Nonlinear X-Ray Spectroscopy for Materials Science (Springer Series in Optical Sciences, 246) [1st ed. 2023] 9819967139, 9789819967131

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Springer Series in Optical Sciences 246

Iwao Matsuda Ryuichi Arafune   Editors

Nonlinear X-Ray Spectroscopy for Materials Science

Springer Series in Optical Sciences Volume 246

Springer Series in Optical Sciences is led by Editor-in-Chief William T. Rhodes, Florida Atlantic University, USA, and provides an expanding selection of research monographs in all major areas of optics: • • • • • • • •

lasers and quantum optics ultrafast phenomena optical spectroscopy techniques optoelectronics information optics applied laser technology industrial applications and other topics of contemporary interest.

With this broad coverage of topics the series is useful to research scientists and engineers who need up-to-date reference books.

Iwao Matsuda · Ryuichi Arafune Editors

Nonlinear X-Ray Spectroscopy for Materials Science

Editors Iwao Matsuda The Institute for Solid State Physics The University of Tokyo Kashiwa, Chiba, Japan

Ryuichi Arafune Research Center for Materials Nanoarchitectonics (MANA) National Institute for Materials Science (NIMS) Tsukuba, Ibaraki, Japan

ISSN 0342-4111 ISSN 1556-1534 (electronic) Springer Series in Optical Sciences ISBN 978-981-99-6713-1 ISBN 978-981-99-6714-8 (eBook) https://doi.org/10.1007/978-981-99-6714-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

There has been a rise of a new analysis method of nonlinear X-ray spectroscopy for materials science. This book provides the basic principles for understanding the issue and reviews the frontier topics, followed by the future prospects. Readers are expected to have elementary knowledges of electromagnetism and quantum mechanics. A spectrum is essentially a set of signals taken with parameters. For example, it is the photon intensity signal at different wavelengths (photon energies). The dependence provides information of a sample that is characteristic to the materials. Interpretations of experimental spectra can naturally be done when amounts of the signal are proportional to the incident intensity. However, there are varieties of the optical phenomena that show the nonlinear dependence. Moreover, they are associated with events, such as second harmonic generation, that are unusual in everyday life but critical for our scientific development. These special phenomena are typically induced with the ultrashort pulse light source, i.e., laser, and the recent technical innovations have pushed the controllable wavelength down to the X-ray region. “X-ray” was named after a ray of “something unknown (X)” by Röntgen, and it has been a significant experimental probe to investigate structure and electronic states of materials. Today, there rises a new research field of the nonlinear X-ray experiments that have led to the significant discoveries. The understandings have also opened a new analysis method of the nonlinear X-ray spectroscopy that are found to be useful for materials science. The frontier topics deserve known widely among researchers, but the experimental principles are based on the unusual phenomena with the mysterious light. Thus, the issue is likely hard to get started with, and it is desirable to have a guide for those who are interested in. This situation has motivated us to develop the book. This book is composed of six chapters (Fig. 1). Chapter 1 introduces the basic principles of the nonlinear responses and the related issues. Chapter 2 guides various linear X-ray spectroscopies for materials science and provides the useful knowledges for understanding the following chapters. Chapters 3–5 review the frontier topics of nonlinear spectroscopy at various wavelength (energy) regions of photons. At the end of the book, future prospects of this new research field are given. The editing of this book was done at two sites: (i) Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science, and (ii) Laser v

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Preface

Fig. 1 Structure of the book

and Synchrotron Research Center (LASOR), the Institute for Solid State Physics (ISSP), the University of Tokyo. We would like to express our sincere appreciations to all the authors and those acknowledged in the individual chapters. Colleagues who contributed directly in providing the wonderful illustrations and the critical reading of the entire manuscript are: Nanami Arafune, Yurina Matsuda, Akiko Itakura, Tasha Welling, Masafumi Horio, Yuki Tsujikawa, and the ISSP Public Relation Office. We would like to appreciate you for reading this book and wish to enjoy it.

Ryuichi Arafune Tsukuba, Ibaraki, Japan

Iwao Matsuda Kashiwa, Chiba, Japan

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryuichi Arafune 1.1 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Classical Model of Nonlinear Optical Process . . . . . . . . . . . . . . . . . . 1.3 Pragmatic Importance of Nonlinear Spectroscopy for Materials Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Synopsys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iwao Matsuda 2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Optical Responses of Materials in the X-Ray Region . . . . . . 2.1.2 X-Ray Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Light–Matter Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 X-Ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Photoelectron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 X-Ray Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Time-Resolved X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Measurement Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Examples of Time-Resolved Photoemission Spectroscopy Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Researcher’s Guide to Material Characterization with X-Rays . . . . . 2.4.1 The Guiding Chart for Experiments . . . . . . . . . . . . . . . . . . . . . 2.4.2 Application for Beamtime at the X-Ray Facility . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 9 12 13 15 15 15 20 22 28 28 33 36 39 39 45 49 49 51 52 53

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Contents

3 Probing Nonlinear Light–Matter Interaction in Momentum Space: Coherent Multiphoton Photoemission Spectroscopy . . . . . . . . . Marcel Reutzel, Andi Li, Zehua Wang, and Hrvoje Petek 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Experimental Setup: The Photoelectron Analyzer . . . . . . . . . 3.2.2 Experimental Setup: The Light-Source . . . . . . . . . . . . . . . . . . 3.3 mPP: Highly Nonlinear Mapping of the Energyand Momentum-Dispersive Electronic Band Structure . . . . . . . . . . . 3.3.1 Static mPP in Threshold Order of Photoemission . . . . . . . . . 3.3.2 Above Threshold Photoemission . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Toward Full Surface Brillouin Zone Mapping by Coherent mPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Coherent Two-Dimensional Photoelectron Spectroscopy . . . . . . . . . 3.4.1 Coherent 2D FT Photoelectron Spectroscopy—Optical Bloch Equation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Coherent 2D FT Photoelectron Spectroscopy of Ag(111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Coherent Above Threshold Photoemission . . . . . . . . . . . . . . . 3.5 Ultrafast Quasiparticle Dressing by Light . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonlinear Soft X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Craig P. Schwartz and Walter S. Drisdell 4.1 Nonlinear Spectroscopy—Development with Visible Light . . . . . . . 4.2 Ultrafast X-Ray Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Family of Soft X-Ray Nonlinear Spectroscopy . . . . . . . . . . . . . . . . . . 4.4 Nonlinear Soft X-Ray Optics and Spectroscopies . . . . . . . . . . . . . . . 4.4.1 Multiphoton Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Stimulated Emission/Forward Scattering . . . . . . . . . . . . . . . . 4.4.3 Stimulated Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Four Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Soft X-Ray Second Harmonic Generation/Sum Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Theoretical Calculations for the Spectral Analysis . . . . . . . . . . . . . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nonlinear X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuya Kubota and Kenji Tamasaku 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Basic Theory of Nonlinear Optics in the Hard X-Ray Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Nonlinear Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 X-Ray Second Harmonic Generation . . . . . . . . . . . . . . . . . . .

57 58 60 60 60 62 63 63 65 66 67 69 72 74 76 77 83 84 90 92 95 95 97 99 100 106 110 112 113 119 119 122 122 127

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5.2.3 Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Sum Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Featuring Examples of Nonlinear X-Ray Spectroscopy . . . . . . . . . . . 5.3.1 X-Ray Two-Photon Absorption Spectroscopy . . . . . . . . . . . . 5.3.2 Saturable Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Atomic X-Ray Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Stimulated X-Ray Emission Spectroscopy . . . . . . . . . . . . . . . 5.3.5 X-Ray Transient Grating Spectroscopy . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128 130 132 132 136 136 139 139 141 141

6 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iwao Matsuda, Craig P. Schwartz, Walter S. Drisdell, and Ryuichi Arafune 6.1 Toward Multi-dimensional Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 6.2 Phase Sensitive Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Vacuum Nonlinear X-Ray Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Developments in Experimental Stations for Materials Science . . . . . 6.5 Into the Deep: Nonlinear Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

147 149 152 154 155 156

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Contributors

Ryuichi Arafune Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan Walter S. Drisdell Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA Yuya Kubota RIKEN SPring-8 Center, Hyogo, Japan Andi Li Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh, Pittsburgh, PA, USA Iwao Matsuda The Institute for Solid State Physics, The University of Tokyo, Chiba, Japan Hrvoje Petek Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh, Pittsburgh, PA, USA Marcel Reutzel I. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany Craig P. Schwartz Nevada Extreme Laboratory, University of Nevada, Las Vegas, USA Kenji Tamasaku RIKEN SPring-8 Center, Hyogo, Japan Zehua Wang Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh, Pittsburgh, PA, USA

xi

Abbreviations

4WM AED AES ARPES ASE ATP CDI CLS DFG EXAFS FEL FT HHG IP ITR-mPP KKR LCLS mPP NEXAFS OBE OPA OPA OPO PDC PED RIXS RTTDDFT SA SACLA SARPES

Four-wave mixing Auger electron diffraction Auger electron spectroscopy Angle-resolved photoelectron (photoemission) spectroscopy Amplified spontaneous emission Above threshold photoemission Coherent diffraction imaging Core-level photoelectron (photoemission) spectroscopy Difference frequency generation Extended X-ray absorption fine structure Free electron laser Fourier transform High harmonic generation Image potential Interferometrically time-resolved multi-photon photoemission Korringa–Kohn–Rostoker Linac Coherent Light Source Multi-photon photoemission Near-edge X-ray absorption fine structure Optical Bloch equation One-photon absorption Optical parametric amplification Optical parametric oscillation Parametric down-conversion Photoelectron diffraction Resonant inelastic X-ray scattering Real-time time-dependent density functional theory Saturable absorption SPring-8 Angstrom compact free-electron laser Spin- and angle-resolved photoelectron (photoemission) spectroscopy xiii

xiv

SASE SFG SHG SR SS SXES SXSHG TG THG TPA tr2PP trARPES UPS VG-RTTDDFT VUV XAFS XAS XES XFEL XMCD XMLD XPS XRD XTG

Abbreviations

Self-amplified spontaneous emission Sum frequency generation Second harmonic generation Synchrotron radiation Surface state Stimulated X-ray emission spectroscopy Soft X-ray second harmonic generation Transient grating Third harmonic generation Two-photon absorption Time-resolved two-photon photoemission Time- and angle-resolved photoelectron spectroscopy Ultraviolet photoelectron (photoemission) spectroscopy Velocity gauge real-time time-dependent density functional theory Vacuum ultraviolet X-ray absorption fine structure X-ray absorption spectroscopy X-ray emission spectroscopy X-ray free electron laser X-ray magnetic circular dichroism X-ray magnetic linear dichroism X-ray photoelectron (photoemission) spectroscopy X-ray diffraction X-ray transient grating

Chapter 1

Introduction Ryuichi Arafune

Abstract This book is primarily intended for students and researchers who are actively engaged in experimental work within the field of materials science. It offers an in-depth exploration of cutting-edge techniques and tools designed for measurement of electron dynamics through nonlinear spectroscopy. While it serves as an invaluable resource for thouse already immersed in this area of study, it also aims to inspire readers of nonlinear optical spectroscopy within the field of material science. In this intoroductory chapter, we have underscored the significance of nonlinearity in material science, elucidated the common basic aspects of nonlinear optical response, and outlined how nonlinear optical spectroscopy contributes to the understanding of the dynamics of materials.

1.1 Nonlinearity There are a surprisingly large number of terms and concepts with negative prefix in all fields of science. Even with in the realm of the physical sciences, we encounter terms like non-equilibrium system, non-holonomic system, non-local potential, irreversible process, anharmonicity, non-Abelian theory, asymmetry, inelastic process, non-integral system of coordinates, and many more. Initially, all the concepts in various fields of science without the negative prefix “Non-” such as equilibrium system, reversible process, Abelian theory, and many more were developed first. However, as the field expanded, the need for “Non-” fields also became increasingly important. Due to the rapid pace of development and the vast diversity of the field, it has continued to progress under the name “Non-(something)” without being given a new name. It is important to emphasize that the presence of the negative prefix does not imply that the discipline is dealing with exceptional or uncommon matters. On the contrary, it is a broader and deeper field of study, offering a treasure trove R. Arafune (B) Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_1

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R. Arafune

of interesting research subjects. Nonlinearity, the subject of this book is no exception; rather, it serves as a prime example and the foundation concept underlying the various “Non-(something)” fields mentioned above. The most crucial characteristic of nonlinearity is the absence of the superposition principle. Linear phenomena have developed a simple, easy-to-understand, and elegant system of logic based on the validity of the superposition principle, which means that they can be subdivided into individual elements and analyzed. However, on the other hand, it is easy to imagine how boring the world would be if all science could be described in linear terms. Each element would exist independently and would not be able to cooperate. The primary goals of materials science are to create and discover novel functionality of materials. Furthermore, understanding “What is a novel functionality?” is also essential for the development of science. The coordination and interaction of individual elements play a pivotal role in developing the functions of materials, rendering nonlinearity the essence of material science. In condensed matter physics, nonlinearity has been a significant focal point, including its engineering application. Its significance is anticipated to grow even futher in the future.

1.2 Classical Model of Nonlinear Optical Process X-ray nonlinear spectroscopy, one of the major techniques introduced in this book, has attracted much attention as a practical spectroscopy method due to the recent development of the X-ray free electron laser. The nonlinear optical spectroscopy begun with the infrared/visible laser technology. As the importance of nonlinear optics has been recognized, efforts have been made to shorten the wavelengths. The X-ray nonlinear spectroscopy can be regarded as a milestone in this direction. This great deal of attention is due to the importance and effectiveness of nonlinear spectroscopy developed with the short-pulse laser technology. On the other hand, it should be noted that X-ray nonlinear spectroscopy is not a simple short-wavelength version of ordinary optical nonlinear spectroscopy. In the context of material science, X-rays are not merely shorter-wavelength electromagnetic radiation, but that their wavelength (λ) is of the order of magnitude of the bonding distance of matter, especially for hard X-ray region. An X-ray has been conventionally classified into two groups, “soft” or “hard”, by the range of photon energy (ℏω). A soft X-ray is named for the photon energy region below ℏω = 2 keV (λ = 0.6 nm), while a hard X-ray above ℏω = 2 keV. This feature makes a decisive difference in their principles and applications. As a result, classical/macroscopic treatment in the analysis might be difficult in X-ray nonlinear spectroscopy. The macroscopic nonlinear response, which is based on classical electromagnetism, can be described because we implicitly assume that the spatial scale over which the electric field fluctuates is essentially large compared to the size of the atoms and molecules, which are the fundamental building blocks of matter. It would be essentially important to understand X-ray nonlinear spectroscopy by describing

1 Introduction

3

the interaction between X-ray and matter from a semiclassical/quantum mechanical viewpoint. However, if there is no knowledge of nonlinear optical responses described from the classical viewpoint, it may be difficult to understand the theoretical framework and its direction. Therefore, we will introduce the fundamental and essential features of the nonlinear optical response from a classical viewpoint to establish a basis for understanding nonlinear spectroscopy [1]. The interaction between light and matter, such as reflection, refraction, scattering, emission, and absorption, can be treated as a linear response between the incident light field and the material when the incident radiation field is weak. The intensity of the output signal is linearly proportional to that of the input light. Furthermore, the energy of the output photon or electron is identical to that of the input photon. From a quantum mechanical view, it is often conceived as a situation in which a single photon excites a ground state to another state. Nonlinear optical response (Fig. 1.1) is used to refer to cases that fall outside this view, including: (1) the response of material subjected to interactions with two or more independent incident fields and (2) the situation where linear response theory is inadequate for treating how the material behaves, as in the case of very intense incident radiation. From a macroscopic point of view, nonlinear optical response means that D = ε(E)E, i.e., the permittivity is described as a function of the electric field. Now, let us describe the optical properties of a material in terms of a series of electric fields (power series) to the polarization: . P = χ (1) · E + χ (2) : EE + χ (3) .. EEE + · · · .

(1.1)

The first term represents the normal linear response, while the second term represents the lowest order of nonlinear susceptibility. In many cases, this optical nonlinearity is small (which is why such a series expansion makes sense), and thus, it requires a short-pulse light that can produce a high electric field. It should be noted that nonlinear optical responses are also realized that are not well described by perturbative expansions at present. Next, let us describe the physical significance of the second-order nonlinear susceptibility in a simple but realistic model. In this context, we examine how a

Fig. 1.1 Linear and nonlinear optical response

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R. Arafune

single electron in a plasma is affected by a linearly polarized electric field described as: By = E x =

) 1( E exp(ikz − i ωt) + E ∗ exp(−ikz + i ωt) , 2

(1.2)

where k = ωc . Equations of motion of the electron are: e m x˙ m x¨ = eE x − z˙ B y − c τ m y˙ m y¨ = − τ e m z˙ m z¨ = z˙ B y − c τ

(1.3)

Here, the phenomenological collision time τ is introduced, which describes the damping of the motion. Assuming the velocity of the electron is sufficiently slow compared with the velocity of the light and that the force by the magnetic field can be treated as a perturbation, these equations can be solved by successive approximation in the form of a Fourier series. The first or linear approximation term is a well-known result: x(ω) =

−eE exp(ikz − iωt) ( ) m ω2 + i ω/τ

(1.4)

From this, one can obtain the (linear) dipole moment: ex(ω), and this dipole moment describes the Rayleigh scattering from the free electron gas. By substituting the linear solution into the equation of motion, one can obtain the lowest order of the nonlinear approximation of the electron position. z(2ω) =

−ie2 E 2 exp(2ikz − 2i ωt) )( ) ( m 2 c 4ω + 2iτ ω2 + iω τ

(1.5)

As analogous to the linear term, ez(2ω) describes the scattering of light with a frequency twice that of the incident light. The above idea is extended to the plasma with an average density of No . The linear polarization is: Px = χ (1) (ω)E x (ω) = No ex(ω)

(1.6)

Assuming ωτ « 1, one can deduce the familiar result for the (linear) susceptibility of a plasma: ε − 1 = 4π χ (1) = −4π No e2 /mω2

(1.7)

1 Introduction

5

Similarly, the nonlinear polarization at the second harmonic frequency can be given by Pz (2ω) = No ez(2ω).

(1.8)

By considering the direction of the polarization to the incident electric field, one would see that no coherent radiation at 2 ω is generated. This is because we have assumed that the plasma is centrosymmetric. Next, we go one step further from the free electron plasma model. The frequency dependence of the nonlinear susceptibility based on the Lorenz model is described. One would see that the nonlinear susceptibility is sensitive to the anharmonicity. Let us consider the anharmonic oscillator with damping: x¨ + ⎡ x˙ + ωo2 x + ax 2 + bx 3 =

e E(t), M

(1.9)

driven by the electric field. Here, ω0 is the resonant frequency, and ⎡ is the damping constant. And we assume that the electric field is the summation of the fields with frequencies ω1 and ω2 : E(t) = Re{(E 1 exp(ik1 z − i ω1 t) + E 2 exp(ik2 z − iω2 t)}.

(1.10)

To proceed with the analysis, one expands the displacement x by the power series: x(t) = x (1) (t) + x (2) (t) + x (3) (t) + · · ·

(1.11)

Thus, the linear term can be described by x¨ (1) + ⎡ x˙ (1) + ωo2 x (1) =

e E(t), M

and its solution can be written as: [ 1 1 1 e E 1 exp(ik1 z − i ω1 t) + x (1) (t) = 2 M D(ω1 ) D(ω2 ) ] E 2 exp(ik2 z − i ω2 t) + c.c.,

(1.12)

(1.13)

where the notation D(ω) = ω0 − ω − i⎡ω = D ∗ (−ω)

(1.14)

is introduced. Note that the x (1) (t) term is proportional to the electric field. The lowest nonlinear-order term can be described by: ( )2 x¨ (2) + ⎡ x˙ (2) + ωo2 x (2) + a x (1) = 0

(1.15)

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R. Arafune

By substituting x (1) into the above equation, one can obtain x (2) (t). x (2) (t) consists of many terms, and each term can be represented by the following: x (2) (t) = −

E 12 ae2 exp(2ik1 z − 2i ω1 t) + c.c. 4M 2 D 2 (ω1 )D(2ω1 )

(1.16)

x (2) (t) = −

E 22 ae2 exp(2ik2 z − 2i ω2 t) + c.c. 2 2 4M D (ω2 )D(2ω2 )

(1.17)

x (2) (t) = −

E1 E2 ae2 exp(i(k1 + k2 )z − i(ω1 + ω2 )t) + c.c. 2 2M D(ω1 + ω2 )D(ω1 )D(ω2 ) (1.18)

x (2) (t) = −

E1 E2 ae2 exp(i(k1 − k2 )z − i(ω1 − ω2 )t) + c.c. 2 2M D(ω1 − ω2 )D(ω1 )D(ω2 ) (1.19)

(and there is the DC component of the displacement which is independent of time). One would understand the meanings of each term intuitively. In this model, the dipole moment is represented by ex(t). Thus, each term corresponds to the second-order nonlinear optical response. More specifically, each term corresponds: • • • •

The second harmonic generation (SHG) of the light with frequency ω1 . The second harmonic generation of the light with frequency ω2 . The sum frequency generation (SFG) of the light with ω1 and ω2 . The difference frequency generation (DFG) of the light with ω1 and ω2 . The respective excitation processes are also shown in Fig. 1.2.

Fig. 1.2 Energy level scheme for the second nonlinear optical response as described in (1.16)−(1.19)

1 Introduction

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One application of the nonlinear optical process is the energy conversion of light. For example, SHG can convert low-energy infrared light into visible light while keeping the coherent characteristics as well as the original incident light. This means that the nonlinear processes can extend the wavelength range of coherent light. From a viewpoint of material science, a basic question arises here “What are the benefits of probing nonlinear optical responses from a materials science perspective?” Starting from (1.1), let us see how nonlinear optical responses relate to other “non(something)” characteristics. Generally, experiments that use second-order nonlinear optical effects are conducted using crystalline media and surface interfaces. In other words, isotropic gases and liquids are unlikely to be used, although the liquid/gas interface is an important target for nonlinear optics. Let us confirm simply that the second-order nonlinear response does not occur in symmetric media. In a system with inversion symmetry, the generated dipole must be − P(r) = P(− r), regardless of its order. On the other hand, a nonlinear dipole of the second order must satisfy both −P(r) = −χ (2) : E(r)E(r) and P(−r) = χ (2) : E(−r)E(−r) = χ (2) : E(r)E(r). Thus, χ (2) must be zero. In other words, the second-order nonlinear optical response occurs at the point where the symmetry is broken. From this brief discussion, one would see that nonlinear spectroscopy is a powerful tool for probing symmetry. The third-order term in (1.1) can be treated in a similar manner to the second-order term. One can obtain the third-order term of the displacement x (3) . x (3) contains more terms than x (2) (t). Here, let us introduce an interesting term of x (3) . The frequency of the term is 2ω1 − ω2 : 1 e2 x (3) (t) = − 4M 2 D(2ω1 − ω2 )D(ω1 )2 D(ω2 )∗ ] [ 2a 2 1 a2 3 b− − × 2 D(ω1 − ω2 ) D(2ω1 ) × E 12 E 2∗ exp(i(2k1 − k2 )z − i(2ω1 − ω2 )t) + c.c.

(1.20)

This term describes the non-degenerate four-wave mixing. In this process, there are two interactions: one is between waves with frequency ω1 and another with frequency ω2 . As a result, the wave with frequency 2ω1 − ω2 is generated. Note that for a = 0 and b = 0, the above terms are zero. The anharmonicity is the key to generating the nonlinear optical response. Note that the dispersion of the nonlinear susceptibility is described by a set of more than one frequency. The dispersive property is characterized by the denominators in (1.16)−(1.19). The profile of the 1/D(ω) is shown in Fig. 1.3a. Here, let us examine the dispersive feature of the sum frequency generation, which is a resultant of the two input waves with frequency ω1 and ω2 . By examining (1.18), one would see that nonlinear susceptibility is resonantly enhanced when ω1 or ω2 is equal or nearly equal as well as when the ω1 +ω2 is matched with the resonant frequency (Fig. 1.3b). To make this chapter a more comprehensive introduction to nonlinear optical response, it is important to highlight that current laser technology can generate strong electric fields for which the perturbation theory no longer applies. Nonlinear optical

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Fig. 1.3 a Profile of 1/D(ω) (1.14) which represents the dispersion characteristics of the nonlinear susceptibility. b Schematic diagram of energy level for the possible resonant processes (resonant frequency: ω0 ) involved in SFG. The solid horizontal line and dotted line represent the stationary state and the virtual state, respectively

responses originating from these strong electric fields are also important in science. As noted above, the polarization created by the incident light field is sufficiently described with the linear response alone when the electric field is very low. The perturbation expansion expressed in (1.1) is valid when the electric field is not sufficiently weak for a linear response, but we can still consider that the electric field is sufficiently weak in the case of a nonlinear response. As a criterion for judging whether the perturbation expansion is reasonable or not, the electric field felt by an electron in an atom, a molecule, or a solid would be appropriate, which is approximate 109 V/m. When considering the interaction of matter with the light field whose intensity is comparable to it, the perturbation expansion become inapplicable. Instead, as an effective approach to describe the nonlinear responses for an atom under such a high electric field, a model in which the laser electric field deforms the Coulomb potential of an atom, and the atom is ionized by the electrons tunneling through the deformed potential (tunnel ionization) is known [2]. While the atom is ionized, the electron driven by the laser field can remain in the vicinity of the ion, and then the electron can recollide with the ionized atom. As the result, the electron loses the energy and is emitted as radiation. Harmonic generation originating from the tunnel

1 Introduction

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ionization is referred to as high harmonic generation (HHG) and can create highenergy photons (> 100 eV) and light pulses with the ultrashort temporal duration (< 100 as). Thus, the HHG-related technique potentially provides the tabletop light source that generates extreme ultraviolet light and leads to the technology for investigating material properties within an extremely short time scale. Indeed, there is an active research field in which HHG is used. It is called attoscience [3]. At this time scale, nuclear motion is almost negligible (recall that, within classical approximation, the period of the electron motion in the ground state of the hydrogen atom occurs over a time of 150 as). With few exceptions, there is currently vigorous research on the direct observation and control of electron wave packets using isolated atoms and molecules in a vacuum as the sample (this topic will not be covered in this book).

1.3 Pragmatic Importance of Nonlinear Spectroscopy for Materials Science Through the nonlinear optical response, one can reveal significant deviations from the symmetry and/or harmonicity of the system. Here, let us discuss another important aspect of nonlinear spectroscopy. It is difficult to achieve the high electric field that leads to nonlinear optical responses by irradiating the conventional continuous laser on the sample. It is well known that the pulsed light generated through techniques such as mode-locking and Q-switching is suitable to achieve a higher electric field. At present, pulsed light sources are essential equipment for nonlinear spectroscopy. Indeed, nonlinear optical spectroscopy had its origins with the invention of the pulsed laser. The utilization of pulsed light which is an inevitable feature of nonlinear optical responses caused an essential revolution in spectroscopy adding the temporal resolution with very high resolution. And that revolution continues to develop to the present day. One of the interesting aspects of solid-state properties is their dynamical nature, which unfolds on exceedingly brief time scales. Time, a fundamental variable in physics and chemistry, plays a pivotal role. The temporal evolution of a particle or wave packet is the center of the fundamental equations which describe the phenomena, whether in classical or quantum mechanics. Comprehending these properties is not only crucial for fundamental scientific understanding but also holds significant importance for engineering applications. For example, gaining essential knowledge about the high-speed response of electrons, which carries information bits in current information-processing devices, is important for the development of new processing devices. The short time scales discussed in this book are mainly those with a time resolution of tens of femtoseconds to tens of picoseconds. What would a time scale of this order (10 fs–10 ps) correspond to in materials science? One of the most typical examples is nuclear motion. Consider a molecule as a simple system. Apart from translational motion, molecules have several periodic degrees of freedom of motion,

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vibration, and rotation. Depending on the potential between atoms and the mass of the atoms, the periodicity is of the order of picoseconds to femtoseconds. Similar characteristics can be observed in solids, where the atoms remain in relatively welldefined positions in a lattice. Their motion is limited as a phonon, which appears with time on the order of subpicoseconds. “How does a change in the position of the nucleus affect the electronic state?” is a naturally exciting and interesting question. Furthermore, questions such as how long does the photoexcited electron relax its energy on a given time scale? How does the electron move in real and reciprocal space during the relaxation process? What symmetry does the electronic state of the photoexcited system have? How is its time evolution described? These questions are also fundamental and important. Pulsed light is an ideal tool for studying ultrafast processes of various origins. It provides a “snapshot” measurement of the time evolution of how a system triggered by pulsed light relaxes. This technique, known as the pump-probe experiment, is one of the most fundamental and widely used techniques for measuring physical properties using pulsed light. In the pump-probe experiment, a pulsed light labeled “pump pulse” or “excitation pulse” interacts with the sample and excites it to a new non-equilibrium state. The sample then relaxes to an equilibrium state, which may or may not be the same as before the pulsed light excitation. The relaxation process to equilibrium can be mapped using a second pulse of light, often referred to as the “probe pulse” or “test pulse”. In the typical pump-probe experiment, the probe pulse is intentionally delayed relative to the pump pulse, and the signal from the probe pulse is measured as a function of the delay time betwen two pulses. The temporal limit of the experiment is determined by the duration of both the pump and probe pulses. If the system relaxes to the equilibrium state before the pulse light is applied, the repetition time of the pulse light can be adjusted to be longer than the time for the system to relax to the equilibrium state, allowing repeated measurements and obtaining spectra with a high signal-to-noise ratio. Here, let us take transient absorption spectroscopy as an example, which was developed at an early stage and is probably the most widely used time-resolved optical measurement using the pump-probe technique. While it is classified under the third-order nonlinear optical spectroscopy, the use of the formalism of nonlinear spectroscopy is not required to interpret the experiment in most cases, because of its brevity. To illustrate this technique as simple as possible, let us set up an ensemble of two-level systems as our sample (Fig. 1.4). Here, we assume that all electrons occupy the ground state at thermal equilibrium. Two pulses separated by a delay τ are focused on the sample. One would see that the sample act as a saturable absorber. In this experiment, the pump pulse will saturate the absorber (if the intensity of the pump pulse is sufficiently high, the sample will be transparent). The delayed weak probe pulse light will provide the absorption coefficient α in the current example: α = σ ΔN ,

(1.21)

where σ is the absorption cross section and ΔN is the population difference between the upper and lower level over the transition. After pump-pulse excitation, the sample

1 Introduction

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Fig. 1.4 Schematic diagram of transient absorption spectroscopy

then relaxes to an equilibrium state. By considering a rate equation, the absorption coefficient follows: ) ( τ , (1.22) α(τ ) = α0 + Δα exp − T1 where Δα describes the change in the absorption produced by the pump-pulse light, T1 is the energy relaxation time of the electron in the transition state. Thus, the dynamics of the relaxation to equilibrium can be mapped by measuring the absorbance as the function of the delay. Based on the above discussion, it can be understood that transient absorption spectroscopy measures the light-triggered dynamics of the bulk properties of a sample. However, with advancements in materials science and solid-state physics, the non-bulk properties of materials, i.e., surfaces and interfaces, have become more important. The second-order nonlinear optical spectroscopy is ideal for characterizing surface and interface properties because they are inherently broken in their central inversion symmetry. When the bulk properties of the sample have central inversion symmetry, measuring the SHG and SFG signals allows for the selective investigation of surface properties [4]. These nonlinear optical spectroscopies can be readily integrated with the pump-probe technique. This allows for the examination of surface dynamics, such as symmetry change of the surface structure and the diffusion of adsorbed molecules on the surface, with high temporal resolution. The findings gained from these techniques are important for understanding the electron/ phonon dynamics at surfaces of solids, which are related to chemical reactions at surfaces including catalytic reactions. The pump-probe technique can be universally applied to investigate photoinduced dynamics and has been combined with many spectroscopic methods. A noteworthy combination is that with electron spectroscopy [5]. In this book, the combination of angle-resolved photoemission spectroscopy (ARPES) [6] and the pump-probe technique will be discussed, especially in terms of electron dynamics such as the relaxation process of hot electrons near solid surfaces and coherence in

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multiphoton excitation processes. It should be noted that this “time-resolved photoemission spectroscopy” often requires a compromise between energy and time resolution. The energy resolution of current ARPES has reached the sub-100 meV order [7]. On the other hand, as noted above, current techniques can provide light pulses with sub-100 attosecond durations. Certainly, combining these two techniques would be a very interesting subject. However, it should be noted that the resulting data from this combination does not simply mean both ultra-high temporal resolution and high energy resolution. Time-resolved spectroscopy often provides beautiful (both scientifically and graphically) data, which can significantly enhance our understanding of the decay dynamics and coherence of excited electrons. The reader will recognize that many of the interesting findings revealed in the following chapters are attributed to the properties of the excitation light source. In this context, time-resolved nonlinear X-ray spectroscopy promises to offer distinctive insights into material dynamics. Realizing the full potential of this technique hinges on further advancements in the light sources. Hence the continued development of light sources is of paramount importance for its successful implementation and continued progress.

1.4 Synopsys This book is composed of six chapters. In this introductory chapter, the importance of “non-something “ research field, the classical description of the nonlinear optical response, and the significance of the nonlinear spectroscopy for material sciences have been introduced, which are assumed to be prescribed. After this chapter, Chap. 2 guides various analyses of linear X-ray spectroscopy. These measurement methods have been useful in materials science, and they have played a role as the foundation of nonlinear X-ray spectroscopy. The journey into state-of-the-art in nonlinear spectroscopy commences in Chap. 3, and the photon energy used increases as the chapter progresses. Chap. 3 provides the first example of the nonlinear spectroscopy, coherent multiphoton photoemission spectroscopy. Photoemission spectroscopy has been one of the standard methods to probe electronic states in matters, and the chapter reviews its evolution toward the nonlinear regime. Integrating extremely highly controlled laser technology with photoemission spectroscopy techniques has demonstrated the ability to quantify the coherent response of solid-state materials. In Chaps. 4 and 5, the frontier research of nonlinear spectroscopy with X-ray will be reviewed. Chapter 4 showcases the results of various experiments on nonlinear optical response using soft X-rays. It is of note that the nonlinear response with a soft X-ray is essentially understood by simply extending knowledge of those with an infrared ~ visible ray. On the other hand, the macroscopic description for the nonlinear optical response for hard X-ray is not fully useful, as noted above. Therefore, Chap. 5 that focuses on nonlinear hard X-ray spectroscopy starts from the detailed theoretical description. After that the important features discovered only through hard X-ray will be reviewed. Finally,

1 Introduction

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in Chap. 6, we look ahead and provide insights into the future prospects of this emerging field of research. Before closing this chapter, it should also be essential to highlight that nonlinear spectroscopy serves as a valuable domain for exploring nonlinear physics. Nonlinear optics is strongly related to the important topics of nonlinear physics, such as Chaos and Solitons. Many of these phenomena cannot be fully explained by solving linear equations or by using perturbation theory. The powerful point of nonlinear optics is to experimentally “visualize” the fundamental properties of a phenomenon that are difficult to interpret analytically. Since material science itself is a complex field, the integration of Chaos phenomena and material science remains a relatively unexplored area and is still limited. We believe that this will become an increasingly important research theme in the future.

Bibliography 1. Y.R. Shen, The Principles of Nonlinear Optics (Wiley, London, 1984). 2. T. Brabec, F. Krausz, Intense few-cycle laser fields: frontiers of nonlinear optics. Rev. Modern Phys. 72 (2), 545 (2000); J. Eden, High-order harmonic generation and other intense optical field-matter interactions: review of recent experimental and theoretical advances. Prog. Quantum Electron. 28 (3–4), 197 (2004); H.C. Kapteyn, M.M. Murnane, I.P. Christov, Extreme nonlinear optics: coherent X rays from lasers, Phys. Today 58(3), 39 (2005) 3. F. Krausz, M. Ivanov, Attosecond physics. Rev. Mod. Phys. 81(1), 163 (2009) 4. H.-L. Dai, W. Ho, Laser Spectroscopy and Photochemistry on Metal Surfaces (World Scientific Publishing Company, Singapore, 1995) 5. U. Bovensiepen, H. Petek, M. Wolf, Dynamics at Solid State Surfaces and Interfaces (WILEYVCH Verlag GmbH & Co. KGaA, Weinheim, 2010) 6. S. Hüfner, Photoelectron Spectroscopy (Springer, Berlin, 2003) 7. K. Okazaki, Y. Ota, Y. Kotani, W. Malaeb, Y. Ishida, T. Shimojima, T. Kiss, S. Watanabe, C.-T. Chen, K. Kihou, C.H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa, Y. Kohori, K. Hashimoto, T. Shibauchi, Y. Matsuda, H. Ikeda, H. Miyahara, R. Arita, A. Chainani, S. Shin, Octet-line node structure of superconducting order parameter in KFe2As2. Science 337(6100), 1314 (2012)

Chapter 2

Linear X-Ray Spectroscopy Iwao Matsuda

Abstract This chapter starts with the basics of light–matter interactions, followed by explanations of the linear X-ray spectroscopy, such as X-ray absorption spectroscopy, photoelectron spectroscopy, and X-ray emission spectroscopy, with the experimental examples. The chapter also focuses on the time-resolved measurement of X-ray spectroscopy as one of the operando experiments. In this chapter, various analyses with X-ray are summarized in the guide chart. Nowadays, experiments of X-ray spectroscopy can be made at the X-ray facilities, such as synchrotron radiation and X-ray free electron laser. At the end of the chapter, a typical procedure for applying beamtime is introduced as a flow diagram.

2.1 Basics 2.1.1 Optical Responses of Materials in the X-Ray Region We see things to understand them, as known from the proverb, “to see is to believe (seeing is believing)”. When we study a material, we actually make a little change (perturbation) to the target and observe the result. Even a direct visualization requires illumination of light to an object (Fig. 2.1). In a general experiment of materials science, the situation is described as [1–3] P = ε0 χ E,

(2.1)

where a physical quantity, P, is linearly generated by a gentle perturbation of E to the sample (the linear response) and ε0 is the vacuum dielectric constant. The response function, χ, contains material-specific information, and, thus, it leads to an understanding of the object itself. For example, if E is the electric field, the polarization P can be measured to derive the susceptibility, χ. When the system is I. Matsuda (B) The University of Tokyo, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_2

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Fig. 2.1 When there is no light, there is no sight. a Light-OFF. b Light-ON

homogeneous both in time and space, the macroscopic polarization can be written as ∑∫ ( ) ( ) (2.2) Pα (r, t) = ε0 dt ' d3 r ' χαβ r − r ' ; t − t ' E β r ' , t ' . β

With the vector and tensor representations, it can be written as ∫ P(r, t) = ε0

( ) ( ) dt ' d3 r ' χ r − r ' ; t − t ' · E r ' , t ' .

(2.3)

) ( Due to the causality, χ r − r ' ; t − t ' = 0 when t – t ' < 0. If the spatial change (temporal (variation ) of the envelope function) of E is smaller (slower) than that of χ, E r ' , t ' can be replaced with E(r, t). By the definition, χ ≡ ) ) ( ( ∫ dt ' d3 r , χ r − r , ; t − t ' = ∫ dt '' d3 r ,, χ r ,, ; t '' , thus, the equation becomes simple by the constant susceptibility tensor χ: P(r, t) = ε0 χ · E(r, t).

(2.4)

By the Fourier conversion, it is: P(k, ω) = ε0 χ (k, ω) · E(k, ω).

(2.5)

Focusing on the time and frequency dependence, χ (ω) can be given by the Fourier transform of χ (t) as ∫∞ χ (ω) =

dtχ (t)eiωt . −∞

(2.6)

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The time dependence, t, is adopted for the causality, χ (t) = 0 when t < 0. In general, χ (ω) is a complex number, χ (ω) = χ 1 (ω) + i χ 2 (ω), and it has a relation, χ (−ω) = χ ∗ (ω). The real numbers, E(t) and P(t), are expressed as: E(r, t) = E ω (r)e−i ωt + E ∗ω (r)eiωt .

(2.7)

P(r, t) = ε0 χ (ω)E ω (r)e−iωt + ε0 χ ∗ (ω) E ∗ω (r)eiωt .

(2.8)

Since χ (t) is a real number, χ 1 (ω) and χ 2 (ω) are even and odd functions of ω, respectively. The appearance is a consequence of light–matter interactions that provide information on the target material. Let us consider an energy transfer between P of a material and E of a light through the interaction. Temporal variation of the energy U p per ∂U unit volume stored during the polarization of the medium is given by: ∂t p = ∂∂tP E. Taking the average over a time sufficiently longer than the time period of the electromagnetic wave, it becomes ∂U p = 2ε0 ωχ 2 (ω)|E ω (r)|2 . ∂t

(2.9)

∂U

By the light absorption in a material, ∂t p becomes positive and, thus, χ 2 (ω) > 0. Since the incident photon intensity, I 0 , can be given by the cycle-averaged Poynting ∂U vector magnitude, I0 = 21 ε0 c|E ω (r)|2 , the absorption intensity, I (∝ ∂t p ), has the following relationship (c is the speed of light): I ∝ I0 .

(2.10)

When the optical properties of a material do not change with the incident light, the response is linear and is described in term of the linear response function. However, when the incident light intensity is extremely large, the polarization P additionally contains the nonlinear terms, P (NL) , with nonlinear susceptibilities [4–6], χ (2) , χ (3) , . . .: ∫ P (NL) (r, t) = ε0 dt1 d3 r 1 dt2 d3 r 2 χ (2) (r − r 1 , t − t1 ; r − r 2 , t − t2 ) : E(r 1 , t1 )E(r 2 , t2 ) ∫ + ε0 dt1 d3 r 1 dt2 d3 r 2 dt3 d3 r 3 χ (3) (r − r 1 , t − t1 ; r − r 2 , t − t2 ; . r − r 3 , t − t3 )..E(r 1 , t1 ) E(r 2 , t2 )E(r 3 , t3 ) + · · ·

(2.11)

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. It is of note that the expressions, “:” and “..” in (2.11) are the vector and tensor representations, as shown in “·” in (2.3).∑For example, the term with the 2nd-order (2) susceptibility, χ (2) , is generally given as χi(2) jk E j E k . The quantity, χ , is the third(2) order tensor or the 3 × 3 × 3 matrix, χi jk . The symbol “:” represents the interaction with the two E vectors (E j and E k ), and it is adopted to abbreviate the summation and the subscript expressions. The third-order susceptibility, χ (3) , is the fourth-order tensor that reacts with three E vectors and, thus, the polarization term is express with . “..”. In a simple form, the total polarization, P (total) , can be given by a summation of the linear, P, and the nonlinear, P (NL) , terms as

. P (total) = P + P (NL) = ε0 χ (1) · E + ε0 χ (2) : E E + ε0 χ (3) ..E E E + · · · . (2.12) Then, consequently, the optical characteristics depend on the intensity, and the response becomes nonlinear. In this chapter, linear optical responses, such as light absorption or light scattering, are described. Nonlinear optical phenomena, such as nonlinear light absorption and stimulated light scattering, are described in the subsequent chapters. Interactions between light (electromagnetic wave) and matter include light absorption, light emission, and scattering processes. What we visualize with our eyes is a consequence of visible-light optical events that cover the wavelength range between 380 nm (violet) and 770 nm (red). However, the visible-light region is very narrow in nature, as shown in Fig. 2.2. Light that is invisible to our eyes, such as infrared, ultraviolet, and X-rays, has a much wider range. The “invisible light” also exhibits absorption, emission, and scattering processes as a result of light–matter interactions, but the phenomena appear differently from those with visible light. For example, when a material is irradiated with ultraviolet or X-rays, electrons can be excited out of the sample (i.e., the photoelectric effect). It has been also well-known in radiography that X-rays, transmitted through a human body, can unveil the bone structure. These unique characteristics of X-ray have led to developments of various analytical methods in materials science. For example, X-rays can be used to probe electronic states and atomic structures of a material because the photon energy and wavelength match the atomic scale as well as electron-binding energies, as shown in Fig. 2.2. X-ray beams have been categorized into two regions: X-rays or hard X-rays (0.1 nm or shorter) and soft X-rays (0.1 nm or longer). Conventionally, X-rays have been used for structure analyses, while SX have been used to probe electronic states of materials [7–19]. The nomenclature of the detailed wavelength range has been not clear for the invisible regions. Conventionally, the wavelength range over 10–100 nm has been called extreme ultraviolet (XUV, EUV), or vacuum ultraviolet (VUV), while that over 0.1–10 nm is SX. To make it simple for the public, “soft X-ray (SX)” has been used to cover both two ranges and this chapter follows the trend. As shown in Fig. 2.2, X-ray photon energies can exceed the binding energies of core levels, such as the 1s and 2p orbitals of atoms [7]. Because energies of the inner-shell levels depend on the element and its chemical state, X-ray spectroscopy

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Fig. 2.2 Relationship between energy states of light and matter, using molecules as an example. The interaction depends on the energy level of each state

has been typically used to probe these levels. Because an X-ray beam exhibits light absorption, emission, and scattering with materials, there are many experimental techniques, such as X-ray absorption spectroscopy, X-ray emission spectroscopy, and X-ray scattering [7–19]. At X-ray facilities over the world, such as synchrotron radiation (SR), these methods are used for elemental/chemical analysis, structure determination, electronic structure analysis, and spin/magnetic analysis. For a comprehensive understanding in this chapter, I would like to classify these analysis methods based on the semi-classical quantum theory. Focusing on spectroscopy, some of the methods are described in detail with the experimental spectra. A summary chart of the methods is provided so that readers can seek for the appropriate one for their research. A flow for applying the user experiment at the X-ray facility or the “beamtime” application is also introduced. This chapter has subsections, starting with explanations of X-ray sources (Sect. 2.1.2). When there is no light, a matter is invisible (Fig. 2.1). We have neither light–matter interactions nor spectroscopy. Typical descriptions of light–matter interaction in experiments of materials science are based on semi-classical quantum mechanics (Sect. 2.1.3). Spectroscopy measurements have negligible effects on the target sample; thus, the interactions can be quantitatively described by perturbation theory that supports linearity of the optical response of the sample. Section 2.2 deals with the principles of absorption

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and emission, with examples of X-ray absorption spectroscopy (Sect. 2.2.1), photoelectron spectroscopy (Sect. 2.2.2), and X-ray emission spectroscopy (Sect. 2.2.3). Section 2.3 presents a topic of the time-resolved experiment as an example of the advanced spectroscopy that is related to the issue of the book. In the last section (Sect. 2.4), materials analyses with X-ray are summarized as a guide chart based on the fundamental light–matter interactions. I believe the chart should be useful for readers to seek for the appropriate experimental approach.

