the raw image, Figs. 12.21b and 12.23b exhibits an increased frequency content along the horizontal direction, i.e. the direction in which the stepping was performed. Moreover, the strong Moiré-fringes in the horizontal line pattern around a line width of 2.5 µm, i.e. a period of 5 µm which is equal to the period of the illumination, vanish in the reconstructed image indicating a successful retrieval of the high-frequency information. However, the bridges stabilising the grating lines get imprinted into the retrieved image, cf. Fig. 12.23c, d. Since their horizontal spatial frequency is half the one of the grating itself (and harmonics) they could be filtered out with an appropriate filter in frequency space. Alternatively, different grating designs could be used that do not require bridges for their stability. One example of such gratings are the two-dimensional gratings simulated in Sect. 12.3.1.2, which have been successfully fabricated and employed in another study by colleagues at the Chair of Biomedical Physics (Gustschin et al. 2021). The main goal of this brief outlook was to demonstrate the feasibility of this analysis technique for X-ray-based structured-illumination microscopy. Nevertheless, only a linear reconstruction algorithm was employed in the preceding demonstration which limits the improvement of resolution to a factor of two at maximum (Gustafsson 2005). Considering the fact that both, the triangular line grating as well as binary two-dimensional gratings contain non-linear terms in their Fourier series, the resolution could be further improved with reconstruction algorithms that exploit this fact. One such procedure is detailed in Ingerman et al. (2019), for example. Although yielding good results in optical microscopy, this kind of analysis has not been transferred to the X-ray regime, so far. Therefore, it may be worthwhile to continue with a profound investigation of X-ray-based non-linear structured-illumination microscopy in the future. Currently, the best choice for these kind of studies may be synchrotrons, like Petra III, due to their high flux. One good beamline may be P05 of Petra III since the necessary infrastructure, a precise system for two-dimensional stepping of a grating, exists already.

12.5 Contributions Sections 12.1 and 12.2 of this chapter have been realised with contributions by Lorenz Hehn, Christoph Jud, Alexander Hipp and Martin Dierolf. The proof-of-principle experiment was carried out at PETRA III at DESY, a member of the Helmholtz Association (HGF). I would like to thank the Helmholtz Zentrum Geesthacht (HZG), namely Felix Beckmann, Fabian Wilde and Jörg Hammel for their assistance in using P05. Alexander Hipp, HZG, assisted in adjusting the grating stepper system at P05

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to the needs of our experiment. The experiments were performed by Martin Dierolf, Benedikt Günther, Lorenz Hehn and Christoph Jud. Lorenz Hehn also contributed to the signal extraction part of the multi-beam scanning transmission X-ray microscopylike data analysis for Sect. 12.2.3. Section 12.3 of this chapter has been realised with contributions by Johannes Brantl and Martin Dierolf. They helped constructing the set-up for the measurements and performing the measurements presented in this section.

References Bech M et al (2009) Hard X-ray phase-contrast imaging with the Compact Light Source based on inverse Compton X-rays. J Synchrotron Radiat 16:43–47 Bleuet P et al (2009) A hard X-ray nanoprobe for scanning and projection nanotomography. Rev Sci Instrum 80:056101 Borca CN et al (2009) The microXAS beamline at the swiss light source: towards nano-scale imaging. J Phys Conf Ser 186:012003 Cosslett VE, Nixon WC (1951) X-ray shadow microscope. Nature 168:24 DESY P05 (2021) DESY Petra III, P05, Unified data sheet 2021. http://photon-science.desy.de/ facilities/petra_iii/beamlines/p05_imaging_beamline/unified_data_sheet_p05/index_eng.html Eberle AL et al (2015) High-resolution, high-throughput imaging with a multibeam scanning electron microscope. J Microsc 259:114–120 Eggl E et al (2015) X-ray phase-contrast tomography with a compact laser-driven synchrotron source. Proc Natl Acad Sci 112:5567–5572 Goodman J (2005) Introduction to Fourier optics, 3rd edn. McGraw-Hill Guizar-Sicairos M et al (2014) High-throughput ptychography using Eiger-scanning X-ray nanoimaging of extended regions. Opt Exp 22:14859 Günther B et al (2019) Full-field structured-illumination super-resolution X-ray transmission microscopy. Nat Commun 10:2494. https://doi.org/10.1038/s41467-019-10537-x Gustafsson MG Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy. J Microsc 198:82–87 Gustafsson MG (2005) Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution. Proc Natl Acad Sci USA 102:13081–13086 Gustafsson MG et al (2008) Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination. Biophys J 94:4957–4970 Gustschin A et al (2021) High-resolution and sensitivity bi-directional X-ray phase contrast imaging using 2D Talbot array illuminators. Optica 8:1588–1595 Heintzmann R, Cremer CG (1999) Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating. In: Proceedings of the SPIE optical biopsies and microscopic techniques III, vol 3568, pp 185–196 Ikematsu K et al (2017) A study of phase-grating shapes to improve spatial resolution in X-ray Talbot interferometry. In: Book of Abstracts XNPIG 2017 Ingerman EA et al (2019) Signal, noise and resolution in linear and nonlinear structured-illumination microscopy. J Microsc 273:3–25 Jud C et al (2017) Trabecular bone anisotropy imaging with a compact laser-undulator synchrotron X-ray source. Sci Rep 7:14477 Kirkpatrick P, Baez AV (1948) Formation of optical images by X-rays. J Opt Soc Am 38:766–774 Mamyrbayev T et al (2019) Super-resolution scanning transmission X-ray imaging using single biconcave parabolic refractive lens array. Sci Rep 9:14366

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Mamyrbayev T et al (2021) Staircase array of inclined refractive multi-lenses for large field of view pixel super-resolution scanning transmission hard X-ray microscopy. J Synchrotron Radiat 28:732–740 Martínez-Criado G et al (2012) Status of the hard X-ray microprobe beamline ID22 of the European synchrotron radiation facility. J Synchrotron Radiat 19:10–18 Masters BR (2020) Super resolution optical microscopy. Springer Nature Switzerland AG. ISBN: 978-3-030-21691-7 Mohammadi-Gheidari A, Hagen CW, Kruit P (2010) Multibeam scanning electron microscope: experimental results. J Vac Sci Technol B Nanotechnol Microelectron Materials, Process Meas Phenom 28:1071–1023 Mohr J et al (2012) High aspect ratio gratings for X-ray phase contrast imaging. In: AIP conference proceedings, vol 1466, pp 41–50 Momose A et al (2003) Demonstration of X-ray Talbot interferometry. Jpn J Appl Phys 42:L866– L868 Niemann B, Rudolph D, Schmahl G (1976) X-ray microscopy with synchrotron radiation. Appl Opt 15:1883–1884 Pfeiffer F et al (2008) Hard-X-ray dark-field imaging using a grating interferometer. Nat Mater 7:134–137 Pfeiffer F et al (2006) Phase retrieval and differential phase contrast imaging with low-brilliance X-ray sources. Nat Phys 2:258–261 Rarback H et al (1998) Scanning X-ray microscope with 75-nm resolution. Rev Sci Instrum 59:52– 59 Rayleigh L (1881) On copying diffraction gratings and on some phenomenon connected therewith. Philos Mag 11:196 Rössl E, Köhler T (2013) Differential phase-contrast imaging with focusing deflection structure plates. Patent US 2013/0315373 Al Rutishauser S et al (2013) Fabrication of two-dimensional hard X-ray diffraction gratings. Microelectron Eng 101:12–16 Sakdinawat A, Attwood D (2010) Nanoscale X-ray imaging. Nat Photon 4:840–848 Schroer CG et al (2003) Nanofocusing parabolic refractive X-ray lenses. Appl Phys Lett 82:1485– 1487 Schröter TJ et al (2017) Large field-of-view tiled grating structures for X-ray phase-contrast imaging. Rev Sci Instrum 88:015104 Suleski TJ (1997) Generation of Lohmann images from binary-phase Talbot array illuminators. Appl Opt 36:4686–4691 Talbot HF (1836) Facts relating to optical science. Philos Mag 9:401 Vora KD et al (2007) Fabrication of support structures to prevent SU-8 stiction in high aspect ratio structures. Microsyst Technol 13:487–493 Weitkamp T et al (2005) X-ray phase imaging with a grating interferometer. Opt Exp 13:6296–6304 Wilde F et al (2016) Micro-CT at the imaging beamline P05 at PETRA III. In: AIP conference proceedings, vol 1741, p 030035 Yaroshenko A et al (2014) Non-binary phase gratings for X-ray imaging with a compact Talbot interferometer. Opt Exp 22:547–556 Zanette I et al (2010) Two-dimensional X-ray grating interferometer. Phys Rev Lett 105:248102

Chapter 13

X-ray Techniques and Applications at the MuCLS

This chapter is published as Sects. 5–8 in “The versatile X-ray beamline of the Munich Compact Light Source: design, instrumentation and applications”, Journal of Synchrotron Radiation, 27, 1395–1414 (2020) (Günther et al. 2020). Content licensed under CC BY 4.0. This chapter provides an overview of the different X-ray applications and techniques that can be performed at the Munich Compact Light Source. First, the available imaging techniques are introduced and evaluated with a strong focus on phasecontrast imaging. Afterwards, X-ray absorption spectroscopy as well as microbeam radiation therapy at the Munich Compact Light Source are explained. Finally, a brief outlook on planned upgrades to the beamline of the Munich Compact Light Source is provided.

13.1 X-ray Imaging at the MuCLS This section provides a brief overview on the different X-ray imaging techniques that are currently available at the MuCLS and presents some application examples thereof.

