Progress in Ultrafast Intense Laser Science III

Springer Series in chemical physics 89 Springer Series in chemical physics Series Editors: A.W. Castleman, Jr. J. P...

Author: Kaoru Yamanouchi | Kaoru Yamanouchi | See Leang Chin | Pierre Agostini | Gaetano Ferrante

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Springer Series in

chemical physics 89

Springer Series in

chemical physics Series Editors: A.W. Castleman, Jr. J. P. Toennies

K. Yamanouchi W. Zinth

The purpose of this series is to provide comprehensive up-to-date monographs in both well established disciplines and emerging research areas within the broad fields of chemical physics and physical chemistry. The books deal with both fundamental science and applications, and may have either a theoretical or an experimental emphasis. They are aimed primarily at researchers and graduate students in chemical physics and related fields. 75

Basic Principles in Applied Catalysis By M. Baerns 76 The Chemical Bond A Fundamental Quantum-Mechanical Picture By T. Shida 77 Heterogeneous Kinetics Theory of Ziegler–Natta– Kaminsky Polymerization By T. Keii 78 Nuclear Fusion Research Understanding Plasma–Surface Interactions Editors: R.E.H. Clark and D.H. Reiter 79 Ultrafast Phenomena XIV Editors: T. Kobayashi, T. Okada, T. Kobayashi, K.A. Nelson, S. De Silvestri 80 X-Ray Diffraction by Macromolecules By N. Kasai and M. Kakudo 81 Advanced Time-Correlated Single Photon Counting Techniques By W. Becker 82 Transport Coefficients of Fluids By Byung Chan Eu,

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Quantum Dynamics of Complex Molecular Systems Editors: D.A. Micha, I. Burghardt Progress in Ultrafast Intense Laser Science I Editors: K. Yamanouchi, S.L. Chin, P. Agostini, P.G. Ferrante Progress in Ultrafast Intense Laser Science II Editors: K. Yamanouchi, S.L. Chin, P. Agostini, P.G. Ferrante Free Energy Calculations Theory and Applications in Chemistry and Biology Editors: C. Chipot and A. Pohorille Analysis and Control of Ultrafast Photoinduced Reactions Editors: O.Kühn, L.Wöste Ultrafast Phenomena XV Editors: P.Corkum, D.Jonas, D.Miller, A.M. Weiner Progress in Ultrafast Intense Laser Science III Editors: K.Yamanouchi, S.L. Chin, P. Agostini, G.Ferrante Thermodynamics and Fluctuations far from Equilibrium Editor: J.Ross

Kaoru Yamanouchi · See Leang Chin Pierre Agostini · Gaetano Ferrante

Progress in Ultrafast Intense Laser Science Volume III With 155 Figures, 9 in Colour

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Professor Kaoru Yamanouchi

Professor See Leang Chin

University of Tokyo, Department of Chemistry Hongo 7-3-1, 113-0033 Tokyo, Japan e-mail: [email protected]

Université Laval Center for Optics, Photonics and Laser (COPL) & Department of Physics, Engineering Physics and Optics Quebec, Qc G1K 7P4 , Canada e-mail: [email protected]

Professor Pierre Agostini

Professor Gaetano Ferrante

Ohio State University Department of Physics 191 W Wodruff Ave Columbus OH 43210 e-mail: [email protected]

Università di Palermo Dipto. di Fisica e Tecnologie Relative Viale delle Scienze, 90128 Palermo, Italy e-mail: [email protected]

Series Editors: Professor A.W. Castleman, Jr.