2.1.2 X-Ray Sources There are many (soft) X-ray sources today, such as synchrotron radiation (SR), X-ray free electron lasers, and high harmonic generation in lasers (Fig. 2.3). These sources have been fully incorporated in routine methods essential for materials science [20– 26]. A source has optical characteristics, such as monochromaticity, wavelength tunability, coherence, intensity, brightness, degree of polarization, and pulse width, that should be selected appropriately based on purpose of the experiment. In materials science, most samples have mm-scale sizes; however, only a μm ~ nm region on the surface are typically irradiated with light during an experiment. Thus, we need a bright beam source that irradiates this small area, rather than a point source that emits light radially (such as the sun). It is thus useful to adopt the quantity brilliance to compare the performance of light sources:

Fig. 2.3 Photographs of light sources and facilities: Laser at LASOR (Laser and Synchrotron Research Center, The Institute for Solid State Physics, the University of Tokyo) and undulator at SPring-8 BL07LSU. Synchrotron radiation (SPring-8) and X-ray free electron laser (SACLA). By courtesy of Takeshi Suzuki, Kozo Okazaki, and Jiro Itatani. By permission of RIKEN

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[

] a number of photons [brilliance] = ]2 [ ] [ ]2 [ [time] length angle divergence energy resolution In addition to the photon flux ([a number of photons][time]−1 [length]−2 ), the two-dimensional angular divergence of the beam and the energy resolution (ΔE/ E) are also included in the definition. The number of particles for “high brilliance” corresponds to the number traveling with high directionality under conditions of constant time, area, and energy resolution. For most experiments, it is desirable that the light source has high brilliance. At the SR beamline, the brilliance is generally expressed as [photons/s/mm2 /mrad2 with a 0.01% band width] or [photons/s/mm2 / mrad2 /0.01% BW]. The beamline brilliance at the SPring-8 synchrotron radiation facility is a factor of 1010 larger than that of sunlight on the earth’s surface, although the emitting power of the sun is much enormous. Recently, it has become necessary to define peak brilliance that represents brilliance per light pulse to characterize ultrashort pulse lasers, such as X-ray free electron lasers. Experiments with such a light source are associated with data-acquisition using a single shot, where information in individual shots is critical. Conventional brilliance can be termed the average brilliance to distinguish it from the peak brilliance. Because of the steady development of light sources, average or peak brilliance is updated over time; some of the data are found in references or webpages of the SR facilities. A laser light source is based on light amplification by stimulated emission of radiation (LASER). The optical characteristics are (1) monochromaticity, (2) spatial and temporal coherence, (3) femtosecond pulse widths in the case of mode-locked oscillations, (4) high intensity, and, recently, (5) soft X-ray wavelengths from higherorder harmonic generation. Synchrotron radiation (SR) is light (orbital radiation) generated with charged particles (electrons) at nearly the speed of light that change the motion (acceleration) by bending electromagnets or periodic magnet arrays (undulators or wigglers). As the light source, (1) it can continuously emit from infrared ray to X-ray; (2) it can control linear and circular polarizations of the light; (3) it provides highly oriented beam; (4) it has pulse widths of several tens of picoseconds; and (5) it can generate the spatially quasi-coherent beam. In a free electron laser (FEL), light generation and amplification are sequentially made by interactions between electrons and light (SR) meandering in a periodic magnet array. The FEL sources can be broadly divided into two types: a multi-pass type that combines a laser resonator and an undulator, and a single-pass type that is based on self-amplified spontaneous emission (SASE) in a long undulator. The former FEL requires an appropriate optical mirror for the resonator, and the photon energy is typically used up to the level of visible light. In contrast, the latter FEL requires no mirror and can thus generate photon energies up to the X-ray region. Existing SASE-FEL facilities provide the light sources of (1) wavelengths up to Xray, (2) monochromaticity, (3) femtosecond pulse duration, (4) ultrahigh intensity, and (5) full spatial coherence. It is worth mentioning that the performance has been

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Fig. 2.4 Typical specifications of the light sources. Pulse width/ wavelength characteristics of lasers, synchrotron radiation, and free electron lasers. Example of the SPring-8 soft X-ray beamline is indicated in the diagram. SXFEL and XFEL are the abbreviations for soft X-ray free electron laser and X-ray free electron laser, respectively, while HHG for high harmonic generation

improved further by development of the FEL source that is seeded by a laser (seeded FEL). Figure 2.4 summarizes these light sources with respect to wavelength and pulse width. The sources are for soft X-ray experiments, but the actual ranges are quite distinctive. Lasers generate ultrashort optical pulses and extend photon energies to the soft X-ray region by higher-order harmonic generation. Synchrotron radiation (SR) produces ultrashort wavelengths, but the pulse width is in the 10 ps scale (typically 50 ps) due to the limitation of the electron bunch in the storage ring. Soft X-ray FEL (SXFEL) and X-ray FEL (XFEL) cover both the ultrashort pulse and ultrashort wavelength regions. As shown in Fig. 2.4, characters of the light sources are compensating each other and also share some regions. Experimentalists select the most appropriate X-ray source for their purposes. Nowadays, the cutting-edge researchers conduct advanced measurement by a combination of the different light sources. For example, one can make a series of time-resolved X-ray experiments using both XFEL and SR to trace dynamical events from femtoseconds (fs) to microseconds (μs), capturing a whole picture of the phenomenon. Knowledges of light sources have been significant in optical science.

2.1.3 Light–Matter Interactions Here, we focus on unique soft X-ray and X-ray regions. As noted above in Sect. 2.1.1, soft X-rays include the vacuum ultraviolet (VUV) region.

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When materials are irradiated with soft X-rays, particles (electrons, ions) with various wavelengths and energies are generated, as shown in Fig. 2.5. These phenomena are caused essentially by light absorption and scattering. A magnitude of the light–matter interaction is expressed by a “cross section” that means the event probability. Figure 2.6 shows the dependence of photo-induced events in the X-ray energy range. Absorption dominates in the soft X-ray (VUV) region, but the absorption cross sections decrease with increasing photon energy. However, at a certain energy, there is an increase (a jump) in the cross-sectional area. This is the absorption edge of a core–shell, such as a K-shell, L-shell, or M-shell, and energy position of the edge-jump is determined by each element. In the soft X-ray region, the contribution of scattering, such as Rayleigh scattering, appears. Furthermore, in the X-ray region, Compton scattering becomes dominant, and the scattering cross section becomes larger than the absorption cross section. Absorption and scattering events can be distinguished in terms of processes that are associated with the vector potential A of the electromagnetic wave (light). A is a vector quantity introduced to quantitatively describe the electromagnetic field and is adopted as a parameter in the Hamiltonian of the light–electron interaction, as given below. In the electromagnetic theory, the electric field E and magnetic field B have the following relationships: E = −∇φ −

∂A , ∂t

B = ∇ × A,

(2.13) (2.14)

where φ is the scalar potential. Because physical properties of materials are mainly determined by the electrons, the interaction between light and matter essentially corresponds to the interaction between electromagnetic waves and electrons. The

Fig. 2.5 Interactions between matter and light (vacuum ultraviolet rays, soft X-rays, and X-rays): The events can be classified as the one or two A process, where A is a vector potential of the light

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Fig. 2.6 Schematic drawing of the total optical cross section of carbon atom at various photon energy with the contribution processes of absorption (photoemission), coherent scattering (Rayleigh scattering), and incoherent scattering (Compton scattering) [7]

essence can be captured by the semiclassical theory that treats electronic states in a material quantum-mechanically and electromagnetic fields classically [27–31]. The Schrödinger equation of the interaction between the electromagnetic field and the electron is expressed in terms of A and φ: { HΨ =

} 1 ∂Ψ (p + eA)2 − eφ + V Ψ = i h , 2m ∂t

(2.15)

where V is the potential in the material that electrons sense other than the electromagnetic wave. By rearranging the terms in the equation, p = −ih∇, the Hamiltonian H transforms as: H=

e p2 e2 +V + A·A (p · A + A · p) − eφ + 2m 2m 2m

= H0 +

e e2 A · A, (2A · p − i h∇ · A) − eφ + 2m 2m

(2.16)

(2.17)

where H 0 is the Hamiltonian of an electron in a material that does not contain an electromagnetic field. If we choose the radial gauge condition eφ = 0 and ∇ · A = 0 from the gauge invariance, we can make the Hamiltonian simpler and treat that the electromagnetic wave is spatially uniform. This situation is realized in soft Xray experiments because the wavelengths are a factor of 10–100 greater than the spread of electrons around atoms. Thus, A can be regarded as a constant in actual e2 A · A term makes the dominant experiments, and the ∇ · A term is negligible. The 2m

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contribution in the light scattering events in X-ray region (Fig. 2.6) [7]. To keep our argument simple, we neglect the term at a moment. Then, the Hamiltonian becomes: H = H0 +

e (A · p) = H0 + Hop . m

(2.18)

When the magnitude of the Hamiltonian H op is sufficiently small relative to H 0 , perturbation theory can be applied to deal with H op . This treatment corresponds to the fact that light–matter interactions during measurements affect the sample trivially and guarantee linearity in the optical response. In quantum mechanics, the occurrence of phenomena is expressed as a probability. Application of the perturbation theory provides a transition probability P from an initial state |i⟩ to a final state |f⟩ by light–matter interactions with different orders. First-Order Perturbation |/ | e | \|2 | | | | P1 ∝ | f | (A · p)|i | . (2.19) m Second-Order Perturbation | (⟨ | )| |∑ f | e (A · p)||n ⟩⟨n || e (A · p)||i ⟩ ⟨ f || e (A · p)||n ⟩⟨n || e (A · p)||i ⟩ |2 | | m m m m + P2 ∝ | | . | | n E i − E n + hν E f − E n − hν (2.20) In these equations, E i , E f , and E n are the energies of the initial (i), final ( f ), and intermediate (n) respectively. ⟨f |H op |i⟩ is a matrix element that corresponds ˝states, Ψ ∗f Hop Ψi dxdydz, using the expression of the wave functions Ψ to the integral (Ψ f ) for the initial (final) state. P1 in (2.19) corresponds to Fermi’s Golden Rule, |⟨ | | ⟩| | f | Hop |i |2 ρ, where ρ is density of states at the energy of the final state. The P1 = 2π h transition probability P1 is described by the square of the matrix element. P2 in (2.20) is characterized by the appearance of an intermediate state |n⟩, and it corresponds to the Kramers–Heisenberg–Dirac dispersion equation. It is of note that hν in the equation is the incident photon energy. Using quantum field theory, A is linear with respect to photon creation and annihilation operators [27–31]. That is, the transition probability of a first-order perturbation with one A (2.19) corresponds to the absorption or emission (e.g., fluorescence) of light. Whereas the transition probability of the second-order perturbation contains two A (2.20), and it corresponds to the scattering process. The terms in the formula contain the energy difference in the denominator, and thus, the processes describe the resonant scattering and resonant inelastic X-ray scattering (RIXS). It is of note e2 A · A term, neglected earlier that application of the first-order perturbation of the 2m from (2.17), results in giving the probability of the non-resonant scattering, such as Thomson scattering. In a scattering process in a crystal, the term contributes to the Bragg peaks in diffraction. It is also responsible for non-resonant inelastic scattering.

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Fig. 2.7 Feynman diagrams of interactions between a photon and an electron. Time passes from bottom to top, a and b correspond to light absorption and emission in the one A process, c–e are the examples of the two A process corresponding to the light scattering. (c) is scattering in the first perturbation (Thomson scattering), and d, e are scattering process by the second perturbation of p·A via a virtual intermediate state n

The effect appears in high-energy X-rays, as shown in Fig. 2.6. Figure 2.7 summarizes these processes with Feynman diagrams. Figure 2.7a, b correspond to (2.19), while Fig. 2.7c represents Thomson scattering. Figure 2.7d, e describe the first and second terms in (2.20), respectively. Figure 2.8a–e illustrates events associated with the absorption process. When the optical transition energy is below the vacuum level (a), electrons remain in the transition to the unoccupied level, but when it exceeds the vacuum level (b), they are emitted into the vacuum (photoelectric effect). Holes are generated at the core level in processes (a, b), and both are accompanied by the secondary processes shown in (c, d). Fluorescence emission (c) occurs when light is generated by the transition of electrons in the valence band to the core level, and (d) occurs when electrons are emitted into the vacuum by the non-radiative Auger process. Process (e) is called the resonant photoelectric effect, which involves (a), (b), and (d). Thus, when the incident light matches the energy difference between the core and the unoccupied levels, there are electrons emitted in vacuum by the photoelectric effect and the Auger process that have the same kinetic energy and share the same final states. In this case, the final-state wavefunctions interfere constructively, resulting in enhancement of the photoelectron intensity. Therefore, one can use the resonant photoelectric effect to extract information on an element or an orbital of the probing electronic states. Figures 2.8f–h are events of the scattering process. In Rayleigh scattering (elastic X-ray scattering) in (f), electrons at the core level eventually make transitions to the

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Fig. 2.8 Absorption/scattering processes, where A and k mean a vector potential and a wavenumber of the light wave, respectively. a Absorption (electronic transition to an unoccupied state), b absorption (photoelectric effect), c fluorescence (secondary process of absorption), d Auger process (secondary process of absorption), e resonant photoelectric effect [(a), (b), and (d) mixed process], f Rayleigh scattering (elastic X-ray scattering), g resonant elastic X-ray scattering, h resonant Raman scattering (resonant inelastic X-ray scattering)

original level through intermediate states. As a result, photon energy (wavenumber) of the incident light and the emitted light are the same. When the intermediate state coincides with the unoccupied state, as shown in (g), the resonance term in (2.20) increases, resulting in significant enhancement of the scattering process. This process is the resonance (resonant) scattering or resonant elastic X-ray scattering. There is a case in the resonant scattering process, shown in (h), that the scattered light has

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smaller photon energy (lower wavenumber) than the incident light. This event is the resonance Raman scattering or resonant inelastic X-ray scattering. The energy loss (Raman shift) carries information on a material, such as band gap in the electronic states or the collective modes of phonons. The interaction of light and electrons also yields information on the spin state of the electrons. In that case, we consider the interaction Hamiltonian of the electron and the electromagnetic field, including the spin σ, as follows [16]. These derivations incorporate relativistic effects. ' Hop

eh e2 h e e2 A·A+ σ · (∇ × A) − = (A · p) + σ· m 2m 2m (2mc)2

(

) ∂A × A . (2.21) ∂t

The transition probability of absorption and scattering process, for example, can be obtained by assembling the perturbation terms of the one and two A process, respectively. The principle is simple, but the calculation may be complicated. It is worth mentioning that, in X-ray scattering, there is a magnetic scattering process because of the magnetic interaction between the electromagnetic field and the electron spin or the electron orbital angular momentum, as described in (2.21). The cross section is reduced by a factor of hω/mc2 or (hω/mc2 )2 with respect to the photon energy hω, relative to that for scattering attributed to charge. Note that mc2 = 511 keV, where m is the electron rest mass. Therefore, the scattering intensity itself is extremely small. Recently, however, high-intensity X-rays have been used to analyze the magnetic structure of crystals. In the soft X-ray region, resonance magnetic scattering occurs with X-rays having energies matching the absorption edge and the large scattering signal can be obtained. A simple description of semi-classical quantum mechanics can classify varieties of X-ray analysis methods, based on absorption/emission (one A-process) and scattering (two A-process). The relationships are summarized in Sect. 2.4 as a chart in an easyto-understand manner that will help one find the most appropriate approach. A deeper understanding and details can be found in the references of the chapter. Some of the X-rays spectroscopy methods are discussed in the following sections.

2.2 X-Ray Spectroscopy 2.2.1 X-Ray Absorption Spectroscopy In an absorption experiment, the incident light intensity I 0 on the sample and the intensity I of the light transmitted through the sample are measured. Based on the Lambert–Beer law, absorption satisfies the relation: I = I0 e−μρd .

(2.22)

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The exponent μρd corresponds to absorption coefficient μ (m2 /kg), density ρ (kg/m3 ), and thickness d (m) of a sample. One obtains: I ∝ I0 ,

(2.23)

which confirms the linearity of the optical response. Because ρ and d can be known in advance, μ can be obtained experimentally. The value can be expressed as the amount per atom μ = NAwA σabs , where N A is the Avogadro coefficient, Aw is the atomic weight, and σabs is the absorption scattering cross section per atom, introduced in Fig. 2.6 [7]. The penetration of soft X-rays through materials is very small; thus, it is difficult to measure the intensity of transmitted X-rays in normal measurements. However, when a material absorbs X-rays and emits mainly electrons, the emission is proportional to the amount of absorption. Therefore, by measuring with the electron yield, it is possible to obtain a soft X-ray absorption spectrum of a sample even in a case of no transmission (Fig. 2.9). The total electron yield is used in many soft X-ray absorption measurements because it is relatively easy to obtain spectra by measuring the current flowing through the sample (specimen current or drain current). The emitted electrons can include Auger electrons and secondary electrons in addition to photoelectrons. The partial electron yield method selects these electrons at the specific energy range. The fluorescence yield captures fluorescent X-rays emitted from the sample. It is worth mentioning that yields of the electron and the photon provide different probing depths of the signal. The measurement can be surface-sensitive (electron-probe) or the bulk-sensitive (photon-probe). Thus, one can choose one of the yield methods to select the probing sample region (surface or bulk) or one can combine the different yield methods to capture a whole picture of the sample (surface and bulk). X-ray absorption fine structure is a technique that can directly investigate unoccupied states of a material [9–11]. Absorption increases when a certain energy is reached, as seen for the absorption edge in Fig. 2.6. This corresponds to the photon Fig. 2.9 Measurement set-ups for X-ray absorption spectroscopy

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energy becoming large enough to excite an electron in a certain core level, and the energy position depends on the element. In addition, a fine structure appears near the absorption edge, as shown schematically in Fig. 2.10. X-ray absorption spectroscopy can be used to analyze spectral features such as near-edge X-ray absorption fine structure (NEXAFS) or the X-ray absorption near-edge structure (XANES). When the investigation ranges up to several hundred eV, it is called extended X-ray absorption fine structure (EXAFS). In NEXAFS, the transition matrix has the core level as the initial state |i⟩. The final states |f ⟩ are (1) the unoccupied levels, (2) the Rydberg levels, or (3) the continuum level, exhibiting the characteristic spectral shapes in each energy range (Fig. 2.10).

Fig. 2.10 Spectral appearance of the photoabsorption near the absorption edge for a binuclear molecule in near-edge X-ray absorption fine structure (NEXAFS) and X-ray absorption fine structure (EXAFS). E vac represents the vacuum level

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Compared to Fig. 2.8a, b, the discussion has now extended to states above the vacuum level. In region (1), the final state can be an antibonding orbital (σ *, π *) of a molecule or a conduction band of a crystal. The Rydberg state in region (2) appears uniquely near the vacuum level. These electrons far away from the nucleus are mainly affected by interactions with the potential of the ion core, as shielded by the other electrons; thus, they have orbits with high principal quantum numbers. In region (3), the final state corresponds to an electron with a continuum energy that is partially trapped by a potential of the material (molecule) or an electron wave that is scattered by surrounding atoms. In solids, regions (1) and (3) are focused for analyzing a sample. In the case of the region (1), Fig. 2.11 shows examples of NEXAFS spectra of fullerene (C 60 ), highly oriented pyrolytic graphite (HOPG), and diamond. All the materials show absorption edges at the carbon K-shell, and the spectral structures appear differently. The features depend on the material and have reproducibility. Because of this unique character, NEXAFS spectra have been used as fingerprints to identify the material. Machine learning or big data analysis has been performed for accurate material identifications [32, 33]. Figure 2.12a shows incidence-angle-dependent NEXAFS results for (a) HOPG and (b) hexagonal boron nitride (h-BN), which are highly oriented because of the stacking two-dimensional structures [34–37]. The peak structure near the absorption edge corresponds to the unoccupied states (π *, σ *) of the material. By changing the incident angle of the SR beam on the sample surface, one can find that the π * peak at 285 eV and the σ * peak at 291 eV for HOPG changes significantly with respect to the incident angle, and shows the opposite behavior. This is because the π *-orbital spreads in a perpendicular direction to the honeycomb plane of HOPG, while the Fig. 2.11 For various carbon materials, near-edge X-ray absorption fine structure (NEXAFS) spectra of fullerene (C 60 ), graphite, and diamond, taken at the K-shell absorption edge and at normal incidence. By courtesy of Masato Niibe

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Fig. 2.12 Incidence-angle dependence of X-ray absorption spectra of a carbon K-shell absorption edge in highly oriented pyrolytic graphite (HOPG) and b nitrogen-K-shell absorption edge in hexagonal boron nitride (h-BN). The angle is measured from the sample surface normal with 0° corresponding to normal incidence and 60° corresponding to oblique incidence. By courtesy of Masato Niibe

σ *-orbital spreads in the plane. In another example, Fig. 2.12b shows the incidentangle dependence of the K-shell soft X-ray absorption spectrum of nitrogen (N) in h-BN. The π * and σ * peaks correspondingly exhibit the same behavior as seen for graphite. In this way, the orientation of the molecular orbitals (or the direction of chemical bonding) can be analyzed from the incident-angle dependence of the soft X-ray absorption spectrum. The electron, excited to the continuum level of (3), behaves as a photoelectron wave and scatters between the X-ray absorbing atom (photoelectron emitter) and surrounding atoms. This causes the electron interference in the final state, and the EXAFS spectrum shows the intensity modulation with photon energy (Fig. 2.10). By converting from a spectrum of the photon energy to that of the electron wavenumber (k), its modulation structure ξ is generally expressed as: ξ (k) ∝

∑ j

} { ( ) 2r j 1 sin 2kr j + δ . Nj exp − (kr j )2 λ(k)

(2.24)

Here, the λ(k) is the decay length. The phase, kr, of the photoelectron wave changes between atoms at distance r, and the absorption is modulated spectroscopically. This is the EXAFS method, and, by analyzing this modulated structure, it is possible to obtain information the local atomic structure, such as the coordination number N and the bond distance r around the absorbing atom. The phase shift, δ, is determined from the results of a standard sample using the same element as the scatterer. Because ξ (k) is a sine function, the Fourier transform of EXAFS can also be used to obtain a radial distribution function for structural analysis.

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2.2.2 Photoelectron Spectroscopy Photoelectron spectroscopy directly probes occupied valence states and core-level states of a material [12, 13]. It is a spectroscopic application of the photoelectric effect, in which a material is irradiated with light that has energy higher than the work function W, and the emitted electrons or photoelectrons are analyzed (Fig. 2.13). From the conservation of energy, kinetic energy (E k ) of the electrons in vacuum given as a function of the photon energy hυ: E k = hν − E B − W,

(2.25)

where E B is the binding energy of electrons in the material based on the Fermi level E F , and W generally has a value of 4–5 eV. The higher E B (deeper energy level) electron requires the higher photon energy hυ. Ultraviolet (UV) to vacuum ultraviolet (VUV) ray can emit valence electrons, and soft X-ray to X-ray can emit core-level

Fig. 2.13 Principle of photoelectron spectroscopy

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electrons around the nucleus (Fig. 2.13). Electrons in the valence bands of solids or in the highest-occupied molecular orbitals (HOMO) of molecules are critical for processes such as electrical conduction and chemical reactions, and photoelectron spectroscopy can investigate these states. The E B values of the core level depend on the element, while the additional E B shifts (chemical shifts) depend on the valence levels and the chemical environment. Therefore, chemical composition can also be determined quantitatively from the core level E B and its photoelectron intensity. Based on the information obtained from a material by photon energy, the former is so-called ultraviolet photoelectron (photoemission) spectroscopy (UPS), and the latter is X-ray photoelectron (photoemission) spectroscopy (XPS) historically or core-level photoelectron spectroscopy (CLS). While an energy level of an electronic state ψ(E) can be probed by photoemission spectroscopy, other parameters, such as momentum k or spin vector σ, can also be directly evaluated by extensions of the methodology. Angle-resolved photoemission spectroscopy (ARPES) maps out band-dispersion curves of the electronic state ψ(E, k), based on the momentum conservation rule. By installing a spin polarimeter, spin-resolved experiments can be performed to determine the spin-polarized band structure of the electronic state ψ(E, k, σ) of a sample. Recently, measurements can also be conducted with spatial and temporal resolutions. Spatial mappings have been performed by scanning focused X-ray beams or by imaging with photoelectron microscopy to determine the electronic states ψ(E, k, σ, r) of a non-uniform sample. The spatial resolution nowadays reaches in the nanoscale. Time-resolved measurements of photoemission spectroscopy have recently been performed to trace temporal evolutions of electronic states ψ(E, k, σ, r, t) during dynamic events in real time. This topic is focused in Sect. 2.3.1 as one of the operando experiments in Xray spectroscopy. Details of the individual measurement methods of photoemission spectroscopy can be found in [12, 13]. The photoelectron intensity I PES ([a number of photoelectrons][time]−1 ) for an electronic state that is characterized by E and k can be simply expressed by taking a product of transition probability and distribution of electrons over the possible range. ( IPES =

2π h

) ¨ |/ | | \|2 ( ) | | | |e | f | (A · p)|i | ρ(E) f FD (E, T )δ E f (k) − E i (k) − hν dkdE. m (2.26)

IPES is described by a matrix transition probability (2.19), the Fermi–Dirac distribution function f FD (E, T ), and density of states ρ(E). In the dipole approximation, (2.26) can transformed to: ¨ )2 ( |⟨ f |(er A )|i⟩|2 IPES ∝ A0 ω f i ( ) × ρ(E) f FD (E, T )δ E f (k) − E i (k) − hν dkdE, (2.27)

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) ( where ω f i = E f − E i /h. When the vector potential A = A0 exp{i (kr − ω f i t}, the electric field E is given by (2.13) and the amplitude E 0 = A0 ω f i . Because the incident photon intensity I 0 is given by I0 = 21 ε0 cE 02 , one obtains: IPES ∝ I0 ,

(2.28)

which confirms the linearity of the optical response. Figure 2.14 collects spectra of atomic sheets, as examples of XPS [38–40]. The sample spectra in Fig. 2.14a, b are obtained from a graphene layer on the hydrogenterminated SiC substrate. A survey XPS spectrum in Fig. 2.14a shows core-level peaks of the carbon (C), silicon (Si), and oxygen (O) atoms. The result quantitatively unveils not only the composing element but also the impurity. Intensity of each peak corresponds to amount of each element in the sample. Figure 2.14b focuses on the C 1s core-level region. The peak is composed of two components that are assigned to carbon atoms of graphene and silicon carbide. The spectrum can be decomposed by curve-fitting with Voigt functions. The function is convoluted by a Lorentzian function that characterizes the material and a Gaussian function that represents the experimental resolution. One can find that two chemical species appear separately at different energy positions and thus XPS is significant for analyzing a material. Figure 2.14c presents an XPS spectrum of a new material, the atomic sheet of hydrogen boride (HB). Photoemission spectroscopy has also played a significant role for material synthesis.

Fig. 2.14 a Survey spectrum of X-ray photoelectron spectroscopy of a graphene layer on the hydrogen-terminated SiC substrate, taken at hυ = 650 eV (synchrotron radiation). b Core-level spectrum at the C 1s peak in (a). Two carbon peaks are assigned to the graphene layer and the SiC substrate. A spectrum is curve-fitted by the Voigt functions and the background. c B 1s core-level photoemission spectrum of the hydrogen boride (HB) sheet, measured with a Mg X-ray tube. By courtesy of Fumio Komori and Takahiro Kondo

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2.2.3 X-Ray Emission Spectroscopy In X-ray emission spectroscopy [9, 10, 14–16], X-rays are incident on a sample using a monochromatic source such as a SR beamline, and the emitted X-rays from the sample are dispersed by a diffraction grating to acquire the spectrum (Fig. 2.15). The emission spectrum, which differs from the incident energy, contains fluorescence and inelastic scattering components. X-ray fluorescence is a secondary process that occurs after absorption, as it is generated by electronic transitions to form holes in the inner shell (Fig. 2.8c). Because it is determined by the position of the core level, the fluorescence energy is unique to the element (characteristic X-ray) and is useful for elemental component analysis. Carbon K-edge X-ray emission spectra of carbon allotropes excited by SR are shown in Fig. 2.16. The excitation energy was 310 eV, a bit larger than the C K-edge. This is the energy distribution of soft X-rays emitted when the 2p-orbital electron transitions to the 1s vacancy according to the selection rule and basically reflects the electron state density of the 2p orbital. Therefore, various electronic states for each compound appear clearly as changes in the shape of the spectrum. From such soft X-ray emission spectra, it is possible to analyze the electronic and chemical states of occupied orbitals, which can be used for fingerprint analyses. Figure 2.17 shows the takeoff-angle dependence of soft X-ray emissions at (a) the C K-shell absorption edge of HOPG and (b) the N K-shell absorption edge of h-BN [41]. In these figures, each emission intensities are normalized to keep the intensity of the tail on the low energy side the same. With such normalization, the intensity of σ emission on the lower energy side becomes approximately equal value. On the other hand, it can be seen that the π emission intensity on the higher energy side shows a large takeoff-angle dependence. This corresponds well with the incident-angle dependence of NEXAFS seen in Fig. 2.12. The reason for the angular dependence Fig. 2.15 X-ray emission spectroscopy arrangement. A typical emission spectrometer consists of a diffraction grating and a detector

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Fig. 2.16 A collection of carbon K-edge X-ray emission (fluorescence) spectra of carbonaceous materials. By courtesy of Masato Niibe

is that the π orbital is perpendicular to the honeycomb plane and the σ orbital is parallel to the plane, as in the same case of NEXAFS. In Fig. 2.18, an example of the resonant inelastic X-ray scattering (RIXS) experiment is given for a case of graphite (HOPG) [42]. Since excitation energy range of RIXS corresponds to that of the absorption edge, the X-ray absorption spectrum of the sample is collected at the C K-shell, as shown in Fig. 2.18a. The spectral peaks at 285.4 and 291.8 eV are assigned to the photoexcitations to the π * and σ * states, respectively, as presented in Figs. 2.11 and 2.12. A measurement of the sample was made with the surface normal at 55° relative to the incident beam. Figure 2.18b is the RIXS map that was recorded at the same geometry with the X-ray spectrometer placed at 90° scattering angle. The prominent features at emission energies between 270 and 280 eV are from emissions (Figs. 2.16 and 2.17) and also from inelastic scattering by the band transitions. In the map, a diagonal line of the elastic peak (zero energy loss) appears as the white line. Focus on a region of the yellow box, one can find inelastic peaks of phonon excitations, as shown in Fig. 2.18c. High-resolution measurements of RIXS nowadays unveil not only electronic states but also collective modes and the associated couplings in a material.

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Fig. 2.17 a Takeoff-angle dependence of X-ray emission spectra of (a) carbon K-shell absorption edge in highly oriented pyrolytic graphite (HOPG) and b nitrogen K-shell absorption edge in hexagonal boron nitride (h-BN). The takeoff angle, α, is measured from the sample surface, with 15° corresponding to very small oblique angle. In this measurement, the X-ray incident and takeoff directions are fixed at 90°, and the takeoff angle can be changed by rotating the sample around the vertical axis. By taking the takeoff angle, α, as shown in the inset, the incident angle and the takeoff angle are always expressed as the same value. By courtesy of Masato Niibe

Fig. 2.18 a An X-ray absorption spectrum at the C K-edge of a HOPG sample. b Results of resonant inelastic X-ray scattering (RIXS), mapped with excitation and emission energies. c RIXS spectra around the elastic peaks, observed at excitation energies between 291 and 292 eV (light yellow box). Reprinted with permission from Feng et al. [42]. Copyright (2020) by the American Physical Society

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2.3 Time-Resolved X-Ray Spectroscopy 2.3.1 Measurement Principles Measurements of X-ray spectroscopy have been used to characterize sample properties, such as structures or electronic states. Today, the information has been measured to examine functionalities of a material by in situ or operando experiments under the working condition. Operando X-ray spectroscopy measurements have been performed, for example, during electrochemical operations in a battery, redox reactions at the catalytic surface, or photovoltaic events in a solar cell [34]. There have been significant developments in sample environments in an experimental chamber of X-ray spectroscopy, such as electrochemical cells and near-ambient-pressure cell, and also in measurement techniques with temporal and/or spatial resolution(s). In this section, the time-resolved measurement is focused since the technique is related to experimental techniques of nonlinear X-ray spectroscopy, which is the issue of this book. For readers who wish to learn about the operando experiments extensively, selected issues of the advanced X-ray spectroscopies, such as ambient-pressure or nanospace spectroscopy, can be found in [34, 43]. A time-resolved measurement is conducted to track temporal evolution of a dynamic event in real time [38, 43–53]. Interests of chronometric research can be ephemeral and also eternal. The clock time can be as short as femtoseconds (fs, 10–15 s) and as long as centuries (100 years). In human society, the former is required for fast communications of information, while the latter is needed for the safe storage, for example. Such an enormously wide range of the possible target is one of the difficulties in designing an experimental set-up for the time-resolved measurement. In general, a researcher adopts a system that covers a certain range of time and experimentally investigates a dynamical phenomenon of the sample that has a critical step in the temporal region. In materials science, research of the fs-time scale phenomena has been conducted to investigate elementary process, such as electron–electron interactions, electron– coherent phonon coupling, and electron–phonon coupling. The temporal region is also significant to capture transient states or bond-breaking/making moments in chemical reaction. Experiments of ultrafast time-resolved spectroscopy have been demanded to directly monitor such ultrafast events. On the other hand, a chemical reaction, in general, is composed of various steps, such as diffusion, formation of reaction intermediates and desorption. A whole reaction process is complicated and takes over time scales that are longer than picoseconds (ps). It is of note that most chemical reactions are governed by kinetics, and it is necessary to reveal the rate-determining step. Therefore, comprehensive and detailed understandings of a chemical reaction require the time-resolved experiment for a wide range of time that could even take over days. It is, thus, necessary to design a time-resolved experiment that can be completed in reasonable but enough time to achieve the research goal. Figure 2.19 summarizes experimental information at various time scales. Individual time is indicated with distance that light travels during the corresponding

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Fig. 2.19 Chronical structures at various time scales (distance that light travels during a period of time). Optical specifications of X-ray sources, dynamic events, and measurement methods are indicted. Abbreviations are as: attosecond, fs: femtosecond, ps: picosecond, ns: nanosecond, μs: microsecond, ms: millisecond, s: second, ks: kilosecond, and Ms: megasecond. HHG, XFEL, and SR indicate high harmonic generation, X-ray free electron laser, and synchrotron radiation, respectively

period. One finds that timing of the fs-ps range can reasonably be controlled by a delay stage. A distance of m ~ km at μs matches with sizes of the SR facilities in the world, and the period corresponds to a lap time of electrons in the storage ring. Pulse widths of X-ray lasers, XFEL and HHG, range around fs, and the sources can be used for time-resolved experiments to trace ultrafast dynamic phenomena. For the measurement, one adopts the pump-probe method that generates a transient state of a sample by an optical pump pulse, followed by an optical probe to detect signals from the sample. To obtain the enough sample information, the measurement is carried out repeatedly. A period of the cycle is needed to be longer than the recovery time of an event to keep the same initial condition (before pumping). The repetition rate typically ranges Hz ~ MHz. It is worth mentioning that there are two types of the XFEL resources that operate at frequencies of the Hz and (sub)-MHz range, indicated as the XFEL-pulse intervals #1 and #2 in Fig. 2.19, respectively. The former is useful for making ultrahigh pulse energy of a pump, while the latter is suitable for increasing detection cycles of a probe. In a case of the synchrotron radiation, the pulse width (50–100 ps) is limited by an operation of the electron storage ring. The SR pulse interval is determined by distance between the electrons (electron bunches) that emit SR pulses, and it ranges typically in ns-scale. By making an operation of the several bunch or single bunch mode, the interval can be tuned up to the lap time (typically μs). During the SR

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beamtime for a user, typically awarded for hours ~ days, time-resolved experiments have been conducted to investigate dynamic events in various time scales (sub-ns ~ ks). In general, designing of a time-resolved experiment starts by choosing time structure of the pump, probe, and detection system. Figure 2.20 shows examples of the combinations. A pump is needed as a trigger to initiate a dynamic event. As shown in Fig. 2.20a, one can trace a sample before pumping (Pump OFF), during pumping (Pump ON), and after pumping (Pump OFF) with a continuous probe and a detection system that operates fast enough to complete the measurement at the various stages of pumping. In this scheme, the temporal resolution is limited by the detection time. The measurement design has been adopted to trace irreversible phenomena. When a pump is a short pulse, one can consider making a single detection after a delay time, as presented in Fig. 2.20b. The scheme is useful when temporal width of the pump pulse is much shorter than the detection time window. In a case of examining reversible phenomena, the measurement can be repeated in cycles to accumulate enough signals in the detector. Inversely, one can set a pulsed probe, instead of continuous one, and keep the detection continuous. This scheme, Fig. 2.20c, is the pump-probe method. Since time resolution of the method is determined by temporal width of the pump and the probe, it is useful to capture ultrafast phenomena when a pump and a probe are ultrashort optical pulses. In a typical time-resolved experiment of such an ultrafast dynamic event, time range of the measurement is set as an order of the optical pulse width. It is thus clever to limit the detection time, Fig. 2.20d, to eliminate noises that are detected during the interval. Comparing the schemes in Fig. 2.20 with the temporal structure of the light sources, shown in Fig. 2.19, one can find that X-ray lasers of XFEL and HHG are suitable for the pump-probe methods, Fig. 2.20c, d, while SR can be used for all the methods. Figure 2.21a shows an experimental system using a laser with the high harmonic generation (HHG). A timing jitter between pulsed beams of a pump and a probe can be suppressed by sharing the same laser source. A soft X-ray probe is generated by injecting the laser pulse in to the HHG system (see Fig. 2.22 for detail). To improve efficiency of HHG, photon energy (wavelength) of the original laser pulse is often doubled (halved) by a β-BaB2 O4 or BBO crystal before the injection, as shown in Fig. 2.22. The delay time is controlled by changing the optical path by a movable stage. Figure 2.21b shows a schematic drawing of an electron storage ring in a SR facility. The storage ring is fundamentally composed of (i) bending magnets that ensure revolutions or cycles of electron bunches and simultaneously induces radiation of continuous spectrum, (ii) an undulator that radiates quasi-monochromatic light, and (iii) a radio frequency (RF) cavity that accelerates the electrons (replenishes energy to the electrons). Since SR originates from individual electron bunches, the emitted SR light has an inherent nature of pulsed light; the temporal width of SR light corresponds to the size of an electron bunch and the repetition rate is determined by integer division of the frequency at the RF cavity. Figure 2.21c presents the basic principle of a free electron laser (FEL) of the oscillator-type or the multi-pass type. Electrons in a bunch gain or lose their energy

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Fig. 2.20 Examples of measurement schemes for time-resolved experiments operated by a combination of the pump, probe, and detection system. a A pulsed pump, continuous probe, and cycles of detection. Operations of the elements are described by “ON” and “OFF.” b A short pump pulse, continuous probe and a single-time detection. The delay time, Δt, is defined between pump and detection time. c Short pulses of a pump and a probe with continuous detection. The delay time, Δt, is defined between the pump and the probe. The scheme corresponds to the pump-probe method. d The pump-probe method with limited time for detection of (c) that has been adopted to reduce the background noise at a detector

through interaction with electromagnetic field, as in the case at the RF cavity. In an undulator, when the oscillation period of electrons matches with a frequency of light (electromagnetic wave) injected through the undulator, the energy is transferred resonantly between electron and light. Choosing a proper phase shift between two oscillations, the intensity of SR light increases when it passes through the undulator again. The SR beam, reflected by appropriate mirrors, is amplified by repeating the cycle and it is, eventually, transmitted to a sample. It is a sort of a resonant cavity, within which the electrons exchange energy with a trapped radiation field. It is of note that a generation scheme of FEL is fundamentally distinct from that of conventional quantum lasers which utilize the population inversion in a laser media. An advantage of FEL is continuous tunability of photon energy by changing the undulator parameters, which is in contrast to the discrete choice of photon energy for quantum lasers. Due to the technical limitation of mirrors, a wavelength of the

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Fig. 2.21 A collection of pulse light sources for X-ray. a A laser system with high harmonic generation (HHG), b undulator in an electron storage ring of synchrotron radiation (SR), c an oscillator-type free electron laser (FEL), d a combination of laser and SR, e the laser-slicing source, and f self-amplified spontaneous emission (SASE)-type X-ray FEL [43]

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Fig. 2.22 Example of a set-up for time-resolved photoelectron spectroscopy experiment with high harmonic generation of laser. Soft X-ray HHG pulses are generated by irradiating the laser beam at noble gases, followed by selections of photons at specific energy by multi-layered mirrors. Pump pulses are guided by another path with delay stage to make the delay time [43, 45, 48]. By courtesy of Takeshi Suzuki

oscillator-type FEL typically ranges from the millimeter wave to the visible light and a pulse width is restricted by the electron bunch. As summarized in Fig. 2.19, there are characteristic chronical parameters for the individual electron storage rings at various SR facilities. The time structure is composed of three key “clocks”: a lap time, an interval, and a duration of the electron bunch. One revolution (a lap time) of an electron bunch in the storage ring, running at the velocity of about light speed, takes an order of microseconds (μs). Time interval between neighboring bunches depends on the electron filling pattern in the ring. The minimum time interval is determined by the frequency of the RF cavity, typically about nanoseconds (ns). The pulse width is determined by the bunch length, and it is in an order of 10 picoseconds (ps). One can recognize that the time structure of electron storage rings is hierarchic and the time layer is almost three orders of magnitude different from the neighbors. Using the SR pulses as a probe, a pumpprobe method has been performed over the SR facilities with a femtosecond laser pulse as a pump. Figure 2.21d illustrates the beam paths from the two light sources toward a sample. It is of note that the timing jitter affect the temporal resolution when two sources are different, especially in fast time scale. Thus, the most critical factor in the pump–probe experiment is the timing control between pump laser and probe SR. Through recent technical developments at the SR facilities, the timing is now adjusted better than 10 ps and the time-resolution is now limited by the SR pulse width. Details of the timing control between SR and laser pulses are described in [43, 46].

