Springer Series in Plasma Scienceand Technology

Series editors

Michael Bonitz, Kiel, GermanyLiu Chen, Hangzhou, ChinaRudolf Neu, Garching, GermanyTomohiro Nozaki, Tokyo, JapanJozef Ongena, Brussel, BelgiumHideaki Takabe, Dresden, Germany

Plasma Science and Technology covers all fundamental and applied aspects of whatis referred to as the “fourth state of matter.” Bringing together contributions fromphysics, the space sciences, engineering and the applied sciences, the topics coveredrange from the fundamental properties of plasma to its broad spectrum ofapplications in industry, energy technologies and healthcare.

Contributions to the book series on all aspects of plasma research and technologydevelopment are welcome. Particular emphasis in applications will be on high-temperature plasma phenomena, which are relevant to energy generation, and onlow-temperature plasmas, which are used as a tool for industrial applications. Thiscross-disciplinary approach offers graduate-level readers as well as researchers andprofessionals in academia and industry vital new ideas and techniques for plasmaapplications.

More information about this series at http://www.springer.com/series/15614

Hideaki Takabe

The Physics of Laser Plasmasand Applications - Volume 1Physics of Laser Matter Interaction

Hideaki TakabeInstitute of Radiation PhysicsHelmholtz Zentrum Dresden RosendorfDresden, Germany

ISSN 2511-2007 ISSN 2511-2015 (electronic)Springer Series in Plasma Science and TechnologyISBN 978-3-030-49612-8 ISBN 978-3-030-49613-5 (eBook)https://doi.org/10.1007/978-3-030-49613-5

© Springer Nature Switzerland AG 2020This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of thematerial 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 informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors, and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, expressed or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

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Snap shot of fusion target irradiated by Gekko XII laser at Osaka Univ.

Preface

This year, 2020, is the 60th anniversary of the birth of the laser. When the laser wasinvented by Maiman in 1960, the news spread as a shocking technology of thefuture. Soon after, the laser was used in a scene from the film “Goldfinger” in the“007 series” in 1964, and this film made people worldwide realize how powerful thelaser is. Soon after, Maiman’s small laser had scaled up the purpose of fusion energyafter the publication of laser fusion concept in 1972. I was very much interesetd inreseraching fusion energy when I was a student at Osaka University and when the oilcrisis of 1973 had crippled the global economy.

Along with research of the laser fusion as graduate student and young facultymember, I faced numerous challenges in integrated physical phenomena to researchon laser fusion and analyze experimental data from many angles with large-scalelasers. Such intergrated physics is the coupled phenomenon of compressible fluid,radiation, dense matter state, atomic physics, kinetic physics, and so on in mm scalefusion target materials. These physical phenomena are also key to research inastrophysics, and there are several connections in the non-dimensional scale oftime and space.

Toward the end of the 1980s, chirped pulse amplification (CPA) technique madeit possible to increase the laser intensity more than 104 times in the intense lasersused for laser fusion and related plasma physics. The electron motion in such anultra-intense laser is highly relativistic and the Lorentz factor becomes roughly γ ¼104. These new lasers have opened the window for relativistic plasma physics andquantum electrodynamics in a small-scale laboratory.

In the last 60 years since Maiman’s invention, the physics of laser plasma hasgrown into a mature science and a challenging subject with the help of precisediagnostics with ultra-short and small-scale resolutions. Honestly speaking,however, greater effort is required to understand the physics in real experimentsand to control the laser plasma for many applications from fundamental toindustrial ones.

The author has worked for laser plasma and related physics for 40 years as acomputational theorist. I was fortunate because my institute in Osaka is a placewhere the majority of research is experimental in nature, and big laser Gekko-XII for

vii

fusion research and LFEX laser for ultra-intense physics are installed there. I havebenefited from discussions with experimentalists from all over the world.

I have tried to write these three books so that the description is always withintuitive understanding as I explain to experimentalists. I tried not to use complicatedequations in order to keep the books easy to read for non-specialists. Many colorfigures are used to help non-experts obtain concrete images about the physics. Sincethe idea of laboratory astrophysics was conceived from the similarity of the physicsof turbulence in laser implosion and supernova explosion, some ideas and examplesof experimental laboratory astrophysics are also described.

While writing this book, I realized that one book is not enough to transfer myknowledge to the readers clearly; therefore, I decided to write three volumes.