13.1.1 X-ray Microtomography With the goal of combining the benefits of the ICS of the MuCLS with fast dynamic imaging, our high-resolution imaging and microtomography (micro-CT) efforts have been focusing on the ∼10 µm resolution range. For this, we use very efficient detec© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_13

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tor systems based on fibre-coupled scintillators instead of optical coupling. Since the X-ray beam divergence is 4 mrad, geometric magnification is very low. On the one hand, this inhibits implementation of projection microscopy for this resolution range as often used in cone-beam micro-CT systems with detectors with large pixels, e.g. hybrid pixel array detectors. On the other hand, a low divergence beam allows for long source-to-sample distances. This increases the coherence length without impairing flux density on the sample and is thus beneficial for phase-contrast imaging. Phase imaging techniques are discussed in more detail in Sects. 13.1.3 and 13.1.4. As a result, X-ray microtomography at the MuCLS mainly benefits from the increased sensitivity (Töpperwien et al. 2018) and flux density, while in addition the X-ray source’s quasi-monochromatic spectrum prevents beam-hardening artefacts (cupping) in computed tomography (CT) (Achterhold et al. 2013). An example where the latter benefits come into play is presented in Fig. 13.1 depicting a sturgeon fish head, which was treated with a molybdenum-based stain (Handschuh et al. 2017) before the micro-CT measurement. At an X-ray energy of 25 keV, 2500 projections, each with an exposure time of 0.4 s, were acquired over 360◦ with the Ximea camera, cf. Table 11.1. An axial and a sagittal slice of the volume reconstructed with

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Fig. 13.1 Micro-CT (2500 projections over 360◦ , exposure time 0.4 s/projection) of a sturgeon fish head acquired at an X-ray energy of 25 keV with the Ximea detector (9 µm pixel size). a and b are axial and sagital slices reconstructed applying filtered back-projection. c and d are volume renderings with direction of view corresponding to a and b. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

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filtered back-projection are shown in Fig. 13.1a and b respectively. Figure 13.1c, d are volume renderings with directions of view corresponding to Fig. 13.1a, b. Absence of beam-hardening artefacts can be inferred from Fig. 13.1a. Moreover, the features of interest of the inner fish anatomy are resolved very well at high contrast, cf. Fig. 13.1a, b, which provides an example for the excellent capabilities of our microtomography system for applications which require a resolution of ∼10 µm. Notwithstanding the MuCLS’s great performance in microtomography, two properties of the X-rays produced at ICS make high-resolution microscopy, which either relies on coherent X-rays, like coherent diffractive imaging and ptychography, or on focusing optics (scanning transmission X-ray microscope, projection microscope and full-field transmission X-ray microscope (TXM)), not that straightforward. First, X-ray flux at an ICS is significantly lower than at synchrotrons, which requires very efficient optics in order to generate high-resolution images at short acquisition times. Second, the divergence angle of X-rays produced at an ICS is orders of magnitude larger (mrad instead of µrad). This requires the optics to be placed either very close to the X-ray source or to be quite large in order to intercept the whole beam. In the case of the MuCLS, this is complicated by the fact that its ICS is based on a storage ring for the electron beam and an enhancement cavity for the laser. Accordingly, no X-ray optic may be placed inside the laser cavity, limiting the minimum distance of the optic to the X-ray source to ∼1.4 m, where the beam diameter is already nearly 6 mm. This is far too large for Fresnel zone-plates and typical compound refractive lenses. Lyncean Technology Inc. has developed a multilayer-based KBmirror system, which allows 1:1 refocusing and in principle could be used as, e.g. a condensor for a TXM or for crystallography. However, if the distance between the machine and the wall of the radiation projection enclosure is increased the closest distance at which the unfocused beam can be used gets also larger, which results in a—potentially undesirable—larger beam diameter. To circumvent this problem, Lyncean’s current design places the optics into the radiation protection enclosure wall itself at a source to mirror distance of ∼2 m. The required large hole in the shielding wall raised radiation protection concerns at our institute, whereupon this system was not acquired together with the source. Without a condensor optic, image acquisition at an X-ray energy of 25 keV took at least 300 s with a Rigaku XSight Micron LC CCD-camera set to an effective pixel size of 0.54 µm in order to get an acceptable signal-to-noise ratio. However, considering the current X-ray flux after several upgrades, the acquisition time should be lower by a factor of three at least. Nevertheless, optical magnification significantly narrows down the field-of-view, in this case to 1.80 × 1.36 mm2 , whereas the X-ray beam itself is 18 mm in diameter. A maximum flux of 3 × 1010 ph/s at 25 keV corresponds to a flux density of ∼118 ph/s/µm2 on the detector which covers only 0.96% of the X-ray beam at this distance. Based on these considerations, high-resolution microscopy with either Xray optics and/or magnifying visible light optics has not been implemented as a standard technique at the MuCLS for now.

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13.1.2 Contrast Enhanced and K-Edge Subtraction Imaging Monochromatic or quasi-monochromatic spectra with tunable X-ray energy provide means to increase the signal-to-noise ratio compared to bremsstrahlung spectra in contrast enhanced imaging, e.g. in coronary angiography, where an iodine contrast agent is necessary to visualise the arteries. The reason for this is that the spectrum of the MuCLS can be adjusted to lie directly above the K-edge of iodine resulting in a higher average absorption coefficient for the MuCLS spectrum than the average one for a bremsstrahlung spectrum. Consequently, the contrast between the iodinecontaining parts and surrounding tissue was increased by 20% in an experiment with a porcine heart ex vivo (Eggl et al. 2017a, b). Another, more sophisticated technique, that exploits the step-increase in absorption resulting from the material-dependent energy of the K-shell transition, is K-edge subtraction imaging (Rubenstein et al. 1981). Image contrast for a specific material can be generated by subtracting two images from each other, one taken at an energy just above this material’s K-edge and the second one at an energy just below. As the material’s absorption coefficient is a continuous function of energy and therefore barely changes between the two images for all materials except the one with a K-edge, only the material of interest remains visible in the subtraction image. Although this technique is very useful, its efficient application in the laboratory is difficult when using X-ray tubes, as the required narrow bandwidth spectra can be generated with Ross-filter-pairs only (Arhatari et al. 2017). This results in very long data acquisition times, because of the tubes’ inherently low brilliance away from their characteristic emission lines. Consequently, this technique has been mainly performed at synchrotrons. However, the MuCLS’s brilliance is orders of magnitude higher than the one for the bremsstrahlung continuum of X-ray tubes, which makes K-edge subtraction imaging compatible with standard diagnostic imaging in a laboratory. In order to switch the mean energy of the X-rays without changing the ICS’s energy configuration, we used an iodine filter, following the approach by Umetani et al. (1993). Being able to perform K-edge subtraction imaging with an ICS source that could in principle fit into a clinical environment is an important perspective for clinical diagnostic imaging in which contrast agents are often employed to increase soft-tissue or blood to soft-tissue contrast. An example is the separation of iodine and calcium which is important, e.g. in the detection of atherosclerosis. In a test experiment for angiography, we demonstrated the separation of iodine, which was injected as contrast agent into the arteries, and bone. This enabled the observation of vessels behind bones, which otherwise would have been oblique (Kulpe et al. 2018). Figure 13.2 shows an example for the separation of iodine from tissue and calcium. A porcine kidney with a human kidney stone placed onto its surface (as it did not contain any kidney stones naturally) was chosen for this purpose. In addition, an iodine solution (Imeron, Bracco Imaging Deutschland GmbH, Germany, 400 mg/ml) was injected into the arteries and the ICS of the MuCLS was set to a peak energy of 33.7 keV. 1000 projections with an acquisition time of 44 ms each were taken

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Fig. 13.2 K-edge subtraction imaging. a Full slice of a porcine kidney containing one kidney stone where an iodine solution (Imeron, Bracco Imaging Deutschland GmbH, Germany, 400 mg/ml) was injected into the arteries. The kidney was placed into a plastic baker glass filled with water. The dynamic range of the image has been adapted for water and soft tissue and the contour of the kidney is plotted in grey for better visibility. b is the same slice cropped to the region indicated in green in a which contains the kidney. Here and in the following sub-figures, the full dynamic range is displayed. c Image of the same sample recorded with an iodine filter. d is the same slice of the resulting K-edge subtraction volume containing only the iodine, i.e. the material with the K-edge. Orange arrows point to vessels containing iodine contrast agent, while the blue arrow indicates the kidney stone. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

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over 360◦ for both tomograms. The unfiltered one, cf. Fig. 13.2a, displays a full reconstructed slice. Figure 13.2b is the same slice cropped to the region containing the kidney and Fig. 13.2c the same region of the same slice for the filtered tomography. A flatpanel detector (Dexela 1512, Perkin Elmer Inc., USA) with a Gd2 O2 S:Tb scintillator was used for image acquisition, which provided an effective pixel size of 70 µm for the projection images. A dark current as well as reference correction of the projections was performed first. The centre shift and tilt of the rotation axis were determined and corrected in a second step. Computed tomograms were reconstructed from these projections by a statistical iterative reconstruction algorithm (Fessler 2000) employing a Huber-penalty (Huber 2011) in the regularisation with a strength of 1−10 and 15 iterations. In Fig. 13.2, the reconstructed slices are cropped to the region containing the kidney. Image reconstruction followed the filter-based approach for K-edge subtraction imaging in Kulpe et al. (2018), which is very well suited to the spectral bandwidth available at the MuCLS. It has been extended to computed tomography (Kulpe et al. 2019) as well as time-sequence imaging (Kulpe et al. 2020). In the KES-image 13.2d, no calcium contribution is visible which enables the separation of contrast containing vessels from kidney stones. Combining information from the three images allows to discriminate materials with similar absorption but different chemical composition, as demonstrated here for the biomedical problem of identifying kidney stones in vessels laced with contrast agent for better visibility. Although all of the mentioned applications have been performed at the iodine Kedge, this technique can be applied to all specimens containing elements whose K-edge is covered by the energy range that is provided by the ICS of the MuCLS (cf. Fig. 13.7c). KES-imaging with an ICS might also be an interesting alternative in contrastenhanced spectral mammography. Consequently, a research programme was initiated to evaluate its potential and performance compared to other techniques, like material decomposition and clinically-employed polychromatic contrast-enhanced mammography (Heck et al. 2019).

13.1.3 Propagation-Based Phase-Contrast Imaging In general, the change in the real part of the refractive index is significantly stronger than the one in the imaginary part for weakly absorbing materials in the X-ray regime. Consequently, their image contrast is improved when applying phase-contrast techniques. In terms of instrumentation, the simplest approach for X-ray phase-contrast imaging is propagation-based phase-contrast imaging (PBI). It relies on interference by free-space propagation of a (partially-) coherent beam in the Fresnel regime, which translates the object’s phase shift into a measurable intensity modulation (Snigirev et al. 1995; Cloetens et al. 1996), see Fig. 13.3. This intensity modulation is proportional to the Laplacian of the phase shift. As a consequence, steep transitions in the refractive index are pronounced, e.g. at air-tissue interfaces of the lung. Therefore, this observed effect is also called edge enhancement.