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To achieve time resolution better than picoseconds, there exist several schemes which generate femtosecond (fs) ultrashort pulses in the vacuum ultraviolet (VUV), soft X-ray, and X-ray region. As an example of fs x-ray pulse generation, the “laserslicing” scheme is shown in Fig. 2.21e. In the modulator section, electron bunch is shot in the magnetic field, such as in a wiggler, by a high-intensity ultrashort laser pulse to induce a very short electron modulation in an electron bunch, forming the non-uniform electron distribution. In the subsequent passage through magnet field in an undulator or a bending magnet (the radiator section), the spiked fraction radiates ultrashort SR pulse, while the remnant part generates the (coherent) SR. This “laser-slicing” technique is currently under operation at various SR facilities in the world [47]. Although the laser-slicing technique stably generates fs light pulses, the actual measurement using the beam is quite challenging due to very low photon flux. The situation is easily understood by a simple estimation. Repetition frequency of the high-intense laser is typically kHz, which is 1/100,000 of the SR frequency (typically on the order of 100 MHz) and a ratio of the spiked region to the total electron bunch is 100 fs/ 50 ps = 1/500. Thus, photon flux of the laser-slicing fs light pulses is about 10–8 of that of the usual SR pulses. Demands of the intense fs pulses of VUV, soft X-ray, and X-ray have been increased recently. It has been discussed that an interaction between light and electron is the key phenomenon to overcome the technical difficulties. As described above, the oscillator-type FEL has not been extrapolated to X-ray wavelengths yet due to the limited efficiencies and power-handling capacities of X-ray mirrors. Instead, a scheme to generate X-ray laser without mirrors has been identified (a single-pass FEL) [23, 24]. Traveling through a sufficiently long undulator, electrons in a bunch are subjected to the combined force of undulator magnets and their own SR beam. Then, they begin to form microbunches, spaced by one wavelength of the dominant SR. Constructive interference of the radiation from these microbunches produces an X-ray beam with full transverse coherence. The instantaneous or peak brilliance is ten billion-fold greater and the pulse width (< 10 to 100 fs) is thousand-fold shorter than SR. Such FEL amplification is traditionally called “self-amplified spontaneous emission” (SASE). Nowadays, the technical developments have been carried out and we have varieties of advanced XFEL sources today.

2.3.2 Examples of Time-Resolved Photoemission Spectroscopy Measurement Figure 2.22 shows an example (Fig. 2.21a) of the time-resolved photoelectron spectroscopy experimental system using a laser with HHG [43, 45, 48]. With the laser that has the pulse width of femtoseconds, the time-resolved photoemission spectroscopy measurement system can capture temporal changes in electronic states during ultrafast phenomena. Titanium sapphire (Ti:S) laser, which is a general ultrashort pulse laser, has a wavelength of 800 nm (hυ = 1.55 eV) and the energy is doubled by BBO

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(hυ = 3.10 eV). The laser is injected into a jet of noble gas in the HHG chamber, followed by monitoring by a spectrometer and by monochromatizing by multi-layer mirrors. In this time-resolved photoemission spectroscopy system, soft X-rays with light energies of hυ = 20–60 eV are used for experiments and electronic states of a sample have been investigated near the Fermi level. Time-resolved photoemission spectroscopy allows us to directly observe changes in the actual carrier dynamics as the occupancy in electronic states or bands. These data can be band images obtained by wavenumber-resolved (angle-resolved) photoemission spectroscopy, measured at various delay time (Δt) between the pump light and the probe light. Figure 2.23 is an example of such temporal evolutions of the band diagrams that are observed near the K point of a graphene layer on a SiC substrate [45]. In general, an energy dispersion curve of the (nearly) free electron is quadratic but in a case of graphene, the curve is linear, as called the Dirac cones, and the crossing point is called the Dirac point. Therefore, electrons in graphene behave as if massless. Such a Dirac band can be directly observed by photoelectron spectroscopy as shown in Fig. 2.23, and it can be confirmed that the Dirac point exists at binding energy = 0.0 eV (Fermi level). Before the pump light irradiation, Fig. 2.23a, only the lower Dirac cone is observed, but after the pump light irradiation, (b–f), the upper Dirac cone can also be seen due to the electron excitation to the unoccupied states. Furthermore, by examining the change in photoelectron intensity for each Δt, the relaxation process of non-equilibrium carriers in the Dirac band can be tracked quantitatively. As shown in Fig. 2.23, the electrons in the upper Dirac cone relax in the femtosecond time scale

Fig. 2.23 Results of femto- to picosecond time-resolved photoemission spectroscopy of graphene layers on a C-face SiC substrate. The measurement was performed by the pump-probe method, where a pump is a laser beam of hυ = 1.55 eV and a probe is the HHG beam of hυ = 27.9 eV. a–f Ultrafast temporal changes of (a–f) the photoemission band diagrams and g–k the difference, taken at various delay time, Δt, between a pump and a probe [45]

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Fig. 2.24 Example of a set-up for time-resolved photoelectron spectroscopy experiment at the SR beamline, SPring-8 BL07LSU [43, 46]

and also accumulate near the Dirac point, where the relaxation slows down. Quantitative understanding of specific relaxation times is important for the development of graphene devices, and this research provides material dynamics information from theory to industrial applications [45]. Figure 2.24 illustrates an example (Fig. 2.21d) of the time-resolved photoelectron spectroscopy system at the SR beamline, SPring-8 BL07LSU [38, 43, 46]. By installing the femtosecond laser system at the beamline, one can conduct the pumpprobe experiment with pump (laser) and probe (SR). Timing controls of the pump and probe are made electrically based on the master oscillator that controls the motion of the electron bunches in the storage ring. Using the soft X-ray beam, one can detect not only valence bands but also core levels in a material. As a result, even in nonuniform samples, it has the characteristic of being able to trace time changes in an element-selective manner. Figure 2.25 shows a series of (sub)nanosecond time-resolved core-level photoelectron spectra of graphene layers on the C-face SiC substrate [38]. By using a photon beam at hυ = 740 eV as a probe, one can measure photoelectron spectra of both the C 1s core level at binding energy of ~ 285 eV and Si 2p with binding energy of ~ 100 eV. Two peaks in the C 1s spectra are assigned to carbon atoms in the graphene (G) layer and the SiC substrate, respectively. In this way, the information of the graphene layer and the substrate can be measured simultaneously. By optical pumping with the laser, there is an energy shifts of all the core-level peaks due to generation of the surface photovoltage effect and to the following relaxation. The measurement unveils temporal variations of the energy shift at components and elements in the functional interface that directly links to understanding of carrier dynamics [38, 43, 49, 50]. This chapter section gave examples of the time-resolved experiment of photoemission spectroscopy with synchrotron radiation (SR) and the laboratory-based laser (HHG). It should be mentioned that such measurements of the time-resolved photoemission spectroscopy can also be conducted at the XFEL facility, FLASH, today [51].

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Fig. 2.25 Results of nanosecond time-resolved photoemission spectroscopy of graphene layers (G) on a C-face SiC substrate. The measurement was performed by the pump-probe method, where a pump is a laser beam of hυ = 3.1 eV and a probe is the SR beam of hυ = 740 eV. a and b are temporal changes of the C 1s and Si 2p core-level spectra with respect to delay time (Δt) [38]

The experiment had been considered difficult due to the space charge effect [52]. The effect is associated with the ultrahigh and ultrashort XFEL pulse, resulting in high densities of photoelectrons. Due to the significant Coulomb interaction (repulsion) between electrons, the photoemission spectra are significantly disturbed. However, with the special mode of the XFEL operation with the MHz-frequency, time-resolved measurement of angle-resolved photoemission and core-level photoemission spectroscopies have been made successfully [51]. Similar to the photoemission spectroscopy stations at beamlines of the SR facilities, the XFEL facility is becoming to open the beamline station for the time-resolved photoemission spectroscopy [53].

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2.4 Researcher’s Guide to Material Characterization with X-Rays 2.4.1 The Guiding Chart for Experiments There are many measurement methods using light and particle signals generated by X-ray irradiation [7–19]. They have often confused researchers who seek for the appropriate methods for their experiments. In Figs. 2.26 and 2.27, methods are summarized with fundamental sample analyses, such as elemental/chemical analysis, structure determination, electronic state analyses, and spin/magnetism analyses. The classifications are absorption (one A-process) and scattering (two A-process), as described in Sect. 2.1. In the optical experiments, such as elemental analysis and electronic-state measurements, it seems that the same information can be obtained by the different methods. However, in practice, we analyze electrons emitted by excitation light and use light scattered by electrons in materials. Because electrons and photons have different probe lengths, the former is more useful for analyzing material surfaces, and the latter is more useful for analyzing material interiors. The latter is also suitable

Fig. 2.26 Interactions between light and electrons (one A-process) and the relationship with each measurement method

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Fig. 2.27 Interactions between light and electrons (two A-process) and the relationship with each measurement method

for measurements under external fields, such as magnetic fields. In this way, spectroscopy and diffraction methods can be used selectively according to the research purpose and, by assembling all the dataset on the same sample, one can perform a diversified analysis. Usage of X-ray beams benefit by adopting electrons as the material signals, as shown in Fig. 2.5. The detected electrons carry various information, like receiving an e-mail from the sample (Fig. 2.28). Fig. 2.28 X-ray spectroscopy is like getting an e-mail from a sample

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Fig. 2.29 A user flow diagram for the beamtime experiment at the X-ray facilities

2.4.2 Application for Beamtime at the X-Ray Facility There are many facilities for X-ray experiments, such as synchrotron radiation and X-ray free electron laser, all over the world. Some are in Japan, as shown in Fig. 2.3. One can use these valuable and cutting-edge experimental resources. Therefore, I would like to introduce the actual “user flow” below and in Fig. 2.29. (1) Consider the measurements and experiments. (2) At the facilities, the incident beam source is a “beamline” that is designed and developed to match with the purpose of the experimental station. The beamline information can be found, for example, in the website of the facilities. Hence, consult with the person in charge of the beamline. (3) Submit a proposal of the research project by the deadline. (4) Project proposals will be reviewed by a committee of the experts in the research fields. Because the operational time (the user beamtime) in the dedicated facility is limited, the proposal can be rejected. In that case, the committee provides the useful feedback comments. Then, restart from (2). The staff in charge of the beamline will be happy to consult with the researcher. (5) When the project proposal is awarded, the researcher will receive a notification of acceptance, and, in general, will be assigned a project number with the beamtime schedule. (6) In a dedicated facility, procedures such as chemical management and radiation safety training are performed so that experiments can be conducted safely and smoothly. After successfully completing these steps, the researcher will go to the beamline where the experimental station is located. (7) The experiment will then be conducted during the designated period (beamtime). (8) Analyze data obtained during the beamtime, and, if necessary, perform theoretical simulations of the data. Then, discuss and summarize the results with co-researchers. (9) Present the research results at academic conferences and publish them as research papers to share the valuable information. Academic communications via presentations and publications provide positive feedbacks in the research, deepen the understanding, and stimulate the subsequent research. In addition, the researcher is requested to submit a report of the beamtime to the facility

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where the experiments were conducted. On the facility side, such information is compiled and used to improve the future usage. At the beginning of this chapter, the well-known proverb, “To see is to believe (seeing is believing),” was mentioned. While the proverb is in English, it is common to the whole world and there is a Japanese version. In Japan, the proverb was extended by people to provide more lessons and can be expressed as “One-eye witness is better than many hearsays.” One may extend it for a researcher as: “One-eye observation is better than many hearsays.” “One consideration is better than many observations.” “One action is better than many considerations.” “One publication is better than many actions.”

It is more important “to observe by yourself” than “to just listen to others a lot,” and “to consider on your own” is more important than “to just observe many things.“ Then, it is more important “to take real actions” than “to only consider in your brain,” and it is worth having publications to share with others than “to just keep it in oneself.” I hope readers will perform more and more research to solve individual problems or mysteries, and then share the findings or knowledges in the public.

2.5 Summary In the present chapter, I introduced varieties of linear X-ray spectroscopy with the basic principles that have been used to analyze electronic (chemical) states and structure of materials. Then, the analyses are summarized in a chart diagram so that a reader can seek for the appropriate ones for the experiment. It is important to mention again that some of the analyses, such as absorption spectroscopy, can be made with different types of the signal detections, such as electrons and photons. Since these signals have the unique probing lengths, one can made the surface-sensitive (electron-probe) and the bulk-sensitive (photon-probe) measurements to capture a whole picture of a sample. A choice of the probe is an important and interesting feature of the X-ray experiments. It is also significant to note that a combination of the different analyses and methods provides the multi-dimensional data that unveils a sample from different aspects. These concepts have led to developments of operando experiments that investigates electronic (chemical) states, for example, of functional materials during the working condition. X-ray spectroscopy has become the useful analysis method in both academic and industrial fields. The experiment can be made as the user beamtime at the X-ray facilities in the world. I wish readers to utilize the chapter and to conduct their research of materials science. Acknowledgements I appreciate Masahito Niibe, Masafumi Horio, Masashige Miyamoto, Junichiro Mizuki, Yoshihito Tanaka, Yasuhisa Tezuka, Fumio Komori, Takeshi Suzuki, and Takahiro

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Kondo for valuable discussion, careful readings, and data usages for this chapter. This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012 and No. JP18H03874) and by JST, CREST Grant No. JPMJCR21O4, Japan.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27.

J. D. Jackson, Classical Electrodynamics (Wiley, London, 1998) N. Mermin, N. Ashcroft, Solid State Physics (Brooks/Cole, Pacific Grove, 1976) C. Kittel, Introduction to Solid State Physics (Wiley, London, 2004) R.W. Boyd, Nonlinear Optics (Academic, Cambridge, 2020) G. New, Introduction to Nonlinear Optics (Cambridge University Press, Cambridge, 2011) V.V. Shtykov, S.M. Smolskiy, Introduction to Quantum Electronics and Nonlinear Optics (Springer, Berlin, 2020) X-RAY DATA BOOKLET Center for X-ray Optics and Advanced Light Source}, Lawrence Berkeley National Laboratory. http://xdb.lbl.gov/ J.A. Samson, D.L. Ederer, Vacuum Ultraviolet Spectroscopy (Academic, Cambridge, 2000) F. de Groot, A. Kotani, Core-Level Spectroscopy of Solids (CRC Press, Boca Raton, 2008) F. de Groot, High-resolution X-ray emission and X-ray absorption spectroscopy. Chem. Rev. 101, 1779 (2001) J. Stohr, NEXAFS Spectroscopy (Springer, Berlin, 2003). S.Hufner, Photoelectron Spectroscopy: Principles and Applications (Springer, Berlin, 2003) S. Suga, A. Sekiyama, Photoelectron Spectroscopy: Bulk and Surface Electronic Structures (Springer, Berlin, 2014) F. Gel’mukhanov, H. Agren, Resonant X-ray Raman Scattering (Elsevier, Amsterdam, 1999) A. Kotani, S. Shin, Resonant inelastic X-ray scattering spectra for electrons in solids. Rev. Mod. Phys. 73, 203 (2001) L.J.P. Ament, M. van Veenendaal, T.P. Devereaux, J.P. Hill, J. van den Brink, Resonant inelastic X-ray scattering studies of elementary excitations. Rev. Mod. Phys. 83, 705 (2011) E. Zolotoyabko, Basic Concepts of X-Ray Diffraction (Wiley, London, 2014) D. McMorrow, J. Als-Nielsen, Elements of Modern X-ray Physics (Wiley, London, 2011) J. Miao, T. Ishikawa, I.K. Robinson, M.M. Murnane, Beyond crystallography: Diffractive imaging using coherent X-ray light sources. Science 348, 530 (2015) D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge University Press, Cambridge, 1999) H. Wiedemann, Synchrotron Radiation (Springer, Berlin, 2002) S. Mobilio, F. Boscherini, C. Meneghini, Synchrotron Radiation: Basics, Methods and Applications (Springer, Berlin, 2014) C. Pellegrini, A. Marinelli, S. Reiche, The physics of X-ray free-electron lasers. Rev. Mod. Phys. 88, 015006 (2016) K.-J. Kim, Z. Huang, R. Lindberg, Synchrotron Radiation and Free-Electron Lasers: Principles of Coherent X-Ray Generation (Cambridge University Press, Cambridge, 2017) P. Jaegle, Coherent Sources of XUV Radiation: Soft X-Ray Lasers and High-Order Harmonic Generation (Springer, Berlin, 2005) D. Popmintchev, B.R. Galloway, M.-C. Chen, F. Dollar, C.A. Mancuso, A. Hankla, L. MiajaAvila, G. O’Neil, J.M. Shaw, G. Fan, S. Ališauskas, G. Andriukaitis, T. Balˇciunas, O.D. Mücke, A. Pugzlys, A. Baltuška, H.C. Kapteyn, T. Popmintchev, M.M. Murnane, Near- and extendededge X-ray-absorption fine-structure spectroscopy using ultrafast coherent high-order harmonic supercontinua. Phys. Rev. Lett. 120, 093002 (2018) J.J. Sakurai, Modern Quantum Mechanics (Benjamin-Cummings Publishing Co., Subs. of Addison Wesley Longman, San Francisco, 1985)

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28. 29. 30. 31. 32.

L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1969) H. Rubin, Landau (Wiley, Quantum Mechanics II, 1995) A. Corney, Atomic and Laser Spectroscopy (Clarendon Press, Oxford, 1977) B. Henderson, G.F. Imbusch, Optical Spectroscopy of Inorganic Solids (Oxford, 2006) T. Mizoguchi, S. Kiyohara, Machine learning approaches for ELNES/XANES. Microscopy 92 (2020) Y. Suzuki, H. Hino, M. Kotsugi, K. Ono, Automated estimation of materials parameter for Xray absorption and electron energy-loss spectra with similarity measures. NPJ Comput. Mater. 5, 39 (2019) I. Matsuda (ed.), Monatomic Two-Dimensional Layers: Modern Experimental Approaches for Structure, Properties, and Industrial Use (Elsevier, Amsterdam, 2019) I. Matsuda, K. Wu ed., 2D Boron: Boraphene, Borophene, Boronene (Springer, Berlin, 2021) X. Zhang, Y. Tsujikawa, I. Tateishi, M. Niibe, T. Wada, M. Horio, M. Hikichi, Y. Ando, K. Yubuta, T. Kondo, I. Matsuda, Electronic topological transition of 2D boron by the ion exchange reaction. J. Phys. Chem. C 126, 12802 (2022) M. Niibe, M. Cameau, N. T. Cuong, O. Ilemona Sunday, X. Zhang, Y. Tsujikawa, S. Okada, K. Yubuta, T. Kondo, I. Matsuda, Electronic structure of a borophene layer in rare-earth aluminum/ chromium boride and its hydrogenated derivative, borophane. Phys. Rev. Materials 5, 084007 (2021) T. Someya, H. Fukidome, N. Endo, K. Takahashi, S. Yamamoto, I. Matsuda, Interfacial carrier dynamics of graphene on SiC, traced by the full-range time-resolved core-level photoemission spectroscopy. Appl. Phys. Lett. 113, 051601 (2018) X. Zhang, M. Hikichi, T. Iimori, Y. Tsujikawa, M. Horio, K. Yubuta, F. Komori, M. Miyauchi, T. Kondo, I. Matsuda, Accelerating synthesis of topological borophane by the HCl assisted ion exchange reaction. Molecules 28, 2985 (2023) H. Nishino, T. Fujita, N.T. Cuong, S. Tominaka, M. Miyauchi, S. Iimura, A. Hirata, N. Umezawa, S. Okada, E. Nishibori, A. Fujino, T. Fujimori, S. Ito, J. Nakamura, H. Hosono, T. Kondo, Formation and characterization of hydrogen boride sheets derived from MgB2 by cation exchange. J. Am. Chem. Soc. 139, 13761 (2017) M. Niibe, N. Takehira, T. Tokushima, Electronic structure of boron doped HOPG: selective observation of carbon and trace dope boron by means of X-ray emission and absorption spectroscopy, e-J. Surf. Sci. Nanotech. 16, 122 (2018) X. Feng, S. Sallis, Y.-C. Shao, R. Qiao, Y.-S. Liu, L.C. Kao, A.S. Tremsin, Z. Hussain, W. Yang, J. Guo, Y.-D. Chuang, Disparate exciton-phonon couplings for zone-center and boundary phonons in solid-state graphite. Phys. Rev. Lett. 125, 116401 (2020) S. Yamamoto, I. Matsuda, Time-resolved photoemission spectroscopy with synchrotron radiation: Past, Present, and Future. J. Phys. Soc. Jpn. 82, 021003 (2013) Y. Hayashi, Y. Tanaka, T. Kirimura, N. Tsukuda, E. Kuramoto, T. Ishikawa, Acoustic pulse echoes probed with time-resolved X-ray triple-crystal diffractometry. Phys. Rev. Lett. 96, 115505 (2006) T. Someya, H. Fukidome, H. Watanabe, T. Yamamoto, M. Okada, H. Suzuki, Y. Ogawa, T. Iimori, N. Ishii, T. Kanai, K. Tashima, B. Feng, S. Yamamoto, J. Itatani, F. Komori, K. Okazaki, S. Shin, I. Matsuda, Suppression of supercollision carrier cooling in high mobility graphene on SiC(0001). Phys. Rev. B 95, 165303 (2017) M. Ogawa, S. Yamamoto, Y. Kousa, F. Nakamura, R. Yukawa, A. Fukushima, A. Harasawa, H. Kondo, Y. Tanaka, A. Kakizaki, I. Matsuda, Development of soft X-ray time-resolved photoemission spectroscopy system with a two-dimensional angle-resolved time-of-flight analyzer at SPring-8 BL07LSU. Rev. Sci. Instrum. 83, 023109 (2012) S. Khan, K. Holldack, T. Kachel, R. Mitzner, T. Quast, Femtosecond undulator radiation from sliced electron bunches. Phys. Rev. Lett. 97, 074801 (2006) T. Suzuki, S. Shin, K. Okazaki, HHG-laser-based time- and angle-resolved photoemission spectroscopy of quantum materials. J. Electron Spectrosc. Relat. Phenom. 251, 147105 (2021) K. Ozawa, M. Emori, S. Yamamoto, R. Yukawa, Sh. Yamamoto, R. Hobara, K. Fujikawa, H. Sakama, I. Matsuda, Electron−hole recombination time at TiO2 single-crystal surfaces: influence of surface band bending. J. Phys. Chem. Lett. 5, 1953 (2014)

33.

34. 35. 36.

37.

38.

39.

40.

41.

42.

43. 44.

45.

46.

47. 48. 49.

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50. K. Ozawa, S. Yamamoto, R. Yukawa, R.-Y. Liu, M. Emori, K. Inoue, T. Higuchi, H. Sakama, K. Mase, I. Matsuda, What determines the lifetime of photoexcited carriers on TiO2 surfaces? J. Phys. Chem. C 120, 29283–29289 (2016) 51. D. Kutnyakhov et al., Time- and momentum-resolved photoemission studies using time-offlight momentum microscopy at a free-electron laser. Rev. Sci. Instrum. 91, 013109 (2020) 52. A. Pietzsch et al., Towards time resolved core level photoelectron spectroscopy with femtosecond X-ray free-electron lasers. New J. Phys. 10, 033004 (2008) 53. I. Matsuda, Y. Kubota, Recent progress in spectroscopies using soft X-ray free-electron Lasers. Chem. Lett. 50, 1336 (2021)

Chapter 3

Probing Nonlinear Light–Matter Interaction in Momentum Space: Coherent Multiphoton Photoemission Spectroscopy Marcel Reutzel, Andi Li, Zehua Wang, and Hrvoje Petek

Abstract Optical fields interacting with metal surfaces can drive collective free electron plasma currents and single-particle dipole excitations. The outcome of these coherent interactions is immediately evident in nearly perfect images that appear in metal mirrors. Yet, performing quantum state-resolved measurements and electronic charge/spin actuations of metals is extremely challenging because of interactiondriven decoherence of the polarization of the electronic system. In this chapter, we describe interferometrically time-resolved multiphoton photoemission spectroscopy as a quantum state-resolved method for investigating coherent responses of solids. We perform 3D (energy, momentum, and time)-resolved nonlinear photoelectron spectroscopy to record the coherent response of single crystal surfaces with attosecond accuracy. The measurements resolve the coherent polarization oscillations in response to the driving light field, the resonant frequencies of the metal sample, and the optical field-induced dressing of the electronic band structure. As a model system, we focus on the coherent responses of single crystal silver and copper surfaces to intense ~20 fs laser pulses in the IR-UV spectral range. Noble metals provide wellknown band structures, deeply studied attosecond collective responses, and simple surface preparation methods that guide the development of experimental and theoretical extension of coherent nonlinear photoemission spectroscopy to more complex materials. To photoexcite electrons above the vacuum level in a nonlinear manner, the excitation creates Floquet ladders of quasi-energy states up to fifth order in the driving light field, including signatures of above threshold photoemission. Finally, we show that at high driving field amplitudes, electrons follow both the space and optical field-dependent periodic potentials causing non-perturbative modifications M. Reutzel (B) I. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany e-mail: [email protected] A. Li · Z. Wang · H. Petek (B) Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_3

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of metal crystal electronic structures. Hence, nonlinear interferometric photoelectron spectroscopy enables to quantify, as well as to transiently modify, the coherent responses of solid-state materials.

3.1 Introduction The microscopic understanding of light–matter interactions in the condensed phase is a key ingredient for the implementation of the next-generation opto-electronic devices. In this context, recent efforts strive for the understanding of coherent light– matter interaction and the creation new and emergent material phases under the irradiation with femtosecond light-pulses [1, 2]. Under intense illumination, light pulses do not only induce dipole transitions and drive intraband currents, but can also significantly perturb the materials electronic band structure, create new photondressed electronic band dispersions, and change the correlated many-body interactions between the electron–phonon-spin subsystems [3–16]. Experimentally, there are multiple approaches to probe such light-dressed material phases. On the one side, all-optical techniques have identified the dominant timescales of ultrafast material control [17]. Complementary, distinct changes in the electronic, phononic, and spin subsystems have been identified using time- and angle-resolved photoelectron spectroscopy (trARPES) [18] and ultrafast electron scattering techniques [19]. In this chapter, we focus on the application of highly nonlinear spectroscopies to address such questions. High-harmonic generation (HHG) has been shown to be extremely powerful to characterize the nonlinear material response [20–25]. In these experiments, intense light pulses in the mid-infrared to terahertz regime impinge onto the sample surface, and the emitted HHG light is detected and analyzed. While the experiment provides a manifold insight onto microscopic processes, as any photon-in-photon-out technique, its information is integrated over the energy– momentum coordinates of the crystalline solid. Hence, it lacks specificity to the energy–momentum-dispersive electronic band structure, or, more generally, the spectral function, in which all the information of the single-particle band structure and many-body interactions is imprinted in the photoemission experiment [26]. In this chapter, we review the complementary approach of multiphoton photoemission spectroscopy (mPP) [27–36]. In contrast to HHG, where the nonlinear response of the condensed matter material is probed by the detection of high-frequency photons, mPP is a photon-in-electron-out technique. As such, mPP measures the system’s response by detecting photoelectrons with energies much larger than the driving light-field frequency. By varying the driving light frequency, the number of photons m necessary to overcome the work function of the material of interest changes systematically (Fig. 3.1). Additional to this mth-order outcome, in a process termed above threshold photoemission (ATP), (m + n)th-order replicas of the main photoemission peak can be detected. Often, experiments detecting highly nonlinear mPP have been performed on sharp metal tips or nanostructured surfaces where the nonlinearity can be increased by plasmonic fields or lightning-rod enhancements

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[36–45] with the goal to analyze non-perturbative light–matter interactions. Significantly less effort has been reported to employ mPP to well-defined crystalline solids, where the parallel momentum information is preserved [27–35]. Importantly, as the highly nonlinear variant of trARPES, mPP is directly sensitive to the energy and the in-plane momentum of the photoelectrons. As such, it carries distinct information on the periodically driven energy–momentum-dispersive band structure. In this chapter, we discuss the experimental basics of interferometrically time-resolved mPP (ITRmPP) and explore its application in the perturbative regime of coherent light–matter interaction where it probes optical dipole transitions. Finally, we discuss the application of ITR-mPP to study the onset of non-perturbative light–matter interaction and provide an outlook toward Floquet engineered material phases.

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3.2 Experimental Setup Angle-resolved photoemission spectroscopy (ARPES) has a long tradition as a probe of the spectral function of a manifold of material classes [18, 46, 47]. While the basics of ARPES are detailed in many monographs, such as the book by Hüfner [48], recent experimental developments and case studies are summarized in the review on the beginners [47] and the expert [18] level. In the following two sections, we will briefly review the necessary experimental requirements from the extension of ARPES to a highly nonlinear driving scheme.

3.2.1 Experimental Setup: The Photoelectron Analyzer Modern photoelectron analyzers are designed to provide direct access to the energy and at least one in-plane momentum coordinate of the photoemitted singleparticle electrons. Here, we will briefly highlight hemispherical energy analyzer and momentum microscope methods and their demonstrated applications in mPP experiments. In mPP, and, more generally, trARPES, the conventional approach employs hemispherical electron energy–momentum analyzers that are combined with a twodimensional photoelectron detector to provide direct access to the emission angle Θ (i.e., the in-plane momentum, k || ) and the energy E of the photoelectrons; exemplarily mPP (m = 2) data obtained on pristine Ag(111) with such an setup are shown in Fig. 3.2a [27, 49]. Complementary, time-of-flight momentum microscopes can be used to simultaneously collect multidimensional data sets that contain information on both in-plane momentum coordinates of the photoelectrons and their energy [50, 51]; exemplary mPP (m = 2) data measured on the pristine Ag(110) surface are shown in Fig. 3.1b [52, 53]. Because of the higher-dimension photoelectron detection scheme, the momentum microscope has some advantages over the hemispherical electron analyzer. The momentum microscope approach, however, is severely constrained in regard to space charge electron repulsion effects, which limit its dynamic detection range [54–56]. In highly nonlinear mPP experiments that involve high photoelectron currents, hemispherical electron analyzer detection may be preferable.

3.2.2 Experimental Setup: The Light-Source On the side of the light source, naturally, there are many parameters that can be optimized for the desired experiment. High-resolution ARPES experiments are optimized for a detailed band mapping of the occupied band structure and the extraction of many-body band renormalizations. These experiments are typically performed at large-scale synchrotron facilities that offer high-photon flux in the VUV-soft X-ray range, tunable photon energy, and small bandwidth [18, 26, 57].

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Fig. 3.2 Exemplary mPP (m = 2) data measured with an (a) hemispherical electron analyzer (b) and a time-of-flight momentum microscope. a In 2PP with 3.32 eV, the hemispherical energy analyzer provides direct access to the final state energy and the in-plane momentum of the photoemitted electrons collected from the Ag(111) surface. The bulk sp-band transition (U sp ←− L sp ), the occupied SS state, and a weak signal from the IP1 state are labeled in the energy-distribution-curve taken at k || =0 Å−1 . b The time-of-flight momentum microscope provides three-dimensional access to the photoemission data as a function of final state energy and two in-plane momenta. When excited with 3.72 eV photons, the accessible momentum range is limited by the photoemission horizon. The spectral signatures of surface states (S1, S2, S3) and bulk transitions (B1, B2, B3, B4) are labeled in the figure and their assignments are reported in Ref. [52]. a and b reproduced from [27, 52]

In order to study ultrafast and coherent light–matter interaction, photoemission spectroscopy has to be performed with femtosecond laser pulses in a pump-probe scheme. Early realizations of such experiments, however, were limited to nanosecond laser pulses to investigate the occupied and unoccupied electronic band structure of pristine metal surfaces [58, 59]. With the advent of femtosecond laser technology, time-resolved two-photon photoemission spectroscopy (tr2PP) has been the experiment of choice to probe the coherent and incoherent electron dynamics in different material classes [60–65]. In the implementation of a two-color experiment, a pump laser pulse of one color occupies an initially unoccupied state that is probed by a timedelayed laser pulse of different color (Fig. 3.1). In this way, the inelastic lifetimes 1/T 1 of, for example, image potential states on pristine and adsorbate covered metal surfaces have been studied in detail [62, 66–68]. Newer implementations of tr2PP, nowadays largely termed trARPES, combined optical pump pulses with extreme ultraviolet probe pulses that facilitate the access to large in-plane momenta and thus to the full momentum range of the surface Brillouin zone [4, 5, 15, 69–73]. In order to access the coherent response of a material to a light stimulus, and, specifically, to experimentally extract the decoherence time 1/T 2ij of a coherent polarization field built up between two states i and j, the mPP experiment has to

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be performed with an interferometric pump-probe scheme, i.e., so-called interferometrically time-resolved mPP (ITR-mPP) [27, 60, 74–80]. In common implementations of a single-color experiment, a phase-locked pump-probe pulse pair is created by passing an ultrashort laser pulse through a Mach–Zehnder interferometer. By varying the delay of one interferometer arm with interferometric precision, sub-50 as resolution movies of the energy- and momentum-dependent coherent electron dynamics can be resolved (Fig. 3.3). As such, this experiment has been applied to study surface plasmon polariton propagation [42, 81, 82], coherent light–matter interaction on metals [49, 75, 80, 83, 84], adsorbate covered surfaces [85–87], and in coherent photoemission electron microscopy [88–95]. In an alternative approach, recent efforts used birefringent wedges [96] to measure ITR-mPP data, promising a more straightforward alignment of the optical setup, and an increased stability of the interferometric measurement.

3.3 mPP: Highly Nonlinear Mapping of the Energyand Momentum-Dispersive Electronic Band Structure Static mPP can be used to map the occupied and unoccupied energy–momentumdispersive band structure of crystalline solids. In this section, we show the potential that it provides on the example of the Shockley surface state and the Rydberg series of image potential states on the pristine Cu(111) and Ag(110) surfaces. First, we describe the mPP process with the threshold nonlinear order of m that is necessary to

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overcome the material’s work function. Second, we systematically characterize the contribution of above threshold photoemission (ATP) to the mPP spectra. Finally, we motivate the application of highly nonlinear mPP to measure coherent light–matter interaction at large in-plane momenta, i.e., throughout the full surface Brillouin zone.

3.3.1 Static mPP in Threshold Order of Photoemission In the (111) projection of the electronic band structure of Cu, the Shockley surface state (SS) and the n = 1 image potential state (IP1) reside in the L-projected band gap with energy minima at the ⎡-point of the surface Brillouin zone; higher-order states of the Rydberg series of image potential states such as the n = 2 image potential state (IP2) lie resonant with the surface project upper sp-band (band diagram in Fig. 3.4b), and thus are not strictly localized at the surface. mPP is capable to directly map the energy–momentum dispersion of these surface states: Using 1.53 eV photons, the lowest order of photoemission to overcome the work function of Cu(111) is m = 4 (Φ ≈ 4.9 eV). For the chosen photon energy, the IP1 state is excited resonantly in a 3-photon process from the occupied SS state, leading to a strong enhancement of photoemission yield at the ⎡-point. Consequently, the SS and the IP1 states are detected at a final state energy of 5.6 eV above the Fermi level. Because of the different effective masses of the SS and the IP1 state, however, the excitation becomes detuned for larger in-plane momenta k || and the parabolic dispersion of both states under near-resonant excitation conditions is observed separately. Notably, in the inplane momentum region where the resonance condition is detuned, the photoemission yield is significantly reduced. If the driving photon energy is increased to 1.57 eV, the excitation becomes detuned over the full accessible in-plane momentum range and, concomitant, the photoemission intensity becomes more smoothly distributed in momentum space. A further increase of the photon energy to 1.67 eV leads to a pronounced enhancement of photoemission yield at large in-plane momenta, which can be attributed to onset of a resonant 3-photon transition from the SS state into the IP2 state. In short, mPP is fully capable to map the electronic band structure in the full accessible momentum range, and its efficiency is strongly dependent on the nonlinear resonance conditions between the occupied and unoccupied electronic bands, which are the SS and IP1 or IP2 states in the example of Cu(111).

3.3.2 Above Threshold Photoemission At sufficiently high driving field strength, higher-order replicas of the lowest-order photoemission signature become accessible. While the photoemission intensity in Fig. 3.4a is dominated by signal in nonlinear order m = 4, higher-order replicas in the order of m + n = 5 and m + n = 6 are clearly detected. At the excitation conditions used for the data shown in Fig. 3.4, each higher order of photoemission is roughly a

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Fig. 3.4 Exemplary mPP and ATP data obtained from Cu(111). a–c From left to right, the energy– momentum-resolved mPP spectra are obtained with increasing photon energy. The IP1(i) and the SS(i) state are observed in threshold order of photoemission i = m = 4. In addition, ATP leads to the formation of higher-order replicas in i = m + n = 4 + 1 and 4 + 2. b Excitation diagram for 1.67 eV photons. Reproduced from [28]

factor 10 less intense. This is a first indication that these experiments are performed in the perturbative regime where still the lowest order of photoemission dominates the signal [28]. The excitation of multiple orders of nonlinear interaction, and consequently photoemission, can be understood in terms of generation of a Floquet quasienergy ladder, where levels excited to energies above the work function can decay into the vacuum-free electron continuum [27, 28], as will be explained further below. In the non-perturbative regime, one might expect to observe that higher-order processes (m + n) become more intense than the signal obtained in threshold photoemission with nonlinear order m [38]. Nevertheless, energy–momentum-resolved spectra of well-ordered, single-crystalline surfaces carry rich information on the allowed dipole transitions under intense optical excitation. Specifically, it is directly apparent that the parabolic dispersion of the surface bands is unchanged. Moreover, the enhanced photoemission yield is clearly detected at the in-plane momenta where the SS to IP1 (IP2) transition occurs in a three-photon resonance, indicating that indeed such resonant conditions lead to higher-order transitions that are observed as ATP signal.

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3.3.3 Toward Full Surface Brillouin Zone Mapping by Coherent mPP So far, we have applied mPP spectroscopy to characterize the energy–momentumdispersive electronic band structure of the (111) crystalline orientation of noble metal surfaces such as silver or copper. On these facets, the surface projected band gap and the surface bands are found around the center of the surface Brillouin zone, i.e., at the ⎡-point. However, there are many modern quantum materials such as graphene, whose important electronic band structure features are found at large inplane momenta, i.e., at the corners of the surface Brillouin zone. If one aims to probe these bands in an ARPES experiment, one has to realize that probing larger inplane momenta requires excitation of electrons to higher energies such that they can perform work against the surface potential [48]. This is because the energy in electron surface parallel motion is conserved and is not available to overcome the work function in photoemission [32]. Consequently, the accessible in-plane momentum range in a photoemission experiment is constrained by the so-called photoemission horizon that is described by E kin = hk|| /2m e (green parabola in Fig. 3.5a; E kin : photoelectron kinetic energy; h: Plank constant, me : electron mass). In a nutshell, a photon energy of approximately 20 eV is necessary to probe the first surface Brillouin zone with a typical extension of < 2 Å−1 . Of course, the necessary excitation is easily accessible in static ARPES experiments at synchrotrons or with conventional gas discharge lamps in the laboratory. It is more challenging for femtosecond lasers excitation. Here, the last years have shown a tremendous development in the installation of table-top high-repetition rate high-harmonic generation beamlines [97–103] that enable generation of sufficiently high photon energies with a few ten femtosecond pulse duration. Complementary, highly nonlinear mPP provides a powerful approach to study the coherent light–matter interaction without the need to first create extreme ultraviolet light. Instead of creating high energy photons in a nonlinear process, it is possible to utilize this nonlinear process in order to photoexcite electrons to a sufficiently high energy in order to overcome the photoemission horizon (Fig. 3.5a). We illustrate the potential of this approach with the example of the Ag(110) surface [104]. The (110) surface occurs at a junction of two (111) planes. Consequently, the band gaps of (110) surface also occur in the ⎡ − L direction, but this is an off-normal direction for (110). As in the case of the (111) facets, a surface projected band gap is formed on this crystal orientation. The band gaps of the (110) surface support Shockley surface bands (S oc labels the occupied SS, and S un labels the unoccupied SS) at the Y-point, corresponding to in-plane momenta of ≈ 0.7 Å−1 (Fig. 3.5a). When performing a mPP experiment with 1.73 eV photons, a m = 3 nonlinear process is sufficiently energetic to overcome the work function of 4.2 eV at the ⎡-point. For example, the Fermi level E F can be probed in 3PP in vicinity of the ⎡-point (Fig. 3.5b). As can be seen in the excitation diagram in Fig. 3.5a, however, at the Y-point, the energy of the photoelectrons probing the occupied surface band is not sufficient to overcome the photoemission horizon. Nevertheless, in mPP, the noted surface bands can be detected in next nonlinear order: For m + n = 4, the

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Fig. 3.5 Illustration of the opportunity to use high-order mPP to probe electronic bands at large in-plane momenta. a While 3PP with 1.73 eV photons is sufficient to overcome the vacuum level at the ⎡-point, the occupied surface state S oc at the Y-point is not accessible because it remains hidden below the photoemission horizon (green parabola). The S oc state, however, can be probed in the next higher order of photoemission, i.e., in 4PP. b Corresponding mPP data in which the occupied surface state is probed in 4PP, but not in 3PP. In contrast, the Fermi level at the ⎡-point is accessible in 3PP and in 4PP. Reproduced from [104]

energy is sufficient to overcome the photoemission horizon, and, indeed, we detect the S un ← S oc resonant transition in (1 + 3)PP (Fig. 3.5b). In direct comparison to data obtained in 1PP [105] and 2PP (Fig. 3.2b) [52], we can thereby confirm that the energy–momentum dispersion (i.e., the effective mass) of the surface band is well-reproduced in (1 + 3)PP, exemplifying thus the potential of highly nonlinear mPP to probe the occupied and unoccupied electronic band structure at large in-plane momenta. Moreover, the full potential of ITR-mPP, as will be discussed in the next section, can then be employed to map these bands.

3.4 Coherent Two-Dimensional Photoelectron Spectroscopy The strength of the time-resolved photoemission technique is the direct access to the energy–momentum-resolved population dynamics in excited states. In conventional two-color pump-probe experiments; typically, the phase information and thus the important insight into coherent light–matter interaction are not accessible. In order to gain access to the momentum-resolved coherent light–matter interaction, the timeresolved photoemission experiment has to be extended to a phase-resolved pumpprobe scheme. This facilitates the evaluation of the interferometrically time-resolved photoemission data akin to a two-dimensional spectroscopy [106, 107], providing direct access to the momentum-resolved coherent light–matter interaction.

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In this context, so far, we have discussed mPP as a static spectroscopy tool to map the energy–momentum-dispersive surface projected band structure, as applied to noble metal surfaces. Its full potential, however, becomes accessible if the experiment is extended into the time-domain in a phase-resolved driving scheme. In the following, first, we will introduce the concept of coherent two-dimensional Fourier transform photoelectron spectroscopy (2D FT photoelectron spectroscopy) on the model example of a three-level system that we simulate within the optical Bloch equation (OBE) approach. Subsequently, we will compare these simple model calculations with actual ITR-mPP data collected on the Ag(111) surface; this comparison of OBE simulations and experimental data on Ag(111) will provide a direct benchmark of the potential of 2D FT photoelectron spectroscopy. Using this experimental approach, subsequently, we will show that the mPP and ATP processes from a discrete Shockley surface state are indeed a coherent process.