Volume 1 discusses the physics of how laser energy is absorbed by plasma andhow electrons obtain laser energy in non-thermodynamic equilibrium process.Importance of non-linear and chaos physics are described. Assuming that readershave preliminary knowledge about this field, I explain the following subjects inVolume 1: Electromagnetism, Mechanics, Analytical mechanics, Quantummechanics, and Relativity.

Volume 2 is devoted to the physics of plasma energized by laser absorption andrelated physics. Abrupt heating by laser generates extremely high pressure, drivingstrong shock waves to compress matter to a high density like the core of planets andstars. The physics of compressible hydrodynamics becomes important, and thesubject of turbulence driven by the hydrodynamic motion is a challenging subject.Volume 2 discusses the following topics of physics to the readers: Fluid dynamics,Thermodynamics, Statistical physics, Quantum statistical physics, and Atomicphysics.

In Volume 2, it is assumed that the plasma is collisional and is in thermodynamicequilibrium locally. On the other hand, the physics of plasma without collision isdiscussed in Volume 3. In relatively low-density region, high-energy electronsgenerated by laser are freely running without collision, and they do not collidewith each other, the so-called collisionless plasma. The electrons interact withelectric and magnetic field fluctuations to generate a variety of plasma instabilities.The field and charged particle interaction provides particle acceleration, anomaloustransport, anti-matter generation, and so on. Volume 3 provides insights into thefollowing areas of physics: Electromagnetism, Physical kinetics, Statistical physics,Relativity, and Quantum electrodynamics.

As you already know, the laser-plasma physics is not a single subject disciplineand the expression of “integrated physics” would be nice expression. As a metaphor,the laser plasma is a kind of “decathlon”, not 100m dash. Note that it requires aconstant study and effort.

Since I am a theoretical plasma physicist, most topics discuss about theoreticalphysics of laser plasmas. Although I have worked with many experimentalists, Ihave not written about the methods of diagnostics for laser plasma. Severalexperimental data are shown in this book, but the accuracy of diagnostics is notdiscussed. Note that I selectively cited a limited number of papers so that thosepapers are also very useful for senior researchers to know more about each topic.

viii Preface

I would like to express my special acknowledgment to Prof. C. Yamanaka whoguided me in the research of laser plasma and its applications, especially in laserfusion, and Prof. K. Mima who navigated me to the research of theoretical plasmaphysics. I thank sincerely Dr. R. Sauerbrey and Dr. T. Cowan for providing me thebest opportunities and constant encouragement to write the current books. Topics onrelativistic lasers in Vol. 1 and on high-density plasma in Vol. 2 are reflection of thesubjects of HiBEF (Helmholtz International Beamline for Extreme Fields) project inDresden where I work. I would like to thank Prof. P. Mulser and Prof. R. L. Morsefor their supervision of my research in Munich and Tucson, respectively.

I also thank the following individuals for their fruitful discussions: R. Kodama,Y. Sakawa, Y. Kuramitsu, S. Yamada, T. Morita, T. N. Kato, R. M. More,D. Saltzmann, F-L Wang, S. Blinnikov, A. Titov, S. Atzeni, B. Remington, H-S.Park, M. E. Campbell, D. Kraus, K. Falk, J. Zhang, Z-M. Sheng, L-G Huang,M. Murakami, H. Nagatomo, T. Yabe, N. Ohnishi, A. Mizuta, J-F Ong, T. Kato,J. Myer-ter-Vehn, S. Witkovski, L. Montieth, and P. McKenty.

I would like to express my thanks to the people involved in HiBEF project atHZDR for their valuable comments on this topic. I also thank H. Niko whosuggested and encouraged me to write these books.

Relating to the proposal of laboratory astrophysics, I would like to express sincerethanks to K. Nomoto who guided me to the world of supernova explosion physicsafter the explosion of SN1987a. I also thank my friends in astrophysics, I. Hachisu,K. Shibata, R. Yamazaki, and M. Hoshino, who introduced me to the concept oflaboratory astrophysics and shared their ideas to find topics on astrophysics to bepossibly modeled by laser plasma.

Finally, I would like to thank my family, Yoko, Yugo, Ayana, and Ayako, fortheir unwavering support to my research for a long time and for understanding mydecision to go to Germany by taking early retirement from Osaka University.