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Fig. 13.3 Effect of increasing the sample-to-detector distance from 2 to 50 cm when imaging the respiratory system of a mouse. The magnified region clearly shows the enhanced contrast at the edges of the lung. Inside the lung, a speckle pattern is seen which arises from the scattering from the small air sacs in the lung. These chest images show that the intensity projection image (edge-enhanced image), recorded at a certain sample-to-detector distance, contains a mixture of contributions from both the absorption (e.g. the contrast of the bones) and the phase shifts (e.g. the contrast of the lung tissue) in the sample. This means that, in general, the edge-enhanced images only deliver qualitative data. Detector: Hamamatsu 6.5 µm pixel size, 0.5 s exposure time. Scale bars: 2 mm, magnified area: 0.5 mm. 25 keV X-rays, flux 1.7 × 1010 ph/s. Flat-field and dark-current corrected. Linear gray-scale. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

After PBI with an ICS had been demonstrated first by Ikeura-Sekiguchi et al. in 2008 (Ikeura-Sekiguchi et al. 2008), we could show at the MuCLS that the coherence provided by the ICS is sufficient for quantitative PBI at short acquisition times using a phantom of a single material (Gradl et al. 2017). Moreover, PBI at the MuCLS can be advantageous compared to liquid-metal jet sources, as demonstrated by Töpperwien et al. (2018). Once the feasibility of this method at the MuCLS was demonstrated, our efforts have been focused on enabling dynamic phase-contrast imaging in a laboratory environment. A very important application which requires dynamic imaging is pre-clinical research on the progression of diseases and response to drug treatment. For these scientific questions, longitudinal studies are indispensable, so regular beamtime over an extended period of time is required, which is hard or impossible to realise at standard synchrotrons. As PBI allows for single-shot data acquisition, it is extremely well suited for nonrepetitive motion and time-sequence imaging. Considering the X-ray energy range at the MuCLS, the beam diameter and the corresponding flux density at the first experimental end-station, a focus has been on dynamical studies of small animal models in vivo. More specifically, we tried to assess indicators for the health of the respiratory system as a first prerequisite for successful longitudinal studies using this dynamical information. In this context, the velocity distribution of particles injected into the trachea has been identified as such an indicator in synchrotron experiments, e.g. Donnelley et al. (2014), as the mucociliary transport or clearance of particles from the airways is impaired in some conditions, such as cystic fibrosis (Knowles and Boucher 2002). We could show that this type of experiment can also be carried out at the MuCLS (Gradl et al. 2018). This can be used to track disease development or treatment response which is foreseen to be an important application in the future.

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In order to investigate treatment response in the respiratory system, it is very important to identify (i) whether the drug is delivered to the inflamed region and (ii) how much of the amount applied actually reaches the inflammation. Both are difficult to predict, especially when aerosols are applied. In a proof-of-principle experiment, the capability of dynamically capturing liquid instillation into the lung of a mouse was investigated and successfully demonstrated (Gradl et al. 2019b; Yang et al. 2019). To this end, the benefit of PBI for generation of soft tissue contrast was combined with contrast-enhanced imaging in which the visibility of the tiny amount of liquid was increased by adding iodine-based contrast agent. However, sometimes projection imaging is not sufficient in order to determine material location, e.g. particle distributions in lungs or airways. In such a case, computed tomography (CT) becomes necessary. Due to rapid degradation of biological substances like tissue, which would lead to motion artefacts in the reconstruction, these measurements have to be fast. As an example, a chest tomography of a recently deceased non-fixated mouse is shown in Fig. 13.4. 1500 projections were captured over 360◦ within 5 min, with 0.2 s exposure per projection and a continuous rotation. The air-tissue contrast was enhanced by 50 cm of free-space propagation and the corresponding edge-enhanced reconstruction is depicted in Fig. 13.4a and free of motion artefacts. Clear identification of bones and air-filled lung structures is possible. Applying the single-distance phase-retrieval algorithm (Paganin 2002) to the edge-enhanced data results in an improved differentiation between lung tissue and air, which is shown in Fig. 13.4b. However, the homogeneity assumption (i.e. a constant ratio of δ and β for the whole specimen) within this algorithm introduces a slight blurring of the bony structures, since the ratio of δ and β was optimised for lung soft tissue (δ/β = 1891 at 25 keV). This ratio deviates from the one of bones (δ/β = 156 at 25 keV). The ratios are calculated from the compositions of lung tissue and bone in Mohammadi et al. (2014). Figure 13.4c is a volume rendering combined from both data sets. The lung rendering was performed using the phase-retrieved data set, while rib cage and beads (∅98 µm) were rendered using the edge-enhanced one. Figure 13.4d is the upper part of the left lung rendered from the phase-retrieved data set demonstrating the high resolution achievable for small airways in the fast CT-scans. Additional motion renders acquisition of the same high resolution tomography data in vivo more challenging. Often they suffer from poor resolution and/or long scan times (e.g. at the Skyscan small animal scanner (Velroyen et al. 2015)). In particular, the respiratory system is challenging because the lung is a continuously moving organ, close to the beating heart. Here, we present such an in vivo CT of a mouse which was acquired within 15 min, cf. Fig. 13.5. The scan took longer in this case compared to the 5 min in the ex vivo one because image acquisition had to be synchronised to a short breath hold to minimise motion blur, which in turn resulted in a reduction of the frame rate from 5 to 1.1 fps due to the duration of the respiration cycle (including the breath hold) of 0.9 s. 1000 projections were taken over 360◦ . As the heart of the mouse is beating freely in this type of experiment, some motion blur is introduced. Moreover, fine airways and alveoli may still be displaced slightly in

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Fig. 13.4 Fast tomography of the chest region of an ex vivo mouse, displayed with a edgeenhancement and b the result obtained using phase retrieval with the Paganin algorithm with a δ-to-β ratio optimised for lung soft tissue (thus bony structures are blurred, e.g. Beltran et al. (2010)). c Combined rendering of the data-set. Only the lungs are rendered using the phase-retrieved reconstruction, the other parts are from the edge-enhanced reconstruction. The high refractive index glass beads in the respiratory system are shown in blue. d Segmentation and rendering of the upper part of the left lung lobe of the phase-retrieved reconstruction. Note that 2 × 2 binning was used, resulting in an effective pixel size of 13 µm of the Hamamatsu detector. The sample-to-detector distance was 0.5 m. 1500 projections were acquired over 360◦ in 5 min while rotating continuously at a speed of 1.2◦ s−1 with 0.2 s exposure time per projection. The input value for phase retrieval was δ/β = 1891, an approximation for lung tissue described by H10 C0.83 O5 (Mohammadi et al. 2014). This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

each image, even if a gating technique is used. Consequently, the spatial resolution is reduced in the in vivo case, Fig. 13.5, compared to the ex vivo case shown in Fig. 13.4. The total time required for an in vivo propagation-based phase-contrast CT depends on the number of projections and thus the resolution, the ventilation frequency and the acquisition time for a single frame. These parameters can vary tremendously depending on the employed X-ray source and intention of the experiment. At synchrotrons and liquid-metal jet sources, single-frame acquisition times around 20 ms have been demonstrated for mechanically ventilated small animal models (Stahr et al. 2016; Murrie et al. 2020). In these two set-ups, respiratory cycles lasted 0.45 and 0.5 s during the acquisition of the CT. Consequently, data was taken at multiple positions in the respiratory cycle within the duration of a single CT scan, while only one frame per respiratory cycle was recorded in the proof-of-principle

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experiment at the MuCLS. For a sharp CT result, all projection images of a single CT have to be recorded at the same position in the respiratory cycle. Considering this boundary condition, a full tomography can be acquired about a factor of two faster with the parameters reported in Stahr et al. (2016) or Murrie et al. (2020) compared to the MuCLS. However, pushing the total acquisition time to the smallest possible value was not the objective of the proof-of-principle experiment presented above. With the flux currently available at the MuCLS, acquisition times down to 30 ms offer sufficient image quality. This enables dynamic imaging of the respiratory cycle at the MuCLS similar to liquid-metal jet sources or synchrotrons. The final choice of the X-ray source for such measurements might be influenced by the attainable soft-tissue contrast and source availability. Synchrotron sources outperform both laboratory sources in the former, yet ICSs provide better soft-tissue contrast than liquid-metal jet sources (Töpperwien et al. 2018). However, the acquisition time for a single frame might be slightly higher at the MuCLS.

13.1.4 Grating-Based Phase-Contrast Imaging One drawback of PBI compared to grating-based imaging (GBI) with a quasimonochromatic beam is that the former is not quantitative because the phase is typically reconstructed using the single material approximation. However, the most common implementation of GBI requires several images to be taken in order to extract the desired information and consequently is not suited for dynamics that require single-shot imaging, but may be applied for time-resolved imaging of a repeatable motion. A Talbot interferometer is employed in GBI which translates the object’s (differential) phase shift and the so-called dark-field signal, which can be related mainly to small-angle scattering (Bech et al. 2012), into measurable intensity signals (Momose et al. 2003; Weitkamp et al. 2005; Pfeiffer et al. 2008). GBI with an ICS has been demonstrated first by Bech et al. (2009). Dynamic GBI with an ICS has been demonstrated very recently by resolving the respiration cycle of mice in vivo (Gradl 2019a). This could enable identification of the best point in time for singleshot diagnostic lung imaging thereby reducing the dose significantly. Additionally, the dark-field signal ratio between exhalation and inhalation could be a non-invasive biomarker for local lung function allowing for better differentiation and detection of lung diseases, especially where alveoli are affected. The dynamic GBI capability is demonstrated in Fig. 13.6 for one point of the breathing cycle of an in vivo mouse. Image acquisition followed the procedure described in Gradl (2019a) with individual exposure times of 140 ms and averaging over six exposures. The complementary information of the three image modalities obtained by GBI, namely the absorption image on the left, differential phase-contrast image in the centre and dark-field image on the right, is very well visible. While bones show up prominently in absorption, this is the case for soft tissue, e.g. airways, in the differential phase-contrast image and structures with a high density of scattering structures, e.g. the lung, in the darkfield image. Furthermore, at the prototype of the Lyncean Compact Light Source,