3.4.1 Coherent 2D FT Photoelectron Spectroscopy—Optical Bloch Equation Modeling The general motivation of using OBE simulations to model time-resolved mPP and ARPES experiments is to simplify the complex dynamics in a condensed matter system to a more atom-like system described only by a few discrete levels. This is schematically illustrated in the three-level system shown in Fig. 3.6b. All loss and decoherence process, which in the solid state are given by electron–electron and electron–phonon scattering processes, are added phenomenologically to the diagonal and off-diagonal elements of the density matrix, respectively, as 1/T 1i and 1/T 2ij rates (i, j: levels) [27, 108, 109]. Details on the application of the optical Bloch equations to the evaluation of time-resolved photoemission data can be found in many references, e.g., [27, 70, 108–111]; here, we want to focus on the evaluation of the OBE simulation akin to a two-dimensional spectroscopy. For the actual modeling of the mPP process with m = 2, we construct a three-level system assembling an occupied SS state, an unoccupied intermediate state (IP), and a final state continuum (Fig. 3.6b). Note, that the continuum can only be modeled by OBE by adding a dense discrete energy level structure [109]. Dipole transitions between all states are induced by the coherent driving with a 20 fs laser pulse, which is consistent with the experimental driving field in related experiments.The photon energy is chosen such that the intermediate IP state can be occupied only in a largely detuned transition. The photoemission spectrum is modeled by evaluating the occupation of the discretized final state continuum, such as shown in the energydistribution-curve in the top of Fig. 3.6a. Under the detuned excitation condition between SS and IP, only negligible occupation is found at the probing energy of the IP state, but SS state occupation is efficiently excited into the final state continuum in a two-photon process. In order to model the coherent light–matter interaction, we repeat the OBE calculation with a phase-locked pulse pair and tune the delay

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between both in pulses in sub-100 as steps; in this way, energy-resolved interferometric two-pulse correlation traces are collected (main panel of Fig. 3.6a). In this data representation, the occupation of the final state continuum is color-coded and we can clearly identify how the occupation is modulated as a function of pump-probe delay on the timescale of the optical cycle of the laser pulses due to constructive and destructive interferences. Prominently, at a pump-probe delay of > 20 fs, we find a tilting of the interference fringes with increasing final state energy (inset in Fig. 3.6a). This time-domain tilting of the interference fringes can be more intuitively understood when evaluating the ITR-mPP simulation akin to a two-dimensional spectroscopy. In related work, the fringe tilting was said to have a hyperbolic dependence on the relative phase [112]. For this, we Fourier transform the delay axis in order to relate the polarization energy (excitation energy) with respect to the final state energy of the detected occupation (Fig. 3.6c); the Fourier amplitude is shown on the color-coded heatmap. As we simulate a correlation experiment of second order, we find Fourier amplitude at polarization frequencies that correspond to the harmonics of the driving frequency ω, i.e., at 0ω, 1ω, and 2ω. This observation directly highlights that indeed, the mPP experiments measure the rectification of the nonlinear coherent polarization field to a photoelectron current. The slowly varying (decaying) contribution to the occupation dynamics at zero polarization energy (0ω) can be attributed to the phase-averaged intermediate state population dynamics; this is the common transient that is typically accessed in a

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two-color mPP (m = 2) experiment [66–68]. From this signal Fourier component, the inelastic decay times 1/T 1i of level i can be extracted [60, 113]. The 1ω and 2ω components represent the linear and nonlinear coherent polarizations, respectively, which the driving field induces to excite the nonlinear transitions in the sample. Notably, from evaluation of the 1ω and 2ω coherences, one can obtain physical insights onto the linear coherences that are coupled by linear superposition (SS & IP and IP & final state continuum) and the nonlinear coherences (SS and final state continuum), respectively, and thus extract their dephasing 1/T 2ij rates. Hence, in order to explore the coherence of the two-photon transition from the discrete SS state into the final state continuum in more detail, Fig. 3.7a shows expanded elements of the 2D FT spectra corresponding to the 0ω, 1ω, and 2ω components. Strikingly, we find that the Fourier amplitude profiles show very distinct dependence on the polarization and the final state energy: While the 0ω is not dispersive, the 1ω and 2ω components show a distinct linear dispersion, where the slope increases from the 0ω to the 2ω-component. This distinct shape of the 2D FT spectra is a direct result of the coherent excitation process and the conservation of energy in the photoexcitation of a discrete initial state with a spectrally broad laser pulse (Fig. 3.7b). Consequently, the slope of the 2D FT spectra can be described by the ratio k/m, where m is the order of the photoexcitation process and k is the nonlinear polarization order. Specifically, for the case of the three-level model simulation of occupation transfer from the SS state into the final state continuum in a coherent twophoton process, we find slopes of 0/2, 1/2, and 2/2 for the 0ω, 1ω, and 2ω-components, respectively. Naturally, these fingerprints of coherent light–matter interaction identified in the 2D FT photoelectron spectra derive from the signal evolution in the time domain. In the expanded part of Fig. 3.6a, one clearly finds the already mentioned hyperbolic tilting of the interference fringes toward time zero with increasing pump-probe delay. This tilting results from the coherent excitation process as the optical cycle frequency follows the spectral component of the broadband laser pulse [114–116].

3.4.2 Coherent 2D FT Photoelectron Spectroscopy of Ag(111) The Shockley surface band of Ag(111) provides a simple model system to systematically study the impact of the nonlinear order of photoemission (m = 4) onto the coherence of the photoemission process. In this section, in direct comparison to the OBE simulations, we will show how it provides a benchmark response for application of 2D-FT to more complex systems. For this purpose, it is important to note that this band probably has the longest dephasing time among occupied states of metal surfaces [66].

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nonlinear photoemission process at approximately the optical cycle (≈ 2.7 fs) of the optical field as the pump-probe delay is advanced in subcycle steps. The interference fringes record a direct signature of coherence in the 4-photon promotion of SS electrons to above the vacuum level: At a pump-probe delay of around 30 fs, we find the interference fringes to tilt toward time zero with increasing final state energy, fully consistent with the OBE model simulations of the ITR-mPP experiment. Actually, this tilting of the interference fringes then directly reflects the apparent upshift of

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the energy–momentum-dispersive SS state in Fig. 3.8a, i.e., as the delay is advanced within a single optical cycle. Having compared the time-domain appearance of the ITR-mPP data between experiment and OBE simulation, in the next step, we evaluate the coherent response akin to 2D spectroscopy [106, 107]. We Fourier transform the time-domain interferometric data to obtain 2D FT photoelectron spectra in the conjugate frequency domain, as shown in Fig. 3.8c. Again, we find Fourier amplitude at the driving laser frequency and its harmonics, indicating that coherent polarization fields up to fourth order are detected in the 4-photon photoexcitation process of the SS state. Significantly, the 2D FT spectra have distinct shapes: Their slopes take on the ratio k/m where k is the polarization order and m is the photoemission order (here: m = 4). In direct comparison to the OBE simulation, this data analysis directly shows that the 4PP process from the SS state is a coherent process. For further verification of this statement, we repeat the OBE modeling for a 5-level system in order to describe the 4-photon process in a ladder of transitions via excitation that is detuned from the IP state and proceeds through virtual intermediate states up to the photoemission continuum (Fig. 3.8d). As detailed in Ref. [27], we can reproduce the overall shapes of the experimental 2D FT spectra. Note that the energy width of the 2D FT spectra contains information on the decoherence rate of the coherent polarization [117]. Moreover, by systematically varying the photoemission order from m = 4 to m = 3 and 2, the shape of the 2D FT spectra behave as expected [27, 117]. Related interferometric time-resolved nonlinear photoemission experiments have been applied to studies of dephasing and coherent control of electronic bands in surface and bulk metals [11, 27, 75–77, 83], metal surfaces with organic/inorganic adsorbates [85– 87], metal/semiconductor interfaces [113, 118], ultrafast microscopy of trivial [42, 88, 119] and topological plasmonic fields [89, 90, 93, 120–122].

3.4.3 Coherent Above Threshold Photoemission Above threshold photoemission describes the detection of photoelectrons with energies that can only be reached if the electron absorbs more photons than necessary to overcome the material’s work function [123, 124]. In general, this process of ATP can be envisaged in two ways: On the one-side, it might be possible that a free photoelectron is created in front of the metal surface, which is subsequently dressed by the laser field. Otherwise, the driving laser field can directly induce coherent polarization field in a Floquet ladder oscillating at the frequency where the photoelectrons are rectified, and, as such coherently lead to photoemission at above threshold energy. If both processes occur with similar amplitudes, one might expect the two processes to show evidence of two-path interference. In this section, we employ the ITR-mPP experiment in the manner of the 2D FT photoelectron spectroscopy in order to show that we indeed find a direct contribution from higher-order coherent polarization field at the order of ATP.

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For this, we repeat the ITR-mPP experiment on Ag(111), but now evaluate the photoemission signal not in lowest order of photoemission (m = 4), but focus on the m + n = (4 + 1)PP data (Fig. 3.9). As discussed in the 4PP experiment and the OBE simulations, for ATP from the occupied Shockley surface band, we clearly find tilted interference fringes in the time domain and the distinct shape of the 2D FT photoelectron spectra in Fourier space. Specifically for the ATP process that occurs in the order of m = 5, we find that the slopes of the 2D FT spectra are nicely described by the ratio k/5. This experimental observation indeed shows the mPP experiment detects highly nonlinear polarization fields created in the sample as the system is driven by intense near-infrared laser pulses. Consequently, under our experimental conditions, we conclude that ATP is a one-step process where the highly nonlinear coherent polarization field is directly rectified such that photoelectrons are detected in an above threshold process.

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3.5 Ultrafast Quasiparticle Dressing by Light In the final section of this chapter, we will explore the parameter regime where the driving electric field becomes sufficiently strong to modify the equilibrium quasiparticle band structure of the solid-state system. In the context of Floquet theory [2, 125, 126], it is proposed that a time-periodic oscillating field can lead to the coherent formation of light-dressed band structure whose properties are composed of the solid itself and those of the light field [126]. In this regime, if a resonance condition between an occupied and an initially unoccupied state is driven at a sufficiently high field, band structures with light-induced band gaps can emerge, as initially found in semiconductors [127, 128], and more recently on metal surfaces [11], topological insulators [9], graphene [10], and black phosphorus [12]. Likewise, in atomic physics, it is well established that a light field can lead to the renormalization of the atomic eigenstates, requiring the formulation of new light-dressed eigenstates [129], which have been identified in all-optical spectroscopies of atomic gas species as the Autler–Townes splitting [130]. So far, we have explored that the coherent polarization fields imprinted onto the solid-state material by a time-periodic external perturbation can be directly measured with energy- and momentum resolution with ITR-mPP. In the following, using ITR-mPP and 2D FT photoelectron spectroscopy, we will show that it is indeed possible to extract information on the coherent response leading to light-dressed band structure of Cu(111) surface bands [11]. Figure 3.10a shows an overview of ITR-4PP data collected with 1.54 eV pulses on pristine Cu(111). In contrast to the last section where we focused on non-resonant excitation pathway of the SS state, the photon energy is now chosen such that it drives a three-photon resonant excitation from the SS to the IP1 state at the ⎡-point; with increasing in-plane momentum k || , the resonance condition detunes (Fig. 3.3a). When considering the IP1 ← SS resonance as a coherently driven twolevel system, Rabi oscillations will be induced, if the driving field is sufficiently strong and the decoherence time is sufficiently long [129]. The strong driving field causes the mixing of the coupled states such that this coherently driven system becomes strongly perturbed, and needs to be described by a new set of light-dressed eigenstates E ± , corresponding to Autler–Townes doublet [130]. The energetic splitting of the transiently dressed states is described by the generalized Rabi splitting √ ∼ hΩ R (τ, k) = h μeff E3 (τ ) + Δ2 (k), (μeff : effective dipole moment of the threephoton transition; E(τ ): electric field strength; Δ(k): in-plane momentum dependent detuning). The challenge in the ITR-4PP experiment now is the experimental access to the energy–momentum dependence of the coherent polarization fields and how they carry information on the light-induced band renormalizations. In order to access the energy–momentum-dispersive coherent polarization fields in different harmonic orders of the driving laser’s harmonics, we Fourier analyze the ITR-4PP data as detailed in Ref. [11]: In short, in the 2D FT spectra, we band-pass filter the frequency components oscillating with frequencies close to 2ω and inverse Fourier transform the data back into the time domain. In the data in Fig. 3.10b, the amplitude of the coherent polarization field is then plotted on the color scale as

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function of pump-probe time delay, energy, and in-plane momentum. Notably, we find that the amplitude of the coherent polarization field oscillating at 2ω mimics the energy–momentum dispersion of the SS and the IP1 state at pump-probe delays of 65 fs, i.e., when the local electric field strength inducing the photoexcitation process is dominantly given by a single-laser pulse (Fig. 3.10c, right panel). When the pumpprobe laser pulses come into temporal overlap, however, the electric field strength is significantly increased, leading to a stronger AC Stark splitting of the SS band, i.e., a larger generalized Rabi frequency and thus a measurable light-induced band gap opening between the light-dressed eigenstates E ± . During the time period of overlap of the pump- and the probe-laser pulses, we then indeed find a systematic opening and closing of the light-induced band gap (Fig. 3.10c, left panels). Notably, as expected, the detuning of the three-photon resonance toward larger in-plane momenta leads

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to a weaker influence of the driving electric field strength on the observed band dispersion (Fig. 3.10c). We consider this observation of the gap opening in the energy–momentum dispersion that appears in the 2ω coherent polarization field as a direct evidence for the coherent engineering of the electronic band structure of the noble metal surface. Notably, the light-dressed electronic band structure is obscured, i.e., not yet detectable, in the phase-averaged data shown in Fig. 3.3a, highlighting that the analysis of the phase-stable mPP experiment has significant advantages to probe light-induced phenomena as it approaches the non-perturbative regime. As a photoemission experiment, it will be applicable to further material systems, providing, in addition to two-color trARPES [9, 12], a powerful energy- and momentum-sensitive technique to probe the coherent control of condensed matter material phases.

3.6 Conclusion The coherent response of metals to optical fields is familiar in our daily lives as coherent reflections of optical fields from metallic mirrors. Measuring such coherent responses to optical fields by optical methods is extremely challenging [131], because it is defined by the coherent response of free electrons occupying electronic bands with several eV bandwidth. Consequently, the coherent response leading to the screening, and therefore, optical reflection at a metal surface occurs on the attosecond time scale, as defined by the collective plasmon response of metals [132]. The energy and momentum resolution of photoemission spectroscopy enables measurements of quasiparticle decoherence from photoemission linewidths, though this strongly depends on the sample quality [133], and can easily lead to incorrect conclusions with poorly prepared samples. The nonlinear ITR-mPP spectroscopy enables the direct measurement of coherence of surface and bulk electronic bands of metals particularly in the electron [27, 75] and hole [77] excited states, as illustrated in this chapter. Beyond probing the linear and nonlinear polarization dephasing, the applied Coulomb field of femtosecond laser excitation can be sufficiently strong to drive the Rabi frequency near the carrier wave frequency and thereby modify the electronic band structure of a solid [11], and even exert fields that exceed the binding energy of electrons in metals. Moreover, the optical fields can impose topology that transiently modifies the time inversion and parity symmetry of coherent responses of metals [89, 121]. Therefore, the coherent ITR-mPP spectroscopy is a route to probe the non-perturbative responses of metals and to transiently impress electronic fields that modify the electronic properties of solids by Floquet and Poincaré engineering methods [27, 28, 134, 135]. Acknowledgements We acknowledge fruitful discussions with Namitha Ann James. M. R. acknowledges support through the Alexander von Humboldt Foundation for his Feodor Lynen

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PostDoc fellowship at the University of Pittsburgh in the group of H.P. In addition, M.R. acknowledges funding through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—217133147/SFB 1073, project B10, and inspiring discussions with the newly established ITR-mPP team in Göttingen: Hannah Strauch, Marco Merboldt, and Stefan Mathias. H.P. thanks partial financial support from the National Science Foundation Grant No. CHE-2102601, and promotion of ITR-mPP methods by Shijing Tan.

References 1. D.N. Basov, R.D. Averitt, D. Hsieh, Towards properties on demand in quantum materials. Nat. Mater. 16, 1077 (2017) 2. A. de la Torre, D.M. Kennes, M. Claassen, S. Gerber, J.W. McIver, M.A. Sentef, Colloquium: nonthermal pathways to ultrafast control in quantum materials. Rev. Mod. Phys. 93, 041002 (2021) 3. D. Fausti et al., Light-induced superconductivity in a stripe-ordered cuprate. Science 331, 189 (2011) 4. T. Rohwer et al., Collapse of long-range charge order tracked by time-resolved photoemission at high momenta. Nature 471, 490 (2011) 5. C.W. Nicholson, A. Lücke, W.G. Schmidt, M. Puppin, L. Rettig, R. Ernstorfer, M. Wolf, Beyond the molecular movie: Dynamics of bands and bonds during a photoinduced phase transition. Science 362, 821 (2018) 6. F. Schmitt et al., Transient electronic structure and melting of a charge density wave in TbTe3 . Science 321, 1649 (2008) 7. L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, D. Mihailovic, Ultrafast switching to a stable hidden quantum state in an electronic crystal. Science 344, 177 (2014) 8. E.J. Sie et al., An ultrafast symmetry switch in a Weyl semimetal. Nature 565, 61 (2019) 9. F. Mahmood, C.-K. Chan, Z. Alpichshev, D. Gardner, Y. Lee, P.A. Lee, N. Gedik, Selective scattering between Floquet-Bloch and Volkov states in a topological insulator. Nat. Phys. 12, 306 (2016) 10. J.W. McIver, B. Schulte, F.U. Stein, T. Matsuyama, G. Jotzu, G. Meier, A. Cavalleri, Lightinduced anomalous Hall effect in graphene. Nat. Phys. 16, 38 (2020) 11. M. Reutzel, A. Li, Z. Wang, H. Petek, Coherent multidimensional photoelectron spectroscopy of ultrafast quasiparticle dressing by light. Nat. Commun. 11, 2230 (2020) 12. S. Zhou et al., Pseudospin-selective Floquet band engineering in black phosphorus. Nature 614, 75 (2023) 13. T. Frigge et al., Optically excited structural transition in atomic wires on surfaces at the quantum limit. Nature 544, 207 (2017) 14. J.G. Horstmann, H. Böckmann, B. Wit, F. Kurtz, G. Storeck, C. Ropers, Coherent control of a surface structural phase transition. Nature 583, 232 (2020) 15. M. Düvel et al., Far-from-equilibrium electron–phonon interactions in optically excited graphene. Nano Lett. 22, 4897 (2022) 16. A. Kogar et al., Light-induced charge density wave in LaTe3 . Nat. Phys. 16, 159 (2020) 17. A. Zong, B.R. Nebgen, S.-C. Lin, J.A. Spies, M. Zuerch, Emerging ultrafast techniques for studying quantum materials. Nat. Rev. Mater. 8, 224 (2023) 18. J.A. Sobota, Y. He, Z.-X. Shen, Angle-resolved photoemission studies of quantum materials. Rev. Mod. Phys. 93, 025006 (2021) 19. D. Filippetto, P. Musumeci, R.K. Li, B.J. Siwick, M.R. Otto, M. Centurion, J.P.F. Nunes, Ultrafast electron diffraction: visualizing dynamic states of matter. Rev. Mod. Phys. 94, 045004 (2022)

78

M. Reutzel et al.

20. S. Ghimire, D.A. Reis, High-harmonic generation from solids. Nat. Phys. 15, 10 (2019) 21. S. Ghimire, A.D. DiChiara, E. Sistrunk, P. Agostini, L.F. DiMauro, D.A. Reis, Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138 (2011) 22. F. Langer et al., Lightwave-driven quasiparticle collisions on a subcycle timescale. Nature 533, 225 (2016) 23. C.P. Schmid et al., Tunable non-integer high-harmonic generation in a topological insulator. Nature 593, 385 (2021) 24. C. Heide et al., Probing topological phase transitions using high-harmonic generation. Nat. Photon. 16, 620 (2022) 25. J. Freudenstein et al., Attosecond clocking of correlations between Bloch electrons. Nature 610, 290 (2022) 26. A. Damascelli, Probing the electronic structure of complex systems by ARPES. Phys. Scr. T109, 61 (2004) 27. M. Reutzel, A. Li, H. Petek, Coherent two-dimensional multiphoton photoelectron spectroscopy of metal surfaces. Phys. Rev. X 9, 011044 (2019) 28. M. Reutzel, A. Li, H. Petek, Above-threshold multiphoton photoemission from noble metal surfaces. Phys. Rev. B 101, 075409 (2020) 29. W.S. Fann, R. Storz, J. Bokor, Observation of above-threshold multiphoton photoelectric emission from image-potential surface states. Phys. Rev. B 44, 10980 (1991) 30. M. Aeschlimann, C.A. Schmuttenmaer, H.E. Elsayed-Ali, R.J.D. Miller, J. Cao, Y. Gao, D.A. Mantell, Observation of surface enhanced multiphoton photoemission from metal surfaces in the short pulse limit. J. Chem. Phys. 102, 8606 (1995) 31. F. Banfi, C. Giannetti, G. Ferrini, G. Galimberti, S. Pagliara, D. Fausti, F. Parmigiani, Experimental evidence of above-threshold photoemission in solids. Phys. Rev. Lett. 94, 037601 (2005) 32. F. Bisio, M. Nývlt, J. Franta, H. Petek, J. Kirschner, Mechanisms of high-order perturbative photoemission from Cu(001). Phys. Rev. Lett. 96, 087601 (2006) 33. G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M.M. Murnane, H.C. Kapteyn, Laser-assisted photoemission from surfaces. Phys. Rev. A 77, 022903 (2008) 34. F. Bisio, A. Winkelmann, C.-T. Chiang, H. Petek, J. Kirschner, Band structure effects in above threshold photoemission. J. Phys. Condens. Matter 23, 485002 (2011) 35. F. Sirotti, N. Beaulieu, A. Bendounan, M.G. Silly, C. Chauvet, G. Malinowski, G. Fratesi, V. Véniard, G. Onida, Multiphoton k-resolved photoemission from gold surface states with 800-nm femtosecond laser pulses. Phys. Rev. B 90, 035401 (2014) 36. P. Dreher, D. Janoschka, A. Neuhaus, B. Frank, H. Giessen, M. Horn-von Hoegen, F.-J. Meyer zu Heringdorf, Quantitative determination of the electric field strength in a plasmon focus from ponderomotive energy shifts. Nanophotonics 11, 3687 (2022) 37. R. Bormann, M. Gulde, A. Weismann, S.V. Yalunin, C. Ropers, Tip-enhanced strong-field photoemission. Phys. Rev. Lett. 105, 147601 (2010) 38. M. Schenk, M. Kruger, P. Hommelhoff, Strong-field above-threshold photoemission from sharp metal tips. Phys. Rev. Lett. 105, 257601 (2010) 39. P. Dombi et al., Strong-field nano-optics. Rev. Mod. Phys. 92, 025003 (2020) 40. M. Merschdorf, W. Pfeiffer, A. Thon, S. Voll, G. Gerber, Stimulated emission induced by exciton–exciton scattering in ZnO/ZnMgO multiquantum wells up to room temperature. Appl. Phys. A 71, 547 (2000) 41. P. Dombi, A. Hörl, P. Rácz, I. Márton, A. Trügler, J.R. Krenn, U. Hohenester, Ultrafast strong-field photoemission from plasmonic nanoparticles. Nano Lett. 13, 674 (2013) 42. M. D˛abrowski, Y. Dai, H. Petek, Ultrafast microscopy: imaging light with photoelectrons on the nano-femto scale. J. Phys. Chem. Lett. 8, 4446 (2017) 43. M. Sivis, N. Pazos-Perez, R. Yu, R. Alvarez-Puebla, F.J. García de Abajo, C. Ropers, Continuous-wave multiphoton photoemission from plasmonic nanostars. Commun. Phys. 1, 13 (2018) 44. A. Li, M. Reutzel, Z. Wang, D. Novko, B. Gumhalter, H. Petek, Plasmonic photoemission from single-crystalline silver. ACS Photon. 8, 247 (2021)

3 Probing Nonlinear Light–Matter Interaction in Momentum Space …

79

45. H. Petek, A. Li, X. Li, S. Tan, M. Reutzel, Plasmonic decay into hot electrons in silver. Prog. Surf. Sci. 98, 100707 (2023) 46. A. Damascelli, Z. Hussain, Z.-X. Shen, Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473 (2003) 47. H. Zhang, T. Pincelli, C. Jozwiak, T. Kondo, R. Ernstorfer, T. Sato, S. Zhou, Angle-resolved photoemission spectroscopy. Nat. Rev. Methods Primers 2, 54 (2022) 48. S. Hüfner, Photoelectron Spectroscopy: Principles and Applications (Springer, Berlin, 2003) 49. M. Reutzel, A. Li, B. Gumhalter, H. Petek, Nonlinear plasmonic photoelectron response of Ag(111). Phys. Rev. Lett. 123, 017404 (2019) 50. B. Krömker, M. Escher, D. Funnemann, D. Hartung, H. Engelhard, J. Kirschner, Development of a momentum microscope for time resolved band structure imaging. Rev. Sci. Instrum. 79, 053702 (2008) 51. K. Medjanik et al., Direct 3D mapping of the fermi surface and fermi velocity. Nat. Mat. 16, 615 (2017) 52. A. Li et al., Multidimensional multiphoton momentum microscopy of the anisotropic Ag(110) surface. Phys. Rev. B 105, 075105 (2022) 53. M. Keunecke et al., Electromagnetic dressing of the electron energy spectrum of Au(111) at high momenta. Phys. Rev. B 102, 161403 (2020) 54. J. Maklar et al., A quantitative comparison of time-of-flight momentum microscopes and hemispherical analyzers for time- and angle-resolved photoemission spectroscopy experiments. Rev. Sci. Ins. 91, 123112 (2020) 55. B. Schönhense et al., Multidimensional photoemission spectroscopy—the space-charge limit. New J. Phys. 20, 033004 (2018) 56. G. Schönhense et al., Suppression of the vacuum space-charge effect in fs-photoemission by a retarding electrostatic front lens. Rev. Sci. Instrum. 92, 053703 (2021) 57. E. Rotenberg, A. Bostwick, microARPES and nanoARPES at diffraction-limited light sources: opportunities and performance gains. J. Synch. Radiati. 21, 1048 (2014) 58. K. Giesen, F. Hage, F.J. Himpsel, H.J. Riess, W. Steinmann, Two-photon photoemission via image-potential states. Phys. Rev. Lett. 55, 300 (1985) 59. W. Steinmann, Magneto-optical Kerr effect and perpendicular magnetic anisotropy of evaporated and sputtered Co/Pt multilayer structures. Appl. Phys. A 49, 365 (1989) 60. H. Petek, S. Ogawa, Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals. Prog. Surf. Sci. 56, 239 (1997) 61. M. Wolf, Femtosecond dynamics of electronic excitations at metal surfaces. Surf. Sci. 377– 379, 343 (1997) 62. M. Bauer, A. Marienfeld, M. Aeschlimann, Hot electron lifetimes in metals probed by timeresolved two-photon photoemission. Prog. Surf. Sci. 90, 319 (2015) 63. U. Höfer, I.L. Shumay, C. Reuss, U. Thomann, W. Wallauer, T. Fauster, Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces. Science 277, 1480 (1997) 64. U. Bovensiepen, H. Petek, M. Wolf, Dynamics at Solid State Surfaces and Interfaces, Vol. 1: Current Developments (Wiley-VCH Verlag Gmbh & Co., Weinheim, 2010), Vol. 1: Current developments 65. U. Bovensiepen, H. Petek, M. Wolf, Dynamics at Solid State Surfaces and Interfaces, Vol. 2: Fundamentals (Wiley-VCH Verlag Gmbh & Co., Weinheim, 2012), Vol. 2: Fundamentals 66. P.M. Echenique, R. Berndt, E.V. Chulkov, T. Fauster, A. Goldmann, U. Höfer, Decay of electronic excitations at metal surfaces. Surf. Sci. Rep. 52, 219 (2004) 67. T. Fauster, M. Weinelt, U. Höfer, Quasi-elastic scattering of electrons in image-potential states. Prog. Surf. Sci. 82, 224 (2007) 68. M. Weinelt, Time-resolved two-photon photoemission from metal surfaces. J. Phys. Condens. Matter 14, R1099 (2002) 69. D. Schmitt et al., Formation of moiré interlayer excitons in space and time. Nature 608, 499 (2022) 70. R. Wallauer et al., Tracing orbital images on ultrafast time scales. Science 371, 1056 (2021)

80

M. Reutzel et al.

71. J. Madéo et al., Directly visualizing the momentum-forbidden dark excitons and their dynamics in atomically thin semiconductors. Science 370, 1199 (2020) 72. J.C. Johannsen et al., Direct view of hot carrier dynamics in graphene. Phys. Rev. Lett. 111, 027403 (2013) 73. I. Gierz et al., Snapshots of non-equilibrium Dirac carrier distributions in graphene. Nat. Mater. 12, 1119 (2013) 74. J. Lehmann, M. Merschdorf, W. Pfeiffer, A. Thon, S. Voll, G. Gerber, Surface plasmon dynamics in silver nanoparticles studied by femtosecond time-resolved photoemission. Phys. Rev. Lett. 85, 2921 (2000) 75. H. Petek, A.P. Heberle, W. Nessler, H. Nagano, S. Kubota, S. Matsunami, N. Moriya, S. Ogawa, Optical phase control of coherent electron dynamics in metals. Phys. Rev. Lett. 79, 4649 (1997) 76. X. Cui, C. Wang, A. Argondizzo, S. Garrett-Roe, B. Gumhalter, H. Petek, Transient excitons at metal surfaces. Nat. Phys. 10, 505 (2014) 77. H. Petek, H. Nagano, S. Ogawa, Hole decoherence of d bands in copper. Phys. Rev. Lett. 83, 832 (1999) 78. J. Güdde, M. Rohleder, T. Meier, S.W. Koch, U. Höfer, Time-resolved investigation of coherently controlled electric currents at a metal surface. Science 318, 1287 (2007) 79. T. Eul, E. Prinz, M. Hartelt, B. Frisch, M. Aeschlimann, B. Stadtmüller, Coherent response of the electronic system driven by non-interfering laser pulses. Nat. Commun. 13, 3324 (2022) 80. M. Bauer, S. Pawlik, M. Aeschlimann, Decay dynamics of photoexcited alkali chemisorbates: real-time investigations in the femtosecond regime. Phys. Rev. B 60, 5016 (1999) 81. A. Kubo, N. Pontius, H. Petek, Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface. Nano Lett. 7, 470 (2007) 82. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F.J. Garcia de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, F. Steeb, Adaptive subwavelength control of nano-optical fields. Nature 446, 301 (2007) 83. S. Ogawa, H. Nagano, H. Petek, A.P. Heberle, Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission. Phys. Rev. Lett. 78, 1339 (1997) 84. M. Merschdorf, C. Kennerknecht, W. Pfeiffer, Collective and single-particle dynamics in time-resolved two-photon photoemission. Phys. Rev. B 70, 193401 (2004) 85. H. Petek, M.J. Weida, H. Nagano, S. Ogawa, Real-time observation of adsorbate atom motion above a metal surface. Science 288, 1402 (2000) 86. S. Ogawa, H. Nagano, H. Petek, Phase and energy relaxation in an antibonding surface state: Cs/Cu(111). Phys. Rev. Lett. 82, 1931 (1999) 87. M. Aeschlimann et al., Observation of optical coherence in a disordered metal-molecule interface by coherent optical two-dimensional photoelectron spectroscopy. Phys. Rev. B 105, 205415 (2022) 88. M. Dabrowski, Y. Dai, H. Petek, Ultrafast photoemission electron microscopy: imaging plasmons in space and time. Chem. Rev. 120, 6247 (2020) 89. Y. Dai, Z. Zhou, A. Ghosh, R.S.K. Mong, A. Kubo, C.-B. Huang, H. Petek, Plasmonic topological quasiparticle on the nanometre and femtosecond scales. Nature 588, 616 (2020) 90. Y. Dai, Z. Zhou, A. Ghosh, S. Yang, C.-B. Huang, H. Petek, Ultrafast nanofemto photoemission electron microscopy of vectorial plasmonic fields. MRS Bull. 46, 738 (2021) 91. E. Prinz, M. Hartelt, G. Spektor, M. Orenstein, M. Aeschlimann, Orbital angular momentum in nanoplasmonic vortices. ACS Photon. 10, 340 (2023) 92. P. Dreher, D. Janoschka, B. Frank, H. Giessen, F.-J. Meyer zu Heringdorf, Focused surface plasmon polaritons coherently couple to electronic states in above-threshold electron emission. Commun. Phys. 6, 15 (2023) 93. G. Spektor et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices. Science 355, 1187 (2017) 94. G. Spektor, D. Kilbane, A.K. Mahro, M. Hartelt, E. Prinz, M. Aeschlimann, M. Orenstein, Mixing the light spin with plasmon orbit by nonlinear light-matter interaction in gold. Phys. Rev. X 9, 021031 (2019)

3 Probing Nonlinear Light–Matter Interaction in Momentum Space …

81

95. B. Huber et al., Space- and time-resolved UV-to-NIR surface spectroscopy and 2D nanoscopy at 1 MHz repetition rate. Rev. Sci. Ins. 90, 113103 (2019) 96. J. Réhault, M. Maiuri, A. Oriana, G. Cerullo, Two-dimensional electronic spectroscopy with birefringent wedges. Rev. Sci. Ins. 85, 123107 (2014) 97. C.M. Heyl, J. Güdde, A. L’Huillier, U. Höfer, High-order harmonic generation with μJ laser pulses at high repetition rates. J. Phys. B 45, 074020 (2012) 98. C.-T. Chiang, A. Blättermann, M. Huth, J. Kirschner, W. Widdra, High-order harmonic generation at 4 MHz as a light source for time-of-flight photoemission spectroscopy. Appl. Phys. Lett. 101, 071116 (2012) 99. S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, A. Tünnermann, High photon flux table-top coherent extreme-ultraviolet source. Nat. Photon. 8, 779 (2014) 100. M. Puppin et al., Time- and angle-resolved photoemission spectroscopy of solids in the extreme ultraviolet at 500 kHz repetition rate. Rev. Sci. Instrum. 90, 023104 (2019) 101. M. Keunecke et al., Time-resolved momentum microscopy with a 1 MHz high-harmonic extreme ultraviolet beamline. Rev. Sci. Instrum. 91, 063905 (2020) 102. A. Kunin et al., Momentum-resolved exciton coupling and valley polarization dynamics in monolayer WS2 . Phys. Rev. Lett. 130, 046202 (2023) 103. X. Li, M.A.R. Reber, C. Corder, Y. Chen, P. Zhao, T.K. Allison, High-power ultrafast Yb:fiber laser frequency combs using commercially available components and basic fiber tools. Rev. Sci. Instrum. 87, 093114 (2016) 104. A. Li, N.A. James, T. Wang, Z. Wang, H. Petek, M. Reutzel, Towards full surface Brillouin zone mapping by coherent multi-photon photoemission. New J. Phys. 22, 073035 (2020) 105. A. Gerlach, G. Meister, R. Matzdorf, A. Goldmann, High-resolution photoemission study of ¯ surface state on Ag(110). Surf. Sci. 443, 221 (1999) the Y 106. S.T. Cundiff, S. Mukamel, Optical multidimensional coherent spectroscopy. Phys. Today 66, 44 (2013) 107. S. Mukamel, Multidimensional femtosecond correlation spectroscopies of electronic and vibrational excitations. Annu. Rev. Phys. Chem. 51, 691 (2000) 108. T. Klamroth, P. Saalfrank, U. Höfer, Open-system density-matrix approach to image-potential dynamics of electrons at Cu(100): energy- and time-resolved two-photon photoemission spectra. Phys. Rev. B 64, 035420 (2001) 109. M.J. Weida, S. Ogawa, H. Nagano, H. Petek, Ultrafast interferometric pump–probe correlation measurements in systems with broadened bands or continua. J. Opt. Soc. Am. B 17, 1443 (2000) 110. S. Dong et al., Direct measurement of key exciton properties: energy, dynamics, and spatial distribution of the wave function. Nat. Sci. 1, e10010 (2021) 111. N. Pontius, V. Sametoglu, H. Petek, Simulation of two-photon photoemission from the bulk sp-bands of Ag(111). Phys. Rev. B. 72, 115105 (2005). http://link.aps.org/abstract/PRB/v72/ e115105 112. J. Mauritsson et al., Attosecond electron spectroscopy using a novel interferometric pumpprobe technique. Phys. Rev. Lett. 105, 053001 (2010) 113. S. Tan, Y. Dai, S. Zhang, L. Liu, J. Zhao, H. Petek, Coherent electron transfer at the Ag/ graphite heterojunction interface. Phys. Rev. Lett. 120, 126801 (2018) 114. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Springer US, New York, 2000) 115. A. Anderson, K.S. Deryckx, X.G. Xu, G. Steinmeyer, M.B. Raschke, Few-femtosecond plasmon dephasing of a single metallic nanostructure from optical response function reconstruction by interferometric frequency resolved optical gating. Nano Lett. 10, 2519 (2010) 116. M. Mascheck, S. Schmidt, M. Silies, T. Yatsui, K. Kitamura, M. Ohtsu, D. Leipold, E. Runge, C. Lienau, Observing the localization of light in space and time by ultrafast second-harmonic microscopy. Nat. Photon. 6, 293 (2012) 117. H. Strauch, Georg-August-Universität Göttingen (2022)

82

M. Reutzel et al.

118. S. Tan, A. Argondizzo, J. Ren, L. Liu, J. Zhao, H. Petek, Plasmonic coupling at a metal/ semiconductor interface. Nat. Photon. 11, 806 (2017) 119. D. Podbiel, P. Kahl, F.-J. Meyer zu Heringdorf, Analysis of the contrast in normal-incidence surface plasmon photoemission microscopy in a pump–probe experiment with adjustable polarization. Appl. Phys. B 122, 90 (2016) 120. A. Ghosh, S. Yang, Y. Dai, Z. Zhou, T. Wang, C.-B. Huang, H. Petek, A topological lattice of plasmonic merons. Appl. Phys. Rev. 8, 041413 (2021) 121. Y. Dai, Z. Zhou, A. Ghosh, K. Kapoor, M. D˛abrowski, A. Kubo, C.-B. Huang, H. Petek, Ultrafast microscopy of a twisted plasmonic spin skyrmion. Appl. Phys. Rev. 9, 011420 (2022) 122. T. J. Davis, D. Janoschka, P. Dreher, B. Frank, F.-J. Meyer zu Heringdorf, H. Giessen, Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution. Science 368, eaba6415 (2020) 123. R.R. Freeman, P.H. Bucksbaum, H. Milchberg, S. Darack, D. Schumacher, M.E. Geusic, Above-threshold ionization with subpicosecond laser pulses. Phys. Rev. Lett. 59, 1092 (1987) 124. W. Becker, F. Grasbon, R. Kopold, D.B. Miloševi´c, G.G. Paulus, H. Walther, in Advances in Atomic, Molecular, and Optical Physics. ed. by B. Bederson, H. Walther (Academic Press, New York, 2002), p.35 125. J.H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979 (1965) 126. T. Oka, S. Kitamura, Floquet engineering of quantum materials. Ann. Rev. Condens. Matter Phys. 10, 387 (2019) 127. O.D. Mücke, T. Tritschler, M. Wegener, U. Morgner, F.X. Kärtner, Signatures of carrier-wave Rabi flopping in GaAs. Phys. Rev. Lett. 87, 057401 (2001) 128. Q.T. Vu, H. Haug, O.A. Mücke, T. Tritschler, M. Wegener, G. Khitrova, H.M. Gibbs, Lightinduced gaps in semiconductor band-to-band transitions. Phys. Rev. Lett. 92, 217403 (2004) 129. M. Fleischhauer, A. Imamoglu, J.P. Marangos, Electromagnetically induced transparency: optics in coherent media. Rev. Mod. Phys. 77, 633 (2005) 130. S.H. Autler, C.H. Townes, Stark effect in rapidly varying fields. Phys. Rev. 100, 703 (1955) 131. W.H. Knox, Dispersion measurements for femtosecond-pulse generation and applications. Appl. Phys. B 58, 225 (1994) 132. V.M. Silkin, P. Lazi´c, N. Došli´c, H. Petek, B. Gumhalter, Ultrafast electronic response of Ag(111) and Cu(111) surfaces: from early excitonic transients to saturated image potential. Phys. Rev. B 92, 155405 (2015) 133. F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, S. Hüfner, Direct measurements of the L-gap surface states on the (111) face of noble metals by photoelectron spectroscopy. Phys. Rev. B 63, 115415 (2001) 134. B. Gumhalter, D. Novko, H. Petek, Electron emission from plasmonically induced Floquet bands at metal surfaces. Phys. Rev. B 106, 035422 (2022) 135. Y. Dai, A. Ghosh, S. Yang, Z. Zhou, C.-B. Huang, H. Petek, Poincaré engineering of surface plasmon polaritons. Nat. Rev. Phys. 4, 562 (2022)

Chapter 4

Nonlinear Soft X-Ray Spectroscopy Craig P. Schwartz and Walter S. Drisdell

Abstract The history and current state of the field of soft X-ray nonlinear optics is discussed. The development of soft X-ray nonlinear optics has been largely based upon the earlier work done with visible and infrared spectroscopy. While nonlinear optics in those frequency regimes are relatively mature today, soft X-ray nonlinear optics is a young field. This delayed development was due to the lack of intense ultrafast X-ray light sources, but with the recent rise of X-ray free electron lasers and other intense X-ray sources, there has been an explosion in the field. The specific soft X-ray nonlinear optics techniques that have been employed so far are discussed in detail and the underlying physical principles are discussed. These include non-sequential multiple-photon absorption, stimulated emission to drive lasing, and various forms of stimulated Raman scattering. These techniques have enabled new measurements that were previously infeasible, providing access to dipole forbidden states, driving anisotropic X-ray emission, and allowing for the understanding of excited state surfaces in an unprecedented way. Additionally, four wave mixing has been demonstrated and a dedicated four wave beamline built for the study of complex phenomena such as diffusional dynamics. By performing four wave mixing in the soft X-ray region, one accesses particularly large momentum transfer and thus spatially small volumes. Second harmonic generation has also been demonstrated in the soft X-ray region, allowing for the study of symmetry breaking and surfaces in materials ranging from batteries to graphite. This technique has recently been extended to a variation of sum frequency generation, where the soft X-ray beam is mixed with a visible laser pulse and has been shown to be a sensitive probe of charge localization. Finally, the available computational methods which can be used to model and understand these phenomena are discussed.

C. P. Schwartz (B) Nevada Extreme Laboratory, University of Nevada, Las Vegas, Las Vegas, NV, USA e-mail: [email protected] W. S. Drisdell (B) Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_4

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4.1 Nonlinear Spectroscopy—Development with Visible Light Over the course of an average day, a person may see all sorts of wondrous things, but they will all be in the regime of linear optics. That is to say, almost all everyday visual occurrences happen in a regime where the electric field associated with light can be considered weak. This means that if one makes the light brighter, a material illuminated by that light will appear brighter, but otherwise its appearance will not fundamentally change [1]. This was the state of the observable world for most of human existence, and only the relatively recent development of bright light sources (notably lasers) has allowed for the development of nonlinear optics [2, 3]. While it may seem like magic, strong fields enable phenomena to be seen which otherwise are not possible as shown in Fig. 4.1. Specifically, the advent of a strong electric field can lead to all sorts of phenomena which otherwise would not be observable. Strong laser fields lead to a strong polarization density in a material which then reacts nonlinearly with the electric field. One of the first applications of lasers was to demonstrate second harmonic generation (SHG) on a crystal, a phenomenon which combines two photons of the same energy to generate a single photon of twice the energy [3]. While the energy is linear (two photons combine to give identical energy as from the single generated photon), SHG is a nonlinear effect as the signal scales quadratically with respect to input power. Unfortunately, even with the advent of lasers, the signal was so weak compared to the much stronger input energy (fundamental) that the small signal shown in the first manuscript figure was believed by the journal editors to be a speck of dust and was removed. The electric fields required are typically quite large (> 108 V/m) but can be readily achieved with modern lasers [4, 5]. Given the small signal, it is reasonable to ask why there has been so much work on nonlinear optics. This is because, in a variety of cases, the only way to determine certain properties of materials is by using nonlinear optics. Many types of information cannot be obtained by other methods, such as surface specific phenomena, access to diffusion rates for phenomena such as heat transfer, self-focusing, generation of different frequencies of light, and spectroscopy with different selection rules; these have provided insight in fields ranging from biology to electrical engineering. Most of these phenomena and the related information simply cannot be obtained in any other manner besides using nonlinear optics. The exploitation of nonlinear optics is therefore essential in many fields. For instance, to study buried interfaces, the only widely accessible measurements use nonlinear optics. An optical event is when as an incoming light wave induces electric polarization in a material that generates the outgoing light wave. Nonlinear optics are defined by being sensitive to the electric field. In general, the instantaneous polarization can be represented as ( ) P(t) = ε0 χ 1 E 1 (t) + χ 2 E 2 (t) + χ 3 E 3 (t) + · · · ,

(4.1)

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Fig. 4.1 Second harmonic generation (SHG), can seem like magic, combining two photons to generate a single photon at twice the energy. As the signal (2ω) varies as the square of the input field (ω), a strong field is required. This particular technique is often used to probe surfaces and symmetry breaking

where P(t) is the polarization, ε0 is the dielectric of free space, χ n is the nth order susceptibility in terms of the electric field E(t). χ n is usually a tensor of rank n + 1 that is sensitive to the polarization and symmetry of the system. A second order process such as second harmonic generation would have a χ 2 consisting of a rank 3 tensor, and the higher order terms are not necessary in that case. There are many varieties of nonlinear optical effects that have been observed experimentally. Some of them are listed in Table 4.1 with characteristic information. It is worth mentioning that some nonlinear optical phenomena, such as multiple ionization or saturable absorption, have also been observed with a linear response of the polarization with a linear susceptibility, χ 1 . Such optical events themselves are “nonlinear” but the

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equation defines them as “linear”. These examples are given in Table 4.2 and the description is given in Sect. 4.3. The case of wave mixing (where two or more photons are combined to generate a single photon) has shown use in both the generation of tailored light and the study of materials, as it enables generation of light at different wavelengths [25, 26]. A whole host of related techniques can be used to mix multiple photons. The mixing of two degenerate photons is SHG, as mentioned above, and mixing three degenerate photons to generate a 4th is called third harmonic generation (THG). Mixing two photons of different energy, where the resultant photon is the sum of the energies of the input photons, is called sum frequency generation (SFG), and mixing two photons where the resultant photon energy is the difference between the input photon energies is called difference frequency generation (DFG). The mixing of an even number of photons (generally two) has the special property of requiring symmetry breaking in a material for signal to occur, which requires either a noncentrosymmetric material, or an interface. This is not true with an odd number of photons which is bulk sensitive, as will be the case in four wave mixing, which provides similar information to third harmonic generation—both probe χ 3 , described below—but have different phase matching conditions. Combining larger numbers of photons is just generally termed high harmonic generation (HHG) [5, 27]. These techniques have shown a host of applications in the study of materials and in preparing tailored laser light to probe other materials through the generation of specific frequencies. This idea of generating different frequencies to probe matter is frequently used with other wave mixing techniques such as optical parametric amplification (OPA), optical parametric oscillation (OPO), and optical parametric generation (OPG) [28–30]. Indeed, much of modern laser spectroscopy would not be possible without these techniques. The details of performing these techniques have been reviewed extensively before and the reader is encouraged to seek out those manuscripts if more detailed knowledge is necessary [4, 26]. Nonlinear optics, however, is not limited to wave mixing and includes a host of other processes that are only possible as the field becomes strong enough to lead to nonlinear effects. These include but are not limited to: transient grating (i.e., 4-wave mixing), where diffusional processes can be mapped spectroscopically; Raman amplification which enhances the Raman signal under large fields; the optical Kerr effect which leads to a change in the refractive index with field; and multiphoton absorption (i.e., two-photon absorption), wherein multiple (two) photons are absorbed simultaneously, among many other techniques [2, 31–34]. Many of these processes provide ways of probing matter in a manner for which there are either no available alternatives or no options with all the benefits of these methods. It is important to note that just because an effect can require a strong field to occur, does not mean the process is nonlinear—the signal must change nonlinearly with respect to the electric field. For instance, pump-probe spectroscopy where an initial pump prepares an excited state which is then probed by a second pulse is not necessarily a nonlinear phenomenon even though typically somewhat large fields are required to drive a noticeable fraction of the system into an excited state [27].