Dresden, Germany Hideaki TakabeMarch 10, 2020

Preface ix

About the Book

After 60 years of the invention of laser, it has grown as a new light sourceindispensable for our life. The application to industrial and research has spreadover wide fields. The book discusses how the power lasers are absorbed by matterand how high temperature and high density plasmas are produced. It is notedthat such plasmas are mimics of the plasmas in the Universe, namely, plasmainside of stars, exploding plasma like supernovae, accelerating plasma like cosmicray. The book (Vol. 1) gives the physics of the intense laser and materialinteraction to produce such plasmas. The physics is explained intuitively so thatnon-specialists can obtain clear images about how the laser energy is converted tothe non-thermal plasma energy; simply saying, how laser heats matter. For intenselaser of 1013–16 W/cm2, the laser energy is mainly absorbed via collisional process,where the oscillation energy is converted to thermal energy by non-adiabaticCoulomb collision with the ions. Collisionless interactions with the collectivemodes in plasma are also described. The main topics are the interaction of ultra-intense laser and plasma interaction for the intensity near and over 1018 W/cm2. Insuch regime, relativistic dynamics become essential. A new physics appears due tothe relativistic effects, such as mass correction, relativistic nonlinear force, chaosphysics of particle motions, and so on. The book provides clearly the theoretical basefor challenging the laser-plasma interaction physics in the wide range of powerlasers.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Brief History of Intense Lasers . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Invention of Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 High-Power Lasers for Nuclear Fusion . . . . . . . . . . . . . 51.1.3 Ultra-Intense and Ultra-Short Lasers . . . . . . . . . . . . . . . 8

1.2 What Is Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 High-Density Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Magnetic Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Electron Motion in Laser Field . . . . . . . . . . . . . . . . . . . 141.3.3 Normalized Laser Strength . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Non-relativistic Laser-Plasma Interaction . . . . . . . . . . . . . . . . . . 161.4.1 Collisional Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 Collisionless Absorption and Ponderomotive Force . . . . 18

1.5 Relativistic Laser-Plasma Interaction . . . . . . . . . . . . . . . . . . . . . 201.5.1 Relativistic Electron Oscillation . . . . . . . . . . . . . . . . . . 201.5.2 Relativistic Laser and Solid Interaction . . . . . . . . . . . . . 211.5.3 Theory of Chaotic and Stochastic Heating . . . . . . . . . . . 21

1.6 PIC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Part I Non-relativistic Lasers

2 Laser Absorption by Coulomb Collision . . . . . . . . . . . . . . . . . . . . . . 292.1 Plasma Generation by Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.1 Field-Induced Electron Emission . . . . . . . . . . . . . . . . . . 292.1.2 Ionization by Multiphoton Absorption . . . . . . . . . . . . . . 322.1.3 Tunneling and Over-Threshold Ionizations . . . . . . . . . . . 342.1.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xiii

2.2 Laser as Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . 392.2.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.2 Electromagnetic Waves in Vacuum . . . . . . . . . . . . . . . . 432.2.3 Lasers as Coherent Electromagnetic Waves . . . . . . . . . . 44

2.3 Electron Current Induced by Laser Fields . . . . . . . . . . . . . . . . . . 452.3.1 Antenna and Thomson Scattering . . . . . . . . . . . . . . . . . 482.3.2 Electron Current in Matters . . . . . . . . . . . . . . . . . . . . . . 50

2.4 Electron Coulomb Collision by Ions in Plasma . . . . . . . . . . . . . . 532.4.1 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.2 Yukawa Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4.3 Coulomb Logarithm (Log) . . . . . . . . . . . . . . . . . . . . . . 582.4.4 Collision Frequency and Electrical Resistivity . . . . . . . . 612.4.5 Relaxation Time to Thermal Equilibrium . . . . . . . . . . . . 62

2.5 Lasers in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5.1 Classical Absorption of Laser Energy . . . . . . . . . . . . . . 69

2.6 Laser Absorption in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.6.1 Physical Image of Classical Absorption . . . . . . . . . . . . . 722.6.2 Simple Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . 732.6.3 Kinetic Derivation by Dawson and Oberman . . . . . . . . . 752.6.4 Quasi-Linear Model of Absorption . . . . . . . . . . . . . . . . 78