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Fig. 13.5 Fast tomography of the chest region of an in vivo mouse, displayed with a edgeenhancement and b the respective result from the Paganin reconstruction with a δ-to-β ratio optimised for lung soft tissue. c Combined rendering of the data set. One side of the lung is segmented and rendered using the phase-retrieved reconstruction, the bones and beads are from the edgeenhanced reconstruction. The high refractive index glass beads in the respiratory system are shown in blue. d Zoomed-in region, showing that even in an in vivo mouse small airways remain visible, albeit with some motion blur. 2 × 2 binning is used for reconstruction, resulting in an effective pixel size of 13 µm of the Hamamatsu detector. 1000 frames were acquired over 360◦ at an acquisition time of 0.2 s. Since the images had to be taken during a short breath hold, the rotation speed was reduced to 0.4◦ s−1 resulting in a total time of 15 min. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

the dark-field signal has been demonstrated to be a good parameter for the diagnosis of emphysema in a pre-clinical study with ex vivo mice (Schleede et al. 2012a). A detailed overview over the different methods for dynamic respiratory X-ray imaging with living animals can be found in Morgan et al. (2020). As mentioned earlier, (differential) phase-contrast imaging provides improved soft-tissue contrast. Therefore, phase contrast is expected to improve diagnostic value of the images, e.g. in mammography. A first patient study was carried out at the Elettra synchrotron in Trieste, Italy, which demonstrated increased diagnostic value, but at the same time emphasised that the main limitation for a wide-spread use would be the availability of synchrotron radiation at clinics (Castelli et al. 2011). Consequently, this is an interesting application for compact synchrotron sources, like the MuCLS. First, a phantom study was performed using GBI at a dose compatible with clinical mammography which yielded improved contrast over classical attenuation images (Schleede et al. 2012b). A follow up study explored tomosynthesis at ICSs (Eggl et al.

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Fig. 13.6 Projection images of an in vivo mouse obtained with grating-based imaging. On the left the absorption image is depicted, in the centre the differential phase-contrast one and on the right the dark-field image. Six images with individual exposure times of 140 ms were averaged improving the signal-to-noise ratio. The complementary information obtained by this technique can be seen very well, as different structures generate the most prominent contrast for the three different properties: bones in absorption, soft tissue in the differential phase contrast and dense scattering structures (e.g. lungs) in the dark field. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0

2016). Very recently, GBI-mammography at the MuCLS of mastectomy specimens has been shown to improve the delineation of tumorous lesions at an X-ray dose compatible with clinical mammography (Eggl et al. 2018). Combined with quasi-monochromatic X-rays, GBI allows for quantitative determination of the absorption coefficient as well as the phase shift. The complex refractive index can be reconstructed from this information, which enables quantitative material discrimination (Eggl et al. 2015; Braig et al. 2018). Therefore, this technique can be used for similar purposes as dual-energy imaging, while at the same time providing a reliable differentiation for both, low absorbing material, like soft tissue, as well as strongly absorbing materials, like bone or contrast agents. While the former vary mainly in electron density and consequently the real part of the refractive index, the latter exhibit considerable differences in their effective atomic number.

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13.1.5 X-ray Vector Radiography and X-ray Tensor Tomography In the preceding discussion, the fact was omitted that gratings employed in GBI typically are line gratings. Consequently, wave modulation occurs in one direction only which generates a preferential sensitivity direction perpendicular to the grating lines for X-ray scattering as well as differential phase shifts. In order to employ this directional sensitivity for determination of structure orientations within the specimen, coined directional dark-field imaging, either the sample or the interferometer has to be rotated around the systems optical axis and images have to be acquired at each rotation position (Jensen et al. 2010a, b). By these means, it is possible to determine the orientation of scattering structures with dimensions smaller than the detector resolution, e.g. fibrils in bones. We demonstrated at the MuCLS that microfractures, occult in classic radiography, can be detected applying this technique which would be of diagnostic value in clinical practice (Jud et al. 2017a, b). Nevertheless, this technique is not limited to (pre-)clinical diagnostics, but can also be applied in material science or quality assurance. Implementation of a 3-circle Eulerian cradle expands this technique to three-dimensional reconstruction, so called Anisotropic Xray Dark-Field Tomography (AXDT) (Wieczorek et al. 2016), the generalised form of X-ray Tensor Tomography (Maleki et al. 2014). AXDT is currently not possible at the MuCLS, as the sample stage lacks one rotational degree of freedom, but is going to be available in the near future.

13.2 X-ray Absorption Spectroscopy X-ray absorption spectroscopy (XAS) is an element-selective probe of the surroundings of an atom of interest which provides information on its chemical state, geometric and electronic structures among others (Koningsberger and Prins 1988). The main advantage of this technique compared to crystallography is that it can be applied to non-crystalline materials. At the MuCLS, an energy-dispersive set-up was implemented which exploits the provided X-ray bandwidth and divergence most efficiently. Due to the natural beam divergence of 4 mrad, the angle of incidence on a silicon wafer oriented in Bragg condition is spatially varying which translates into a spatiallydependent energy of the diffracted beam. The sample is placed into the converging beam path and the X-rays are recorded by a 2D-detector placed right after the sample, on which the X-ray energy is encoded in one spatial dimension. Since the X-rays penetrate the sample at different positions, a concentration correction is applied to the data. Details on the set-up, data acquisition and data processing can be found in Huang et al. (2020). With this set-up located in the first experimental end-station and depicted in Fig. 13.7a, b, X-ray absorption near-edge structure (XANES) and extended X-ray absorption fine structure (EXAFS) spectra have been collected at an energy resolu-

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Fig. 13.7 A picture of the XAS set-up is depicted in a with a schematic of the beam path shown in b. The elements whose K- & L-edges can be covered by the energy range of the ICS of the MuCLS are highlighted in c in blue and green, respectively. Elements for which no data is available from the NIST database (NIST 2020) are coloured in grey. As an example, a measurement of a silver foil and a silver-nitrate solution is depicted in d. For comparison, synchrotron measurements of the same substances are included and shifted upwards. This figure has been published in Günther et al. (2020), licensed under CC BY 4.0.

tion down to ∼4 eV at 25.5 keV. Examples of XANES spectra for a silver foil and a silver-nitrate solution are depicted in Fig. 13.7d. The total acquisition time for each of the spectra recorded at the MuCLS and depicted here was 10 min. Recently, data acquisition times as short as 1 min have been successfully demonstrated at a quality comparable to synchrotrons (Huang et al. 2020). Other laboratory XAS set-ups, based on X-ray tubes (Seidler et al. 2014; Néemeth et al. 2016; Bès et al. 2018; Błachucki et al. 2019; Jahrman et al. 2019), laser-plasma sources (Uhlig et al. 2013), laserbetatron ones (Mahieu et al. 2018) or high-harmonic generation (Pertot et al. 2017; Popmintchev et al. 2018) typically operate at a lower X-ray energy below 10 keV and typically require exposure times of tens of minutes to hours. This elongated acquisition time originates from the lower brilliance of X-ray tubes compared to ICSs, like at the MuCLS. The difference in brilliance becomes more prominent at higher X-ray energies, which makes ICSs advantageous in this regime. In contrast to attosecond to few femtosecond pulses available from high harmonic generation or at laser-driven betatron sources, the pulse length at the MuCLS of ∼60 ps does not allow for ultrafast time-resolved spectroscopy. However, the X-ray energy range at the MuCLS allows to perform XAS measurements on a large range of elements of the periodic table complementary to the typical sub-10 keV-range of other laboratory XAS systems by

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probing either K-edges from bromine to xenon or L-edges from osmium at least to fermium, cf. Fig. 13.7c. The absorption edge’s positions were retrieved from NIST (2020). The energy range covered at once in the energy-dispersive XAS set-up is E range = E 0 · (4 mrad − Δθ) · cot θBragg .

(13.1)

E 0 denotes the central energy, θBragg the corresponding Bragg angle and Δθ = l R −1 , where l is the diameter of the beam on the curved crystal and R is the crystal’s radius of curvature. In case the energy range is too small for the acquisition of a full EXAFS spectrum, it can be extended either by scanning the Bragg angle of the crystal in fine steps while recording images continuously or by stitching several exposures at more widely-spaced Bragg angles. Switching to a different crystal plane which diffracts a larger energy range is another possibility. The energy resolution of the system can be approximated by (13.2) ΔE = (ΔE cryst ∗ ΔE det. ) ∗ ΔE src . ΔE cryst is the intrinsic energy resolution of the chosen crystal plane. ΔE det is the contribution of the detector point-spread-function which can be calculated from the energy resolution per pixel (determined by the energy bandwidth E range and the number of illuminated pixels) and the detector point-spread-function of the used camera, cf. Table 11.1. ΔE src is the contribution of the X-ray source size which can be determined from the measured source size, typically ∼50 µm (σ-value), and the energy resolution per pixel. The ∗ denotes a convolution. Using these equations, the energy range and energy resolution can be optimised for the desired experiment.

13.3 Microbeam Radiation Therapy Finally, proof-of-principle studies on microbeam radiation therapy (MRT) have been performed at the MuCLS. Contrary to classical radiation therapy, where a spatially homogeneous dose is applied, the intensity is redistributed into microbeams with very high peak doses sparing tissue in between (Slatkin et al. 1992). Brilliant X-rays are a requirement for this technique to work properly which means that MRT has been restricted to synchrotron facilities until now. Since ICSs provide a quasimonochromatic, low divergence beam and in principle are capable of producing very hard X-rays on a small footprint, they are well suited for this technique and currently seem to be the most promising option to transfer this technique into clinical practice in the long run. At the MuCLS, generation of microbeams with the required peakto-valley dose ratio has been demonstrated at an X-ray energy of 25 keV (Burger et al. 2017). This is much lower than typical energies of ∼100 keV used for studies at synchrotron facilities (Crosbie et al. 2015). Treatment research at the MuCLS is therefore limited to superficial tumours, e.g. skin tumours. A dedicated set-up and small animal model have been developed for this research. Details on the set-up are

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given in the publication by Burger et al. (2020) and about the small animal model by Dombrowsky et al. (2020). At other ICSs like ThomX, which aim to generate higher X-ray energies, this limitation to superficial tumours does not apply.

13.4 Planned Upgrades to the MuCLS Beamline Although the MuCLS is extremely well suited for imaging applications, a few limitations have been identified which are currently addressed or are going to be addressed in the near future.