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Table 4.1 A list of nonlinear optical phenomena with nonlinear susceptibility, optical properties, and representative references Optical phenomena

Nonlinear susceptibility

Notes on the phenomena

Selected references for soft X-ray experiment and section in this chapter

Second harmonic generation (SHG) 2ω1 → ω2

χ (2)

Sensitive to symmetry [6, 7] breaking as an even Section 4.4.5 order process. Generates a photon at twice the input photon energy

Third harmonic generation (THG) 3ω1 → ω3

χ (3)

Generates a photon with three times the input energy. Has been used in the optical regime as a form of microscopy

Sum frequency generation (SFG) ω1 + ω2 → ω3

χ (2)

Similar selection rules A variation with 3 input to SHG, but with photons is shown in [9] input photons of Section 4.4.5 different energy

Difference frequency generation (DFG) ω1 − ω2 → ω3

χ (2)

Similar selection rules Has been discussed to SHG and SFG theoretically in the hard X-ray regime, but not yet demonstrated with two photons in the soft X-ray regime [10]

High harmonic generation (HHG)

Varies

Used in the optical regime to generate X-ray pulses

[11, 12] Section 4.2

Optical parametric amplification (OPA)

χ (2)

Used in the optical regime to tune laser frequencies

Similar technique used in the hard X-ray regime as X-ray parametric down-conversion [13, 14]

Optical parametric oscillation (OPO)

χ (2)

Can be thought of as a Not demonstrated in the X-ray way to improve the regime focus and power of light generated with an OPA

4-wave mixing

χ (3)

A series of related techniques where 3 waves interact to generate a 4th wave. Amongst the most common is transient grating

Not yet demonstrated in the X-ray regime. See [8] for more discussion from the optical regime

[15, 16] Section 4.4.4

(continued)

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Table 4.1 (continued) Optical phenomena

Nonlinear susceptibility

Notes on the phenomena

Selected references for soft X-ray experiment and section in this chapter

Raman amplification

Can vary

Method of amplifying Not demonstrated in the X-ray a Raman signal regime

Optical Kerr effect

Can vary

A variety of related Not demonstrated in the X-ray techniques where the regime refractive index of a material varies with the input electric field

Two-photon absorption (TPA)

χ (3)

Two (or in certain cases more) photons are absorbed, allowing the probing of different selection rules

[17, 18] Section 4.4.1

Stimulated emission

N/A

Drives a population inversion which then undergoes a stimulated emission process, like lasing

[19, 20]

Superfluorescence

N/A

Ensemble of emitters coupling to emit an intense burst of light

[21, 22] Section 4.4.2

Stimulated Raman

χ (3)

A process where a species is driven into a higher excited state before being driven back down into a lower state and emitting a photon

[23, 24]

Table 4.2 A list of nonlinear optical phenomena with linear susceptibility, optical properties and representative references Technique name

Nonlinear susceptibility

Notes on the technique

Selected references and section in this chapter

Multiple ionization

χ (1)

Multiple electrons are removed from a bound system simultaneously or nearly simultaneously

[17] Section 4.3

Saturable absorption

χ (1)

At high fluxes, absorption [19, 20] saturates Section 4.3

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To use nonlinear optics experimentally, a variety of complications must be carefully understood and potentially optimized to maximize signal. One example is phase matching. The phase of light must be considered because the index of refraction of a material generally changes as a function of wavelength, meaning the angle of the crystal being studied where the incoming and outgoing waves interfere will have a large effect on the signal. For a second order process such as SFG, at position x in a crystal system, the second order polarization is P 2 (x, t) ∝ E 1n1 E 2n2 ei[(k1 +k2 )x−ω3 t] + (complex conjugate),

(4.2)

where E 1 and E 2 are two components of the electric field at frequencies ωa and indices na , with the wave vector ka = na (ωa ) ωa /c where c is the speed of light, and na (ωa ) is the index of refraction at frequency ωa . For SFG, definitionally ω1 + ω2 = ω3 . Therefore, phase matching, which requires constructive interference, can only be true if the output wave vector (k→3 ) is equal to the input wave vectors (k→1 , k→2 ). k→3 = k→1 + k→2 .

(4.3)

This statement can be generalized for a system of n − 1 input vectors such that k→n = k→1 + · · · + k→n−1 .

(4.4)

This is known as the phase matching conditions, albeit in certain cases the vectors may subtract instead of adding, as is the case for difference frequency generation. For certain systems, for instance those with geometric constraints, such matching is not always possible, making measurements challenging [5]. Perhaps the largest challenge in general, is that the demands on the light sources are quite high, requiring an intense light source of monochromatic and coherent light. Even today, these sources simply do not exist for every frequency range, and even when they do exist they can be quite expensive to use and maintain. It is almost a certainty that the quality and availability sources will continue to improve in the future, as they have for the past 50 years. The key to development of nonlinear optics has been the development of powerful light sources [5]. The development of bright X-ray light sources has lagged significantly behind those in the visible, leading to a similar lag in the development of X-ray nonlinear optics. The current state of the art is significantly less advanced than that in the visible. In addition, X-rays are generally harder to work with than visible light, as they are harder to focus and reflect, and require specialized and expensive optics for these purposes [35]. Only the recent rise of X-ray free electron lasers, with their high peak photon flux, has enabled significant work and development in the field of X-ray nonlinear optics [36]. These light sources are discussed in more detail in the next section. Visible nonlinear optics and X-ray nonlinear optics in the so-called hard X-ray regime (photon energies greater than 3000 eV) are fundamentally different. In the visible the spectroscopies are dipole selected, but in the hard X-ray regime the dipole approximation breaks down leading to different effects, such as SHG being bulk

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sensitive in centrosymmetric media, and more accessibility of the nonlinear Compton effect [37–39]. It appears, however, that in the soft X-ray regime (photon energy less than 1000 eV) nonlinear optics can be described well by the dipole approximation and the previous equations can be readily applied, and as such visible nonlinear optics can be used to guide the development of soft X-ray nonlinear optics. The region below approximately 150 eV is sometimes referred to as extreme ultraviolet (XUV), although the cutoff between XUV and soft X-ray is not well defined. We wish to note the description of the development of visible nonlinear optics and its detailed uses are covered only in a cursory fashion here. For a more thorough treatment, the reader is encouraged to read the several books that have been published on the topic [5, 27].

4.2 Ultrafast X-Ray Light Sources In almost all cases, to observe nonlinear effects one needs a strong field. (The case of X-ray parametric down-conversion is skipped here, but it is a hard X-ray phenomenon and not relevant at the energies discussed in this chapter [13, 14].) The original X-ray sources, dating back to their invention by Roentgen in 1895, were based on X-ray tubes. While these were improved over the course of decades by developing rotating anodes which allowed for increased load, the continuous nature of the X-ray emission and the relatively low flux precludes their use in generating the large fields necessary to perform nonlinear optics. Additionally, these sources are not generally and readily tunable in terms of the strongest energy available [40]. The next step in the development of X-ray sources was the development of synchrotron light sources. These machines were discovered by Veksler in 1944 [41], built by McMillan in 1945 [42] and the observation of synchrotron radiation was made in 1947 by General Electric [43]. These machines move electrons at relativistic speeds around a ring, and as the electrons are accelerated, they emit radiation, a phenomenon which was not initially expected. Following this discovery, synchrotron light sources have undergone continuous improvements, from their early use being run parasitically from early synchrotron particle accelerators, to their current designs which approach the diffraction limit. Synchrotron light sources may now be close to their ultimate performance limits [44], having become the workhorses of X-ray science. They deliver stable X-ray light with a high average flux, but their design is not optimized for high field science. This is because while the machines generally run at a high enough pulse rate to approach a continuous source (MHz), their peak power (or equivalently their power per shot) is quite low, with pulse lengths typically in the hundreds of picoseconds and the number of monochromatic photons per pulse often approaching 106 [45]. While this is many orders of magnitude higher than an X-ray tube (depending on how this is counted, 15 orders of magnitude may be a conservative estimate for the increase in monochromatic flux), the field strength is still much too low to perform most nonlinear optical experiments.

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The advent of X-ray free electron lasers (FELs), however, brought even more intense X-ray sources with the potential to drive nonlinear optical phenomena. These lasers use unbound (‘free’) electrons as the lasing medium [46]. Because the Xray radiation they generate does not reflect well, a reflection-based lasing cavity has not yet been built and the radiation must be generated on a single pass of the electron beam, leading these machines to be large (European XFEL is 3.4 km long), and therefore quite expensive. Despite the cost of these machines, their tremendous power (up to mJ per pulse), short pulse length (as short as hundreds of attoseconds), and relatively high repetition rate (now approaching MHz), makes these machines ideal for performing nonlinear optics. Indeed, the vast majority of nonlinear Xray experiments that have ever been performed have been performed with these machines [47–49]. They are not without their drawbacks, however, as the typical lasing method, self-amplified spontaneous emission (SASE), in essence amplifies noise and leads to a broad and non-Gaussian spectral and temporal profile [50]. There has been work to improve this by performing self-seeding, which has shown improved spectral quality and has seen rapid improvements recently [48, 49]. Perhaps the optimal solution would be to use external seeding, where an initial laser pulse is amplified, as is done at FERMI in Italy [51]. While this provides a very coherent and monochromatic source, it does so at the cost of laser power, which is lower than SASE FELs. Additionally, as amplifying an external laser source amplifies the noise, using this technique to develop pulses into the hard X-ray regime and maintaining decent quality pulses has proved challenging so far and will require the developments of several new technologies [52]. The details of how X-rays from X-ray tubes, synchrotrons and FELs are generated has been handled elsewhere [44]. The overall progress can be seen in a graph plotting photon energy versus peak brilliance—which is roughly proportional to monochromatic photon flux [45]. As shown in Fig. 4.3, conventional lasers can be quite intense in terms of photon power, but struggle to produce high energy photons. This is contrasted with synchrotrons which are quite good at generating high energy photons but struggle to produce high peak flux. With synchrotrons near their performance limits, free electron lasers are opening entirely new fields of study by enabling high flux and high photon energy. An example of a modern X-ray free electron laser is shown in Fig. 4.2, taken at the soft X-ray FEL at SACLA in Japan. Finally, it is worth acknowledging recent advances in lab-based X-ray sources. Beyond X-ray tubes, X-rays can be generated in a variety of ways, such as by impacting a metal with an intense laser pulse, but this provides inadequate photon flux for nonlinear experiments [53]. Perhaps more likely to be of use for X-ray nonlinear optics in the future is HHG-based sources, which upconvert a laser pulse into the soft X-ray region [11, 12]. While at present these sources provide inadequate peak power for nonlinear optics, they are very attractive in that they maintain excellent pulse quality and are significantly smaller financial investments than FELs. Current work has largely focused on increasing the average power of the machine by increasing the repetition rate, increasing the photon energy to hundreds of eV and decreasing the pulse length of these sources, but perhaps if more work is performed to optimize

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Fig. 4.2 A photo of soft XFEL at SACLA. By permission of RIKEN

peak flux performing nonlinear optical experiments at soft X-ray energies will be possible with HHG sources [54].

4.3 Family of Soft X-Ray Nonlinear Spectroscopy Well-Established Linear Spectroscopies Before describing nonlinear X-ray spectroscopies already developed, it is perhaps beneficial to briefly describe the uses of linear X-ray spectroscopy. X-ray spectroscopy uses a high energy photon to excite core or semi-core electrons in an atom, molecule, or material system. These core or semi-core electrons are generally well separated in energy for each element, allowing X-ray spectroscopy to probe different elements in a system individually [55]. This contrasts with visible spectroscopy, which typically probes a delocalized state which can extend over an entire system [56]. Also, as the excitation is high energy, the spectroscopy is often sensitive to neighboring molecules as the excited state is often quite diffuse spatially. It therefore can be a very sensitive probe; for instance the X-ray spectrum of water is significantly more feature rich than the infrared (IR) spectrum of water [57]. X-ray spectroscopy generally takes one of several forms. In X-ray absorption spectroscopy (XAS), the input energy is scanned and the amount of photons absorbed by the sample is measured [55]. The absorption transition can be described (as was detailed in Chap. 2) by Fermi’s golden rule as

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Fig. 4.3 Plot of photon energy versus peak brilliance, roughly proportional to peak photon flux. Conventional lasers are high flux but low photon energy whereas synchrotrons are high energy but low flux. The X-ray free electron laser is uniquely high flux and high energy. X-ray tubes can produce high energy X-rays, but their flux is so low it is off the scale of the plot. HHG sources are at present above synchrotron peak brightness (up to ~ 1025 photons/s/mrad2 /mm2 /0.1%/-BW at 100 eV). The pace of their improvement is significantly faster than that of synchrotrons, however. Taken from Boutet and Yabashi [45]. Reprinted with permission from Boutet and Yabashi [45]. Copyright by Springer Nature

⎡i→ f =

⟩|2 2π ||⟨ f |H ' |i | ρ(E f ), h

(4.5)

where the transition probability from the initial state (i) to the final state ( f ) is given by ⎡i→ f . H ' is the Hamiltonian describing the transition from initial to final, ρ(E f ) is the density of states, and h is Planck’s constant divided by 2π. In the case of XAS, the transition is from a relaxed state (i) to a core excited state ( f ). As this is a single photon process, there must be a single unit of angular momentum difference between i and f , often meaning a transition from an s to a p-type orbital. As f is unoccupied, this provides a sensitive probe of the unoccupied projected density of states (for instance K-shell excitations are from the 1s shell into unoccupied p-type states). In X-ray emission spectroscopy (XES), the input energy is well above the core or semi-core electron binding energy, and the emitted photons are measured as a function of energy. Many of these are elastic, emitted when the excited electron relaxes directly back to its original core or semi-core state. But there is also inelastic

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emission, from electrons transitioning from valence states into the empty core hole left by the X-ray excitation. This provides a measure of the projected density of states of the occupied density of states [58, 59]. In terms of Fermi’s golden rule, i is a core hole state and f is the relaxed state it transitions into. As the process is single photon, the same selection rules a XAS apply. X-ray emission spectra can also be measured when the input X-rays are resonant with an excitation to a specific final state, known as resonant inelastic X-ray scattering (RIXS). This in essence combines X-ray absorption and X-ray emission spectroscopy [60]. It is also possible to examine the electrons emitted from a system. X-ray photoelectron spectroscopy probes the energies of the initial occupied electronic states by measuring the energy of the photoelectrons excited from these states by an incident X-ray photon. Introducing incident electrons of constant energy and measuring the emitted photon energy is called inverse photoemission and is sensitive to the unoccupied states [61]. All of these techniques would be classified as linear, but all nonlinear soft X-ray spectroscopy are built upon linear techniques in some way. Linear X-ray spectroscopy is described in detail in Chap. 2. High Field Linear Soft X-Ray Spectroscopies Beyond measurements that can be made at synchrotrons, there are other linear effects that can be observed at FELs. The most common of these are discussed briefly here. Among the first papers to be produced from an X-ray free electron laser (XFEL) was a demonstration of the multiple ionization of neon [62]. In terms of Fermi’s golden rule, this means it goes through a series of excitations from the ground state, where after the first excitation the ground state has become some ionized state. It was shown that the highly intense X-ray pulses would drive the formation of core holes by removing the inner core electrons through direct excitation; the valence electrons would then fall into the core within the duration of the X-ray pulse, which then removed these formerly valence electrons as well. This meant the way to maximize the ionization—as was demonstrated for both an atom (Ne) and a molecule (N2 )— was to increase the length of the laser pulse, not the field strength [62, 63]. So while this is an interesting high field phenomenon, it is not an example of nonlinear optics. Other related high field phenomena in the soft X-ray (and demonstrated as well in the hard X-ray regime on Fe) include the phenomena of saturable absorption (SA) and X-ray induced transparency [64, 65]. In SA, the amount of absorption is so high that the final states are filled and absorption is saturated, manifesting as a decreasing amount of the rate of absorption with increasing laser power. When the ground state is fully depopulated by a phenomenon such as SA, this will give rise to the related phenomenon of X-ray induced transparency, wherein X-rays which would normally be absorbed by the material are instead fully transmitted, because all the core electrons have been depopulated. This may have use in futuristic advanced materials where X-ray transparency can be turned on and off. This has been demonstrated at the Ledge of solid Al [66], and the N-edge of tin [67]. While both these phenomenon are dependent on the pulse power per area, the effect is not nonlinear with respect to the electric field, so here we will not consider it a nonlinear optical effect, although such a term is used elsewhere.

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4.4 Nonlinear Soft X-Ray Optics and Spectroscopies This section will deal primarily with various soft X-ray nonlinear optical techniques. These include multiphoton absorption, stimulated phenomena, stimulated Raman, four wave mixing, and second harmonic generation. Some of the key results are summarized in Table 4.1, listing nonlinear susceptibility and a few representative papers.

4.4.1 Multiphoton Absorption One of the most used nonlinear optical techniques in the visible is two-photon absorption (TPA) [68]. With this technique, rather than a single photon being absorbed, followed by another photon, two photons (generally of the same energy for practical reasons) are absorbed simultaneously. This has shown use in the visible as a method of photolithography for generating narrow etched lines, as only the most intense part of the laser pulse will have the requisite intensity for two-photon absorption, ultimately leading to higher resolution linewidths [69, 70]. TPA also has different selection rules, and can therefore access states that are not normally allowed [71]. Fermi’s golden rule to the first order (as shown previously in Eq. (4.5)) is inadequate for this two-photon process. This is because the simultaneous absorption of two photons allows for transitions to states that are dipole forbidden with a single photon, or that require a change of two units of angular momentum. In the X-ray region, TPA forbids s → p transitions but allows (for instance) s → d transitions, or s → s transitions (allowed angular momentum changes for two photons are 0, ± 2) generating an entirely different spectrum than the low field single photon absorption spectrum [72]. This allows for matter to be characterized in a way that is not possible at lower fields; these otherwise inaccessible states can have interesting properties such as different symmetry and different lifetimes. TPA also has a different absorption rate compared to linear absorption. Linear absorption allows transmission proportional to Beer–Lambert law, corresponding to an exponential decay in intensity with regards to sample thickness I (x) = I0 e−αx ,

(4.6)

where I(x) is the intensity, I 0 is the input intensity, α is the single photon absorption coefficient, and x is the sample thickness. By contrast two-photon absorption is described by I (x) =

I0 , 1 + βx I0

(4.7)

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with β being the two-photon absorption coefficient and the other terms the same as in the single photon absorption case. The linear intensity is dominant at low field strength, but this is not always true. In the limit of incredibly high fields, two-photon absorption and other multiple-photon absorption processes can become dominant. Indeed, this is how the evidence of TPA in the soft X-ray regime was noticed. The production of Ne9+ was observed at lower photon energies than what was required for Ne8+ [17]. Following careful theoretical analysis, Doumy et al. assigned it to two-photon absorption; they cleverly changed the input energy for ionization and showed that the absorption rate was different as the energy was moved from 1110 to 1225 eV, going from a quadratic scaling factor with respect to pulse power to one that was roughly linear (exponent of 2.0 vs. 1.1, respectively). 1110 eV should not be enough energy to ionize Ne8+ into Ne9+ , allowing for TPA without the possibility of single photon absorption. The reason the scaling factor is not 1.0 for the case of 1225 eV is probably related to other processes beyond simple absorption occurring, including TPA amongst many other possible effects, as shown in Fig. 4.4. The cross section of TPA in the case of Ne9+ was many orders of magnitude higher than what had been expected, which is likely why it was observable. Two-photon absorption had previously been seen for helium at 41.5 eV [73, 74]. In this case, light generated from an HHG source was incident upon helium, generating a variety of ions [18]. Because the spectra of helium are so well understood and the quality of the data was so high, the research team was able to determine the cross section and compare it to theory, with reasonable agreement, although there is some error related to unknown amounts of 83.0 eV contamination from the source. While the importance of this specific result is perhaps limited, the ability to do these measurements in a lab setting is a significant achievement and opens the possibility of future nonlinear measurements in lab settings.

Fig. 4.4 Normalized ratio of Ne9+ to Ne8+ for excitation with 1110 eV (filled circles) and 1225 eV (open squares). The fit for the 1110 eV showed a quadratic dependence with flux (E 2.0 ) whereas the fit for 1225 eV showed a roughly linear dependence (E 1.1 ) [17]. Reprinted with permission from Doumy et al. [17]. Copyright (2011) by the American Physical Society

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Finally, there have been two sets of measurements on graphite. The first measurement [75] showed two-photon absorption near the carbon K-edge. However, this view has been revised with more recent data and is now ascribed to shifting in the edge energy over the finite length of the pulse [65]. In the newer work, if is found that TPA is fundamentally competing with SA, described in Sect. 4.3. What is seen is that at relatively low fluences SA occurs, and at higher fluences the importance of TPA is observable. In solids the processes occur at different rates than in the case of gases, due to the very different local environment when comparing a gas to condensed matter. This all indicates that spectroscopy at very high fluences will need to be carefully considered going forward, as multiple effects can occur. This complexity inherently grows as the systems grow more complicated. However, it ultimately provides unique insight into materials and offers unique abilities to control light and matter in ways that are otherwise not possible.

4.4.2 Stimulated Emission/Forward Scattering While the generation of laser light from a material via stimulated (rather than spontaneous) emission is generally not considered a nonlinear optical technique as the high fields do not drive the process, the process requires intense X-ray sources and leads to a significant nonlinear enhancement in the number of photons generated as compared to spontaneous emission. In the case of Ne, an X-ray laser was made by pumping a gas with very intense X-ray pulses which causes a population inversion of the neon single excited ion. This transition to stimulated emission from spontaneous emission for 960 eV input photons occurred at roughly 0.15 mJ power [76]. For an infinitesimally thin sample with interaction length dx, the emission probabilities P can be described as P = σρatom dx,

(4.8)

P = σρch dx,

(4.9)

and

where σ is the cross section of either absorption or stimulation, ρatom and ρch are the number density of absorbing atoms and core holes, respectively. As such, more core holes linearly increase the probability of stimulated emission. In the limit where stimulated emission is large (P ~ 1), the ratio of densities can be expressed as ρch λ , = ρatom dx

(4.10)

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where λ is the absorption length. For high efficiency, the absorption length relative to the interaction length has to be similar to the ratio of excited atoms. For comparable interaction and absorption lengths it would require full excitation of the system, that is to say a population inversion will be required, i.e., lasing will occur. The ability to drive this type of lasing phenomenon should open the way to future nonlinear X-ray studies. This was later extended to show that in certain situations superfluorescence, a quantum effect leading to the collective emission of fluorescent light, similar to but distinct from amplified stimulated emission, could also occur [21, 22]. There has been a variety of work expanding upon this in the hard X-ray regime with potential spectroscopic applications [77–79]. The demonstration of stimulated emission for materials science with soft X-rays was made in 2013 [19]. This work was motivated by X-ray emission generally being an isotropic process with the efficiency for low energy excitations often below one percent [80]. This means that sample damage from nonradiative decay, such as Auger decay, limits the ability to make measurements on samples that are sensitive to radiation damage, notably including biological and organic molecules. Using stimulated emission instead allows for a higher conversion efficiency and thus less sample damage by mitigating processes such as Auger decay. This is identical to what drove the lasing phenomenon seen in neon. Beye et al. propose that stimulated emission can be used with nonlinear optical techniques to enable measurements such as X-ray emission on samples that are highly susceptible to damage. This work was expanded upon more recently by studying the emission from Co/Pd multilayers. Through careful design of the X-ray pulses, Stöhr and coworkers were able to observe an enhancement of 106 for stimulated versus spontaneous emission [81]. This large enhancement was found for both inelastic X-ray scattering as well as elastic X-ray scattering [81]. However, they also found that there were effects caused by the temporally spiky nature of the SASE radiation used in this study, as the electrons would diffuse between spikes within a pulse causing unwanted spectral effects. Thus, they suggest that the ultimate way to minimize the unwanted effect and enhance the stimulated RIXS spectrum would be for a single short (possibly hundreds of attosecond) spike. This would enable linear spectroscopy efficiently in the high photon flux regime often associated with nonlinear optics. On the same Co/Pd system at the Co L-edge, differences in the absorption and the magnetic circular dichroism spectra were observed when comparing XFEL measurements to synchrotron measurements. It appears that over the intensity range available at XFELs near absorption edges, the intense X-ray pulse changes the valence electronic structure, which can then be observed by differences in absorption [20]. This paper also saw enhancement in elastic X-ray scattering. Such an enhancement has been observed previously in a series of manuscripts. It was first proposed that this would occur in high intensity coherent X-ray pulses leading to a deviation from the Beer–Lambert law [82]. The results verifying this theory showed up only a year later, when high fluence led to a loss of diffraction signal in Co/Pd systems [83]. This was definitively shown two years later by a loss of diffraction signal with a simultaneous rise in forward elastic scattering for the same system [84]. Stöhr and

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coworkers were able to explain how the ratio of stimulated to spontaneous emission affected transmission and diffraction contrast as a function of intensity.

4.4.3 Stimulated Raman Scattering A critical milestone in the development of 2D X-ray spectroscopy is a demonstration of the ability to control the electronic state of molecules as well as control the motion of the constituent atoms such that they can be driven in specific relative motion. This can in principle be used to drive highly specific reactions as well as understand the motion of atoms. While this was theorized for a long time with X-rays [85], the cutting-edge experiment to coherently drive matter from an excited state to another excited state is relatively recent, using stimulated electronic X-ray Raman scattering [23]. In normal Raman scattering, a single pump photon is input ωPump , and some of the energy is lost (or gained) inelastically, corresponding to a difference of ωSignal . In contrast, in stimulated Raman, two photons are input, the pump photon ωPump , and the Stokes photon ωStokes . When the difference between the two photons (ωPump − ωStokes ) corresponds to a Raman transition ωSignal , a large enhancement is seen due to stimulated Raman scattering. In the frontier X-ray experiment of stimulated Raman scattering, an X-ray pulse pumped Ne gas into an excited state and then resonantly deexcited the atom into a lower energy state accompanied by photon emission. This is possible due to the wide energies available in a SASE pulse and its finite pulse length, ultimately leading to an emission band in this case, slightly lower then the input photon energy. In this case the input energy was roughly 870 eV and the emitted energy was roughly 850 eV. This pushing of an atom from a strongly excited state into a less excited state by resonant deexcitation is a demonstration of control of an excited state. This type of exquisite experimental control will be necessary for the ultimate realization of 2D X-ray spectroscopy. Unfortunately, despite substantial effort to extend this technique [86], advances have been slow due to a range of experimental challenges. One pulse schema suffer from the inherent jitter and irreproducibility of individual shots, making data analysis and definitive proof hard at SASE beamlines, and the spectral width of seeded lasers are too low to easily observe Raman deexcitation in a single pulse. More advanced techniques require two pulses with well-tuned parameters and high power. This is the kind of problem which is challenging for current FELs as the FELs with excellent spectral properties have limited energy tunability for two pulses and lower power, and the FELs with tunability for a second pulse and high power produce noisy pulses. It does seem inevitable that newer machines will enable these measurements eventually. In 2020 while studying nitric oxide, researchers were able to drive NO into a core hole excited state, before pushing another electron back down into the core hole leaving the element in a relatively stable excited state [87]. This excited state could be used to drive chemical reactions and otherwise prepare molecules into specifically chosen states and extends upon the previous work by demonstration atomic control in a molecular system. It also provides the opportunity to track the motion of electrons

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within a molecule with higher spatial and temporal resolution than can be obtained with direct excitation of the same energetic states. This technique requires fast pulses and high fields, so the increasing availability of higher intensity X-ray pulses should make such experiments more accessible. Finally, it was noticed that when exciting neon in a particularly strong resonant X-ray field, it was possible to drive it from an excited state (hundreds of eV) into a slightly excited yet stable neutral state (few eV) which does not undergo recoil after excitation due to no emission of an electron [24]. This is done by exciting the electron and then driving it back down into a neutral but excited ground state that is below the binding energy of neon, preventing the possibility of the of emission an electron. When imaging an atomic beam of xenon, it is quite apparent that these excited atoms exist, as the excited xenon molecules show up as a solid line on the 2-dimensional imaging detector, tracking the path of the beam of atoms which was inserted. Interestingly, this phenomenon is seen only at 855 eV excitation, not 860 eV excitation, indicating it is sensitive to the specific nature of the excited state. Regardless, such exquisite control is a demonstrative step toward 2D spectroscopy of neon atoms. This is shown in detail in Fig. 4.5.

4.4.4 Four Wave Mixing There is only a single beamline in the world dedicated to the study of nonlinear phenomena: TIMER at FERMI. That beamline is dedicated specifically to the four wave mixing (4WM) technique, a powerful method demonstrated with optical pulses before its more recent push into the X-ray regime [88]. In this technique, two (typically degenerate) pulses are incident upon a surface with some angle relative to each other. The positive and negative interference between the two waves generate areas which are excited and other areas which are not excited in the form of a transient grating. This allows for time scale of properties from attoseconds (if the initial pulses are fast enough) to dynamics of hundreds of picoseconds as the system returns to initial conditions. Over time this energy will diffuse, and it can be used to study various dynamic properties such as thermal transport, coherent phonon diffusion, and ultrafast magnetism by probing it with a third wave, which scatters off the transient grating and in the process generates a fourth wave which is then detected [31]. This technique is quite important in the visible and has been used to study a variety of materials such as radiation materials, quantum materials, and magnetic materials [89–91]. It is important that phase matching conditions be met such that the total momentum of the four pulses cancels out. In the case of the geometry used in typical transient grating experiments k→FWM = k→1 − k→2 + k→3 ,

(4.11)

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Fig. 4.5 a Excitation and decay pathway in neon for X-ray scattering near 855 eV. The initial pulse excited the laser into a state Ne[1s−1 3p] state, followed by either Raman emission to the Ne[2p−1 3p] state or auger emission (blue line to gray horizontal line). The Ne[2p−1 3p] state can decay to a long lived metastable Ne[2p−1 3s] state which can be detected. b Experimental setup for this experiment including position-sensitive MCP detector to observe excited Ne atoms. c Distribution of excited neon atoms deflected as a result of spontaneous X-ray scattering leading to momentum transfer with an incident photon energy of 855 eV. The narrow vertical line is not seen when excited at 860 eV and is due to the mechanism described in a. Adapted from Eichmann et al. [24]. Reprinted with permission from Eichmann et al. [24]. Copyright by Science AAAS

where k→FWM is the signal, k→1 and k→2 form the grating, and k→3 is the probe pulse. In the general case where the grating is made up 1 frequency, k→1 and k→2 will be the same energy. The signal is generally time dependent due to both a long-term exponential decay of the signal and the sinusoidal ‘beating’ of the signal. This generally takes the form of | |2 Signal ∝ |e−Δt/τi sin 2π vi Δt | ,

(4.12)

where Signal is the signal, Δt is the time delay, and τi and vi are the frequencies and damping times of the impulsive stimulated mode. There are a variety of prefactors

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which can be dependent on the specific nature of the measurement and will otherwise affect the absolute signal but not the general shape. A higher frequency grating, and thus a higher momentum transfer and energy, allows one to go from studying molecular excitations to studying electronic excitations, in addition to the thermal and acoustic modes. This means that properties on a smaller length scale, approaching the size of a unit cell can be studied. If pulses as high energy as hard X-ray pulses were used as a grating the length scales that could be studied approach nanometers and would allow for determination of phenomena like thermal diffusion on an unprecedentedly small length scale. The length scale is limited by the wavelength of the pulses that make up the grating, so shorter wavelength allows for more microscopic information. Specifically, the spacing of the grating is given by L = λ/2 sin θ,

(4.13)

where L is the spatial periodicity, λ is the frequency of light used in the grating, and θ is the angle between the two pulses that make up the grating. For that reason, there has been tremendous effort put into the development of soft X-ray 4WM and thus lower λ, as soft X-rays have significantly shorter wavelengths than the UV wavelengths used previously, and is a necessary step to shorter frequencies, as well as allowing for the study of phenomena on the tens of nanometers length scale. The key to these measurements is that the transient grating must be made from relatively coherent light, as otherwise the grating will be of insufficiently high quality. Thus a seeded FEL with a beam splitter was key to making the cutting-edge measurements [15]. The measurements were therefore conducted at FERMI, an externally seeded FEL, on SiO2 , likely chosen because it is easy to work with and made for a strong demonstration [15]. Part of the challenge is overlapping two XFEL pulses in space and time while also meeting the phase matching conditions. In this demonstration, the probe pulse used was an optical pulse, which allows for high intensity and a frequency that exploits an optical resonance, allowing for one to examine the destruction of a specific state due to the existence of the grating. The experimental and conceptual setup is shown in Fig. 4.6. The phase matching conditions are also shown in Fig. 4.6. Because the FEL is inherently ultrafast, it is quite simple to demonstrate the timedependent nature of the technique. The ultrafast beating as the transient grating sinusoidally oscillates was readily detected even in these initial measurements. Using an X-ray probe pulse with an X-ray transient grating, however—a so called all X-ray transient grating experiment—would allow for one to exploit the element specific nature of X-rays as a probe, so it was perhaps inevitable that such a measurement would be accomplished [92]. By cleverly exploiting the harmonics generated at FERMI, the same group demonstrated an all X-ray experiment from both a thin and a thicker sample which allows for different experimental geometries. They were able to estimate the effective cross section for 4WM in the all XUV setup of 6 × 10−24 m2 V−2 . While in this case the probe pulse was not used in a resonant scheme, this experiment clearly opens the possibility of exploiting that phenomenon.

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Fig. 4.6 a Experimental schematic of FWM experiment. ɅEUV = 27.6 nm and λopt = 392.8 nm were used here. The CCD is placed in the expected direction of the FWM signal beam (kFWM ), as determined by the phase matching conditions. b Phase matching conditions used here where kopt , kEUV1, and kEUV2 are the wavevectors of the optical and of the two FEL pulses, respectively [15]. Reprinted with permission from Bencivenga et al. [15]. Copyright by Springer Nature

Silicon nitride has become the most reliable standard for X-ray 4WM experiments because it’s cheap, readily available, and reasonably robust. In fact, after SiO2 , the next measurement published was on Si3 N4 [93]. These initial measurements established that 4WM is a general technique and could be applied to a wide variety of systems, and that the systems—not surprisingly—will show different cross sections and different time constants depending on the specific material being studied. It was also discovered that these systems can have quite complicated dynamics, with Si3 N4 displaying both a slow (extending over 100 ps) and rapid component (as fast as 100 fs can be resolved) to its time decay following excitation. The power dependence was also investigated, and the system behaved as expected indicating that four wave mixing in the X-ray regime will behave similarly to that of optical 4WM. Perhaps the most important result of this study was showing that sample damage is going to be a problem for X-ray 4WM experiments. While certain samples can work, at present there is always a risk of sample damage before signal can be obtained and experiments must be carefully planned. However, silicon nitride is a particularly robust sample and this makes it an excellent tool for the curious scientist. Because it is so robust, Si3 N4 was used as an example for a resonant transient grating (in this case, resonant with the Si L-edge) [94]. The probe pulse remained optical, and the work was otherwise similar to the previous work on Si3 N4 with an optical probe, the large difference being the switched pump energy. By resonantly exciting the silicon, the method of thermalization was changed dramatically as the silicon can relax by newly accessible channels involving core hole decay such as Auger and fluorescence. This led to faster dynamics when the silicon nitride is resonantly excited. The most recent and most advanced published FEL work using 4WM concerned optimization of grating resolution. Using both Si3 N4 and crystalline silicon as samples, the authors were able to show different dynamics between the two samples

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[16]. Of more significance in the long term, they used grating wavelengths of 13.3 nm spacing, producing period spacings of 28 nm. This means it is possible to observe thermal and coherent phonon dynamics on length spacings as small as tens of nanometers. At this length scale, it becomes possible to study the effects of defects in materials on phenomena such as heat flow and magnetic dynamics. It may even be possible to investigate structural effects in liquids, although for simple liquids such as water the ideal grating spacing will be even smaller, which will require using higher X-ray energies for the grating. The most comprehensive X-ray 4WM study compared diamond, glass, and Bi4 Ge3 O12 [95]. By performing Fourier analysis on the resulting spectrum, the observed signal could be assigned to Rayleigh surface waves and bulk longitudinal waves moving throughout the sample, with frequencies ranging from THz to GHz. Waves that were combinations of these waves were also observed—a combined Rayleigh and longitudinal wave for instance. Bi4 Ge3 O12 was found to have a coherent optical phonon mode at approximately 3 THz. The authors point out that this method can be used to generate waves of a wide variety of acoustic frequencies without the need of making a transducer. While this is true, a transducer is generally more practical and cost effective. It will be more immediately useful in explaining particularly interesting phenomena such as microscopic heat diffusion. Finally, there has been work reported on the 4WM of silicon carbide [96]. In this case, a transient grating with a period of 84 nm was used. At these length scales the Fourier heat equations break down, but the system can be fully described by Lamb waves, elastic waves that travel along the surface. Additionally, there was a slight angular difference in the signal corresponding to the signal strength. Although not definitively shown, the authors speculate it is related to time-dependent non-isotropic changes in refractive index. There has also been extensive work in demonstrating four wave mixing with gaseous samples using lab-based HHG systems. While these sources lose the pulse power and often the energy range of FELs, they have large advantages in that the dedicated equipment is comparatively cheap and optimized for the measurements, and signal can be averaged indefinitely without needing to accommodate other measurements. Amongst the first to make such measurements were Leone and Neumark, who led a series of measurements mixing XUV with near IR in a 4WM setup. They first demonstrated that they could drive neon into an electronically resonant state (the ns/ nd state) which would then transition to the ground state [97]. They also successfully demonstrated the expected behavior with regards to signal versus input power intensities and the expected sinusoidal decay of signal as is seen in other frequency ranges. One interesting aspect of this study was that the signal changed monotonically with increasing gas pressure, suggesting phase matching conditions are important even in these conditions. A similar result demonstrating electronic sensitivity was shown by a separate research group demonstrating ultrafast oscillations of neon [98]. The work on phase matching has been extended to really explore the detailed effects of phase matching conditions [99].

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Eventually, a noncollinear setup where the HHG pulse was offset from the two Near IR (NIR) pulses was demonstrated [100]. In this arrangement, the HHG pulse intercepts the NIR pulse, combines with two NIR photons, and generates a photon offset angularly from both the NIR and HHG pulse. This setup is highly desirable because it removes the background and also allows one to see various four wave mixing process and their different excitation pathways, and in certain cases weaker 6 wave mixing processes were observed. The research team were able to show coherent Rydberg wave packets, which could be prepared in different ways depending upon the experimental details. They point out this is a significant step toward multidimensional spectroscopy with coherent dynamics. This would be similar to 2D-infrared but with X-ray pulses, allowing for 2D-X-ray spectroscopy. This work has now been extended to self-heterodyne, enabling even cleaner spectra, and up to eight distinct states in the case of helium [101]. The more states that can be seen, the more accurately the electronic structure of the material can be understood. It was subsequently demonstrated that dynamics as fast as attoseconds could be probed. First, krypton was studied and dynamics as fast as 22 fs were demonstrated [102]. However, because the technique is so sensitive, multiple different decay channels with different decay times are all simultaneously accessible. Because the signal is convoluted by all these decay channels determining the ultimate time resolution is difficult, but probably was noticeably faster than 22 fs. This was later proven by some of the same authors [103]. Ultimately, by working with attosecond XUV pulses in helium mixed with few cycle NIR pulses, they demonstrate that the resolution approaches hundreds of attoseconds. It is certainly better than 1.5 fs which can be determined from the higher order diffraction peaks from the transient grating and could be assigned based on excellent matching with the Maxwell–Schrodinger equation. One clever variation of four wave mixing is to use three nondegenerate pulses. While this makes the experimental setup more difficult, it enables the effect of each individual pulse to be separated and studied in a simpler manner. This has been accomplished and was used to study lifetimes of argon as short as 200 fs [104]. Such a setup may ultimately have limited time resolution as mixing three separate pulses inherently increases jitter and uncertainty. However, this is not a theoretical limit, and subsequent work did claim substantially higher time resolution (probably approaching 10 fs) [105]. The ability to have three separate pulses may prove important in the demonstration of future multidimensional measurements. Four wave mixing has already shown uses beyond simple demonstration spectroscopy. It has been used to narrow the bandwidth of XUV pulses, and such a scheme should be extendable to the X-ray regime [106]. However, this is probably not necessary for seeded FELs which have intrinsically narrow bandwidth. Finally, we wish to note that 4WM been extended into the hard X-ray regime, albeit using a very different setup [107]. For matters requiring investigations on the truly atomic scale, hard X-ray 4WM provides a unique probe for which there are no currently existing analogous techniques.

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4.4.5 Soft X-Ray Second Harmonic Generation/Sum Frequency Generation As discussed above, one of the most common uses of nonlinear optics is to combine waves. When one combines two waves through either SFG or SHG the signal will only be generated when there is no center of inversion, i.e., the measurement occurs at a place that lacks centrosymmetry. This is generally, although not always, surface and interfaces. This centrosymmetry argument does not hold when one combines three photons to generate a fourth, as in 4WM. Broken inversion symmetry can be shown to be essential for even order nonlinear processes by considering the expansion, P→ = ∈0

n ∑

χ (i ) E→ i ,

(4.14)

i=1

where P→ is the polarization, ∈0 is the permittivity of free space, χ (i) is the nonlinear susceptibility of order i, and E→ i is the electric field associated with order i and n is the arbitrarily large polarization expansion. If the inversion is defined as Iinv P→ = − P→ then it can be shown that ( ) n n ∑ ∑ (i ) → i → → Iinv P = − P = Iinv ∈0 χ E = ∈0 (4.15) (− 1)i χ (i) E→ i , i=1

i=1

and adding the first equation to the second yields P→ − P→ = 0 = 2∈0

n/2 ∑

χ (2i) E→ 2i .