2.7 Bremsstrahlung and Collisional Absorption . . . . . . . . . . . . . . . . 79References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Ultra-Short Pulse and Collisionless Absorption . . . . . . . . . . . . . . . . . 813.1 Ultra-Short Pulse in Non-relativistic Intensity . . . . . . . . . . . . . . . 813.2 Self-Consistent Analysis of Short Pulse Absorption . . . . . . . . . . . 883.3 Quantum Theory of Electric Conductivity in Dense Plasmas . . . . 953.4 Nonlinear Inverse Bremsstrahlung (IB) Absorption (ve < vos) . . . . 993.5 Electron Plasma Waves and Collisionless Absorption . . . . . . . . . 1043.6 Resonance Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.6.1 Collisional Absorption Model . . . . . . . . . . . . . . . . . . . . 1103.6.2 Singular Point Integral Model . . . . . . . . . . . . . . . . . . . . 111

3.7 Resonance in Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.8 Linear Mode Conversion in Resonance Absorption . . . . . . . . . . . 115

3.8.1 Absorption Rate and Pump Depletion . . . . . . . . . . . . . . 1173.9 Large Amplitude Electron Plasma Waves . . . . . . . . . . . . . . . . . . 118

3.9.1 Wave-Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.10 Vacuum Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.10.1 Physical Image and Absorption Rate . . . . . . . . . . . . . . . 1263.10.2 Skin Depth at Sharp Boundary . . . . . . . . . . . . . . . . . . . 129

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4 Nonlinear Physics of Laser-Plasma Interaction . . . . . . . . . . . . . . . . . 1314.1 Ponderomotive (PM) Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.2 Nonlinear Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 133

xiv Contents

4.3 Filament Instability of Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.4 Ion Fluid and Ion Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . 139

4.4.1 Charge Neutral Plasma Fluids . . . . . . . . . . . . . . . . . . . . 1404.4.2 Ion Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.5 Density Profile Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.6 Principle of Parametric Instabilities . . . . . . . . . . . . . . . . . . . . . . 146

4.6.1 Coupled Oscillator Model-1 . . . . . . . . . . . . . . . . . . . . . 1494.6.2 Thermal Noise of Electrostatic Fluctuations . . . . . . . . . . 1504.6.3 Coupled Oscillator Model-2 . . . . . . . . . . . . . . . . . . . . . 151

4.7 Stimulated Raman and Brillouin Scattering . . . . . . . . . . . . . . . . . 1524.8 Decay-Type Parametric Instabilities . . . . . . . . . . . . . . . . . . . . . . 1544.9 Experimental Data for SRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.10 Physics of Saturation of SRS Instability . . . . . . . . . . . . . . . . . . . 157

4.10.1 Effect of Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . 1594.10.2 Nonlinear Saturation of SRS . . . . . . . . . . . . . . . . . . . . . 160

4.11 Broadband Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Part II Relativistic Lasers

5 Relativistic Laser-Electron Interactions . . . . . . . . . . . . . . . . . . . . . . 1675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.2 Special Relativity for Electron Motion . . . . . . . . . . . . . . . . . . . . 167

5.2.1 Equation of Relativistic Electron Motion . . . . . . . . . . . . 1695.2.2 Lorentz Transformation of Time and Space . . . . . . . . . . 1725.2.3 Lorentz Transformation of Velocities . . . . . . . . . . . . . . . 1745.2.4 Lorentz Transformation of Fields . . . . . . . . . . . . . . . . . 1755.2.5 Plane Electromagnetic Waves in Vacuum . . . . . . . . . . . 177

5.3 Electron Motion in a Relativistic Strong Field . . . . . . . . . . . . . . . 1795.3.1 Constant of Motion in Vacuum . . . . . . . . . . . . . . . . . . . 1805.3.2 Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.3.3 Electron Motion in Vacuum . . . . . . . . . . . . . . . . . . . . . 1825.3.4 Free Electron Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.3.5 Electrons in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.3.6 Linear Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.3.7 Circular Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.4 Nonlinear Radiation Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 1885.4.1 Linear Thomson Scatterings . . . . . . . . . . . . . . . . . . . . . 1895.4.2 Compton and Inverse Compton Scatterings . . . . . . . . . . 1915.4.3 Nonlinear Thomson Scattering . . . . . . . . . . . . . . . . . . . 1945.4.4 Relativistic Beaming Effect and Doppler Shift . . . . . . . . 1975.4.5 Nonlinear Compton Scattering . . . . . . . . . . . . . . . . . . . 199

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Contents xv

6 Relativistic Laser Plasma Interactions . . . . . . . . . . . . . . . . . . . . . . . . 2036.1 Charge Separation in Low-Density Plasma . . . . . . . . . . . . . . . . . 203

6.1.1 Charge Separation by Photon Force . . . . . . . . . . . . . . . . 2036.1.2 Wake Field Generation and Energy Deposition . . . . . . . 205