13.4.1 Grating-Based Imaging A stepper motor (LTA-HL, Newport Corp., Irvine, USA) in conjunction with a flexure-based nanoconverter developed at the Paul Scherrer Institute has been used for phase stepping in GBI (Henein et al. 2007). Although this concept is very reliable, grating movement is inherently slow due to the employed stepper motor and because it is operated in uni-directional motion to achieve the desired position accuracy. As a consequence, dynamic imaging with the grating interferometer is limited by the motor movement overhead at the moment. Therefore, a new stepping mechanism was implemented recently by replacing the nanoconverter with a fast piezo-electric linear actuator. This speeds up motor movement for the distance of one step of the phase stepping by a factor of 18. Including typical acquisition times, communication overheads of the control system and detector readout, the speed-up with the piezo is still a factor of 6–8 for typical phase-contrast tomography measurements compared to using the nanoconverter. Nevertheless, this new set-up is a significant improvement for future dynamic experiments, e.g. for in vivo respiratory dark-field imaging. Furthermore, in Sect. 13.1.5 it was mentioned that AXDT is currently not available due to the missing third rotational degree of freedom. In order to overcome this limitation, a 7-axis robot arm (Panda, Franka Emika, Munich, Germany) was acquired and is currently being commissioned. The advantage of a robot arm over a conventional Eulerian cradle is that it can cover more angles without obstructing the X-ray beam itself. Moreover, it allows more flexible sample movement which might reduce data acquisition time for an AXDT measurement compared to one performed with an Eulerian cradle.

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13.4.2 Replacement of the Second End-Station At the moment, the second end-station of the MuCLS offers only a very limited space for experiments. The main reason for this is that, at the time this end-station was installed, another large laboratory X-ray set-up was located right next to it. Therefore, the X-ray hutch neither could be placed any closer to the source nor be wider as it would have blocked access to this other set-up. However, this other set-up was relocated to another laboratory recently in order to replace the existing endstation in 2022 with a new one which will be 6–7 m long and 2.5–3 m wide. This is going to enable access to a much larger range of beam diameters as well as much more flexible set-ups, thereby enhancing the MuCLSs capabilities significantly.

13.5 Contributions The research presented in this chapter provides an overview about all capabilities of the MuCLS. Accordingly, it covers topics of various other dissertations, mainly the ones by Elena Eggl, Regine Gradl, Juanjuan Huang, Christoph Jud and Stephanie Kulpe. In the following only the contributions to the data discussed in this chapter are listed, always in alphabetic order • X-ray microtomography: – – – –

set-up: Martin Dierolf, Regine Gradl, Benedikt Günther experiment: Martin Dierolf, Regine Gradl, Benedikt Günther tomographic reconstruction: Lorenz Hehn sample: The sturgeon fish head was prepared by Brian Metscher of the University of Vienna

• K-edge subtraction imaging: – set-up: Madleen Busse, Martin Dierolf, Benedikt Günther, Stephanie Kulpe – experiment: Martin Dierolf, Benedikt Günther, Stephanie Kulpe – data analysis: Benedikt Günther, Stephanie Kulpe • Propagation-based and grating-based phase contrast: – in vivo set-up: Regine Gradl; Martin Donnelley, David Parsons (both from Women’s and Children Hospital Adelaide) – experiments: Martin Dierolf, Regine Gradl, Benedikt Günther, Kaye Morgan; Martin Donnelley, David Parsons (both from Women’s and Children Hospital Adelaide) – data analysis: Regine Gradl, Benedikt Günther – renderings in Figs. 13.4c, d and 13.5c, d: Regine Gradl – animal handling: Helena Haas, Melanie Kimm (both from Klinikum rechts der Isar)

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• X-ray absorption spectroscopy: – – – –

set-up: Martin Dierolf, Benedikt Günther, Juanjuan Huang experiment: Martin Dierolf, Benedikt Günther, Juanjuan Huang data analysis: Juanjuan Huang reference XAS-spectra: Yitao Cui and the SPring-8 BL14B2 XAFS database provided the Ag, AgNO3 standard spectra.

References Achterhold K et al (2013) Monochromatic computed tomography with a compact laser-driven X-ray source. Sci Rep 3:1313 Arhatari BD, Gureyev TE, Abbey B (2017) Elemental contrast X-ray tomography using Ross filter pairs with a polychromatic laboratory source. Sci Rep 7:218 Bech M et al (2009) Hard X-ray phase-contrast imaging with the compact light source based on inverse Compton X-rays. J Synchrotron Radiat 16:43–47 Bech M et al (2012) Experimental validation of image contrast correlation between ultra-smallangle X-ray scattering and grating-based dark-field imaging using a laser-driven compact X-ray source. Photon Lasers Med 1:47–50 Beltran M et al (2010) 2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance. Opt Exp 18:6423–6436 Bès R et al (2018) Laboratory-scale X-ray absorption spectroscopy approach for actinide research: experiment at the uranium L3-edge. J Nucl Mater 507:50–53 Błachucki W et al (2019) A laboratory-based double X-ray spectrometer for simultaneous X-ray emission and X-ray absorption studies. J Anal Atom Spectrom 34:1409–1415 Braig E et al (2018) Direct quantitative material decomposition employing grating-based X-ray phase-contrast CT. Sci Rep 8:16394 Burger K et al (2017) Increased cell survival and cytogenetic integrity by spatial dose redistribution at a compact synchrotron X-ray source. PLoS ONE 12:e0186005 Burger K et al (2020) Technical and dosimetric realization of in-vivo X-ray microbeam irradiations at the Munich compact light source. Med Phys 47:5183–5193 Castelli E et al (2011) Mammography with synchrotron radiation: first clinical experience. Radiology 259:684–694 Cloetens P et al (1996) Phase objects in synchrotron radiation hard X-ray imaging. J Phys D: Appl Phys 29:133–146 Crosbie JC et al (2015) Energy spectra considerations for synchrotron radiotherapy trials on the ID17 biomedical beamline at the European synchrotron radiation facility. J Synchrotron Radiat 22:1035–1041 Dombrowsky AC et al (20202) A proof of principle experiment for microbeam radiation therapy at the Munich compact light source. Radiat Environ Biophys 59:111–120 Donnelley M et al (2014) Tracking extended mucociliary transport activity of individual deposited particles: longitudinal synchrotron X-ray imaging in live mice. J Synchrotron Radiat 21:768–773 Eggl E et al (2015) X-ray phase-contrast tomography with a compact laser-driven synchrotron source. Proc Natl Acad Sci 112:5567–5572 Eggl E et al (2016) X-ray phase-contrast tomosynthesis of a human ex vivo breast slice with an inverse Compton X-ray source. EPL 116:68003. ISSN: 12864854 Eggl E et al (2017a) Mono-energy coronary angiography with a compact light source. In: Proceedings of the SPIE, medical imaging 2017: physics of medical imaging, vol 10132, 101324L

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Mohammadi S et al (2014) Quantitative evaluation of a single-distance phase-retrieval method applied on inline phase-contrast images of a mouse lung. J Synchrotron Radiat 21:784–789 Momose A et al (2003) Demonstration of X-ray Talbot interferometry. Jpn J Appl Phys 42:L866– L868 Morgan KS et al (2020) Methods for dynamic synchrotron X-ray imaging of live animals. J Sychrotron Radiat 27:164–175 Murrie RP et al (2020) Real-time in vivo imaging of regional lung function in a mouse model of cystic fibrosis on a laboratory X-ray source. Sci Rep 10:447 Néemeth Z et al (2016) Laboratory von Hámos X-ray spectroscopy for routine sample characterization. Rev Sci Instrum 87:103105 NIST (2020) X-ray transition energies database 2020. https://doi.org/10.18434/T4859Z Paganin D et al (2002) Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J Microsc 206:33–40 Pertot Y et al (2017) Time-resolved X-ray absorption spectroscopy with a water-window highharmonic source. Science 355:264–267 Pfeiffer F et al (2008) Hard-X-ray dark-field imaging using a grating interferometer. Nat Mater 7:134–137 Popmintchev D et al (2018) Near- and extended-edge X-ray-absorption fine-structure spectroscopy using ultrafast coherent high-order harmonic supercontinua. Phys Rev Lett 120:093002 Rubenstein E et al (1981) Synchrotron radiation and its application to digital subtraction angiography. Proceedings of the SPIE, digital radiography 0314:42–49 Schleede S et al (2012a) Emphysema diagnosis using X-ray dark-field imaging at a laser-driven compact synchrotron light source. Proc Natl Acad Sci 109:17880–17885 Schleede S et al (2012b) Multimodal hard X-ray imaging of a mammography phantom at a compact synchrotron light source. J Synchrotron Radiat 19:525–529 Seidler GT et al (2014) A laboratory-based hard X-ray monochromator for high-resolution X-ray emission spectroscopy and X-ray absorption near edge structure measurements. Rev Sci Instrum 85:113906 Slatkin DN et al (1992) Microbeam radiation therapy. Med Phys 19:1395–1400 Snigirev A et al (1995) On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation. Rev Sci Instrum 66:5486–5492 Stahr CS et al (2016) Quantification of heterogeneity in lung disease with image-based pulmonary function testing. Sci Rep 6:29438 Töpperwien M et al (2018) Propagation-based phase-contrast X-ray tomography of cochlea using a compact synchrotron source. Sci Rep 8:4922 Uhlig J et al (2013) Table-top ultrafast X-ray microcalorimeter spectrometry for molecular structure. Phys Rev Lett 110:138302 Umetani K et al (1993) Iodine filter imaging system for subtraction angiography using synchrotron radiation. Nucl Instrum Methods Phys Res Sect A: Accelerators, Spectrom Detect Assoc Equip 335:569–579 Velroyen A et al (2015) Grating-based X-ray dark-field computed tomography of living mice. EBioMedicine 2:1500–1506 Weitkamp T et al (2005) X-ray phase imaging with a grating interferometer. Opt Exp 13:6296–6304 Wieczorek M et al (2016) Anisotropic X-ray dark-field tomography: a continuous model and its discretization. Phys Rev Lett 117:158101 Yang L et al (2019) Multimodal precision imaging of pulmonary nanoparticle delivery in mice: dynamics of application, spatial distribution, and dosimetry. Small 15:1904112

Part IV

Conclusion

Chapter 14

Conclusion and Outlook

This thesis presents research on both aspects relevant to the advancement of inverse Compton X-ray sources: performance improvements of the Compact Light Source itself as well as development of beamline instrumentation and experimental techniques. Additionally, Chap. 6 contains a brief history of inverse Compton scattering X-ray sources and pictures the various modern inverse Compton scattering X-ray sources that were realised. The Compact Light Source is introduced in detail as well.