(4.16)

i=1

Proving that the signal will be zero when the process is even order and inversion symmetry is present. This argument does not hold in the hard X-ray regime for both SFG and SHG [38, 39]. This is because in this regime, as the wavelength is on the order of the density fluctuations in the electron density of a material, in essence everything becomes a surface and SFG/SHG signal is generated in the bulk, causing the previous equation to break down at small wavelengths. At these particularly short wavelengths the dipole approximation can also become inadequate, as the electric field may not be constant over the length scale of the excited transition. What would happen in the soft X-ray regime, with photon wavelengths between visible and hard X-ray, was not obvious before the soft X-ray SHG measurements were made. Typically, soft X-rays processes are dipole selected, but this had not been proven for even order nonlinear processes. The pioneering demonstration of SHG in the soft X-ray regime used graphite samples in transmission mode, exploiting the coherence of the FERMI FEL source [6]. Signal was observed at incident photon energies below the carbon K-edge, resonant with the carbon K-edge and above the carbon K-edge, but generally close to

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the carbon K-edge. A quadratic power dependence in accordance with theory was seen. DFT calculations were also performed to try to simulate the spectrum, but with only three experimental energy points the simulations could not be fully validated by experiment. More significantly, both the calculations and the experiments showed that the measurements were surface sensitive. This was done theoretically by determining the signal as a function of slab thickness and noting that is saturates after about 4 atomic layers. Experimentally, SHG signal was shown to be constant for transmission samples of varying thickness, implying that signal was generated only from the front and back surfaces. This measurement was performed in a single shot mode, where the graphite films were destroyed with each shot. The experimental geometry (transmission) is detailed in Fig. 4.7a. A different research team simultaneously conducted separate SXSHG measurements at SACLA, studying a noncentrosymmetric crystal, GaFeO3 , at the Fe M-edge [7]. This contrasts with the previous experiments at FERMI, but by using a bulk crystal these measurements were made in a reflection geometry, and perhaps most significantly, below the damage threshold, meaning the measurements can be made on rare and hard to produce samples. In this case, several energy points near the Fe M-edge in double resonance were measured and with enough energetic data points to be able to show definitively that SHG occurred. Once again the expected quadratic signal was observed. The reflection geometry and associated theory is shown in part b of Fig. 4.7. These two measurements collectively demonstrated second harmonic

Fig. 4.7 a Experimental setup of a transmission experiment on graphite performing SXSHG at FERMI free electron laser (left). In this experiment, pulses were transmitted through sheets of graphite at roughly resonant with the carbon K-edge (right) generating signal at roughly 600 eV and a quadratic dependence with no thickness effect was seen (middle). This suggests SXSHG is largely surface sensitive [6]. Reprinted with permission from Lam et al. [6]. Copyright (2018) by the American Physical Society. b Experimental setup of a reflection experiment on GaFeO3 performing SXSHG at SACLA (left). In this experiment, pulses at half resonance (right) were reflected off the sample. Once again a quadratic signal dependence on input power was observed [7]. Reprinted with permission from Yamamoto et al. [7]. Copyright (2018) by the American Physical Society

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generation at input frequencies that were resonant with the one photon transition, generating a photon at twice the resonant frequency (2ω), as well as input frequencies that are half the resonant energy of a single electron transition, generation photons at a frequency of ω. A study of boron films covered by a thin 100 nm layer of a polymer was conducted in transmission mode [108]. This boron/polymer interface was chosen as an example of a buried interface, which are generally difficult to probe with existing experimental methods. In this case, by tuning the FEL source to the boron K-edge, the X-rays passed through the polymer which contained no boron and were thus only probing the boron layer. Distinct spectra were observed for the boron/polymer (B/P) interface compared to a bare boron control sample, and the researchers were able to determine the separation between the polymer and the boron layer to within ~ 0.2 Å by comparing calculations to experiments. In comparison, XAS of boron/vacuum and boron/polymer were roughly identical as shown in Fig. 4.8. One of the advantages of this study over the previous carbon work was more spectral points, enabling a better interpretation of the data. This is a measurement which is only currently possible with soft X-ray SHG (SXSHG). No other techniques can both penetrate to the buried interface and offer the high sensitivity demonstrated in the SXSHG data. Using the SACLA setup, several more measurements have been performed that take advantage of operating below the damage threshold. Lithium osmate is a socalled polar metal below a transition temperature [109], meaning it no longer has centrosymmetry and SHG was easily observable. However, when the temperature was raised and the crystal undergoes a phase transition into a centrosymmetric state, no signal was observable. The SHG spectra were used to determine the lithium bond

Fig. 4.8 SXSHG spectra of boron/vacuum (B/V) and boron/polymer (B/P) interfaces. B/V (dark red triangle) and B/P (dark blue circle) interfaces, shown along with the linear XAS of boron (light red line) and boron/polymer (light blue line), with arrows showing the appropriate y axis. The difference in XAS is believed to be due to differences in the background, whereas the SXSHG signal is significantly different [108]. Reprinted with permission from Schwartz et al. [108]. Copyright (2021) by the American Physical Society

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distances in the crystal based upon comparison to calculations. While this information can also be obtained from other techniques like neutron scattering, performing this experiment spectroscopically allows for the possibility of studying the system following an excitation laser pulse. It should be possible to watch the system ‘recover’ into the polar metal state following excitation, as the detailed way in which this transition occurs is not presently known. Such a measurement should now be possible with SXSHG. Another measurement was made on lithium niobate [110]. This is a standard noncentrosymmetric crystal used in nonlinear optical measurements, for example as an optical parametric oscillator [111]. In this case, the lithium and niobium peaks overlap spectrally, so the resultant SHG spectrum was quite complicated. However, with the aid of calculations even spectra with contributions coming from more than one element can be disentangled. The most important result from this study is that, as is seen in the visible, there is a polarization dependence to the signal at a given energy [112]. This can be used to understand the symmetry of a crystal in the visible, but at X-ray energies it reveals the symmetry around a given element. By measuring a wide variety of different angles, it is possible to obtain detailed polarization analysis. It additionally confirms that SXSHG is dipole selected. This method of using a polarizer to detect SHG has potential in that it efficiently suppresses the background, but the mirrors only function within a limited spectral range. This can be mitigated using multiple sets of different mirrors. This of course increases the equipment cost, but given the expense of FEL experiments, the expense could be considered minor. More recent studies have moved beyond fundamental physics to probe chemistry. A recent work demonstrated that in a solid-state battery system, a large spectral shift was seen between the surface SHG measurements and the bulk absorption measurements [113]. Because of the sensitivity of these measurements, the shift could be assigned to the lithium having more freedom to move (fewer librational constraints) at the surface than in the bulk. This is an important structural property for battery designs and is the cause of the previously unassigned large surface resistance seen in a variety of solid-state batteries. This implies a means of mitigating this problem through careful battery design. There has also been a successful demonstration of two NIR photons and one XUV photon combining in a variation of SFG on LiF [9]. Working below damage required using the three photon scheme so the signal would be bulk sensitive, as the two-photon transition was likely too weak to be measured at the relatively low fluences used here. The authors demonstrated both SFG (XUV + two NIR photons) and difference frequency generation (DFG) (XUV − two NIR photons). They also showed that the overlap between the NIR and XUV photon must be relatively precise, or else the signal drops to zero as is expected. Interestingly, the observed line width was sharper than the bandwidth of the FEL pulse, which suggest the spectral limit is due to the excitonic LiF state probed. This has potentially use in the development of high-resolution spectroscopy at FELs, although may not be particularly useful at seeded FELs. Finally, we should note there has been a SXSHG measurement made without an FEL [114]. Using a large laser system with a power of ~ 1011 W/cm2 , which is roughly

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5 orders less power density then typical from an FEL, SXSHG was successfully demonstrated in lab at the titanium M-edge. This opens the possibility of performing these types of measurements in labs going forward and hopefully one day will negate the need for a national facility such as an FEL. This particular system, however, is substantially larger than an HHG source and lacks the tunability of a FEL. However, there has been large progress in the development of lab-based X-ray sources. Thus, SXSHG represents a technique with great potential for studying surfaces and buried interfaces, dynamics of noncentrosymmetric materials, or dynamic symmetry breaking in centrosymmetric materials. It is maturing as new detection schemes become available and a variety of methods to understand the resultant spectra have been developed. New FELs such as LCLS-II and developments in lab-based light sources may make this technique a standard technique for surface studies in the future.

4.5 Theoretical Calculations for the Spectral Analysis It was only shortly after the laser was invented that the first nonlinear optics measurements were made [3]. This was possible because theoretical predictions made well in advance of the development of the laser had predicted the phenomena, and to some extent even how to perform such measurements. This has been true to some extent also in the field of X-ray nonlinear optics, with measurements such as hard X-ray SHG having been guided by earlier theoretical work [39]. The guiding theory has taken on a variety of forms, and the most widely used conceptual frameworks are discussed further here. Since nonlinear optics are well established in the visible, theoretical frameworks used in the visible can be applied in the soft X-ray regime in certain cases as both are dipole selected and have similar selection rules. Perhaps the clearest example of this was the study of LiF by four wave mixing [9], where the measured single photon X-ray absorption data was combined with reflectivity and known nonlinear optical constants to predict the sum and difference frequency generation spectra [5]. This type of technique, where well-established laws of physics in the visible can be applied to a problem in the soft X-ray regime, is quite powerful when adequate information about a system is known. But well-established values and equations must exist, and this is not always, or even generally, the case. Because of limited information in most cases in X-ray nonlinear optics, such an approach is not generally applicable. More general approaches such as those that propagate motions of atoms, electrons, and fields are available, including techniques such as velocity gauge real-time time-dependent density functional theory (VGRTTDDFT) [115–117]. This method is, like many used for calculations involving soft X-ray nonlinear optics, incredibly expensive, but it gives accurate results and can describe a wide variety of phenomena, including second harmonic generation, sum frequency generation and multiphoton absorption and can potentially be extended to simulate other measurements. This method works by propagating the fields, the

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motions of the electrons, and the motions of the atoms in a real-time formalism. This means in principle it is possible to describe the electronic structure and the resulting change in electronic structure as a function of time as fields interact with the system. There is another approach, based on the first principles calculation that uses the Korringa–Kohn–Rostoker (KKR) method. The calculation can be made by the software, such as AkaiKKR (Machikaneyama) [118, 119]. In the method, the second order susceptibility for SHG is expressed with the Green’s functions and the dipole matrix element that are calculated by the density functional calculation. The calculation successfully reproduced result of the soft X-ray SHG experiment [7]. The technique is highly feasible and it can easily be applied to the non-uniform system, such as impurity or superlattice. Such a technique is inherently powerful, but the computational expense means it requires large amounts of supercomputing time and is limited to relatively simple systems. For this reason, there has been significant effort put into extending cheaper methods that exist in the visible but did not yet exist in the X-ray region, particularly using linear response [6, 120, 121]. While linear response techniques are almost universally cheaper than RTTDDFT, they come with their own costs. First, the formalism must be built to solve a relatively specific problem or class of problems. Secondly, the formalism must be tested to ensure that it works and provides reasonably accurate results. Since nonlinear X-ray optics is a new field, there is often no data to test against, and the ability to perform tests at high fields is limited by FEL time which is difficult to obtain [36]. This is particularly true if the experiment is seen as risky, as may happen if it is based on highly original and possibly controversial theoretical work. There is at present no easy solution to this other than for more free electron lasers and more supercomputers to be built, but this is an expensive solution and one that will likely not be readily championed by governments around the world. Still, some interesting and powerful methods have been developed. Perhaps some of the nicest examples of this have been in the variety of methods used to describe advanced X-ray Raman techniques [23, 85, 86, 122, 123]. These have proven very useful in the development of novel and X-ray specific forms of Raman spectroscopy, as the specific measurements appear to have been simulated in detail before they were initially attempted, leading ultimately to the initial demonstration of stimulated Raman. However, several of the suggested forms of Raman either have not been made to work yet, or may prove ultimately impossible to perform due to some yet unknown issue with the calculations. Still, attempting to develop theories which can be tested is an important aspect of development of nonlinear X-ray spectroscopy and should be encouraged. Going beyond this, several authors, among them Mukamel and Santra, have been engaged in forward looking theory, supporting experiments that may be feasible in 10, 20, or 50 years but are not feasible currently and do not look to be feasible in the immediate future [124–128]. This can serve the purpose of inspiring long-term experiments and goals for large national and international facilities. It also promotes thinking about dream experiments and the ultimate limit of what could be achieved with X-ray nonlinear optics.

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However, one opportunity for real advancement in X-ray spectroscopy in general, and in nonlinear optics in particular is the development of predictive actionable theories. While there is certainly value in predicting experiments with four rotatable, phase and mode locked attosecond, infinitely tunable Fourier-limited FEL pulses, a real opportunity for growth where new theorists could enter is in performing calculations to guide and inform currently feasible experiments. Even for problems where there exists adequate theory to calculate spectra, the methods are often too expensive to be performed in advance of the experiment. For instance, performing VG-RTTDFT is quite expensive, and therefore impractical for performing calculations of spectra in advance of an experiment, when it is still not known whether the measurement will work. In practice, the theorists who have worked on SHG as of the time of this writing have always only offered to calculate results after the experiment was completed. Thus, there certainly would be a welcome niche for potentially less accurate, but significantly cheaper and therefore more predictive calculations, because they can be used to guide experiments. Given the incredible expense of FEL time (approaching $10,000 an hour [129]), such an investment in theory might also make sense financially even if it requires a relatively large investment in supercomputing time.

4.6 Summary The development of high intensity X-ray sources has opened the door into a whole new field of X-ray physics. In the soft X-ray region, nonlinear optics has been shown to be quite similar conceptually to work that is already present in the visible with the added benefit of shorter wavelengths and atomic specificity. Rapid progress has been demonstrated on a wide variety of fronts, ranging from multiphoton absorption to wave mixing. As progress on both sources and experimental setups continue and are in their early stages, we expect to see rapid progress. Theoretical efforts in certain cases have guided many of the existing experimental methods and hopefully will be even more important in the future. The early work laying the foundations of a variety of soft X-ray nonlinear optics measurements has now largely been done, and now the community must begin the process of making more useful and less proof of principle measurements. The ability to do this will depend on a variety of experimental and theoretical developments. Of particular benefit to the experimentalists would be phase locked coherent attosecond lasers with high pulse rates and high repetition rates, beam splitters, flexible mirrors and generally anything which makes soft X-rays more flexible to work with. This will enable measurements which are not just analogous to the measurements made in the visible but can push beyond that, exploiting X-ray photons sensitivity to core excitations and the ability to control electronic structure in a novel manner. Acknowledgements We thank Hisazumi Akai for his comment on theoretical calculation of nonlinear spectral analysis. We appreciate Tomoaki Senoo, XingYu Su, Toshihide Sumi and Iwao

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Matsuda for editing and revising the chapter. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC-0023397.

References 1. J.M. Hollas, Modern Spectroscopy, 4th edn. (Wiley, Chichester, Hoboken, NJ, 2004) 2. W. Kaiser, C.G.B. Garrett, Two-photon excitation in CaF2 : Eu2+ . Phys. Rev. Lett. 7, 229 (1961) 3. P.A. Franken, A.E. Hill, C.W. Peters, G. Weinreich, Generation of optical harmonics. Phys. Rev. Lett. 7, 118 (1961) 4. U. Keller, Ultrafast Lasers: A Comprehensive Introduction to Fundamental Principles with Practical Applications (Springer, Cham, 2021) 5. R.W. Boyd, Nonlinear Optics, 3rd edn. (Academic Press, Amsterdam, Boston, 2008) 6. R.K. Lam et al., Soft X-ray second harmonic generation as an interfacial probe. Phys. Rev. Lett. 120, 023901 (2018) 7. S. Yamamoto et al., Element selectivity in second-harmonic generation of GaFeO3 by a soft-X-ray free-electron laser. Phys. Rev. Lett. 120, 223902 (2018) 8. B. Weigelin, G.-J. Bakker, P. Friedl, Third harmonic generation microscopy of cells and tissue organization. J. Cell Sci. (2016). https://doi.org/10.1242/jcs.152272 9. H. Rottke et al., Probing electron and hole colocalization by resonant four-wave mixing spectroscopy in the extreme ultraviolet. Sci. Adv. 8, eabn5127 (2022) 10. E. Shwartz, S. Shwartz, Difference-frequency generation of optical radiation from two-color X-ray pulses. Opt. Express 23, 7471 (2015) 11. T. Helk, M. Zürch, C. Spielmann, Perspective: towards single shot time-resolved microscopy using short wavelength table-top light sources. Struct. Dyn. 6, 010902 (2019) 12. Y. Fu, K. Nishimura, R. Shao, A. Suda, K. Midorikawa, P. Lan, E.J. Takahashi, High efficiency ultrafast water-window harmonic generation for single-shot soft X-ray spectroscopy. Commun. Phys. 3, 1 (2020) 13. A. Schori, C. Bömer, D. Borodin, S.P. Collins, B. Detlefs, M. Moretti Sala, S. Yudovich, S. Shwartz, Parametric down-conversion of X rays into the optical regime. Phys. Rev. Lett. 119, 253902 (2017) 14. B. Adams, Y. Nishino, D.V. Novikov, G. Materlik, D.M. Mills, Parametric down conversion of X-rays, recent experiments. Nucl. Instrum. Methods Phys. Res. Sect. A 467–468, 1019 (2001) 15. F. Bencivenga et al., Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature 520, 7546 (2015) 16. F. Bencivenga et al., Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses. Sci. Adv. 5, eaaw5805 (2019) 17. G. Doumy et al., Nonlinear atomic response to intense ultrashort X rays. Phys. Rev. Lett. 106, 083002 (2011) 18. H. Hasegawa, E.J. Takahashi, Y. Nabekawa, K.L. Ishikawa, K. Midorikawa, Multiphoton ionization of He by using intense high-order harmonics in the soft-X-ray region. Phys. Rev. A 71, 023407 (2005) 19. M. Beye, S. Schreck, F. Sorgenfrei, C. Trabant, N. Pontius, C. Schüßler-Langeheine, W. Wurth, A. Föhlisch, Stimulated X-ray emission for materials science. Nature 501, 7466 (2013) 20. D.J. Higley et al., Femtosecond X-ray induced changes of the electronic and magnetic response of solids from electron redistribution. Nat. Commun. 10, 1 (2019) 21. L. Mercadier et al., Evidence of extreme ultraviolet superfluorescence in xenon. Phys. Rev. Lett. 123, 023201 (2019)

114

C. P. Schwartz and W. S. Drisdell

22. J.R. Harries et al., Superfluorescence, free-induction decay, and four-wave mixing: propagation of free-electron laser pulses through a dense sample of helium ions. Phys. Rev. Lett. 121, 263201 (2018) 23. C. Weninger, M. Purvis, D. Ryan, R.A. London, J.D. Bozek, C. Bostedt, A. Graf, G. Brown, J.J. Rocca, N. Rohringer, Stimulated electronic X-ray Raman scattering. Phys. Rev. Lett. 111, 233902 (2013) 24. U. Eichmann et al., Photon-recoil imaging: expanding the view of nonlinear X-ray physics. Science 369, 1630 (2020) 25. J. Chen, C.-L. Hu, F. Kong, J.-G. Mao, High-performance second-harmonic-generation (SHG) materials: new developments and new strategies. Acc. Chem. Res. 54, 2775 (2021) 26. F.J. Duarte (ed.), Tunable Laser Applications, 3rd edn. (CRC Press, Taylor & Francis Group, Boca Raton, FL, 2016) 27. Y.R. Shen, The Principles of Nonlinear Optics, Wiley classics library edn. (Wiley-Interscience, Hoboken, NJ, 2003) 28. J.A. Giordmaine, R.C. Miller, Tunable coherent parametric oscillation in LiNbO3 at optical frequencies. Phys. Rev. Lett. 14, 973 (1965) 29. S.A. Akhmanov, A.I. Kovrigin, A.S. Piskarskas, V.V. Fadeev, R.V. Khokhlov, Observation of parametric amplification in the optical range. Sov. J. Exp. Theor. Phys. Lett. 2, 191 (1965) 30. L. Carrion, J.-P. Girardeau-Montaut, Development of a simple model for optical parametric generation. J. Opt. Soc. Am. B JOSAB 17, 78 (2000) 31. F. Hofmann, M.P. Short, C.A. Dennett, Transient grating spectroscopy: an ultrarapid, nondestructive materials evaluation technique. MRS Bull. 44, 392 (2019) 32. E. Poem, A. Golenchenko, O. Davidson, O. Arenfrid, R. Finkelstein, O. Firstenberg, Pulsedpump phosphorus-doped fiber Raman amplifier around 1260 nm for applications in quantum non-linear optics. Opt. Express OE 28, 32738 (2020) 33. P. Weinberger, John Kerr and his effects found in 1877 and 1878. Philos. Mag. Lett. 88, 897 (2008) 34. M. Protopapas, C.H. Keitel, P.L. Knight, Atomic physics with super-high intensity lasers. Rep. Prog. Phys. 60, 389 (1997) 35. C.A. Hoffman, R.G. Driggers, Encyclopedia of Optical and Photonic Engineering, 2nd edn. (CRC Press, Boca Raton, FL, 2016) 36. C. Bostedt, S. Boutet, D.M. Fritz, Z. Huang, H.J. Lee, H.T. Lemke, A. Robert, W.F. Schlotter, J.J. Turner, G.J. Williams, Linac coherent light source: the first five years. Rev. Mod. Phys. 88, 015007 (2016) 37. M. Fuchs et al., Anomalous nonlinear X-ray Compton scattering. Nat. Phys. 11, 11 (2015) 38. T.E. Glover et al., X-ray and optical wave mixing. Nature 488, 7413 (2012) 39. S. Shwartz et al., X-ray second harmonic generation. Phys. Rev. Lett. 112, 163901 (2014) 40. V. Honkimäki, J. Sleight, P. Suortti, Characteristic X-ray flux from sealed Cr, Cu, Mo, Ag and W tubes. J. Appl. Cryst. 23, 5 (1990) 41. V. Veksler, Concerning some new methods of acceleration of relativistic particles. Phys. Rev. 69, 244 (1946) 42. E.M. McMillan, The synchrotron—a proposed high energy particle accelerator. Phys. Rev. 68, 143 (1945) 43. F.R. Elder, A.M. Gurewitsch, R.V. Langmuir, H.C. Pollock, Radiation from electrons in a synchrotron. Phys. Rev. 71, 829 (1947) 44. J. Als-Nielsen, D. McMorrow, Elements of Modern X-Ray Physics, 2nd edn. (Wiley, Hoboken, 2011) 45. S. Boutet, M. Yabashi, X-ray free electron lasers and their applications, in X-Ray Free Electron Lasers: A Revolution in Structural Biology, ed. by S. Boutet, P. Fromme, M.S. Hunter (Springer International Publishing, Cham, 2018), pp. 1–21 46. G. Margaritondo, P. Rebernik Ribic, A simplified description of X-ray free-electron lasers. J. Synchrotron Radiat. 18, 2 (2011) 47. J. Duris et al., Tunable isolated attosecond X-ray pulses with gigawatt peak power from a free-electron laser. Nat. Photon. 14, 1 (2020)

4 Nonlinear Soft X-Ray Spectroscopy

115

48. E. Allaria et al., Two-stage seeded soft-X-ray free-electron laser. Nat. Photon. 7, 11 (2013) 49. I. Nam et al., High-brightness self-seeded X-ray free-electron laser covering the 3.5 keV to 14.6 keV range. Nat. Photon. 15, 6 (2021) 50. I. Inoue et al., Generation of narrow-band X-ray free-electron laser via reflection self-seeding. Nat. Photon. 13, 5 (2019) 51. E. Allaria et al., The FERMI free-electron lasers. J. Synchrotron Radiat. 22, 3 (2015) 52. G. Penco et al., Nonlinear harmonics of a seeded free-electron laser as a coherent and ultrafast probe to investigate matter at the water window and beyond. Phys. Rev. A 105, 053524 (2022) 53. B.W. Adams, D. DeCiccio, M. Michon, P. Chmielniak, T. Parkman, G. Diebold, K. Bisogno, C. Rose-Petruck, LPXS: a high-performance, recirculating liquid-metal laser-driven plasma X-ray source, in Advances in X-Ray/EUV Optics and Components XVI, vol. 11837 (SPIE, 2021), pp. 59–69 54. T. Saule et al., High-flux ultrafast extreme-ultraviolet photoemission spectroscopy at 18.4 MHz pulse repetition rate. Nat. Commun. 10, 1 (2019) 55. J. Stöhr, NEXAFS Spectroscopy (Springer, Berlin, London, 2011) 56. G. Gauglitz, D.S. Moore (eds.), Handbook of Spectroscopy, 2nd, completely revised and enlarged edn. (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2014) 57. J.D. Smith, C.D. Cappa, K.R. Wilson, B.M. Messer, R.C. Cohen, R.J. Saykally, Energetics of hydrogen bond network rearrangements in liquid water. Science 306, 851 (2004) 58. U. Bergmann, P. Glatzel, X-ray emission spectroscopy. Photosynth. Res. 102, 255 (2009) 59. G. Smolentsev, A.V. Soldatov, J. Messinger, K. Merz, T. Weyhermüller, U. Bergmann, Y. Pushkar, J. Yano, V.K. Yachandra, P. Glatzel, X-ray emission spectroscopy to study ligand valence orbitals in Mn coordination complexes. J. Am. Chem. Soc. 131, 13161 (2009) 60. F. de Groot, A. Kotani, Core Level Spectroscopy of Solids (CRC Press, Boca Raton, 2008) 61. S. Hüfner, Photoelectron Spectroscopy: Principles and Applications (Springer Berlin Heidelberg, Berlin, Heidelberg, 2013) 62. L. Young et al., Femtosecond electronic response of atoms to ultra-intense X-rays. Nature 466, 7302 (2010) 63. M. Hoener et al., Ultraintense X-ray induced ionization, dissociation, and frustrated absorption in molecular nitrogen. Phys. Rev. Lett. 104, 253002 (2010) 64. H. Yoneda, Y. Inubushi, M. Yabashi, T. Katayama, T. Ishikawa, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, Saturable absorption of intense hard X-rays in iron. Nat. Commun. 5, 1 (2014) 65. L. Hoffmann et al., Saturable absorption of free-electron laser radiation by graphite near the carbon K-edge, arXiv:2112.12585 [Cond-Mat, Physics:Physics] (2022) 66. B. Nagler et al., Turning solid aluminium transparent by intense soft X-ray photoionization. Nat. Phys. 5, 9 (2009) 67. H. Yoneda et al., Ultra-fast switching of light by absorption saturation in vacuum ultra-violet region. Opt. Express OE 17, 23443 (2009) 68. M. Pawlicki, H.A. Collins, R.G. Denning, H.L. Anderson, Two-photon absorption and the design of two-photon dyes. Angew. Chem. Int. Ed. 48, 3244 (2009) 69. C.N. LaFratta, J.T. Fourkas, T. Baldacchini, R.A. Farrer, Multiphoton fabrication. Angew. Chem. Int. Ed. 46, 6238 (2007) 70. A. Hayat, A. Nevet, P. Ginzburg, M. Orenstein, Applications of two-photon processes in semiconductor photonic devices: invited review. Semicond. Sci. Technol. 26, 083001 (2011) 71. M. Rumi, J.W. Perry, Two-photon absorption: an overview of measurements and principles. Adv. Opt. Photon. 2, 451 (2010) 72. J.H. Eberly, P. Lambropoulos (eds.), Multiphoton Processes: Proceedings of an International Conference at the University of Rochester, Rochester, N.Y., June 6–9, 1977 (Wiley, New York, 1978) 73. Y. Nabekawa, H. Hasegawa, E.J. Takahashi, K. Midorikawa, Production of doubly charged helium ions by two-photon absorption of an intense sub-10-fs soft X-ray pulse at 42 eV photon energy. Phys. Rev. Lett. 94, 043001 (2005)

116

C. P. Schwartz and W. S. Drisdell

74. N. Miyamoto, M. Kamei, D. Yoshitomi, T. Kanai, T. Sekikawa, T. Nakajima, S. Watanabe, Observation of two-photon above-threshold ionization of rare gases by XUV harmonic photons. Phys. Rev. Lett. 93, 083903 (2004) 75. R.K. Lam et al., Two-photon absorption of soft X-ray free electron laser radiation by graphite near the carbon K-absorption edge. Chem. Phys. Lett. 703, 112 (2018) 76. N. Rohringer et al., Atomic inner-shell X-ray laser at 1.46 nanometres pumped by an X-ray free-electron laser. Nature 481, 488 (2012) 77. T. Kroll et al., Observation of seeded Mn Kβ stimulated X-ray emission using two-color X-ray free-electron laser pulses. Phys. Rev. Lett. 125, 037404 (2020) 78. T. Kroll et al., Stimulated X-ray emission spectroscopy in transition metal complexes. Phys. Rev. Lett. 120, 133203 (2018) 79. Y. Zhang et al., Generation of intense phase-stable femtosecond hard X-ray pulse pairs. Proc. Natl. Acad. Sci. 119, e2119616119 (2022) 80. A.C. Thompson, X-Ray Data Booklet (Lawrence Berkeley National Laboratory, University of California, 2001) 81. D.J. Higley et al., Stimulated resonant inelastic X-ray scattering in a solid. Commun. Phys. 5, 1 (2022) 82. J. Stöhr, A. Scherz, Creation of X-ray transparency of matter by stimulated elastic forward scattering. Phys. Rev. Lett. 115, 107402 (2015) 83. B. Wu et al., Elimination of X-ray diffraction through stimulated X-ray transmission. Phys. Rev. Lett. 117, 027401 (2016) 84. Z. Chen et al., Ultrafast self-induced X-ray transparency and loss of magnetic diffraction. Phys. Rev. Lett. 121, 137403 (2018) 85. N. Rohringer, X-ray Raman scattering: a building block for nonlinear spectroscopy. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 377, 20170471 (2019) 86. V. Kimberg et al., Stimulated X-ray Raman scattering—a critical assessment of the building block of nonlinear X-ray spectroscopy. Faraday Discuss. 194, 305 (2016) 87. J.T. O’Neal et al., Electronic population transfer via impulsive stimulated X-ray Raman scattering with attosecond soft-X-ray pulses. Phys. Rev. Lett. 125, 073203 (2020) 88. C. Masciovecchio et al., EIS: the scattering beamline at FERMI. J. Synchrotron Radiat. 22, 3 (2015) 89. M.P. Short, C.A. Dennett, S.E. Ferry, Y. Yang, V.K. Mishra, J.K. Eliason, A. Vega-Flick, A.A. Maznev, K.A. Nelson, Applications of transient grating spectroscopy to radiation materials science. JOM 67, 1840 (2015) 90. J. Janušonis, T. Jansma, C.L. Chang, Q. Liu, A. Gatilova, A.M. Lomonosov, V. Shalagatskyi, T. Pezeril, V.V. Temnov, R.I. Tobey, Transient grating spectroscopy in magnetic thin films: simultaneous detection of elastic and magnetic dynamics. Sci. Rep. 6, 1 (2016) 91. A.A. Maznev et al., Propagation of THz acoustic wave packets in GaN at room temperature. Appl. Phys. Lett. 112, 061903 (2018) 92. L. Foglia et al., First evidence of purely extreme-ultraviolet four-wave mixing. Phys. Rev. Lett. 120, 263901 (2018) 93. F. Bencivenga et al., Four-wave-mixing experiments with seeded free electron lasers. Faraday Discuss. 194, 283 (2016) 94. R. Bohinc et al., Nonlinear XUV-optical transient grating spectroscopy at the Si L2,3–edge. Appl. Phys. Lett. 114, 181101 (2019) 95. A.A. Maznev et al., Generation of coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment. Appl. Phys. Lett. 113, 221905 (2018) 96. D. Naumenko et al., Thermoelasticity of nanoscale silicon carbide membranes excited by extreme ultraviolet transient gratings: implications for mechanical and thermal management. ACS Appl. Nano Mater. 2, 5132 (2019) 97. W. Cao, E.R. Warrick, A. Fidler, S.R. Leone, D.M. Neumark, Near-resonant four-wave mixing of attosecond extreme-ultraviolet pulses with near-infrared pulses in neon: detection of electronic coherences. Phys. Rev. A 94, 021802 (2016)

4 Nonlinear Soft X-Ray Spectroscopy

117

98. T. Ding et al., Time-resolved four-wave-mixing spectroscopy for inner-valence transitions. Opt. Lett. 41, 709 (2016) 99. K.A. Tran, K.B. Dinh, P. Hannaford, L.V. Dao, Phase-matched nonlinear wave-mixing processes in XUV region with multicolor lasers. Appl. Opt. 58, 2540 (2019) 100. W. Cao, E.R. Warrick, A. Fidler, D.M. Neumark, S.R. Leone, Noncollinear wave mixing of attosecond XUV and few-cycle optical laser pulses in gas-phase atoms: toward multidimensional spectroscopy involving XUV excitations. Phys. Rev. A 94, 053846 (2016) 101. A.P. Fidler, E.R. Warrick, H.J.B. Marroux, E. Bloch, D.M. Neumark, S.R. Leone, Selfheterodyned detection of dressed state coherences in helium by noncollinear extreme ultraviolet wave mixing with attosecond pulses. J. Phys. Photon. 2, 034003 (2020) 102. A.P. Fidler, H.J.B. Marroux, E.R. Warrick, E. Bloch, W. Cao, S.R. Leone, D.M. Neumark, Autoionization dynamics of (2P1/2)ns/d states in krypton probed by noncollinear wave mixing with attosecond extreme ultraviolet and few-cycle near infrared pulses. J. Chem. Phys. 151, 114305 (2019) 103. A.P. Fidler, S.J. Camp, E.R. Warrick, E. Bloch, H.J.B. Marroux, D.M. Neumark, K.J. Schafer, M.B. Gaarde, S.R. Leone, Nonlinear XUV signal generation probed by transient grating spectroscopy with attosecond pulses. Nat. Commun. 10, 1384 (2019) 104. L. Van Dao, K.A. Tran, P. Hannaford, Cascaded four-wave mixing in the XUV region. Opt. Lett. 43, 134 (2018) 105. K.B. Dinh, K.A. Tran, P. Hannaford, L.V. Dao, Four-wave mixing of extreme ultraviolet pulses and infrared pulses for studies of atomic dynamics. J. Opt. Soc. Am. B 36, 3046 (2019) 106. L. Drescher, O. Kornilov, T. Witting, V. Shokeen, M.J.J. Vrakking, B. Schütte, Extremeultraviolet spectral compression by four-wave mixing. Nat. Photon. 15, 263 (2021) 107. J.R. Rouxel et al., Hard X-ray transient grating spectroscopy on bismuth germanate. Nat. Photon. 15, 7 (2021) 108. C.P. Schwartz et al., Angstrom-resolved interfacial structure in buried organic-inorganic junctions. Phys. Rev. Lett. 127, 096801 (2021) 109. E. Berger et al., Extreme ultraviolet second harmonic generation spectroscopy in a polar metal. Nano Lett. 21, 6095 (2021) 110. C.B. Uzundal et al., Polarization-resolved extreme-ultraviolet second-harmonic generation from LiNbO3 . Phys. Rev. Lett. 127, 237402 (2021) 111. J. Lu, A.A. Sayem, Z. Gong, J.B. Surya, C.-L. Zou, H.X. Tang, Ultralow-threshold thin-film lithium niobate optical parametric oscillator. Optica 8, 539 (2021) 112. M. Sliwa, K. Nakatani, T. Asahi, P.G. Lacroix, R.B. Pansu, H. Masuhara, Polarization and wavelength dependent nonlinear optical properties of a photo-switchable organic crystal. Chem. Phys. Lett. 437, 212 (2007) 113. C. Woodahl et al., Dynamics of interfacial lithium ions revealed by second harmonic extreme ultraviolet spectroscopy. Nature (2022). Under review 114. T. Helk et al., Table-top extreme ultraviolet second harmonic generation. Sci. Adv. 7, eabe2265 (n.d.) 115. C.D. Pemmaraju, Valence and core excitons in solids from velocity-gauge real-time TDDFT with range-separated hybrid functionals: an LCAO approach. Comput. Condens. Matter 18, e00348 (2019) 116. C.D. Pemmaraju, Simulation of attosecond transient soft X-ray absorption in solids using generalized Kohn–Sham real-time time-dependent density functional theory. New J. Phys. 22, 083063 (2020) 117. M. Noda et al., SALMON: scalable ab-initio light-matter simulator for optics and nanoscience. Comput. Phys. Commun. 235, 356 (2019) 118. H. Akai, Electronic structure Ni–Pd alloys calculated by the self-consistent KKR-CPA method. J. Phys. Soc. Jpn. 51, 468 (1982) 119. H. Akai, Fast Korringa–Kohn–Rostoker coherent potential approximation and its application to FCC Ni-Fe systems. J. Phys. Condens. Matter 1, 8045 (1989) 120. S. Sharma, C. Ambrosch-Draxl, Second-harmonic optical response from first principles. Phys. Scr. 2004, 128 (2004)

118

C. P. Schwartz and W. S. Drisdell

121. K.E. Dorfman, F. Schlawin, S. Mukamel, Nonlinear optical signals and spectroscopy with quantum light. Rev. Mod. Phys. 88, 045008 (2016) 122. J.D. Biggs, Y. Zhang, D. Healion, S. Mukamel, Watching energy transfer in metalloporphyrin heterodimers using stimulated X-ray Raman spectroscopy. Proc. Natl. Acad. Sci. 110, 15597 (2013) 123. S.M. Cavaletto, D. Keefer, S. Mukamel, High temporal and spectral resolution of stimulated X-ray Raman signals with stochastic free-electron-laser pulses. Phys. Rev. X 11, 011029 (2021) 124. A. Karamatskou, R.E. Goetz, C.P. Koch, R. Santra, Suppression of hole decoherence in ultrafast photoionization. Phys. Rev. A 101, 043405 (2020) 125. J.M. Schäfer, L. Inhester, S.-K. Son, R.F. Fink, R. Santra, Electron and fluorescence spectra of a water molecule irradiated by an X-ray free-electron laser pulse. Phys. Rev. A 97, 053415 (2018) 126. J.R. Rouxel, M. Kowalewski, S. Mukamel, Diffraction-detected sum frequency generation: novel ultrafast X-ray probe of molecular dynamics. J. Phys. Chem. Lett. 9, 3392 (2018) 127. G. Batignani, C. Sansone, C. Ferrante, G. Fumero, S. Mukamel, T. Scopigno, Excited-state energy surfaces in molecules revealed by impulsive stimulated Raman excitation profiles. J. Phys. Chem. Lett. 12, 9239 (2021) 128. S.M. Cavaletto, S. Mukamel, Probing delocalized current densities in selenophene by resonant X-ray sum-frequency generation. J. Chem. Theory Comput. 17, 367 (2021) 129. Facts and Figures, https://www.xfel.eu/facility/overview/facts_amp_figures/index_eng.html

Chapter 5

Nonlinear X-Ray Spectroscopy Yuya Kubota and Kenji Tamasaku

Abstract This chapter focuses on nonlinear spectroscopies in the hard X-ray region (wavelength 0.1–1 nm). X-ray nonlinear phenomena differ from those in longer wavelengths because of unique X-ray interactions with matter. Theoretical aspect of Xray-matter interactions is briefly reviewed for calculation of nonlinear polarizability together with some associated experiments. Then, nonlinear X-ray spectroscopy is overviewed by focusing on experimental studies.

5.1 Introduction In this chapter, we discuss nonlinear spectroscopy in the hard X-ray region (wavelength 0.1–1 nm) and related topics. Studies of X-ray nonlinear optics have been developed gradually using synchrotron radiation, but the advent of high-brilliance X-ray free-electron lasers (XFELs) has brought a breakthrough in this field, and been leading to nonlinear X-ray spectroscopy. This trend has been supported by focusing systems, which can realize unprecedented intensity required for the nonlinear processes. First of all, we briefly introduce XFEL and the focusing systems. A free-electron laser (FEL) relies on interaction between electromagnetic wave and relativistic electron beam in alternating magnetic field created by an undulator. The interaction imprints a density modulation with a period of the wavelength on the electron beam, which finally emits electromagnetic wave in phase. Because the interaction is very weak, the FEL requires a cavity, which considerably elongates the effective interaction length. It is lack of X-ray cavities that had prevented researchers from realizing XFEL. This difficulty was cleared by the self-amplified spontaneous emission (SASE) principle [1, 2], where the interaction length was increased by using a very long undulator. A drawback of the SASE principle is that the pulse characteristics, such as pulse energy, wavelength, and spectral shape, fluctuate shot Y. Kubota (B) · K. Tamasaku (B) RIKEN SPring-8 Center, Hyogo, Japan e-mail: [email protected] K. Tamasaku e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_5

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by shot, because the lasing process starts from shot noise. The instabilities of SASEXFEL can be improved by seeding [3, 4]. In this scheme, the undulator is split into two parts. A monochromator set between them selects a single wavelength from the SASE radiation of the upstream undulator. Then, the monochromatic beam is led to the downstream undulator, stabilizing the wavelength and the lasing process. The split undulators can also be used to produce two-color XFEL by operating them at different wavelengths [5]. There are currently five XFEL facilities in operation worldwide, the Linac Coherent Light Source (LCLS) in the USA [6], the SPring-8 Angstrom Compact freeelectron LAser (SACLA) in Japan [7], the hard X-ray free-electron laser at the Pohang Accelerator Laboratory (PAL-XFEL) in the Republic of Korea [8], swissFEL in Switzerland [9], and the European X-Ray Free-Electron Laser Facility (European XFEL) in Germany [10] in chronological order of the first lasing. Figure 5.1 shows some pictures of SACLA. The distinctive features of XFEL compared to other X-ray sources are high brightness, ultrashort pulse duration, and high transverse coherence, which have opened up new frontiers in science. For example, formation and dissociation of chemical bonds after optical laser excitation were revealed by taking advantage of the ultrashort pulse duration on a femtosecond timescale [11–13]. In solid-state physics, ultrafast phenomena in photoexcited quantum materials have also been extensively studied by using the femtosecond pulse duration [14–16]. The highly brilliant XFEL pulses, which enable data acquisition in a single shot, have brought opportunities to perform experiments under instantaneous extreme conditions such as strong magnetic fields [17–19] and high pressure created by a high-power optical laser pulse [20–22]. One of the breakthroughs by XFEL may be a so-called diffract-before-destroy scheme [23]. X-rays have been a powerful tool for crystallography, but radicals and reactants produced during X-ray irradiation inevitably damage radiation-sensitive biological specimens. The intrinsic structures before damage can be obtained from the diffraction pattern taken by a single exposure with the high-brilliant and ultrashort XFEL pulse. This scheme has contributed, for example, to investigation of the oxygen-evolving complex of photosystem II [24, 25]. Single-shot coherent diffraction imaging (CDI) experiments can be performed by combining the high transverse coherence. Similar to the diffract-before-destroy scheme, single-shot CDI is applicable to damage-free observation of non-periodic and nano-sized particles, such as living cells [26]. In the study of X-ray nonlinear optics, the peak intensity is of crucial importance. Benefiting from the high brilliance of XFEL, a number of studies in X-ray nonlinear optics have been reported. In these studies, Kirkpatrick-Baez (KB) mirrors [27] have been used to obtain a high peak intensity. The KB mirror systems developed at SACLA are summarized in Table 5.1. A unique one was the two-stage focusing system, where the first KB mirror expanded the beam and the second focused it [28]. The elaborate optical design allowed a longer working distance for flexible sample environment at a cost of complicated alignment procedure. Hard X-rays have characteristics that the wavelength is close to the interatomic distance of crystals and the photon energy is close to the binding energy of the innershell electrons of metallic elements. Unique interactions between X-rays and matter

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Fig. 5.1 Pictures of a aerial view, b accelerator, and c undulator of SACLA. With permission of RIKEN Table 5.1 KB mirror systems developed at SACLA System Beam size Peak intensity (.W/cm2 ).b General-purpose KB mirror [29] Two-stage focusing system.a [28] 100-exa focusing system [30]

.1 µm

.10

18

1.35

50 nm

.10

20

0.35

100–200 nm

.10

20

0.115

The beam size is in full width at half maximum (FWHM) a Not available b . Estimated for a pink beam .

Working distance (m)

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may produce interesting nonlinear phenomena possible only in the X-ray region. In the following Sect. 5.2, we review a basic theory of nonlinear phenomena in the hard X-ray region and present some associated studies. Then, in Sect. 5.3, we provide highlights of notable studies of nonlinear X-ray spectroscopy. The last Sect. 5.4 gives summary and future prospects.

5.2 The Basic Theory of Nonlinear Optics in the Hard X-Ray Region 5.2.1 Nonlinear Polarizability In general, the interaction between electromagnetic waves and materials can be described using polarizability. We will derive the expression of the polarizability by considering the current density induced by electric fields using elementary quantum mechanics. This is because we cannot directly calculate the polarizability in the X-ray region. Note that we use the Gaussian system of units in this chapter. Electron density in a material at a certain time, .t, can be expressed by ∑

ρ(r, t) =

.