6.2 Laser Propagation in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.2.1 Relativistic Transparency . . . . . . . . . . . . . . . . . . . . . . . 2076.2.2 Higher Harmonic Generation (HHG) . . . . . . . . . . . . . . . 2096.2.3 Electrostatic Field Excitation by vxB Force . . . . . . . . . . 2126.2.4 Density Bunching Current . . . . . . . . . . . . . . . . . . . . . . 2136.2.5 Simulation for a0 ¼ 10 in Low Density . . . . . . . . . . . . . 214

6.3 Ponderomotive Force in Relativistic Field . . . . . . . . . . . . . . . . . . 2176.3.1 PM Force by Electrostatic Wave . . . . . . . . . . . . . . . . . . 2196.3.2 Validity of Electrons in Plasmas Assumption . . . . . . . . . 219

6.4 Relativistic Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.5 Relativistic Self-Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.5.1 Self-Focusing Condition . . . . . . . . . . . . . . . . . . . . . . . . 2236.5.2 Strong Laser Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2246.5.3 Difference of 2D and 3D Focusing . . . . . . . . . . . . . . . . 2256.5.4 Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.5.5 Filamentation Instability . . . . . . . . . . . . . . . . . . . . . . . . 226

6.6 Relativistic Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.7 JxB Force and Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.8 Moving Mirror Model and Higher Harmonic Generation

from Solid Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

7 Relativistic Laser and Solid Target Interactions . . . . . . . . . . . . . . . . 2397.1 Pre-formed Plasma in Laser-Solid Interaction . . . . . . . . . . . . . . . 239

7.1.1 Pedestal of Laser Pulse . . . . . . . . . . . . . . . . . . . . . . . . . 2397.1.2 Model Experiments with Controlled Pre-formed

Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417.2 Laser Absorption at Solid Targets . . . . . . . . . . . . . . . . . . . . . . . 2447.3 Absorption Enhancement by Hot Electron Re-circulation . . . . . . . 2487.4 Hole Boring by Ponderomotive Force . . . . . . . . . . . . . . . . . . . . 252

7.4.1 Density Dependence of Hole Boring . . . . . . . . . . . . . . . 2547.5 Laser Interaction in Long Pre-formed Plasmas . . . . . . . . . . . . . . 2557.6 Absorption Efficiency Based on Conservation Laws . . . . . . . . . . 2607.7 Enhanced Coupling with Foam Layered Targets . . . . . . . . . . . . . 263

7.7.1 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 2717.8 Efficient Absorption in Structured Targets . . . . . . . . . . . . . . . . . 273

7.8.1 Micro-pillar Array Targets . . . . . . . . . . . . . . . . . . . . . . 2737.8.2 Nano-pillar Array Target . . . . . . . . . . . . . . . . . . . . . . . 2757.8.3 Micro-tube Plasma Lenses . . . . . . . . . . . . . . . . . . . . . . 2777.8.4 Common Rule of Better Coupling . . . . . . . . . . . . . . . . . 279

xvi Contents

7.9 Magnetic Field Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2797.10 Multi-dimensional Physics in Pre-formed Plasmas . . . . . . . . . . . . 282References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8 Chaos due to Relativistic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2878.1 Basic Relation of an Electron in Relativistic Laser Field . . . . . . . 2878.2 Laser Direct Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.2.1 Acceleration (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2938.2.2 Acceleration (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2948.2.3 PIC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8.3 Direct Acceleration after Interaction with Longitudinal Field . . . . 2978.3.1 PIC Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8.4 Chaotic Motion due to External Force . . . . . . . . . . . . . . . . . . . . 3058.4.1 Simple Example [1] (Random Force) . . . . . . . . . . . . . . . 3058.4.2 Simple Example [2] (Periodic Force) . . . . . . . . . . . . . . . 3078.4.3 Chaos in Propagating Relativistic Wave . . . . . . . . . . . . . 309

8.5 Electron Heating by Laser Field and Induced Plasma Waves . . . . 3138.5.1 Modeling PIC Simulation . . . . . . . . . . . . . . . . . . . . . . . 3148.5.2 Quasi-linear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 315

8.6 Hot Electron Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3168.6.1 Hot Electrons by Femtosecond Lasers . . . . . . . . . . . . . . 3178.6.2 Hot Electrons by Picosecond Lasers . . . . . . . . . . . . . . . 3188.6.3 Sheath Potential Effect . . . . . . . . . . . . . . . . . . . . . . . . . 3198.6.4 Multi-dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . 320