14.1 Performance Improvements of the Compact Light Source Since the X-ray flux is proportional to the number of laser photons, or in other words, the power contained inside the enhancement cavity, an upgrade of the laser system feeding this optical resonator translates into an increased X-ray photon output. This upgraded laser system was described and characterised in Chap. 7. While the stored power significantly increased from 120 to 350 kW, small losses in the back leg resulted in a noticeable deflection of the mirror assembly deteriorating the overlap between the laser and electron beam. Consequently, an align thermal compensation system was implemented to mitigate these effects. However, a complete stabilisation was only achieved after an X-ray beam position monitoring and stabilisation system, described in Chap. 10 and published in Günther et al. (2019), was designed and constructed. It determines the X-ray source position, source size and flux in parallel to experiments which allows to save these as metadata for the experiment. The X-ray source position is kept constant by adjusting the laser orbit inside the cavity with a piezo-driven tip-tilt mirror mount. The X-ray flux increased by a factor of ∼3 to up to 4.5 × 1010 ph/s as a cumulative result of all the aforementioned efforts (Günther et al. 2020). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2_14

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However, tracking the cavity’s thermal state via the transverse mode range of the closest higher-order resonances revealed a significant elongation of the Rayleigh length in thermal equilibrium at such high powers inside the resonator. This increases the laser focus at the interaction point thereby reducing the X-ray flux in thermal equilibrium. Building on this experience, a deformable cavity mirror was developed in Chap. 8 that actively compensates the thermal deformation of the mirrors’ radii of curvature. The adaptive mirror consists of a thin membrane whose radius of curvature is adjusted by changing the pressure difference in a controlled fashion. Therefore, it prevents elongation of the Rayleigh length during heat-up while the interaction angle between laser and electron beam vanishes at the same time if this optic is implemented into the CLS. First results with a prototype optic operated with a low power laser system yielded promising results. However, the system was so far only tested at a second cavity at the laboratory of Lyncean Technologies Inc. The error signal for a closed-loop feedback would be provided by the aforementioned transverse mode range measurement. While this research was devoted to advance the Compact Light Source and its capabilities, the development of rapid X-ray energy switching was primarily pursued to enable dynamic K-edge subtraction imaging. To this end, two techniques were established at the MuCLS which are described in detail in Chap. 9. The filter-based approach uses the same element as filter material which the contrast agent contains that is applied in the K-edge subtraction imaging experiment. The filter cuts off the high energy part of the spectrum thereby reducing the mean X-ray energy of the spectrum. Such a filter must be solid-state in order to be mounted to a continuously revolving filter wheel. This transfer from the existing liquid filter (Kulpe et al. 2018) to a solid one was realised by my colleague Madleen Busse. Although the filter-based energy switching works reliably, the filter absorbs a significant amount of the X-ray flux which increases the noise level in the final subtraction image. This disadvantage can be overcome if the X-ray spectrum itself is shifted, which is the second approach demonstrated here. However, this is more complex as it requires an adjustment of the electron energy which in turn implies a modification of the magnetic fields, i.e. the currents running through the magnets. Their hysteresis requires dedicated configurations for the two X-ray energies which allow for energy switching on a timescale of seconds. This principle has been further optimised in the Master thesis by Ivan Kokhanovskyi (Kokhanovskyi 2021) and is valuable for the thickness correction in XAS as well (Huang et al. 2020).

14.2 Beamline Instrumentation and Experimental Techniques At the beginning of this project, the Compact Light Source was the only equipment installed inside the laboratory. Accordingly, the first goal was to establish a versatile beamline that enables various types of experiments to be conducted. The result of

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this group effort is described in Chap. 11. Two end-stations have been realised. One located as close as possible to the X-ray source, which is the optimal choice for experiments or techniques that require a high flux density and only medium spatial coherence. The sample size is restricted to small samples with sizes up to 16 mm in the front of this end-station and around 30 mm at its far end. Consequently, this set-up is ideal for micro-tomography with a resolution around 10 µm, propagationbased phase contrast imaging, X-ray absorption spectroscopy or crystallography. An additional optic mildly focussing to about 3 mm in diameter creates a flux density sufficient for micro-beam radiation therapy. The field-of-view at the second end-station reaches 60 mm in diameter which makes this set-up suited for grating-based phasecontrast imaging since gratings of this size can be easily fabricated. This enables investigation of multiple specimens of biomedical interest, for example, research on phase-contrast mammography, or detection of micro-fractures in bones via X-ray vector radiography. Since the grating interferometer can be easily slid out of the X-ray beam path, absorption or spectroscopic imaging, such as K-edge subtraction imaging, can be performed on large specimens. Due to the large beam diameter, the resolution is constrained to the range of 70–170 µm. Application examples of all these techniques are presented in Chap. 13, which highlight the versatility of the installed beamline. Apart from contributing to these results, a new microscopy method is proposed and demonstrated as part of this thesis in Chap. 12. A triangular phase-shifting line grating creates a periodic array of very narrow pencil beams at certain distances behind the grating due to Fresnel-propagation. If the sample is placed there and either the illumination or the sample is stepped over one period while images are acquired, high-resolution images can be generated with sub-pixel resolution as long as the individual pencil-beams are separable on the detector. Data analysis follows a conventional scanning-transmission microscopy approach. One advantage of this technique is that image acquisition can be significantly speed-up since motor movement overhead is the limiting factor in a conventional scanning transmission microscope. The feasibility of this approach was successfully demonstrated with synchrotron radiation at beamline P05 at Petra III at DESY. Simulations indicate that the transfer of this technique to the MuCLS should be feasible. This holds true for appropriate two-dimensional gratings which would enable super-resolution in both detector planes. However, to date only the generation of a Talbot-carpet using the triangular line grating could be experimentally demonstrated for three reasons: First, the lower spatial coherence at the MuCLS requires the experiments to be conducted at the far end-station where the flux density is low. Second, the current high-resolution detector necessary to obtain an appropriate pixel size is less efficient than the one at P05, and has a very inhomogeneous, large point-spread-function. Last, at the time of the experiments, the CLS delivered only about half the specified flux due to cancelled services during the COVID-19 pandemic which increased the anyway long acquisition times significantly. Consequently, it may make sense to resume this experimental study at the MuCLS, once it provides the usual flux and a better camera system is available.

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14.3 Outlook In view of the high demand on X-ray flux (density) of this microscopy technique, transferring the prototype of the mirror assembly with a deformable exit optic to the MuCLS could improve X-ray flux significantly. All flux-hungry experiments, such as the in vivo studies, would benefit from this implementation. In conjunction with this upgrade of the optical resonator, the invasive ring-down technique to determine the power stored inside the enhancement cavity could be replaced with a non-invasive linewidth scan, discussed in Sect. 4.2.5.1. A continuous determination of the stored power would provide valuable feedback when manually optimising the enhancement cavity. Finally, the laser pulse length from the laser characterisation, the power from the ring-down measurement, the X-ray flux and source size information delivered by the X-ray beam position monitor and laser focus from the transverse mode range measurement can be used to determine unknown electron beam parameters. The interaction angle between laser and electron beam can be determined with the laser cavity inserts. These parameters could be used as input into an iterative algorithm modelling the spectrum produced by inverse Compton scattering whose free parameters are the electron beam’s mean energy, energy spread and emittance. These are varied until the measured spectrum is reproduced. This way, the CLS could be completely characterised. These may be the obvious items that could be done next, but there are a lot more possibilities like a new high-power seed laser which would increase the stored power further. But at the same time it induces stronger thermal effects for which new compensation techniques may have to be developed. Another alternative could be a frequency-doubled laser system to double the X-ray energy, which requires an increased stored power to produce the same X-ray flux since the photon energy doubles. Accordingly, all the developments presented in this thesis may just be stepping stones in the development of the next generation of inverse Compton X-ray sources. A similar statement is valid for future developments at the beamline of the MuCLS. It is worth to explore the implementation of multiple further techniques at inverse Compton X-ray sources. Among them are wide-angle X-ray scattering, small-angle X-ray scattering or fluorescence imaging, e.g. using a Maia-384 detector (Ryan et al. 2014) (or its successor). Those methods do not require a very high coherence, but benefit from a tunable narrowband beam of high flux density. The latter requirement favours inverse Compton sources over X-ray tubes. Other recently developed techniques like “Multipoint projection microscopy” (Sowa et al. 2018) should be also suited to the MuCLS and may be an alternative to push the resolution range down to the sub-micrometre level. Furthermore, on the instrumentation side, several improvements can be envisioned. The grating interferometer would benefit if the slow step-and-shoot phasestepping approach would be replaced by a continuous rapid scanning with synchronised data acquisition Alternatively, rapid fringe-scanning approaches could be implemented directly, e.g. by adding an piezoelectric actuator on top of the sample

References

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rotation stage. This way fringe scanning and continuous rotation could be combined allowing for grating-based phase-contrast tomography with durations of only a few minutes. Additionally, implementation of a third rotational degree of freedom at the sample stage in front of the grating interferometer would extend two-dimensional X-ray vector radiography to three-dimensional X-ray tensor tomography (Maleki et al. 2014; Wieczorek et al. 2016). The latter may also be realised with a robot arm instead of an Euler cradle. Moreover, the second end-station is quite tiny and crowded with instrumentation. Thus, it makes sense to replace this end-station with a larger one which also provides space for future additional set-ups or instrumentation. With all these potential developments in mind and a lot more unknown ones ahead, I would like to close my thesis with Richard Feynman’s words1 : “There’s [still] plenty of room at the bottom.”