δ(r − r l (t)),

(5.1)

l

where . r l (t) is the position of the .l-th electron at .t. The current density is written as J (r, t) = −e

∑

.

δ(r − r l (t))vl (t),

(5.2)

l

where .vl (t) is the velocity of the .l-th electron at .t, and -e is the charge of the electron. We calculate the Fourier transform of Eq. (5.2), and obtain .

J˜ (K , ω) = ≃

∫∫ ∑

J (r, t)ei(ωt−K ·r) drdt e−iK ·r l ˜j l (ω).

(5.3)

l

Here, . ˜j l (ω) is the Fourier transform of . j l (t) = −evl (t). We approximated . r l (t) by . r l , because the time scale of electron motion is much longer than the oscillation period of X-rays, i.e., . r l (t) does not change while responding to X-rays. The Hamiltonian for the electron system interacting with electromagnetic fields is given by H=

.

] }2 ∑[ 1 { e pi + A(r i , t) + V (r i ) , 2m c i

(5.4)

5 Nonlinear X-Ray Spectroscopy

123

where . pi and . r i are the momentum and position operators for the .i-th electron, respectively, m is the mass of the electron, c is the speed of light in vacuum, and .V is the Coulomb potential of the nucleus. We ignored spin and interactions among electrons. Below, we employ the Coulomb gauge, where .∇ · A = 0, and write the vector potential: } } 1{ c { A(r, t) + A∗ (r, t) = ∈ E 0 ei(K ·r−ωt) − ∈ ∗ E 0∗ e−i(K ·r−ωt) , 2 2iω (5.5) where .∈ is the polarization vector and . E 0 is the amplitude of the electric field. We consider the perturbed system by . A, and divide Eq. (5.4) into the unperturbed Hamiltonian and the perturbing one: A(r, t) =

.

} ∑ { p2 i + V (r i ) , .H0 = 2m i } ∑{ e e2 2 ' .H = A (r i , t) . p · A(r i , t) + mc i 2mc2 i

(5.6) (5.7)

We used an identity, . p · A = A · p, in the Coulomb gauge. Now, we calculate the current density of the perturbed system using the standard perturbation theory. In this section, we will incorporate up to second-order terms. We consider two X-ray fields with the wave vectors and the angular frequencies, . K 1 , .ω1 , and . K 2 , .ω2 , respectively, and calculate the current density, .J 3 , with . K 3 , .ω3 induced by the nonlinear interaction between these waves and electrons. The perturbation term Eq. (5.7) becomes H' =

.

e2 e {A1 (r, t) + A2 (r, t)}2 . p · {A1 (r, t) + A2 (r, t)} + mc 2mc2

(5.8)

We calculate the Fourier transform of the expectation value of the current density, J = −(e/m) ( p + eA/c), to obtain the nonlinear polarizability:

.

∫

( e) ⟨Ψ(t)|e−iK 3 ·ˆr − m [ }] e{ A1 (ˆr , t) + A2 (ˆr , t) |Ψ(t)⟩eiω3 t dt. × pˆ + c

J˜ 3 (K 3 , ω3 ) =

.

(5.9)

Up to the second order, .|Ψ(t)⟩ is written as |Ψ(t)⟩ =

.

} ∑{ 2 δng + an(1, pA) (t) + an(1,A ) (t) + an(2, pA pA) (t) |n⟩e−iωn t , n

(5.10)

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where .an and .an(1,A ) are coefficients for the first-order perturbed state by the (2, pA pA) 2 . p · A and .A terms in Eq. (5.8), respectively, and .an is for the second-order perturbed state by the . p · A term. In the rest of this section, we focus on a specific nonlinear process with .ω3 = ω1 + ω2 , namely, sum-frequency generation (SFG). Below, we pick up typical matrix elements in Eq. (5.9), which satisfy .ω3 = ω1 + ω2 . 2

( e)e − A− r , t)|Ψ (1, p A1 ) (t)⟩, ⟨Ψ (0) (t)|e−iK 3 ·ˆr − 2 (ˆ c ( m − − e) (0) −iK 3 ·ˆr pˆ |Ψ (1,A1 A2 ) (t)⟩, − .⟨Ψ (t)|e ( m − − e) (0) −iK 3 ·ˆr pˆ |Ψ (2, p A1 p A2 ) (t)⟩, − .⟨Ψ (t)|e m ( e) − (1, p A+ r 1 ) (t)|e−iK 3 ·ˆ pˆ |Ψ (1, p A2 ) (t)⟩. − .⟨Ψ m

.

(5.11) (5.12) (5.13) (5.14)

Here the .± symbols on . A represent the sign of the angular frequency of the field, e.g., . A− 2 for .−ω2 . The relevant coefficients to Eqs. (5.11)–(5.14) are given by a

(1, p A− 1)

. n

a

− (1,A− 1 A2 )

. n

a

− (2, p A− 1 p A2 )

. n

(t) =

ie ⟨n|∈ 1 · pˆ eiK 1 ·ˆr |g⟩E 1 ei(ωng −ω1 )t 2mℏω1 ωng − ω1

(5.15)

(t) =

e2 ∈ 1 · ∈ 2 ⟨n|ei(K 1 +K 2 )·ˆr |g⟩E 1 E 2 ei(ωng −ω1 −ω2 )t , 4mℏω1 ω2 ωng − ω1 − ω2

(5.16)

e2 E 1 E 2 4m 2 ℏ2 ω1 ω2 ∑ ⟨n|∈ 2 · pˆ eiK 2 ·ˆr |l⟩⟨l|∈ 1 · pˆ eiK 1 ·ˆr |g⟩ ei(ωng −ω1 −ω2 )t × . (5.17) ωlg − ω1 ωng − ω1 − ω2 l

(t) = −

Here, .ℏ is the Planck constant divided by 2.π . Finally, we obtain the nonlinear current density for .ω3 = ω1 + ω2 by calculating Eq. (5.9) using Eqs. (5.10) and (5.15)–(5.17): .

˜ 3 , −K 1 , −K 2 , ω1 , ω2 ) : ∈ 1 ∈ 2 E 1 E 2 δ(ω3 − ω1 − ω2 ), J˜ 3 (K 3 , ω3 ) = −iπ ω3 β(K ˜ 3 , −K 1 , −K 2 , ω1 , ω2 ) : ∈ 1 ∈ 2 = β(K

ie3 m 3 ℏ2 ω1 ω2 ω3

(U + B),

(5.18)

where .β˜ is the Fourier transform of the second-order nonlinear polarizability. We( divide ) the contribution into two parts, .U and .B, where .U originates from 2 .( p A) A -type perturbations and .B from .( p A) ( p A) ( p A)-type.

5 Nonlinear X-Ray Spectroscopy

125

{ ∑ ⟨g|∈ 2 ei(K 2 −K 3 )·ˆr |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ .U = −mℏ ωng − ω1 n +

⟨g|eiK 1 ·ˆr ∈ 1 · pˆ |n⟩⟨n|∈ 2 ei(K 2 −K 3 )·ˆr |g⟩ ωng + ω1

+

⟨g|∈ 1 ei(K 1 −K 3 )·ˆr |n⟩⟨n|eiK 2 ·ˆr ∈ 2 · pˆ |g⟩ ωng − ω2

+

⟨g|eiK 2 ·ˆr ∈ 2 · pˆ |n⟩⟨n|∈ 1 ei(K 1 −K 3 )·ˆr |g⟩ ωng + ω2

+

∈ 1 · ∈ 2 ⟨g|e−iK 3 ·ˆr pˆ |n⟩⟨n|ei(K 1 +K 2 )·ˆr |g⟩ ωng − ω3

} ∈ 1 · ∈ 2 ⟨g|ei(K 1 +K 2 )·ˆr |n⟩⟨n|e−iK 3 ·ˆr pˆ |g⟩ + , ωng + ω3 { ∑ ⟨g|e−iK 3 ·ˆr pˆ |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |l⟩⟨l|eiK 2 ·ˆr ∈ 2 · pˆ |g⟩ .B = (ωng − ω3 )(ωlg − ω2 ) n,l +

⟨g|e−iK 3 ·ˆr pˆ |n⟩⟨n|eiK 2 ·ˆr ∈ 2 · pˆ |l⟩⟨l|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ (ωng − ω3 )(ωlg − ω1 )

+

⟨g|eiK 2 ·ˆr ∈ 2 · pˆ |l⟩⟨l|eiK 1 ·ˆr ∈ 1 · pˆ |n⟩⟨n|e−iK 3 ·ˆr pˆ |g⟩ (ωng + ω3 )(ωlg + ω2 )

+

⟨g|eiK 1 ·ˆr ∈ 1 · pˆ |l⟩⟨l|eiK 2 ·ˆr ∈ 2 · pˆ |n⟩⟨n|e−iK 3 ·ˆr pˆ |g⟩ (ωng + ω3 )(ωlg + ω1 )

+

⟨g|eiK 2 ·ˆr ∈ 2 · pˆ |l⟩⟨l|e−iK 3 ·ˆr pˆ |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ (ωng − ω1 )(ωlg + ω2 )

} ⟨g|eiK 1 ·ˆr ∈ 1 · pˆ |n⟩⟨n|e−iK 3 ·ˆr pˆ |l⟩⟨l|eiK 2 ·ˆr ∈ 2 · pˆ |g⟩ + . (ωng + ω1 )(ωlg − ω2 )

(5.19)

(5.20)

The second-order X-ray nonlinear process involves three waves. It is one of unique features of X-ray nonlinear optics that one wave can have longer wavelength than X-rays. However, we limit our discussion here to a special case, where all three waves are X-rays. Other situation, where one wave is in the optical region, will be discussed in Sect. 5.2.4. The three-X-rays case was first calculated by Freund and Levine [31], and then classically by Eisenberger and McCall [32]. We evaluate the first term of .U [Eq. (5.19)], U1 = −mℏ

.

∑ ⟨g|ei(K 2 −K 3 )·ˆr ∈ 2 |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ . ωng − ω1 n

(5.21)

We assume that the nonlinear crystal is composed of light elements, such as beryllium (Be) and carbon. This assumption is not only for convenience in the calculation but also valid for experiments, because the light elements are advantageous for absorp-

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tion. Considering .ωng « ω1 for the light elements, and ignoring the continuum state in the vacuum, we obtain U1 ≃

.

=

mℏ ∑ ⟨g|ei(K 2 −K 3 )·ˆr ∈ 2 |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ ω1 n mℏ ⟨g|e−iS·ˆr ∈ 2 (∈ 1 · pˆ )|g⟩, ω1

(5.22)

where we defined . S = K 3 − K 1 − K 2 . We used the completeness of .|n⟩, i.e., ∑ n |n⟩⟨n| = 1. The second term of Eq. (5.19) can be calculated in the same manner,

.

U2 = − mℏ

.

≃−

∑ ⟨g|eiK 1 ·ˆr ∈ 1 · pˆ |n⟩⟨n|∈ 2 ei(K 2 −K 3 )·ˆr |g⟩ ωng + ω1 n

mℏ ⟨g|∈ 2 (∈ 1 · pˆ )e−iS·ˆr |g⟩. ω1

(5.23)

We can further calculate .U1 + U2 using .∈ 1 · K 1 = 0 and . p = −iℏ∇, and find that we can relate it to a physical quantity: mℏ ∈ 2 ⟨g|[e−iS·ˆr , ∈ 1 · pˆ ]|g⟩ ω1 mℏ2 ∈ 2 (∈ 1 · S)ρ(S). ˜ = ω1

U1 + U2 =

.

(5.24)

Here, .ρ˜ represents the Fourier transform of .ρ(r). Repeating similar calculations for the remaining terms in Eq. (5.19), we finally obtain { U = −mℏ2 ρ(S) ˜

.

} 1 1 1 (∈ 1 · ∈ 2 )S − ∈ 2 (∈ 1 · S) − ∈ 1 (∈ 2 · S) . ω3 ω1 ω2

(5.25)

We find that .U is determined by the Fourier transform of .ρ when all three waves are ˜ X-rays. We note that the X-ray linear polarizability is also determined by .ρ. ˜ Because .U can be small We discuss which mechanism, .U or .B, dominates .β. when . S is small, the contribution of .U is smaller than .B in the visible region. However, it can be shown that the contribution of .B is negligible in the X-ray region. We evaluate the first term of Eq. (5.20), .B1 . In the case of light elements, it can be approximated as .eiK ·ˆr = 1 because .⟨K · r⟩ « 1. Then, .B1 can be approximated by B1 = −

∑

.

im 3 ωgn ωnl ωlg

⟨g|ˆr |n⟩⟨n|∈ 1 · rˆ |l⟩⟨l|∈ 2 · rˆ |g⟩ . (ωng − ω3 )(ωlg − ω2 )

(5.26)

Regarding the size of dipoles as the Bohr radius, .a0 , we evaluate .B1 : B1 ∼ −

.

∑

im 3 ωgn ωnl ωlg

a03 . ω3 ω2

(5.27)

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127

Major resonant frequencies are in the visible and ultraviolet range, which we set to Ωuv . We also set the X-ray frequencies, .ω2 and .ω3 , to .ωx . Note that .Ωuv /ωx ∼ 10−4 . The magnitude of .B1 is approximately given by

.

|B1 | ∼

.

ℏ6 Ω3uv . e6 ωx2

(5.28)

On the other hand, since .|S| is comparable to the wave vector of X-rays, .|U| can be approximated by |U| ∼ mℏ2 ρ/c. ˜

.

(5.29)

We assume .ρ˜ ∼ 10 in crystals, and obtain, | | 3 | B1 | | ∼ 1 (ℏΩuv ) 1 ∼ 10−8 , | . | U | α 3 mc2 (ℏω )2 ρ˜ x

(5.30)

where .α represents the fine structure constant. The above evaluation can be adapted to other terms of .B. Finally, we can conclude that .U dominates .β˜ in the X-ray region. Since the . r-operator appears three times in Eq. (5.26), the sign of .B1 changes under space inversion by . r ' = −r. The same applies to the other terms of .B. Therefore, when a crystal has inversion symmetry, .B is zero within the dipole approximation. On the other hand, .U has no such restriction [Eq. (5.25)]. Because the .U contribution dominates .β˜ in the X-ray region, second-order nonlinear-optical processes are possible even in nonlinear crystals with inversion symmetry.

5.2.2 X-Ray Second Harmonic Generation We consider X-ray second harmonic generation (SHG), which is a special case of SFG, i.e., .ω1 = ω2 . Since the linear and nonlinear polarizabilities are very small even in crystals, we can ignore local-field correction in the X-ray region. Then, the nonlinear polarizability can be regarded as a nonlinear susceptibility. Here, we assume that the crystal is infinitely large. Applying Eqs. (5.18) and (5.25) to the case of the crystal and replacing .ρ˜ with . FH0 /vc , the nonlinear susceptibility is expressed as ∑ (2) (2) .χ (r) = χ H eiH·r , (5.31) H

1 ∗ ˜ ∈ · β (H, ω1 , ω1 ) : ∈ 1 ∈ 1 2vc 2 cell } ie3 FH0 { ∗ ∈ 2 · H − 4(∈ ∗2 · ∈ 1 )(∈ 1 · H) , . = − 4 v 2 8m ω1 c

(2) .χ H

=

(5.32) (5.33)

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where . H, .β˜ cell , and .vc represent the reciprocal lattice vector, the nonlinear polarizability of the unit cell, and the volume of the unit cell, respectively. . FH0 is the crystal structure factor excluding the dispersion corrections and expressed as ∫ .

FH0 =

ρcell (r)e−iH·r dr,

(5.34)

where .ρcell is the electron density of the unit cell. Solving wave equations including the nonlinear terms, we obtain the intensity of SHG: ( ) Δk H l 32π 3 2 2 (2) 2 2 K 2 l |χ H | I1 sinc2 , (5.35) . I2 (l) = c 2 where .l is the crystal length, . I1 is the intensity of the fundamental wave, and we defined .Δk H = 2k 1 − k 2 + H. (5.36) The efficiency of SHG is maximum for .Δk H = 0, which is called a phase-matching condition. In the visible region, birefringence of the nonlinear crystal enables to achieve the phase-matching condition at . H = 0 (collinear geometry, i.e., .2k1 = k2 ). On the other hand, X-ray SHG is observed as nonlinear diffraction associated with . H, because it is difficult to use birefringence in the X-ray region. X-ray SHG was first observed by Shwartz et al. at SACLA [33]. The 7.3-keV monochromatic XFEL beam was focused down to 1.5-.µm size by the generalpurpose KB mirror (Table 5.1), producing a peak intensity of .1016 W/cm2 . A 0.48mm-thick (111) diamond was used for the nonlinear crystal, and aligned to satisfy the phase-matching condition with . H = (0, 2, −2) in the transmission geometry (Fig. 5.2a). Because the .011¯ reflection is forbidden for the fundamental wave, elastic scattering, which can produce strong noise, is suppressed to a very low level. The 14.6-keV SHG signal grows quadratically as the incident pulse energy (Fig. 5.2b) as expected from Eq. (5.35).

5.2.3 Parametric Down-Conversion Parametric down-conversion (PDC) is one of the second-order nonlinear-optical processes. In this process, an incident photon nonlinearly interacts with materials, and spontaneously splits into two photons. The incident photon is called “pump”, while the generated two photons are “signal” and “idler”. PDC can be regarded as an inverse process of SFG and SHG. X-ray PDC was first observed by Eisenberger et al. [32] with a Be crystal in 1971. The synchrotron PDC experiment was performed by Yoda et al. at the Photon Factory, KEK [34]. Figure 5.3 shows the experimental setup and the coincidence events as a function of the angle measured from the Bragg condition. They excited a 0.6 mm-

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129

Fig. 5.2 Second harmonic generation. a Schematic experimental setup. A diamond (111) thin plate placed on the focus was set to satisfy the phase-matching condition (bottom). b Second harmonic signal rate as a function of the incident pulse energy. The inset shows a histogram of the pulse energy. Reprinted with permission from Shwartz et al. [33]. Copyright (2014) by the American Physical Society

Fig. 5.3 X-ray parametric down-conversion. a Schematic experimental setup. b Rocking curves for the diamond 111 nonlinear diffraction (closed circles) and the linear (Bragg) diffraction (open circles). Reprinted with permission from Yoda et al. [34]. Copyright by IUCr Journals

thick diamond by 19-keV X-rays and detected 9.5-keV signal and idler photons. Because the signal and idler photons are generated at the same time, one can recognize the PDC event, when one simultaneously detects two photons. Pulsed nature of synchrotron radiation and fast avalanche photodiode (APD) detectors enabled simultaneous detection at a very low background level (Fig. 5.3b). Since 2000, X-ray PDC experiments have been performed by, for example, Adams et al. [35] and Shwartz et al. [36]. Unfortunately, no new information on the nonlinear medium is available with X-ray PDC, since it depends on . FH0 for the three X-ray waves case [Eq. (5.33)]. However, quantum-optical studies with X-ray PDC have reported successful quantum ghost imaging using entanglement between the signal and idler [37, 38] and generation of heralded X-ray photons [39].

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5.2.4 Sum Frequency Generation This section concludes by introducing an X-ray SFG experiment with infrared light. For this purpose, we first consider the X-ray nonlinear polarizability including one long-wavelength light [40]. We assume .ω1 « ω2,3 and .ωng « ω2,3 in Eq. (5.19), which can be applied to light elements. The first two terms in Eq. (5.19), { ∑ ⟨g|∈ 2 ei(K 2 −K 3 )·ˆr |n⟩⟨n|eiK 1 ·ˆr ∈ 1 · pˆ |g⟩ .U = − mℏ ωng − ω1 n } ⟨g|eiK 1 ·ˆr ∈ 1 · pˆ |n⟩⟨n|∈ 2 ei(K 2 −K 3 )·ˆr |g⟩ + , ωng + ω1

(5.37)

are dominant. Below, we take the dipole approximation and assume a phase-matching condition of . K 3 = K 1 + K 2 + H. (5.38) We calculate Eq. (5.37) and obtain U=−

.

m 2 ℏ2 ω1 ∈2 e2

∫

H · {α(r, ω1 ) · ∈ 1 } e−iH·r dr,

(5.39)

where .α(r, ω) is the linear polarizability for the long-wavelength light. When the response of the material is isotropic (.α = α I), .U can be expressed by U=−

.

m 2 ℏ2 ω1 (H · ∈ 1 ) ∈ 2 e2

∫

α(r, ω1 )e−iH·r dr.

(5.40)

From Eq. (5.18), the nonlinear polarizability is written as β˜ cell (H, ω1 , ω2 ) : ∈ 1 ∈ 2 = −

.

ie (H · ∈ 1 ) ∈ 2 α cell H (ω1 ), mω2 ω3

(5.41)

where .α cell H is defined with the polarizability of the unit cell, .αcell (r), as ∫ α cell H =

.

αcell (r)e−iH·r dr.

(5.42)

The SFG with infrared and X-ray photons was observed by Glover et al. at LCLS [41]. The infrared (1.55-eV) and X-ray (8-keV) pulses were simultaneously illuminated onto a diamond crystal (Fig. 5.4a). The SFG signal was observed at a photon energy of 8 keV + 1.55 eV, as shown in Fig. 5.4b. Figure 5.4c shows the SFG count depending on the polarization direction of the infrared light. Note that both the X-ray and infrared photons are linearly polarized. The signal is maximum when the angle

5 Nonlinear X-Ray Spectroscopy

131

Fig. 5.4 Sum frequency generation. a Schematic experimental setup. b SFG signal as a function of the energy analyzer angle. The inset shows the energy spectrum of the SFG signal. c Optical polarization dependence of the SFG signal. Reprinted with permission from Glover et al. [41]. Copyright by Springer Nature

of optical polarization vector is in the diffraction plane (.0◦ ) and is zero when it is normal to the diffraction plane (.±90◦ ), as expected from the polarization factor of . H · ∈ 1 in Eq. (5.41).

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5.3 Featuring Examples of Nonlinear X-Ray Spectroscopy 5.3.1 X-Ray Two-Photon Absorption Spectroscopy We begin this section by considering nonlinear absorption, i.e., X-ray multi-photon absorption. Young et al. first observed the X-ray multi-photon absorption process using soft X-ray FEL at LCLS [42]. Starting with this research, many experimental and theoretical studies on X-ray multi-photon absorption have been reported in the soft X-ray region [43]. The multi-photon absorption in the hard X-ray region was first reported for xenon (Xe) by Fukuzawa et al., where they observed .Xe24+ caused by absorbing five 5.5-keV photons [44]. Rudenko et al. observed .I48+ , which was accounted for by twenty-photon absorption in .CH3 I at 8.3 keV [45]. One important aim for investigation of hard X-ray multi-photon absorption is its close connection to single-molecule structure analysis with ultra-intense X-rays [23]. However, for the sake of simplicity, we deal with X-ray two-photon absorption (TPA) in this section. There are two processes in X-ray TPA; one is sequential (sequential TPA) where an atom absorbs two photons step by step, and the other is direct (direct TPA) where there is no real intermediate state. First, we discuss the difference between the two processes by comparing the TPA cross-section. The cross-section can be written as [46] σ (2) = σe(1) τe σg(1) ,

.

(5.43)

(1) where .σe,g represents the one-photon absorption (OPA) cross-section in the intermediate and ground states, respectively, and .τe is the lifetime of the intermediate state. The OPA cross-section is given by Fermi’s golden rule

σ (1) (ω) =

.

| 4π 2 αω2f i | |⟨k f |∈ · rˆ |1s⟩|2 δ(ω f i − ω), ω

(5.44)

where .α represents the fine structure constant. We are considering the process where a 1s electron is excited to the continuum state in the vacuum, .|k f ⟩, with an energy (1) are comparable for both the sequential and difference of .ℏω f i . The values of .σe,g direct absorption processes. The intermediate state in the sequential process may have a lifetime of .τe = 0.1−1 fs, while .τe for the direct process is determined by the uncertainty principle. For example, when the X-ray photon energy is 5 keV, we estimate .τe ∼ 1/ω = 1.3 × 10−19 s. It results in that .σ (2) of the direct TPA is about four orders of magnitude smaller than that of the sequential one. Figure 5.5 shows the sequential TPA creating a double core-hole state in krypton (Kr) atom by Tamasaku et al. at SACLA [47]. In this process, the first photon excites a 1s electron to the continuum, and then the second photon does the other 1s electron before the single core-hole state decays. Two weak peaks denoted by . K h α1,2 on the higher energy tail of . K α are hypersatellites, the X-ray fluorescence from the . K -shell double core-hole state. An intensity ratio of . K α to . K h α indicates that about 0.1%

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Fig. 5.5 Double core-hole creation. a X-ray fluorescence spectra of Kr measured with high (circles) and low intensities (triangles). b Pulse-energy dependence of the . K h α2 intensity. Reprinted with permission from Tamasaku et al. [47]. Copyright (2013) by the American Physical Society

of the excited states is in the double core-hole state. Figure 5.5b shows the . K h α2 intensity as a function of the incident X-ray pulse energy, which shows a quadratic dependence expected for the two-photon process. This result indicates that the Kr atoms in the . K -shell single core-hole state interact with X-rays within the lifetime as short as 0.24 fs [48]. Next, we consider direct TPA. For simplicity, we assume excitation by a single beam with . K and .ω. From Eq. (5.7), there can be two contributions to .σ (2) ; one originates from the second-order perturbation of . p · A, another from the first-order perturbation of .A2 . Using Fermi’s golden rule, we obtain | | |∑ ω ω ⟨k |∈ · rˆ |n⟩⟨n|∈ · rˆ |1s⟩ |2 f f n ni | | .σ (ω) = 8π α | | δ(ω f i − 2ω), (5.45) | | n ω ωni − ω | | 2 2 (2,A ) .σ (ω) = 32π 3re2 |⟨k f |v · rˆ |1s⟩| δ(ω f i − 2ω), (5.46) (2, pA)

3 2

where .re is the classical electron radius, and .v is a unit vector in the . K direction written as c (5.47) .v = K. 2ω From Eqs. (5.44) and (5.46), .σ (2,A ) may be expressed by 2

σ (2,A ) (ω, ω) = 2

.

πre2 (1) σ (2ω). αω

(5.48)

We consider which of . p · A and .A2 is dominant in the direct TPA. From Eqs. (5.43) and (5.48), this question can be solved by comparing .re2 /α to .σe(1) . In

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Fig. 5.6 Direct two-photon absorption. a Schematic experimental setup. b Pulse-energy dependence of Ge TPA fluorescence (circles). Dashed line indicates a quadratic function. Reprinted with permission from Tamasaku et al. [50]. Copyright by Springer Nature

the case of the direct TPA near absorption edges, .σe(1) may be evaluated to be −20 .∼ 10 cm.2 . Because this value is about three orders of magnitude larger than 2 −23 .r e /α = 1.1 × 10 cm2 , . p · A is dominant in the direct TPA near absorption edges. On the other hand, Varma et al. performed numerical calculations indicating that the contribution from .A2 to the direct TPA dominates above a photon energy of 6.8 keV in atomic hydrogen [49]. Since the absorption edge of hydrogen is 13.6 eV, this process is above-threshold ionization. An experiment of the direct TPA for germanium (Ge) was performed by Tamasaku et al. at SACLA [50]. The XFEL beam was focused to .110 × 140 nm2 by the two-stage focusing system (Table 5.1 and Fig. 5.6a), yielding a peak intensity of 19 .∼ 10 W/cm2 . The photon energy was set to 5.6 keV half of 11.2 keV, just above the Ge . K -shell binding energy of 11.1 keV. The . K α1 fluorescence was detected by a spectrometer consisting of an analyzer crystal and a charge-coupled device (CCD). Figure 5.6b shows the pulse-energy dependence of two-photon excited . K α1 fluorescence counts. The signal is approximately proportional to the square of the pulse energy, which indicates that the . K α1 fluorescence is due to the direct TPA. Ghimire et al. also observed the direct TPA in zirconium (Zr) [51]. The experimental values of .σ (2) for Ge and Zr were consistent with the theoretically predicted . Z −6 scaling low for hydrogenic ions. Since TPA requires two photons at the same time, it leads to a unique application of X-ray intensity correlator. Osaka et al. constructed an X-ray intensity correlator using a split-delay optics, successfully measured an X-ray autocorrelation function at 9 keV, and determined the pulse duration of only .7.6 ± 0.8 fs [52]. One can see deviation from the quadratic dependence in the high pulse-energy region. When the peak intensity is very high, many Ge atoms could be excited by the OPA process. A Ge atom with a core hole in the . L shell has a higher . K -shell binding

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Fig. 5.7 X-ray two-photon absorption spectroscopy. TPA (squares) and OPA (solid line) spectra of Cu metal. Reprinted with permission from Tamasaku et al. [53]. Copyright (2018) by the American Physical Society

energy due to reduced screening of the nucleus potential by the. L shell. Since the total energy transfer by TPA is just above the . K edge in the ground state, the L-shell corehole states cannot absorb two photons, suppressing TPA. Such a saturable trend will be discussed in Sect. 5.3.2. Although application of TPA to nonlinear spectroscopy may look straightforward, it requires an advanced experimental technique to measure the TPA signal at much lower peak intensities in order to study the undamaged electronic structure. The first TPA spectroscopy was performed for copper (Cu) metal by Tamasaku et al. at SACLA [53]. The peak intensity of .5 × 1015 W/cm2 was carefully determined by checking the threshold where the OPA spectrum started to change. The weak TPA signal due to the lower peak intensity was measured by considerable suppression of higher harmonics of XFEL radiation and a more efficient detecting system using the photon-energy resolution of CCD and omitting an analyzer crystal. The TPA spectrum is quite different from OPA as shown in Fig. 5.7. Not only the spectral shape but the absorption edge differed, which indicates that each spectrum represents different electronic states. The physical background can be readily understood by comparing Eqs. (5.44) and (5.45). The OPA process includes one dipole transition, while TPA does two. Because the dipole transition is allowed between states with different parities, OPA in Cu metal is mainly due to a transition from 1s to 4p (1s.→4p). In the case of TPA, the dominant transition is considered to be 1s.→3d. So, the spectral shape, which relates to a partial density of the final state, differs between the two spectra. The 3d orbital is closer to the nucleus than 4p, and therefore is more tightly bound when a core hole is created in the 1s orbital. This corresponds to the difference in the absorption edge. The sensitivity of the TPA spectroscopy to the 3d orbital is expected to serve for investigation of 3d transition metal compounds, which include various functional materials.

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5.3.2 Saturable Absorption When a sample is irradiated with high-intensity X-rays, the population of atoms in the ground state decreases. Above a certain threshold intensity, few atoms can absorb X-rays in the photon-energy region above the intrinsic absorption edges, and the sample becomes transparent. This phenomenon is called saturable absorption (SA). In the hard X-ray region, Yoneda et al. reported the first observation of SA for iron (Fe) using high-brilliant XFEL pulses with a pulse duration of .∼ 7 fs at SACLA [54]. Figure 5.8a shows the schematic experimental setup. The required X-ray intensity for SA, . ISA , can be roughly estimated from . ISA ∼ n atom ℏω/(μτc ), where .n atom , .ℏω, .μ, and .τc represent the number of atoms in a unit volume, the photon energy, the X-ray linear absorption coefficient, and the core-hole lifetime, respectively. They estimated . ISA to be .2 × 1019 W/cm2 . Such high-intensity XFEL pulse was obtained by the two-stage focusing system (Table 5.1). Figures 5.8b and c show the intensity dependence of absorption coefficient and transmission, respectively, around the Fe 20 . K edge. When the XFEL pulses with .10 W/cm2 were injected, the transmission at 7.13 keV (above the Fe . K edge) increased by a factor of 10 compared to the normal absorption condition (called cold state). Then, SA at the aluminum . K edge has been also observed by Rackstraw et al. at LCLS [55]. In contrast to the above study at SACLA, they discussed relatively slow phenomena in .∼ 100-fs pulse duration of LCLS, such as change in the . L-shell population and collisional ionization in addition to the mechanism discussed above. The time during which SA occurs is determined by .τc . Thus, X-ray pulse duration can be shortened by using SA as an ultrafast shutter, which was demonstrated by Inoue et al. at SACLA [56]. The 9-keV XFEL beam was focused down to 2 .130 × 165 nm size by the 100-exa focusing system (Table 5.1), producing a peak 19 intensity of.10 W/cm2 . The pulse duration was evaluated by an intensity correlation technique [57]. Figure 5.9a shows the result of SA in Cu. The transmittance at 9 keV, above the Cu . K edge (8.979 keV), increases as the incident X-ray fluence. The pulse duration after the saturable absorber decreases by .∼ 35% (Fig. 5.9b).

5.3.3 Atomic X-Ray Laser Most atoms are in the core-hole state under the SA condition. At this instant, there are no atoms in the relaxed states, which will be reached by emitting X-ray fluorescence. Thus, high-intensity X-rays can create population inversion, which is prerequisite for lasing by the conventional scheme. Unlike in the visible region, where a specific threeor four-level system is required to create the population inversion, in the X-ray region, any element can act as the lasing medium. This is because X-ray photon energy can excite inner-shell electrons to the continuum and the absorption photon energy is different from the emission. In addition, the relayed states reached by emission are not the ground state. This atomic X-ray laser was realized in the soft X-ray region

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Fig. 5.8 Saturable absorption. a Schematic experimental setup. b Absorption coefficients for the Fe foil measured in the vicinity of the Fe . K edge (7113 eV) at different X-ray intensities. c Incident intensity dependence of the transmission .T normalized to the cold condition .Tcold at 7130 eV (squares). Reprinted with permission from Yoneda et al. [54]. Copyright by Springer Nature

Fig. 5.9 Pulse shortening by saturable absorber. a Transmittance of 10-.µm-thick Cu foil at 9 keV. Dotted line shows the transmittance in the cold state. b Pulse duration of the incidence and transmission. Reprinted with permission from Inoue et al. [56]. Copyright (2021) by the American Physical Society

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Fig. 5.10 X-ray atomic laser. a, b Schematic experimental setup. XFEL pulses are generated with undulators in single-color mode (the ASE experiment) (a) or two-color mode (the seeding experiment) (b). c, d Output energy of. K α emission as a function of the pump pulse-energy density. c The output energies (blue squares) in the ASE experiment. d The output energies in the seeding experiment (red squares) and those in the non-seeding condition (gray squares). Green squares represent typical energies of the input seeding pulse. e, f Typical spectra of amplified . K α emission in the ASE experiment (e) and the seeding experiment (f). Reprinted with permission from Yoneda et al. [59]. Copyright by Springer Nature

using neon gas for the first time by Rohringer et al. [58]. In the hard X-ray region, Yoneda et al. reported the atomic X-ray laser with Cu at SACLA [59]. Figure 5.10 shows their experimental setup and results. The two-stage focusing system (Table 5.1) generated a 120-nm spot. XFEL pulses with an intensity of about .1019 W/cm2 were tuned to the Cu . K edge and produced sufficient population inversion to generate strong amplified spontaneous emission (ASE) at the Cu . K α lines. In addition, the two-color mode [5] was used to stabilize the lasing process, where one color tuned to the Cu . K edge (pumping) and the other for the seed (starting) light tuned to the Cu . K α. The output power of the seeded atomic laser is larger than that of ASE, as shown in Fig. 5.10d. Furthermore, the bandwidth of the seeded atomic laser (1.7 eV at minimum) is narrower than that of ASE, and its photon energy can be switched between . K α1 and . K α2 by tuning the photon energy of seed light.

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Other unconventional X-ray radiation process is superfluorescence, where coherent radiation is emitted from atoms excited by shorter pulses than the lifetime of the excited state. This quantum optical emission was reported in the vacuum ultraviolet (VUV) region by Harries et al. [60] and for the .57 Fe Mössbauer transition by Chumakov et al. [61].

5.3.4 Stimulated X-Ray Emission Spectroscopy The ASE process found an interesting spectroscopic application, namely, stimulated X-ray emission spectroscopy (SXES). The first SXES in short wavelengths was performed in the VUV region for silicon [62]. In the hard X-ray region, Kroll et al. observed the stimulated . K α emission from solutions of manganese (Mn) complexes at LCLS [63]. They have subsequently succeeded in observing the seeded stimulated emission for the Mn . Kβ [64]. In general, the . Kβ emission is chemically more sensitive but an order of magnitude weaker than the. K α emission. They made use of a two-color XFEL to detect the weak signal of the stimulated . Kβ emission (Figs. 5.11a and b). They used NaMnO.4 solution to perform the Mn. Kβ SXES. The photon energy of the pump pulse was set above the Mn . K edge to 6.6 keV, and the seed pulse were tuned to the . Kβ line at 6.49 keV. The size and peak intensity of pump pulse were 20 .∼ 150 nm and.10 W/cm2 , respectively. Figure 5.11c shows spectra detected around the Mn . Kβ with sample (red) and without (blue). The no-sample curve reflects the averaged SASE spectrum, while the spectrum with the sample indicates the seeded . Kβ SXES signal with a single sharp peak. Comparing the observed seeded stimulated . Kβ emission signal with the spontaneous . Kβ emission into the same solid angle, they estimated a signal enhancement to be more than .105 . Since . Kβ X-ray emission spectroscopy is sensitive to the spin state, oxidation, and ligand environment of 3d transition metal, this demonstration of the . Kβ SXES opens up rich avenues for investigation in chemistry and materials science.

5.3.5 X-Ray Transient Grating Spectroscopy This section concludes with an introduction to X-ray transient grating (XTG) spectroscopy. Transient grating (TG) is one of four-wave-mixing processes where two simultaneous excitation waves generate an interference pattern within a sample. The pattern works as a grating of spatially modulated excitation, which can diffract a controlled time-delayed probe wave. Rouxel et al. demonstrated a TG spectroscopy with XFEL pulses [65]. Short wavelength of X-rays can generate gratings with periods down to nanometer scales, which allows observation of the behavior of excited bulk materials with large momentum transfer, high spatial resolution, and element selectivity.

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Fig. 5.11 Stimulated X-ray emission spectroscopy. a Schematic experimental setup using a partially transparent . K α analyzer and . Kβ analyzer for simultaneous detection of . K α ASE and seeded stimulated . Kβ emission signals. b Schematics of the stimulated emission processes. c The seeded stimulated . Kβ emission spectrum (red), and averaged seed spectrum (blue). Reprinted with permission from Kroll et al. [64]. Copyright (2020) by the American Physical Society

Fig. 5.12 X-ray transient grating spectroscopy. a Schematic experimental setup. b Delay time (.t) dependence of the XTG signal (red circles). Reprinted with permission from Rouxel et al. [65]. Copyright by Springer Nature

Figure 5.12a shows the setup of XTG spectroscopy at SwissFEL [65]. A diamond phase grating with a period of 960 nm diffracted a 7.1-keV XFEL beam to generate an interference pattern with a period of 770 nm in .Bi4 Ge3 O12 (BGO). The size and peak intensity of XFEL beam at the sample were .250 × 150 µm2 and .1011 W/cm2 , respectively. They successfully detected 400-nm probe laser pulses diffracted by TG. Figure 5.12b shows a delay time (.t) dependence of the intensity of diffracted optical wave (XTG signal). The XTG signal rises at the temporal overlap between the excitation and probe waves, i.e.,.t = 0, followed by decay and oscillations. This result indicates generation of a photoexcited A.1 coherent phonon with a frequency of .2.6 ± 0.1 THz in BGO. This research demonstrates usefulness of the XTG spectroscopy for probing properties of bulk materials. The XTG spectroscopy would be a powerful method for investigation of materials science, such as ultrafast dynamics of lattice, charge, and spin states on the nanoscale.

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5.4 Summary This chapter begins with a theoretical description of nonlinear optics in the hard Xray region and provides several noteworthy studies. As mentioned at the beginning of this chapter, we hope the readers have understood that there are many nonlinear phenomena unique to the hard X-ray region. The fact that the .A2 term in the nonrelativistic Hamiltonian plays important role in the linear X-ray scattering, i.e., the Thomson scattering, also has a close relation to the nonlinear scattering. The non-resonant nature of the .A2 interaction makes X-rays see the electron itself, and leads to structural analysis. The dominant terms in the second-order nonlinear susceptibility include the .A2 term, which allows PDC and SFG to unveil the “structure” of the electronic distribution responding to the long-wavelength light. On the other hand, the imaginary part of the linear and the third-order nonlinear susceptibility is determined by the . p · A contribution. Therefore, the linear and the two-photon absorption has the same mechanism as that in the optical region. Only deep in the above-threshold ionization region, the .A2 term may exceed the . p · A absorption mechanism. It is still not clear which term decides the third-order nonlinear susceptibility. The third-order nonlinear process includes four photons, which are connected by various combination of the . p · A and .A2 interactions. Futhermore, one or two of the four photons can be in the longer wavelength region from infrared to soft X-rays, making this problem very complicated. Many basic nonlinear processes have been observed as we saw in this chapter. One of the most important nonlinear processes to be observed might be coherent Raman scattering, which includes stimulated Raman scattering and coherent antistokes Raman scattering. Both are the third-order nonlinear process, however, the former relies on the imaginary part of the third-order nonlinear susceptibility, while the latter on the magnitude. The coherent Raman scattering can be used for nonlinear spectroscopy, and could enhance the sensitivity of the spontaneous Raman spectroscopy. Experimentally, the coherent Raman scattering requires two monochromatic and intense X-ray beams with controllable photon-energy separation around an electronic excitation energy of interest. Since the third-order nonlinear process is observable only in high-intensity region, it may compete with the sample damage. This can be relaxed by using the Fourier-limited pulse, which has the highest intensity at given pulse energy and bandwidth. Advance of XFELs would enable X-ray nonlinear spectroscopy to become a versatile tool to resolve physical, chemical, and biological problems. Acknowledgements We thank Prof. Iwao Matsuda for the valuable discussion.