8.7 Electron Motion in Two Counter-Propagating RelativisticLasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.7.1 Counter-Propagating Two Laser Systems . . . . . . . . . . . . 3248.7.2 One-Electron Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3258.7.3 Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . 3278.7.4 Hot Electron Temperature Scaling . . . . . . . . . . . . . . . . . 329

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

9 Theory of Stochasticity and Chaos of Electrons in RelativisticLasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3319.1 Vlasov-Fokker-Planck Equations . . . . . . . . . . . . . . . . . . . . . . . . 331

9.1.1 Stochastic Diffusion Equation . . . . . . . . . . . . . . . . . . . . 3329.1.2 Vlasov-Fokker-Planck Equation . . . . . . . . . . . . . . . . . . 334

9.2 Stochastic Heating by Laser Filamentation . . . . . . . . . . . . . . . . . 3349.2.1 Numerical Calculation of Test Particles . . . . . . . . . . . . . 3379.2.2 PIC Simulation of Stochastic Diffusion . . . . . . . . . . . . . 339

9.3 Nonlocal Jump in Energy Space . . . . . . . . . . . . . . . . . . . . . . . . 3409.3.1 Levy’s Flights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3409.3.2 Integral Form of Random Walk . . . . . . . . . . . . . . . . . . . 3429.3.3 Fokker-Planck Diffusion Model . . . . . . . . . . . . . . . . . . 3429.3.4 Fractional Fokker-Planck Model . . . . . . . . . . . . . . . . . . 343

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9.4 Time Evolution of Distribution and Fractal Index α . . . . . . . . . . . 3459.4.1 Application to Hot Electron Scaling . . . . . . . . . . . . . . . 3479.4.2 Local and Nonlocal: Gaussian and Lorentzian . . . . . . . . 3499.4.3 Evaluation of Fractional Index α from Experimental Data 351

9.5 Model Experiment of Cosmic Ray Physics in the Universe . . . . . 3549.5.1 Relativistic EM Wave Generation by Relativistic Shocks 3559.5.2 Model Experiments with Relativistic Lasers . . . . . . . . . . 357

9.6 Fractal Index α ¼ 0.8 in Big Tokamak Transports . . . . . . . . . . . . 3589.7 Chaos in Standard Map Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3609.8 Analytical Mechanics of Electron Motions . . . . . . . . . . . . . . . . . 362

9.8.1 One Laser Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3629.8.2 Adiabatic Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3639.8.3 Numerical Solution for Counter Beam Interaction . . . . . 3659.8.4 Perturbation Method in Hamilton Equation . . . . . . . . . . 3679.8.5 Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . 3699.8.6 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Appendix-1: Rutherford Scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Appendix-2: Adiabatic and Nonadiabatic . . . . . . . . . . . . . . . . . . . . . . . 375

Periodic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Derivation of Adiabatic Constants . . . . . . . . . . . . . . . . . . . . . . . 377Nonadiabatic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Appendix-3: PIC Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379Particles in Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379Maxwell Equations in Computer . . . . . . . . . . . . . . . . . . . . . . . . 380Surprising Progress of Computing . . . . . . . . . . . . . . . . . . . . . . . 381

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Appendix 4: Tsallis Statistics at Maximum Entropy . . . . . . . . . . . . . . . 383Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

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About the Author

Hideaki Takabe is a Professor at the Institute of Radiation Physics, HelmholtzZentrum Dresden Rossendorf (HZDR), Dresden, Germany and Professor-Emeritusat Osaka University, Japan. He received a Ph.D. degree from Osaka University andcontinued research at Max-Planck Institute for Plasma Physics, Germany, and theUniversity of Arizona, USA. He continued his research and education activity at theInstitute of Laser Engineering, Osaka University until early retirement in 2015. Hisresearch fields are plasma physics, laser fusion, laboratory astrophysics, and HEDphysics. He is known for Takabe formula in laser fusion and also as a pioneer oflaboratory astrophysics concept. He has taught Plasma Physics and ComputationalPhysics at the School of Engineering and School of Science for more than 20 years.He committed the globalization of Osaka University. He was also a council memberof the Physical Societies of Japan (JPS) and the Association of Asia-Pacific PhysicalSociety (AAPPS), where he founded the Division of Plasma Physics (DPP). Hereceived the Edward Teller Medal in 2003 and John Dawson Award on Excellenceof Plasma Physics Research in 2020. He is a fellow of the American Physical Society(2000�).

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