References Günther B et al (2019) Device for source position stabilization and beam parameter monitoring at inverse Compton X-ray sources. J Synchrotron Radiat 26:1546–1553. https://doi.org/10.1107/ S1600577519006453 Günther B et al. The versatile X-ray beamline of the Munich compact light source: design, instrumentation and applications. J Synchrotron Radiat 27:1395–1414. https://doi.org/10.1107/ S1600577520008309 Huang J et al (2020) Energy dispersive X-ray absorption spectroscopy with an inverse Compton source. Sci Rep 10:8772 Kokhanovskyi I (2021) Rapid keV-switching at the Munich light compact source for K-edge subtraction imaging. Master thesis, Technical University of Munich Kulpe S et al (2018) K-edge subtraction imaging for coronary angiography with a compact synchrotron X-ray source. PLOS ONE 13:e0208446 Maleki A et al (2014) X-ray tensor tomography. Europhys Lett 105:38002 Ryan CG et al (2014) Maia X-ray fluorescence imaging: capturing detail in complex natural samples. J Phys Conf Ser 499:012002 Sowa KM, Jany BR, Korecki P (2018) Multipoint projection X-ray microscopy. Optica 5:577–582 Wieczorek M et al (2016) Anisotropic X-ray dark-field tomography: a continuous model and its discretization. Phys Rev Lett 117:158101

1

Actually, this was part of a title of a lecture Richard Feynman’s at Caltech (https://en.wikipedia. org/wiki/There%27s_Plenty_of_Room_at_the_Bottom) and later evolved into a saying.

Curriculum Vitae

Benedikt Sebastian Günther

Research Fellow School of Natural Sciences & Munich Institute of Biomedical Engineering Technical University of Munich 85748 Garching Germany

Education 05/2015–12/2021

10/2018–02/2019

10/2014–04/2015

06/2014–09/2014

Ph.D. in Physics Technical University of Munich, DE “Storage Ring-based Inverse Compton X-ray Sources Cavity Design, Beamline Development and X-ray applications” Supervisor: Prof. Franz Pfeiffer Grade: Summa cum Laude Research stay at Lyncean Technologies Inc., USA Topic: Development of a deformable laser enhancement cavity optic for the Compact Light Source Internship at Sentech Instruments GmbH, DE Programming in C++ and Optics Topic: Extension of the spectroscopic Müeller-Matrix ellipsometry to anisotropic materials Spanish language course in Madrid, ESP

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. S. Günther, Storage Ring-Based Inverse Compton X-ray Sources, Springer Theses, https://doi.org/10.1007/978-3-031-17742-2

341

342

10/2011–03/2014

10/2008–09/2011

Curriculum Vitae

Master of Science (Physics) Technical University of Munich, DE Thesis: “Design, implementation and characterisation of the 300TW-upgrade of the ATLAS laser system” Supervisors: Prof. Stefan Karsch and Prof. Reinhard Kienberger Bachelor of Science (Physics) Technical University of Munich, DE Thesis: “Study of the charge transfer at the diamond surface for hybrid systems” Supervisor: Prof. Martin Stutzmann

Further Research Experience 03/2012–11/2012

10/2011–02/2012 03/2011–04/2011

Student research assistant Max-Planck-Institute of Quantum Optics, DE Kienberger-Group Student research assistant Walter-Schottky-Institute, DE Garrido-Group

Teaching Experience 10/2020–10/2021 10/2019–08/2021 04/2017–02/2020

Supervisor of a Master Thesis in Physics Seminar Tutoring: Modern X-ray Physics Tutorial: Exercises to Modern X-ray Physics.

Awards Best Poster Award at the combined meeting of the 68th Denver X-ray Conference (DXC) and 25th International Congress on X-ray Optics and Microanalysis (ICXOM) in Lombard, IL, USA (2019) for the contribution “Full-Field Structured Illumination Super-Resolution X-ray Transmission Microscopy”.

Curriculum Vitae

343

First-Authored Publications (Peer-Reviewed) Günther B, Gradl R, Jud C, Eggl E, Huang J, Achterhold K, Gleich B, Dierolf M, Pfeiffer F (2020) The multimodal X-ray beamline at the Munich Compact light source: design, instrumentation and applications. J Synchrotron Radiat 27:1395– 1414 Günther B, Hehn L, Jud C, Hipp A, Dierolf M, Pfeiffer F (2019) Full-field structuredillumination super-resolution X-ray transmission microscopy. Nat Commun 10:2494 Günther B, Dierolf M, Achterhold K, Pfeiffer F (2019) Device for source position stabilization and beam parameter monitoring at inverse Compton X-ray sources. J Synchrotron Radiat 26:1546–1553

Co-authored Publications (Peer-Reviewed) Jud C, Dierolf M, Günther B, Achterhold K, Gleich B, Rummeny E, Pfeiffer F, Pfeiffer D (2018) X-ray vector radiography reveals bone microfractures in an exvivo Porcine Rib model. Sci Rep Huang J, Deng F, Günther B, Achterhold K, Liu Y, Jentys A, Lercher JA, Dierolf M, Pfeiffer F (2021) Laboratory-scale in situ X-ray absorption spectroscopy of a Palladium catalyst on a compact inverse-Compton scattering X-ray beamline. J Anal Atom Spectrosc 36:2649–2659 Huang J, Günther B, Achterhold K, Dierolf D, Pfeiffer F (2021) Simultaneous two-color X-ray absorption spectroscopy using Laue crystals at an inverse-Compton scattering X-ray facility. J Synchrotron Radiat 28:1874–1880 Jud C, Sharma Y, Martins J, Busse M, Günther B, Weitz J, Pfeiffer F, Pfeiffer D (2021) X-ray dark-field tomography reveals tooth cracks. Sci Rep 11:14017 Burger K, Urban T, Dombrowsky AC, Dierolf M, Günther B, Bartzsch S, Achterhold K, Combs SE, Schmid TE, Wilkens JJ, Pfeiffer F (2020) Technical and dosimetric realization of in-vivo X-ray microbeam irradiations at the Munich compact light source. Med Phys 47:5183–5193 Dombrowsky AC, Burger K, Porth A-K, Stein M, Dierolf M, Günther B, Achterhold K, Gleich B, Feuchtinger A, Bartzsch S, Beyreuther E, Combs SE, Pfeiffer F, Wilkens JJ, Schmid TE (2020) A proof of principle experiment for microbeam radiation therapy at the Munich compact light source. Radiat Environ Biophys 59:111–120 Heck L, Eggl E, Grandl S, Dierolf M, Jud C, Günther B, Achterhold K, Mayr D, Gleich B, Hellerhoff K, Pfeiffer F, Herzen J (2020) Dose and spatial resolution analysis of grating-based phase-contrast mammography using an inverse Compton X-ray source. J Med Imaging 7:023505

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Curriculum Vitae

Huang J, Günther B, Achterhold K, Cui Y-T, Gleich B, Dierolf M, Pfeiffer F (2020) Energy dispersive X-ray absorption spectroscopy with an inverse Compton source. Sci Rep 10:8772 Kulpe, S, Dierolf, M, Günther, B, Brantl, J, Busse, M, Achterhold, K, Pfeiffer, F, Pfeiffer, D (2020) Spectroscopic imaging at compact inverse Compton X-ray sources. Physica Medica 79: 137–144 Kulpe S, Dierolf M, Braig E-M, Günther B, Achterhold K, Gleich B, Herzen J, Rummeny E, Pfeiffer F, Pfeiffer D (2020) K-edge subtraction imaging for angiography at a compact synchrotron source. J Med Imaging 7:023504 Kulpe S, Dierolf M, Günther B, Brantl J, Busse M, Achterhold K, Gleich B, Pfeiffer F, Pfeiffer D (2020) Dynamic K-edge subtraction fluoroscopy at a compact inverseCompton synchrotron X-ray source. Sci Rep 10:9612 Gradl R, Dierolf M, Yang L, Hehn L, Günther B, Möller W, Kutschke D, Stoeger T, Gleich B, Achterhold K, Donnelley M, Pfeiffer F, Schmid O, Morgan KS (2019) Visualizing treatment delivery and deposition in mouse lungs using in vivo X-ray imaging. J Controll Release 307:282–291 Gradl R, Morgan KS, Dierolf M, Jud C, Hehn L, Günther B, Möller W, Kutschke D, Yang L, Stoeger T, Pfeiffer D, Gleich B, Achterhold K, Schmid O, Pfeiffer F (2019) Dynamic in vivo chest X-ray dark-field imaging in mice. IEEE Trans Med Imaging 38:649–656 Heck L, Dierolf M, Jud C, Eggl E, Sellerer T, Mechlem K, Günther B, Achterhold K, Gleich B, Metz S, Pfeiffer D, Kröninger K, Herzen J (2019) Contrast-enhanced spectral mammography with a compact synchrotron source. PLoS ONE 14:e0222816 Kulpe S, Dierolf M, Günther B, Busse M, Achterhold K, Gleich B, Herzen J, Rummeny E, Pfeiffer F, Pfeiffer D (2019) K-edge subtraction computed tomography with a compact synchrotron X-ray source. Sci Rep 9:13332 Yang L, Gradl R, Dierolf M, Möller W, Kutschke D, Feuchtinger A, Hehn L, Donnelley M, Günther B, Achterhold K, Walch A, Stoeger T, Razansky D, Pfeiffer F, Morgan KS, Schmid O (2019) Multimodal precision imaging of pulmonary nanoparticle delivery in mice: dynamics of application, spatial distribution, and dosimetry. Small 15:1904112 Braig E-M, Böhm J, Dierolf M, Jud C, Günther B, Mechlem K, Allner S, Sellerer T, Achterhold K, Gleich B, Noël P, Pfeiffer D, Rummeny E, Herzen J, Pfeiffer F (2018) Direct quantitative material decomposition employing grating-based X-ray phase-contrast CT. Sci Rep 8:16394 Eggl E, Grandl S, Sztrókay-Gaul A, Dierolf M, Jud C, Heck L, Burger K, Günther B, Achterhold K, Mayr D, Wilkens JJ, Auweter SD, Gleich B, Hellerhoff K, Reiser MF, Pfeiffer F, Herzen J (2018) Dose-compatible grating-based phase-contrast mammography on mastectomy specimens using a compact synchrotron source. Sci Rep 8:15700