References 1. A.M. Kondratenko, E.L. Saldin, Generation of coherent radiation by a relativistic electron beam in an ondulator. Part. Accel. 10, 207 (1980)

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2. R. Bonifacio, C. Pellegrini, L.M. Narducci, Collective instabilities and high-gain regime in a free electron laser. Opt. Commun. 50, 373 (1984) 3. J. Amann, W. Berg, V. Blank, F.-J. Decker, Y. Ding, P. Emma, Y. Feng, J. Frisch, D. Fritz, J. Hastings et al., Demonstration of self-seeding in a hard-X-ray free-electron laser. Nat. Photonics 6, 693 (2012) 4. I. Inoue, T. Osaka, T. Hara, T. Tanaka, T. Inagaki, T. Fukui, S. Goto, Y. Inubushi, H. Kimura, R. Kinjo et al., Generation of narrow-band X-ray free-electron laser via reflection self-seeding. Nat. Photonics 13, 319 (2019) 5. T. Hara, Y. Inubushi, T. Katayama, T. Sato, H. Tanaka, T. Tanaka, T. Togashi, K. Togawa, K. Tono, M. Yabashi et al., Two-colour hard X-ray free-electron laser with wide tunability. Nat. Commun. 4, 2919 (2013) 6. P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, A. Brachmann, P. Bucksbaum, R. Coffee, F.-J. Decker et al., First lasing and operation of an ångstrom-wavelength free-electron laser. Nat. Photonics 4, 641 (2010) 7. T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi, T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa et al., A compact X-ray free-electron laser emitting in the sub-ångström region. Nat. Photonics 6, 540 (2012) 8. H.-S. Kang, C.-K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S.Y. Baek, H.-J. Choi, G. Mun et al., Hard X-ray free-electron laser with femtosecond-scale timing jitter. Nat. Photonics 11, 708 (2017) 9. E. Prat, R. Abela, M. Aiba, A. Alarcon, J. Alex, Y. Arbelo, C. Arrell, V. Arsov, C. Bacellar, C. Beard et al., A compact and cost-effective hard X-ray free-electron laser driven by a highbrightness and low-energy electron beam. Nat. Photonics 14, 748 (2020) 10. W. Decking, S. Abeghyan, P. Abramian, A. Abramsky, A. Aguirre, C. Albrecht, P. Alou, M. Altarelli, P. Altmann, K. Amyan et al., A MHz-repetition-rate hard X-ray free-electron laser driven by a superconducting linear accelerator. Nat. Photonics 14, 391 (2020) 11. B. Erk, R. Boll, S. Trippel, D. Anielski, L. Foucar, B. Rudek, S.W. Epp, R. Coffee, S. Carron, S. Schorb et al., Imaging charge transfer in iodomethane upon x-ray photoabsorption. Science 345, 288 (2014) 12. K.H. Kim, J.G. Kim, S. Nozawa, T. Sato, K.Y. Oang, T.W. Kim, H. Ki, J. Jo, S. Park, C. Song et al., Direct observation of bond formation in solution with femtosecond X-ray scattering. Nature 518, 385 (2015) 13. J.G. Kim, S. Nozawa, H. Kim, E.H. Choi, T. Sato, T.W. Kim, K.H. Kim, H. Ki, J. Kim, M. Choi et al., Mapping the emergence of molecular vibrations mediating bond formation. Nature 582, 520 (2020) 14. M. Buzzi, M. Först, R. Mankowsky, A. Cavalleri, Probing dynamics in quantum materials with femtosecond X-rays. Nat. Rev. Mater. 3, 299 (2018) 15. T. Suzuki, Y. Kubota, A. Nakamura, T. Shimojima, K. Takubo, S. Ito, K. Yamamoto, S. Michimae, H. Sato, H. Hiramatsu et al., Ultrafast optical stress on BaFe.2 As.2 . Phys. Rev. Res. 3, 033222 (2021) 16. Y. Kubota, Y. Tanaka, T. Togashi, T. Ebisu, K. Tamasaku, H. Osawa, T. Wada, O. Sugino, I. Matsuda, M. Yabashi, Suppression of atomic displacive excitation in photo-induced A.1g phonon mode of bismuth unveiled by low-temperature time-resolved x-ray diffraction. Appl. Phys. Lett. 122, 092201 (2023) 17. S. Gerber, H. Jang, H. Nojiri, S. Matsuzawa, H. Yasumura, D.A. Bonn, R. Liang, W.N. Hardy, Z. Islam, A. Mehta et al., Three-dimensional charge density wave order in YBa.2 Cu.3 O.6.67 at high magnetic fields. Science 350, 949 (2015) 18. H. Jang, S. Song, T. Kihara, Y. Liu, S.-J. Lee, S.-Y. Park, M. Kim, H.-D. Kim, G. Coslovich, S. Nakata et al., Characterization of photoinduced normal state through charge density wave in superconducting YBa.2 Cu.3 O.6.67 . Science Advances 8, eabk0832 (2022) 19. A. Ikeda, Y.H. Matsuda, X. Zhou, S. Peng, Y. Ishii, T. Yajima, Y. Kubota, I. Inoue, Y. Inubushi, K. Tono et al., Generating 77 T using a portable pulse magnet for single-shot quantum beam experiments. Appl. Phys. Lett. 120, 142403 (2022)

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20. D. Kraus, J. Vorberger, A. Pak, N.J. Hartley, L.B. Fletcher, S. Frydrych, E. Galtier, E.J. Gamboa, D.O. Gericke, S.H. Glenzer et al., Formation of diamonds in laser-compressed hydrocarbons at planetary interior conditions. Nat. Astron. 1, 606 (2017) 21. T. Okuchi, Y. Seto, N. Tomioka, T. Matsuoka, B. Albertazzi, N.J. Hartley, Y. Inubushi, K. Katagiri, R. Kodama, T.A. Pikuz et al., Ultrafast olivine-ringwoodite transformation during shock compression. Nat. Commun. 12, 4305 (2021) 22. Z. He, M. Rödel, J. Lütgert, A. Bergermann, M. Bethkenhagen, D. Chekrygina, T.E. Cowan, A. Descamps, M. French, E. Galtier et al., Diamond formation kinetics in shock-compressed C—H—O samples recorded by small-angle x-ray scattering and x-ray diffraction. Science Advances 8, eabo0617 (2022) 23. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, J. Hajdu, Potential for biomolecular imaging with femtosecond X-ray pulses. Nature 406, 752 (2000) 24. M. Suga, F. Akita, K. Hirata, G. Ueno, H. Murakami, Y. Nakajima, T. Shimizu, K. Yamashita, M. Yamamoto, H. Ago et al., Native structure of photosystem II at 1.95 Å resolution viewed by femtosecond X-ray pulses. Nature 517, 99 (2015) 25. M. Suga, F. Akita, M. Sugahara, M. Kubo, Y. Nakajima, T. Nakane, K. Yamashita, Y. Umena, M. Nakabayashi, T. Yamane et al., Light-induced structural changes and the site of O=O bond formation in PSII caught by XFEL. Nature 543, 131 (2017) 26. T. Kimura, Y. Joti, A. Shibuya, C. Song, S. Kim, K. Tono, M. Yabashi, M. Tamakoshi, T. Moriya, T. Oshima et al., Imaging live cell in micro-liquid enclosure by X-ray laser diffraction. Nat. Commun. 5, 3052 (2014) 27. P. Kirkpatrick, A.V. Baez, Formation of optical images by X-rays. J. Opt. Soc. Am. 38, 766 (1948) 28. H. Mimura, H. Yumoto, S. Matsuyama, T. Koyama, K. Tono, Y. Inubushi, T. Togashi, T. Sato, J. Kim, R. Fukui et al., Generation of 10.20 W cm.−2 hard X-ray laser pulses with two-stage reflective focusing system. Nat. Commun. 5, 3539 (2014) 29. K. Tono, T. Togashi, Y. Inubushi, T. Sato, T. Katayama, K. Ogawa, H. Ohashi, H. Kimura, S. Takahashi, K. Takeshita et al., Beamline, experimental stations and photon beam diagnostics for the hard x-ray free electron laser of SACLA. New J. Phys. 15, 083035 (2013) 30. H. Yumoto, Y. Inubushi, T. Osaka, I. Inoue, T. Koyama, K. Tono, M. Yabashi, H. Ohashi, Nanofocusing optics for an X-ray free-electron laser generating an extreme intensity of 100 EW/cm.2 using total reflection mirrors. Appl. Sci. 10, 2611 (2020) 31. I. Freund, B.F. Levine, Parametric conversion of X rays. Phys. Rev. Lett. 23, 854 (1969) 32. P. Eisenberger, S.L. McCall, X-ray parametric conversion. Phys. Rev. Lett. 26, 684 (1971) 33. S. Shwartz, M. Fuchs, J.B. Hastings, Y. Inubushi, T. Ishikawa, T. Katayama, D.A. Reis, T. Sato, K. Tono, M. Yabashi et al., X-ray second harmonic generation. Phys. Rev. Lett. 112, 163901 (2014) 34. Y. Yoda, T. Suzuki, X.-W. Zhang, K. Hirano, S. Kikuta, X-ray parametric scattering by a diamond crystal. J. Synchrotron Radiat. 5, 980 (1998) 35. B. Adams, P. Fernandez, W.-K. Lee, G. Materlik, D.M. Mills, D.V. Novikov, Parametric down conversion of X-ray photons. J. Synchrotron Radiat. 7, 81 (2000) 36. S. Shwartz, R.N. Coffee, J.M. Feldkamp, Y. Feng, J.B. Hastings, G.Y. Yin, S.E. Harris, X-ray parametric down-conversion in the Langevin regime. Phys. Rev. Lett. 109, 013602 (2012) 37. A. Schori, D. Borodin, K. Tamasaku, S. Shwartz, Ghost imaging with paired x-ray photons. Phys. Rev. A 97, 063804 (2018) 38. S. Sofer, E. Strizhevsky, A. Schori, K. Tamasaku, S. Shwartz, Quantum enhanced X-ray detection. Phys. Rev. X 9, 031033 (2019) 39. E. Strizhevsky, D. Borodin, A. Schori, S. Francoual, R. Röhlsberger, S. Shwartz, Efficient interaction of heralded X-ray photons with a beam splitter. Phys. Rev. Lett. 127, 013603 (2021) 40. K. Tamasaku, K. Sawada, E. Nishibori, T. Ishikawa, Visualizing the local optical response to extreme-ultraviolet radiation with a resolution of .λ/380. Nat. Phys. 7, 705 (2011) 41. T.E. Glover, D.M. Fritz, M. Cammarata, T.K. Allison, S. Coh, J.M. Feldkamp, H. Lemke, D. Zhu, Y. Feng, R.N. Coffee et al., X-ray and optical wave mixing. Nature 488, 603 (2012)

144

Y. Kubota and K. Tamasaku

42. L. Young, E.P. Kanter, B. Krässig, Y. Li, A.M. March, S.T. Pratt, R. Santra, S.H. Southworth, N. Rohringer, L.F. DiMauro et al., Femtosecond electronic response of atoms to ultra-intense X-rays. Nature 466, 56 (2010) 43. C. Bostedt, J.D. Bozek, P.H. Bucksbaum, R.N. Coffee, J.B. Hastings, Z. Huang, R.W. Lee, S. Schorb, J.N. Corlett, P. Denes et al., Ultra-fast and ultra-intense x-ray sciences: first results from the Linac Coherent Light Source free-electron laser. J. Phys. B At. Mol. Opt. Phys. 46, 164003 (2013) 44. H. Fukuzawa, S.-K. Son, K. Motomura, S. Mondal, K. Nagaya, S. Wada, X.-J. Liu, R. Feifel, T. Tachibana, Y. Ito et al., Deep inner-shell multiphoton ionization by intense X-ray free-electron laser pulses. Phys. Rev. Lett. 110, 173005 (2013) 45. A. Rudenko, L. Inhester, K. Hanasaki, X. Li, S.J. Robatjazi, B. Erk, R. Boll, K. Toyota, Y. Hao, O. Vendrell et al., Femtosecond response of polyatomic molecules to ultra-intense hard X-rays. Nature 546, 129 (2017) 46. P. Lambropoulos, X. Tang, Multiple excitation and ionization of atoms by strong lasers. J. Opt. Soc. Am. B 4, 821 (1987) 47. K. Tamasaku, M. Nagasono, H. Iwayama, E. Shigemasa, Y. Inubushi, T. Tanaka, K. Tono, T. Togashi, T. Sato, T. Katayama et al., Double core-hole creation by sequential attosecond photoionization. Phys. Rev. Lett. 111, 043001 (2013) 48. M.O. Krause, J.H. Oliver, Natural widths of atomic. K and. L levels,. K α X-ray lines and several . K L L Auger lines. J. Phys. Chem. Ref. Data 8, 329 (1979) 49. H.R. Varma, M.F. Ciappina, N. Rohringer, R. Santra, Above-threshold ionization in the x-ray regime. Phys. Rev. A 80, 053424 (2009) 50. K. Tamasaku, E. Shigemasa, Y. Inubushi, T. Katayama, K. Sawada, H. Yumoto, H. Ohashi, H. Mimura, M. Yabashi, K. Yamauchi et al., X-ray two-photon absorption competing against single and sequential multiphoton processes. Nat. Photonics 8, 313 (2014) 51. S. Ghimire, M. Fuchs, J. Hastings, S.C. Herrmann, Y. Inubushi, J. Pines, S. Shwartz, M. Yabashi, D.A. Reis, Nonsequential two-photon absorption from the . K shell in solid zirconium. Phys. Rev. A 94, 043418 (2016) 52. T. Osaka, I. Inoue, J. Yamada, Y. Inubushi, S. Matsumura, Y. Sano, K. Tono, K. Yamauchi, K. Tamasaku, M. Yabashi, Hard x-ray intensity autocorrelation using direct two-photon absorption. Phys. Rev. Res. 4, L012035 (2022) 53. K. Tamasaku, E. Shigemasa, Y. Inubushi, I. Inoue, T. Osaka, T. Katayama, M. Yabashi, A. Koide, T. Yokoyama, T. Ishikawa, Nonlinear spectroscopy with X-ray two-photon absorption in metallic copper. Phys. Rev. Lett. 121, 083901 (2018) 54. H. Yoneda, Y. Inubushi, M. Yabashi, T. Katayama, T. Ishikawa, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, Saturable absorption of intense hard X-rays in iron. Nat. Commun. 5, 5080 (2014) 55. D.S. Rackstraw, O. Ciricosta, S.M. Vinko, B. Barbrel, T. Burian, J. Chalupský, B.I. Cho, H.-K. Chung, G.L. Dakovski, K. Engelhorn et al., Saturable absorption of an X-ray free-electron-laser heated solid-density aluminum plasma. Phys. Rev. Lett. 114, 015003 (2015) 56. I. Inoue, Y. Inubushi, T. Osaka, J. Yamada, K. Tamasaku, H. Yoneda, M. Yabashi, Shortening X-ray pulse duration via saturable absorption. Phys. Rev. Lett. 127, 163903 (2021) 57. I. Inoue, K. Tamasaku, T. Osaka, Y. Inubushi, M. Yabashi, Determination of X-ray pulse duration via intensity correlation measurements of X-ray fluorescence. J. Synchrotron Radiat. 26, 2050 (2019) 58. N. Rohringer, D. Ryan, R.A. London, M. Purvis, F. Albert, J. Dunn, J.D. Bozek, C. Bostedt, A. Graf, R. Hill et al., Atomic inner-shell X-ray laser at 1.46 nanometres pumped by an X-ray free-electron laser. Nature 481, 488 (2012) 59. H. Yoneda, Y. Inubushi, K. Nagamine, Y. Michine, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, T. Katayama et al., Atomic inner-shell laser at 1.5-ångström wavelength pumped by an X-ray free-electron laser. Nature 524, 446 (2015) 60. J.R. Harries, H. Iwayama, S. Kuma, M. Iizawa, N. Suzuki, Y. Azuma, I. Inoue, S. Owada, T. Togashi, K. Tono et al., Superfluorescence, free-induction decay, and four-wave mixing: propagation of free-electron laser pulses through a dense sample of helium ions. Phys. Rev. Lett. 121, 263201 (2018)

5 Nonlinear X-Ray Spectroscopy

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61. A.I. Chumakov, A.Q.R. Baron, I. Sergueev, C. Strohm, O. Leupold, Y. Shvyd’ko, G.V. Smirnov, R. Rüffer, Y. Inubushi, M. Yabashi et al., Superradiance of an ensemble of nuclei excited by a free electron laser. Nat. Phys. 14, 261 (2018) 62. M. Beye, S. Schreck, F. Sorgenfrei, C. Trabant, N. Pontius, C. Schüßler-Langeheine, W. Wurth, A. Föhlisch, Stimulated X-ray emission for materials science. Nature 501, 191 (2013) 63. T. Kroll, C. Weninger, R. Alonso-Mori, D. Sokaras, D. Zhu, L. Mercadier, V.P. Majety, A. Marinelli, A. Lutman, M.W. Guetg et al., Stimulated X-ray emission spectroscopy in transition metal complexes. Phys. Rev. Lett. 120, 133203 (2018) 64. T. Kroll, C. Weninger, F.D. Fuller, M.W. Guetg, A. Benediktovitch, Y. Zhang, A. Marinelli, R. Alonso-Mori, A. Aquila, M. Liang et al., Observation of seeded Mn K.β stimulated X-ray emission using two-color X-ray free-electron laser pulses. Phys. Rev. Lett. 125, 037404 (2020) 65. J.R. Rouxel, D. Fainozzi, R. Mankowsky, B. Rösner, G. Seniutinas, R. Mincigrucci, S. Catalini, L. Foglia, R. Cucini, F. Döring et al., Hard X-ray transient grating spectroscopy on bismuth germanate. Nat. Photonics 15, 499 (2021)

Chapter 6

Future Prospects Iwao Matsuda, Craig P. Schwartz, Walter S. Drisdell, and Ryuichi Arafune

Abstract This chapter describes the future prospects of research with nonlinear X-ray spectroscopy, based on recent achievements at the XFEL facilities. The measurement techniques have been challenging the cutting edge and, at the same time, the experimental environment has become user-friendly for academic and industrial research. Nonlinear X-ray spectroscopy will soon become one of the standard analysis methods for materials science.

6.1 Toward Multi-dimensional Spectroscopy Some of the most powerful techniques in all of spectroscopy, not merely nonlinear spectroscopy, are the so-called 2-dimensional techniques, including 2D-IR, 2DNMR, and 2D-visible [1]. In these techniques, a material is pumped by two or more separate pulses and the energy transfer between the associated states is observed as a function of time between the pulses. This allows for detailed understanding of ultrafast energy flow and has been used to provide insight into phenomena such as photosynthesis, and in the case of 2D-NMR is even used to solve protein structures [2]. In general the  spectrum can be thought of as a 2D plot of a function, 2Dspectrum ω1 , tdelay , ω2 , where ω1 is the first frequency used, ω2 is the second I. Matsuda The Institute for Solid State Physics, The University of Tokyo, Chiba, Japan e-mail: [email protected] C. P. Schwartz Nevada Extreme Laboratory, University of Nevada, Las Vegas, USA e-mail: [email protected] W. S. Drisdell (B) Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA e-mail: [email protected] R. Arafune Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8_6

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frequency used, and t delay is the time delay between the two pulses. In this way, at a time delay of tdelay the signal that is generated when pumped with frequencies ω1 on the x-axis and ω2 on the y-axis can be plotted, giving a 2D plot with signal intensity the third dimension. This generally enables one to study the energy transfer that occurs between the two pulses in time, for instance in 2D-IR one might watch the energy transfer between two different modes of the water molecule and use this to understand how energy transferred between molecules. In the case of 2D-visible one can watch how the energy redistributes between electronic states. This approach would be particularly powerful in the case of X-ray spectroscopy because it will be possible to watch energy transfer from one particular atom into another by exploiting the atomic sensitivity of X-rays, enabling one to track intramolecular energy flow with unprecedented resolution and understanding. If this could be extended to the X-ray regime, it would provide unprecedented insight into the flow of energy on an atom specific basis and at time scales potentially as short as attoseconds. Such measurements are therefore a motivating factor for the design of future X-ray light sources, and some thought has been put into how to ultimately achieve these measurements. Unfortunately, extending these measurements from the IR to the X-ray will require overcoming numerous logistical difficulties. In an ideal case, as these are coherent spectroscopies, one would like to have a source that is spatially and temporally coherent, which may be true of externally seeded sources such as FERMI but is not true of SASE sources. This benefits all measurements by providing cleaner temporal and spectral resolution but is particularly important for types of spectroscopy where coherence is important, including phenomena such as four wave mixing. Perhaps this is most important for 2D spectroscopy, where it is difficult to achieve such measurements otherwise. There has been progress improving coherence at seeded FELs such as FERMI [3]. Besides coherence, phase stability is also desired. One would like multiple pulses to be phase locked to each other, to eliminate problems with relative phase shifts between pulses as time is scanned. This means the phase of each pulse’s arrival can be controlled besides just the delay time between the pulses. Phase instability would lead to another variable and could make techniques such as 2D spectroscopy significantly more challenging to understand if present. This was demonstrated at a seeded FEL by frequency interferometry [4]. Phase stability has also been accomplished both by using high spatial coherent beams that are then split, [5] or by using clever seeding schemes [6]. As X-ray pulses are inherently ionizing, one must also worry about the core hole lifetime. These can range from tens of femtoseconds to attoseconds depending on the energy of the core hole being probed, with higher energy core holes being more unstable and decaying more rapidly [7]. As the Auger decay process will inherently cause large changes in the system, having a probe pulse occur before Auger emission can occur is a desirable property for 2D-spectroscopy. This would require ultrafast pulses on the time scales of attoseconds, with tunable delays ideally ranging from attoseconds to femtoseconds. Such pulses have been demonstrated previously at LCLS as well as various HHG sources [8–10].

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Since 2D-spectroscopy involves comparing one energy to a second potentially different energy, it requires the ability to separately tune the energy of different pulses. Again, there has been some progress on this at LCLS, demonstrating energy tunability of more than 20% for pulses with separations as short as hundreds of attoseconds [11]. Additionally, one would like to be able to control the angles of incidence of the separate pulses and generally have full control over the experimental geometry. Because X-ray pulses are generally harder to manipulate spatially than infrared and visible pulses, steerable only with limited options such as grazing incidence reflections, Bragg diffractions, and energy specific optics such as zone plates, this is a significant challenge. However, if one is willing to work in restricted geometric conditions that would be necessary only for a single given experiment, such an experiment can often be designed. Finally, all of these measurements are complicated and would benefit from excellent statistics which can be obtained by having a high repetition rate and a high pulse power. There has been substantial progress on this, with new FEL facilities consistently pushing forward with more pulses and often more power. The statistical portion of the problem is perhaps the portion of the problem the global research community is most prepared to solve [12, 13]. The individual pieces which would make such a 2D-X-ray measurement possible exist but putting all these desirable properties together at the same time is a challenge which does not appear likely to be solved soon. However, there has been substantial progress toward making these types of measurements, and it does appear inevitable that such measurements will eventually be made. The potential importance of such measurements to enable us to understand matter on its natural length and time scale is enormous.

6.2 Phase Sensitive Spectroscopy Coherent control is a necessary step in the development of a variety of advanced spectroscopy measurements wherein one can tune the atomic state of an atom. However, coherent control requires exquisite control over light so as to “dress” an atom—in this context dressing means to put the atom into a specific quantum state. By using highly tuned waveforms and pulse shapers, a molecule can be “pushed” and “pulled” through specific transitions to get to a desired end point. These pulses often end up with such complicated forms that there isn’t a simple analytical way to describe the pulse shape. The ability to control matter with light, however, is highly desirable not just as a requirement for spectroscopy, but also because coherent control can potentially be used to drive specific reactions with incredible specificity, such as choosing reaction products of, for example, the dissociation of C2 H2 . Demonstrating the phenomena in the X-ray regime is difficult because the requirements on the laser are sizeable. Coherent control of a seeded beamline was demonstrated on neon ions using two pulses from a coherent FEL, specifically FERMI [14]. When using two different

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pulses, this requires high longitudinal coherence, something which SASE FELs typically struggle with. By using two colors at 63.0 and 31.5 nm, the asymmetry of the angular distribution of photoelectrons could be controlled with a temporal resolution as fine as 3 attoseconds. In this experiment the two pulses were incident upon neon gas and by controlling the relative phase between the two pulses, the asymmetry of emission could be steered. By choosing the properties of the laser light carefully, the excited atom can be controlled and the decay mechanism for photoelectron emission can be tuned to favor the p channels versus s and d channels. Such careful control on a 3 as time scale is unprecedented and could only have been achieved by operating in the X-ray regime. One definitive way of showing coherent control is by demonstrating Rabi oscillations. These occur when an electromagnetic field can drive an atom between two different states as a function of time. While these have been observed for decades, only recently was this demonstrated with X-ray light, exploiting the coherence of the FERMI source [15]. The authors demonstrated an Autler-Townes doublet with the expected spacing from Rabi oscillations at the resonant energy, but when moving slightly above or slightly below the energy, a singlet was restored, as shown in Fig. 6.1. Based on detailed theory they were also able to demonstrate an avoided crossing between the two excited states. This is a significant step toward ultrafast manipulation of matter with atomic specificity. Perhaps even more remarkably, attosecond interferometry was demonstrated from a SASE source. In this case, unlike in the previous example, the beam energy is broad and has very limited longitudinal coherence. By using cleverly designed optics, phase control was demonstrated with time resolution as short as 129 attoseconds [16]. Specifically, with a single pulse and a beam splitter it was not necessary to phase lock two different pulses as a single pulse is intrinsically phase locked with itself. While this is almost two orders of magnitude worse than the resolution demonstrated at FERMI, the X-ray source was significantly less coherent and therefore harder to utilize. In this experiment, by using a split and delay line that simultaneously acts as Fig. 6.1 Experimental kinetic energy of exciting helium with 23.740 eV (red), 23.753 eV (black), and 23.766 eV (blue). The characteristic doublet is only seen at when on resonance at 23.753 eV and is not seen when detuned. This is indicative of a Rabi oscillation and is a powerful demonstration of coherent control. Reprinted from [15] by CC BY 4.0

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a grating due to a cleverly designed optic, the authors were able to choose a specific wavelength, and then delay one pulse relative to the other (of the same energy) generating an interference pattern when incident upon xenon atoms. This shows up as a difference in the ion positions on an imaging detector. Thus, even with relatively impure light some control over matter at ultrafast timescales can be demonstrated, although likely never with as fine control as when the FEL is seeded. This same beam splitter which allows for phase stability between pulses has been used to demonstrate nonlinear coherence effects. Working with neon atoms at the Ledge, Ding et al. [5] demonstrated these effects with pump-probe transient absorption in which both the pump and probe pulses came from the FEL. They demonstrated transient effects on a very fast timescale (2.4 fs) as well as spectral shifts of 10 meV due to the Stark effect. While the resolution of these measurements was limited by the spectrometer employed, in principle there is effectively no limit on the spectral resolution. The temporal resolution is limited by the coherence of the FLASH laser, but also can be improved going forward. These results represented a substantial step forward toward performing 2D (or even higher dimensional) spectroscopy with X-rays. Ding et al. later improved upon their original study, demonstrating a result which is relatively standard for nonlinear optics in the visible, but had eluded scientists in the X-ray regime [17]. Once again working at the L-edge of Ne, they measured the chirp in frequency space of a FEL pulse by using transient absorption spectroscopy. While this technique is well established and relatively standard in the infrared and the visible, making these measurements at ~ 50 eV required the development of a novel model for time-dependent absorption. Using this model, they were able to extract and quantify the chirp. This measurement can in principle be used to characterize and optimize the FEL properties, and in this case they showed that there was a systematic shift in the chirp with increasing pulse energy. As the measurement occurs without external fields the authors claim it can be used in a variety of measurements as it offers a convenient way to characterize the FEL and represents another step on the path toward 2D spectroscopy. Most recently, self-phase modulation has been demonstrated, where the refractive index of a thin film of magnesium changes within a duration of a single incident Xray pulse, leading to a phase shift and a slight change in the spectrum [18]. For soft X-rays, the phase modulation changed as a function of moving over the edge, leading from a nonlinear spectral blue shift broadening below the edge and to a red shift broadening above the edge. This has shown applications in other frequency ranges for pulse compression both spatially and temporally as well as in spectral broadening and the generation of supercontinuum light [19, 20]. These properties will also likely prove useful in the ultimate development of 2D-X-ray spectroscopy.

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6.3 Vacuum Nonlinear X-Ray Optics Through the book, it is shown that interactions between light and matter have led to developments of linear and nonlinear X-ray spectroscopy for materials science. In a case of the measurement with soft X-ray, experiments have been made under the vacuum condition. A vacuum is a space devoid of matter but the vacuum itself can be a target of nonlinear science. One of the characteristic quantities for vacuum science is Schwinger limit. According to quantum electrodynamics (QED), an electron–positron pair is generated in vacuum when the electric field exceeds the Schwinger limit [21], E QED =

m 2 c3 = 1.3 × 1018 V/m e

Thus, neglecting the effect of the magnetic field, no electromagnetic wave exists under the condition. An ultimate goal for X-ray laser development is to push this limit and to examine the nonlinear optical phenomena. So far, the current XFEL source does not have enough optical performance to reach the limit. However, one may observe the nonlinear optical effect in vacuum in the future. As detailed in Chap. 2, a Thomson scattering event is described as the first order process of a Hamiltonian with the A · A or A2 term and it can be recalled in the Feynman diagram in Fig. 6.2a. The diagram depicts a scattering event at a certain time with an electron from p to p’ and a photon from K to K’. This description is an approximate form in the nonrelativistic regime. In relativistic quantum mechanics, the diagram is essentially given as Fig. 6.2b. The intermediate state (p”) is associated with the antiparticle of an electron, a positron, and the electron–positron pair display a polarization in vacuum. Practically, a difference on the order of the photon associations in Fig. 6.2b can be neglected at an X-ray photon energy of 10 keV since the total system lacks the energy of 1 MeV necessary for 2 mc2 . Therefore, Thomson scattering can be described by a single crossing point, as shown in Fig. 6.2a. On the other hand, the existence of virtual polarization indicates there are also nonlinear optical phenomena in vacuum. Such an investigation of vacuum nonlinear X-ray optics directly links to investigation of “vacuum” itself. In QED, the principle of superposition of electromagnetic waves is no longer valid and photons scatter each other, associated with the electron–positron pair. The differential cross section of scattering between the linearly polarized photons are given by [22, 23]: 6   dσγ γ →γ γ α 4 ωCMS = 260 cos4 CMS + 328 cos2 CMS + 580 2 8 d (180π ) m

when a photon energy is much smaller than mc2 . ωCMS and CMS are photon energy and scattering angle in the centroid system. In the formula, one can find that the differential cross section increases by the 6th power with ωCMS and thus detection may

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Fig. 6.2 Thomson scattering event in the a nonrelativistic and b relativistic regimes. Momentum of an electron is given by p, while wavenumber of a photon is by K

be possible with high photon energies such as X-rays. On the other hand, interestingly, quantum mechanics allows for the possibility that the photon-photon scattering event is associated with an undiscovered elementary particle beyond the standard model (Fig. 6.3). A proposed candidate is Axion, which has also been invoked as dark matter. It should be noted that initial experiments have been conducted at SACLA [24, 25], but further technical improvements, such as installation of the higher sensitivity detector, are required to definitively settle the issue. Vacuum nonlinear X-ray optics or spectroscopy will become the intriguing and significant topic in fundamental physics.

Fig. 6.3 Feynman diagrams of photon-photon scattering from (K 1 , K 2 ) to (K 3 , K 4 ) a in the regime of quantum electrodynamics and b in a possible regime with the undiscovered elementary particle

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6.4 Developments in Experimental Stations for Materials Science As described in Chap. 2 X-ray spectroscopy is nowadays used to trace evolutions of electronic states or atomic structures of functional materials under working conditions, such as batteries and catalysts. Such so-called “operando” spectroscopy experiments can also be carried out with nonlinear X-ray spectroscopy, to provide unique information about a sample that cannot be obtained with conventional linear X-ray spectroscopies. Measurements of the nonlinear X-ray optical phenomena, shown in Chaps 4 and 5, are made with a photon-in and photon-out scheme. Therefore, signal can be detected under a variety of external environments for a sample, such as electric current, gate voltage, and magnetic field. At an XFEL facility, such as SACLA, an experimental station for nonlinear X-ray spectroscopy has opened, for both academic and industrial users. The experimental chamber is equipped with sample transfer, temperature control, external field application, ellipsometer, spectrometer, and nanobeam focusing systems. Moreover, recent developments of nanofocusing techniques for the X-ray beams have reached spot sizes of < 100 nm for soft X-ray [26] and < 10 nm for hard X-ray [27]. Technical innovations of nano-focusing mirrors have generated the high-efficient, achromatic and brilliant beam that has led us to probe nano-sized samples for nanotechnology and also to enhance the nonlinear X-ray signals [28–30]. The user-friendly station (Fig. 6.4) has already been used in the industrial area, such as investigations of lithium batteries and solar cells [31– 33]. The recent observation of magnetization-induced soft X-ray second harmonic generation has found to be useful for research of magnetics and spintronics [34]. Nonlinear X-ray spectroscopy will soon be one of the standard analysis methods to characterize a sample and to trace its functionalities. At X-ray beamlines, guiding and shaping of the X-ray beam has been achieved using geometric optics or wave optics based on linear responses of light with optical elements, such as mirrors. It has been the quality of the optical components, such slope error or roughness of a mirror, that has determined the optical specification, such as beam-size and pulse-width. Nowadays, the fabrication is becoming so established that performance is reaching the limit that is expected purely by the optical principles. A rise of nonlinear X-ray optics and spectroscopy opens a new research field to develop the new fundamental principles for beam technology that use the nonlinear optical effect to control the X-ray beam. One example is shortening of the X-ray pulse duration by the saturable absorption that is associated with the inner-shell photoionization [35]. Such a technical development has a high potential to break through the conventional technical limits at the beamline. X-ray free electron laser light sources have been developed significantly with high power, ultrashort pulses, and high repetition rate [36]. The brilliant optical performance also paves new and various ways for nonlinear X-ray spectroscopy. We note that the combination of nonlinear X-ray optical phenomena with coherent imaging methods is also being explored. It is certain that the research field will expand

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Fig. 6.4 Multi-purpose experimental station for nonlinear X-ray spectroscopy at the soft X-ray beamline BL1 at SACLA. By permission of RIKEN

and various mysteries of materials science will be unveiled by the new analysis methods.

6.5 Into the Deep: Nonlinear Science In the previous chapter, we stated that it is important to find new functionalities of materials as well as to consider “what is a new function?” for material science. The target “new function” may differ from researcher to researcher, but we would like to present one of our ideas.

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The difference between “right” and “left” is one of the fundamental concepts that pervade all of science, from particle physics to biological activity. The violation of the law of conservation of parity in particle physics shows that there is an essential left–right distinction in our world. In life science as well, amino acids in DNA have a chiral structure, but only molecules of one chirality constitute life. The synthesis of molecules with only homochiral structures is one of the important topics in chemistry, as exemplified by the study of asymmetric reactions by Noyori [37]. The challenge of creating and controlling the difference between “right and left” in dynamic processes, rather than the static “right and left” in material structures, is now at the forefront of material science. The differences between “rightward” and “leftward” properties are called nonreciprocal phenomena. We believe that the pursuit of nonreciprocal phenomena for quantum (electrons and phonons) and degrees of freedom (spin and valley) will lead to the development of new control methods for materials, information, and energy transfer methods. Since diodes that realize electron rectification are widely used, nonreciprocal phenomena have not been recognized as an important issue in material science. Recently, however, several examples of dynamic control of nonrelativistic performance by new quantum mechanisms have been demonstrated, and this is beginning to be recognized as an interesting issue. This is because, compared to the static “right–left” difference in structure, nonreciprocal phenomena involve not only space-reversal symmetry breaking but also time-reversal symmetry breaking, and thus have richer physics inherent in them. Considering the benefits that diodes have brought to modern society, the impact of facilitating the control of nonrelativistic phenomena in other quantum and degrees of freedom would be extremely large. The readers would have recognized that the mechanisms of photosynthesis and the transport of proteins, information, and energy in the cell are important questions in the life sciences as well. The fundamental question of “What determines the direction in which matter, information, and energy move?” should be closely related to the nonreciprocal phenomena. Readers of this book may think that nonreciprocity is a topic that has come out of nowhere. However, we believe that the experimental techniques discussed in this book can make a significant contribution to the development and advancement of nonreciprocal functions of matter in the future and are therefore highly relevant to this book. Controlling the coherent response of matter with high temporal precision and detecting the asymmetry inherent in matter with high sensitivity, which are discussed throughout this book, are essential for developing the nonreciprocal response of quantum and degrees of freedom in materials.

References 1. P. Hamm, M.T. Zanni, Concepts and Methods of 2D Infrared Spectroscopy (Cambridge University Press, Cambridge, 2011) 2. J. Keeler, Understanding NMR Spectroscopy, 2nd edn. (John Wiley and Sons, Chichester, U.K., 2010)

6 Future Prospects

157

3. A. Fabris et al., Fermi upgrade plans, in Proceedings of the 7th Int. Particle Accelerator Conference, vol. IPAC2016, (2016), p. 4 4. D. Gauthier, P.R. Ribiˇc, G. De Ninno, E. Allaria, P. Cinquegrana, M.B. Danailov, A. Demidovich, E. Ferrari, L. Giannessi, Generation of phase-locked pulses from a seeded free-electron laser. Phys. Rev. Lett. 116, 024801 (2016) 5. T. Ding et al., Nonlinear coherence effects in transient-absorption ion spectroscopy with stochastic extreme-ultraviolet free-electron laser pulses. Phys. Rev. Lett. 123, 103001 (2019) 6. P.K. Maroju et al., Attosecond pulse shaping using a seeded free-electron laser. Nature 578, 7795 (2020) 7. C. Nicolas, C. Miron, Lifetime broadening of core-excited and -ionized states. J. Electron Spectrosc. Relat. Phenom. 185, 267 (2012) 8. P.K. Maroju et al., Complex attosecond waveform synthesis at FEL FERMI. Appl. Sci. 11, 21 (2021) 9. J.T. O’Neal et al., Electronic population transfer via impulsive stimulated X-ray raman scattering with attosecond soft-X-Ray pulses. Phys. Rev. Lett. 125, 073203 (2020) 10. J. Li, J. Lu, A. Chew, S. Han, J. Li, Y. Wu, H. Wang, S. Ghimire, Z. Chang, Attosecond science based on high harmonic generation from gases and solids. Nat. Commun. 11, 1 (2020) 11. Y. Sun, F.-J. Decker, J. Turner, S. Song, A. Robert, D. Zhu, Pulse intensity characterization of the LCLS nanosecond double-bunch mode of operation. J. Synchrotron. Rad. 25, 3 (2018) 12. J. Stohr, Linac Coherent Light Source II (LCLS-II) Conceptual Design Report, No. SLAC-R978, 1029479, 2011 13. I. Georgescu, The first decade of XFELs. Nat. Rev. Phys. 2, 7 (2020) 14. K.C. Prince et al., Coherent control with a short-wavelength free-electron laser. Nature Photon. 10, 3 (2016) 15. S. Nandi et al., Observation of Rabi dynamics with a short-wavelength free-electron laser. Nature 608, 7923 (2022) 16. S. Usenko, A. Przystawik, M.A. Jakob, L.L. Lazzarino, G. Brenner, S. Toleikis, C. Haunhorst, D. Kip, T. Laarmann, Attosecond interferometry with self-amplified spontaneous emission of a free-electron laser. Nat. Commun. 8, 1 (2017) 17. T. Ding et al., Measuring the frequency chirp of extreme-ultraviolet free-electron laser pulses by transient absorption spectroscopy. Nat. Commun. 12, 1 (2021) 18. C. Ferrante et al., Non-linear self-driven spectral tuning of extreme ultraviolet femtosecond pulses in monoatomic materials. Light Sci. Appl. 10, 1 (2021) 19. Y. Yamaguchi, R. Hida, T. Suzuki, F. Kannari, Supercontinuum generation by self-phase modulation and induced phase modulation at fused silica thin plate array, in 2017 Conference on Lasers and Electro-Optics Pacific Rim, (Optica Publishing Group, 2017), p. s1402 20. A.M. Weiner, J.P. Heritage, R.H. Stolen, Self-phase modulation and optical pulse compression influenced by stimulated raman scattering in fibers. J. Opt. Soc. Am. B, JOSAB 5, 364 (1988) 21. J. Schwinger, On Gauge Invariance and Vacuum Polarization. Phys. Rev. 82, 664 (1951) 22. B. Tollis, Dispersive approach to photon-photon scattering. Nuovo Cimento 32, 757 (1964) 23. B. Tollis, The scattering of photons by photons. Nuovo Cimento 35, 1182 (1965) 24. T. Inada, T. Yamaji, S. Adachi, T. Namba, S. Asai, T. Kobayashi, K. Tamasaku, Y. Tanaka, Y. Inubushi, K. Sawada, M. Yabashi, T. Ishikawa, Search for photon–photon elastic scattering in the X-ray region. Phys. Lett. B 732, 356 (2014) 25. Y. Seino, Search for the vacuum polarization using a high-field laser and an XFEL, doctoral thesis, the University of Tokyo, 2021 26. K. Horiba, Y. Nakamura, N. Nagamura, S. Toyoda, H. Kumigashira, M. Oshima, K. Amemiya, Y. Senba, H. Ohashi, Scanning photoelectron microscope for nanoscale three-dimensional spatial-resolved electron spectroscopy for chemical analysis. Rev. Sci. Instrum. 82, 113701 (2011) 27. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, K. Yamauchi, Breaking the 10 nm barrier in hard-X-ray focusing. Nat. Phys. 6, 122 (2010)

158

I. Matsuda et al.

28. H. Motoyama, S. Owada, G. Yamaguchi, T. Kume, S. Egawa, K. Tono, Y. Inubushi, T. Koyama, M. Yabashi, H. Ohashi, H. Mimura, Intense sub-micrometre focusing of soft X-ray free-electron laser beyond 1016 W cm−2 with an ellipsoidal mirror. J. Synchrotron Rad. 26, 1406 (2019) 29. Y. Takeo, H. Motoyama, T. Shimamura, T. Kimura, T. Kume, Y. Matsuzawa, T. Saito, Y. Imamura, H. Miyashita, K. Hiragur, H. Hashizume, Y. Senba, H. Kishimoto, H. Ohashi, H. Mimura, A highly efficient nanofocusing system for soft x rays. Appl. Phys. Lett. 117, 151104 (2020) 30. T. Kume, K. Hiraguri, Y. Imamura, Y. Matsuzawa, T. Saito, H. Miyashita, Y. Takeo, Y. Senba, H. Kishimoto, H. Ohashi, H. Hashizume, H. Mimura, Fabrication of soft x-ray monolithic Wolter mirror based on surface scanning measurement using touch probe. Rev. Sci. Instrum. 93, 063101 (2022) 31. A. Zong, R.R. Nebgen, S.-C. Lin, J.A. Spies, M. Zuerch, Emerging ultrafast techniques for studying quantum materials. Nat. Rev. Mater. 8, 224 (2023) 32. C. Woodahl, S. Jamnuch, A. Amado et al., Probing lithium mobility at a solid electrolyte surface. Nat. Mater. (2023). https://doi.org/10.1038/s41563-023-01535-y 33. M. Horio, T. Sumi, J. Bullock, Y. Hirata, M. Miyamoto, B. Nebgen, T. Wada, T. Senoo, Y. Tsujikawa, Y. Kubota, S. Owada, K. Tono, M. Yabashi, T. Iimori, M.W. Zuerch, I. Matsuda, C.P. Schwartz, W.S. Drisdell, Detecting driving potentials at the buried SiO2 nanolayers in solar cells by chemical-selective nonlinear X-ray spectroscopy. Appl. Phys. Lett. 123, 031602 (2023) 34. T. Sumi, M. Horio, T. Senoo, Y. Kubota, G. Yamaguchi, T. Wada, M. Miyamoto, K. Yamaguchi, Y. Tsujikawa, Y. Sato, M. Niibe, Y. Hirata, Y. Miyauchi, D. Oshima, T. Kato, S. Owada, K. Tono, M. Yabashi, I. Matsuda, Observing soft x-ray magnetization-induced second harmonic generation at a heterojunction interface. Appl. Phys. Lett. 122, 171601 (2023) 35. I. Inoue, Y. Inubushi, T. Osaka, J. Yamada, K. Tamasaku, H. Yoneda, M. Yabashi, Shortening X-ray pulse duration via saturable absorption. Phys. Rev. Lett. 127, 163903 (2021) 36. N. Huang, H. Deng, B. Liu, D. Wang, Z. Zhao, Features and futures of X-ray free-electron lasers. The Innovation 2, 100097 (2021) 37. R. Noyori, Monograph: asymmetric catalysis in organic synthesis (John Wiley & Sons, New York, 1994)

Index

A Above threshold photoemission, 57, 58, 63, 72 Absorption spectroscopy, 10, 11, 15, 19, 20, 28–30, 52, 92, 132, 135, 151 Anharmonicity, 1, 5, 7 Atomic X-ray laser, 136, 138

C Coherent multi-photon photoemission, 12

E Emission spectroscopy, 15, 19, 20, 36, 93, 94, 139

F Floquet engineering, 76 Four wave mixing, 86, 95, 100, 103–105, 110, 148

M Multi-dimensional spectroscopy, 105, 147 Multiphoton absorption, 86, 95, 110, 112

N Nonlinear optics, 2, 7, 13, 83, 84, 86, 89–91, 94, 98, 106, 110–112, 119, 120, 122, 125, 141, 151 Nonlinear optical susceptibility, 3, 87, 88

P Parametric down-conversion, 87, 90, 128, 129, 141 Phase sensitive spectroscopy, 149 Photoelectron spectroscopy, 15, 20, 33–35, 44–47, 57, 58, 66–69, 72, 74, 94 Photoemission, 11, 12, 24, 34, 35, 45–48, 57–67, 69–73, 76, 94 Pump-probe, 10, 11, 40–42, 46–48, 61, 62, 66, 68–71, 73, 75, 86, 151

S Saturable absorption, 85, 88, 94, 136, 137, 154 Second harmonic generation, 6, 83–85, 87, 95, 106, 108, 110, 127, 129, 154 Stimulated Raman scattering, 83, 99, 141 Stimulated X-ray emission spectroscopy, 139, 140 Sum frequency generation, 6, 7, 83, 86, 87, 106, 110, 130, 131 Susceptibility, 3–5, 7, 8, 15–18, 85, 87, 88, 95, 106, 111, 127, 141

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 I. Matsuda and R. Arafune (eds.), Nonlinear X-Ray Spectroscopy for Materials Science, Springer Series in Optical Sciences 246, https://doi.org/10.1007/978-981-99-6714-8

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Index

Synchrotron radiation, 15, 19–22, 35, 40, 43, 47, 51, 90, 119, 129

V Vacuum nonlinear, 152, 153

T Time-resolved, 10, 12, 15, 20, 22, 34, 39–42, 44–48, 57, 59, 61, 62, 66, 67, 72, 73 Two-photon absorption, 86, 88, 95–97, 132, 134, 135, 141

X X-ray free electron laser, 2, 15, 20–22, 40, 51, 83, 89, 91, 93, 94, 154 X-ray spectroscopy, 12, 15, 18, 28, 39, 50, 52, 92, 94, 99, 111, 112, 119, 122, 132, 147, 148, 151, 152, 154, 155