Curriculum Vitae

345

Gradl R, Dierolf M, Günther B, Hehn L, Möller W, Kutschke D, Yang L, Donnelley M, Murrie R, Erl A, Stoeger T, Gleich B, Achterhold K, Schmid O, Pfeiffer F, Morgan KS (2018) In vivo dynamic phase-contrast X-ray imaging using a compact light source. Sci Rep 8:6788 Hehn L, Gradl R, Voss A, Günther B, Dierolf M, Jud C, Willer K, Allner S, Hammel JU, Hessler R, Morgan KS, Herzen J, Hemmert W, Pfeiffer F (2018) Propagationbased phase-contrast tomography of a guinea pig inner ear with cochlear implant using a model-based iterative reconstruction algorithm. Biomed Opt Express 9:5330 Kulpe S, Dierolf M, Braig E, Günther B, Achterhold K, Gleich B, Herzen J, Rummeny E, Pfeiffer F, Pfeiffer D (2018) K-edge subtraction imaging for coronary angiography with a compact synchrotron X-ray source. PLoS ONE 13:e0208446 Burger K, Ilicic K, Dierolf M, Günther B, Walsh DWM, Schmid E, Eggl E, Achterhold K, Gleich B, Combs SE, Molls M, Schmid TE, Pfeiffer F, Wilkens JJ (2017) Increased cell survival and cytogenetic integrity by spatial dose redistribution at a compact synchrotron X-ray source. PLOS ONE 12:e0186005 Eggl E, Mechlem K, Braig E-M, Kulpe S, Dierolf M, Günther B, Achterhold K, Herzen J, Gleich B, Rummeny E, Noël P, Pfeiffer F, Münzel D (2017) Mono-energy coronary angiography with a compact synchrotron source. Sci Rep 7:42211 Gradl R, Dierolf M, Hehn L, Günther B, Yildirim AÖ, Gleich B, Achterhold K, Pfeiffer F, Morgan KS (2017) Propagation-based phase-contrast X-ray imaging at a compact light source. Sci Rep 7:4908 Jud C, Braig E-M, Dierolf M, Eggl E, Günther B, Achterhold K, Gleich B, Rummeny E, Noël P, Pfeiffer F, Muenzel D (2017) Trabecular bone anisotropy imaging with a compact laser-undulator synchrotron X-ray source. Sci Rep 7:14477 Eggl E, Dierolf M, Achterhold K, Jud C, Günther B, Braig E-M, Gleich B, Pfeiffer F (2016) The Munich compact light source: initial performance measures. J Synchrotron Radiat 23:1137–1142

First-Authored Conference Proceedings Günther B, Dierolf M, Gradl R, Jud C, Gleich B, Achterhold A, Pfeiffer F (2020) The versatile X-ray beamline at the Munich compact light source, an inverse Compton synchrotron facility. In: High-brightness sources and light-driven interactions. OSA Technical Digest (online). Optical Society of America, ETuA.2 Günther B, Gradl R, Jud C, Eggl E, Kulpe S, Braig E, Heck L, Brantl J, Achterhold K, Gleich B, Dierolf M, Pfeiffer F (2019) Evaluation and optimization of multimodal X-ray imaging techniques for inverse Compton X-ray sources. In: Proceedings of the SPIE. Advances in laboratory-based X-ray sources, optics, and applications VII, vol 11110, p 1111008

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Günther B, Dierolf M, Gifford M, Eggl M, Gleich B, Achterhold K, Loewen R, Pfeiffer F (2018) The Munich compact light source: flux doubling and source position stabilization at a compact inverse-Compton synchrotron X-ray source. Microsc Microanal 24(S2):312–313 Günther B, Dierolf M, Gradl R, Eggl E, Jud C, Hehn L, Kulpe S, Gleich B, Busse M, Morgan KS, Achterhold K, Pfeiffer F (2018) The Munich compact light source: biomedical research at a laboratory-scale inverse-Compton synchrotron Xray source. Microsc Microanal 24(S1):984–985 Günther B, Dierolf M, Achterhold K, Pfeiffer F (2018) X-ray beam monitoring and source position stabilization at an inverse-Compton X-ray source. In: High-brightness sources and light-driven interactions. OSA Technical Digest (online). Optical Society of America, EM2B.5

Co-authored Conference Proceedings Braig E-M, Dierolf M, Günther B, Mechlem K, Allner S, Sellerer T, Achterhold K, Gleich B, Rummeny E, Pfeiffer D, Pfeiffer F, Herzen J (2019) Single-energy material decomposition with grating-based X-ray phase-contrast CT. In: Proceedings of the SPIE. Developments in X-ray tomography XII, vol 11113, p 111130R Heck L, Eggl E, Grandl S, Dierolf M, Jud C, Günther B, Achterhold K, Mayr D, Gleich B, Hellerhoff K, Pfeiffer F, Herzen J (2019) Dose and spatial resolution analysis of grating-based phase-contrast mammography using an inverse Compton X-ray source. In: Proceedings of the SPIE. Developments in X-ray tomography XII, vol 11113, p 111130M Kulpe S, Dierolf M, Braig E-M, Günther B, Achterhold K, Gleich B, Herzen J, Rummeny E, Pfeiffer F, Pfeiffer D (2019) K-edge subtraction imaging for angiography at a compact synchrotron source. In: Proceedings of the SPIE. Advances in laboratory-based X-ray sources, optics, and applications VII, vol 11110, p 111100J Morgan KS, Gradl R, Dierolf M, Jud C, Günther B, Werdiger F, Gardner M, Cmielewski P, McCarron A, Farrow N, Haas H, Kimm MA, Yang L, Kutschke D, Stoeger T, Schmid O, Achterhold K, Pfeiffer F, Parsons DW, Donnelley M (2019) In vivo X-ray imaging of the respiratory system using synchrotron sources and a compact light source. In: Proceedings of the SPIE. Developments in X-ray tomography XII, vol 11113, p 11130G Dierolf M, Günther B, Gradl R, Jud C, Eggl E, Gleich B, Achterhold K, Pfeiffer F (2018) The Munich compact light source: operating an inverse Compton Source in user mode. In: High-brightness sources and light-driven interactions. OSA Technical Digest (online). Optical Society of America, EM2B.3. Gradl R, Dierolf M, Hehn L, Günther B, Kutschke D, Yang L, Möller W, Stoeger T, Kimm M, Haas H, Roiser N, Donnelley M, Jud C, Geich B, Parsons D, Achterhold

Curriculum Vitae

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K, Schmid O, Pfeiffer F, Morgan KS (2018) Dynamic X-ray imaging at the Munich compact light source. Microsc Microanal 24(S2):350–351 Eggl E, Mechlem K, Braig E-M, Kulpe S, Dierolf M, Günther B, Achterhold K, Herzen J, Gleich B, Rummeny E, Noël P, Pfeiffer F, Münzel D (2017) Mono-energy coronary angiography with a compact light source. In: Proceedings of the SPIE. Medical imaging 2017: physics of medical imaging, vol 10132, p 101324L Jud C, Braig E-M, Dierolf M, Eggl E, Günther B, Achterhold K, Gleich B, Rummeny E, Noël P, Pfeiffer F, Münzel D (2017) X-ray vector radiography of a human hand. In: Proceedings of the SPIE. Medical imaging 2017: physics of medical imaging, vol 10132, p 101325U

Oral Presentation The Versatile X-ray Beamline at the Munich Compact Light Source, an Inverse Compton Synchrotron Facility, High-Brightness Sources and Light-driven Interactions Congress, November 2020, virtual Congress Medical imaging research and spectroscopy at the MuCLS, Advanced Medical Imaging with Synchrotron and Compton X-ray Sources, November 2019, Bologna, I (invited talk) Evaluation and optimization of multimodal X-ray imaging techniques for inverse Compton X-ray sources, SPIE Optics and Photonics: Advances in Laboratory-based X-Ray Sources, Optics, and Applications VII, August 2019, San Diego, USA X-ray beam stabilisation and X-ray imaging at an inverse-Compton X-ray source, Seminar talk at the Monash University, June 3rd, 2019, Clayton, AUS (invited talk) X-ray imaging at an inverse-Compton Synchrotron Source, Seminar talk at the Arizona State University, November 28th, 2018, Tempe, USA (invited talk) The Munich Compact Light Source: Biomedical Research At a Laboratory-Scale Inverse-Compton Synchrotron X-ray Source, Microscopy and Microanalysis 2018, August 2018, Baltimore, USA (invited talk) X-ray Beam Monitoring and Source Position Stabilization at an Inverse-Compton X-ray Source, Annual Meeting of the International Max-Planck Research School of Advanced Photonic Science (IMPRS-APS), May 2018, Ringberg castle, GER X-ray Beam Monitoring and Source Position Stabilization at an Inverse-Compton Xray Source, High-Brightness Sources and Light-driven Interactions Congress, March 2018, Straßburg, F The Munich Compact Light Source -Performance upgrade and biomedical research-, 24th International Congress on X-ray Optics and Microscopy (IXCOM), September 2017, Triest, I

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Curriculum Vitae

The Munich Compact Light Source -Performance and Application at TUM- and The MuCLS: Operational aspects, Compact Light Source Workshop, Australian Synchrotron, November 2016, Melbourne, AUS (invited talk) The Munich Compact Light Source -Performance and Application at TUM-, Workshop on the Sydney Compact Synchrotron, The University of Sydney, November 2016, Sydney, AUS (invited talk) The MuCLS: source properties, characterization and challenges, Workshop “Science at the Munich Compact Light Source”, October 2016, Garching, GER The Munich Compact Light Source—“a truck-size synchrotron”, Annual Meeting of the International Max-Planck Research School of Advanced Photonic Science (IMPRS-APS), November 2015, Ringberg castle, GER

Poster Presentation Full-Field Structured Illumination Super-Resolution X-ray Transmission Microscopy, 68th Denver X-ray Conference (DXC) and 25th International Congress on X-ray Optics and Microanalysis (ICXOM), August 2019, Lombard, USA The Munich Compact Light Source: X-ray Imaging Applications of a Compact Inverse-Compton Synchrotron Source, 10th International Particle Accelerator Conference (IPAC), May 2019, Melbourne, AUS The Munich Compact Light Source: Flux Doubling and Source Position Stabilization At a Compact Inverse-Compton Synchrotron X-ray Source, 14th X-ray Microscopy Conference (XRM), August 2018, Saskatoon, CAN The Munich Compact Light Source -biomedical imaging with a lab-sized laserundulator synchrotron source-, Higher European Research School for Users of Large Experimental Systems (HERCULES), February–March 2017, Grenoble, F Towards high-resolution microscopy with a laboratory-sized quasimonochromatic X-ray source based on inverse Compton scattering, 13th X-ray Microscopy Conference (XRM), August 2018, Oxford, UK