Professor J. P. Toennies

Department of Chemistry, The Pennsylvania State University 152 Davey Laboratory, University Park, PA 16802, USA

Max-Planck-Institut für Strömungsforschung Bunsenstraße 10, 37073 Göttingen, Germany

Professor K. Yamanouchi

Professor W. Zinth

University of Tokyo, Department of Chemistry Hongo 7-3-1, 113-0033 Tokyo, Japan

Universität München, Institut für Medizinischen Optik Öttingerstraße 67, 80538 München, Germany

ISBN 978-3-540-73793-3

e-ISBN 978-3-540-73794-0

DOI 10.1007/978-3-540-73794-0 ISSN 0172-6218 Library of Congress Control Number: 9783540737933 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: WMXDesign GmbH, Heidelberg Typesetting and Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Printed on acid-free paper 987654321 springer.com

Preface

The frontiers in ultrafast intense laser science are rapidly expanding thanks to the progress in ultrashort and high-power laser technologies, and the interdisciplinary nature of this research field is attracting researchers with different expertise and backgrounds. As in the first two volumes of PUILS, each chapter in the present volume, whose page length is in the range of 15 to 25 pages, begins with an introduction in which clear and concise accounts of the significance of the topics is described, followed by a description of the authors’ most recent research results. All chapters are peer-reviewed. The articles of this third volume cover a diverse range of the interdisciplinary research field, and the topics may be grouped in five categories: strong field ionization of atoms (Chaps. 1–3), ionization and fragmentation of molecules and clusters (Chaps. 4–8), generation of high-order harmonics and attosecond pulses (Chaps. 9–11), filamentation and laser plasma interaction (Chaps. 12–14), and the development of ultrashort and ultrahigh-intensity light sources (Chap. 15). The PUILS series has been edited in liaison with the activities of the MEXT Priority Area Program on Control of Molecules in Intense Laser Fields (FY2002–2005), the JSPS Core-to-Core Program on Ultrafast Intense Laser Science (FY2004–), and JILS (Japan Intense Light Field Science Society). Starting with the present volume, JILS has become a sponsor organization for the PUILS series, taking on responsibility for its regular publication. As described in the prefaces of the first two volumes, the PUILS series collaborates also with the annual symposium series of ISUILS (http://www.isuils.jp), designed to stimulate interdisciplinary discussion at the forefront of ultrafast intense laser science. We would like to take this opportunity to thank all the authors who have kindly contributed to the PUILS series by describing frontiers of ultrafast intense laser science. We would also like to thank the reviewers who have served for the book project by reading carefully the submitted manuscripts. One of the co-editors (KY) thanks Ms. Chie Sakuta and Ms. Makiko Oyamada for their help with the editing processes. Last but not least, our gratitude goes out to Dr. Claus Ascheron, Physics Editor of Springer Verlag at Heidelberg, for his kind support of the series.

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We hope this volume will convey the excitement of Ultrafast Intense Laser Science to the readers, and stimulate interdisciplinary interactions among researchers, thus paving the way to explorations of new frontiers.

Co-editors of PUILS III University of Tokyo Laval University Ohio State University University of Palermo May 2, 2007

Kaoru Yamanouchi See Leang Chin Pierre Agostini Gaetano Ferrante

Contents

1 Foundations of the Strong-Field Approximation H.R. Reiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Trajectory Description of Ionization Processes in Strong Optical Fields T. Onishi, A. Shudo, K. S. Ikeda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Many Electron Ionization Processes in Strong and Ultrastrong Fields A. DiChiara, I. Ghebregziabher, S. Palaniyappan, E.L. Huskins, A. Falkowski, D. Pajerowski, B.C. Walker . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Molecular Rearrangements in Intense Laser Fields M. Krishnamurthy, D. Mathur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5 Optical Control of Chiral Molecular Motors K. Hoki, M. Yamaki, Y. Fujimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6 Non-Coulomb Explosions of Molecules in Intense Laser Fields F. Kong, S.L. Chin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7 Excitation, Fragmentation and Control of Large Finite Systems: C60 in Moderately Strong Laser Fields T. Laarmann, C.P. Schulz, I.V. Hertel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8 Theoretical Description of Rare-Gas Clusters Under Intense Laser Pulses I. Georgescu, U. Saalmann, C. Siedschlag, J. M. Rost . . . . . . . . . . . . . . . 149 9 Molecular High Order Harmonic Generation A.D. Bandrauk, S. Barmaki, S. Chelkowski, G. L. Kamta . . . . . . . . . . . . 171 10 Nonlinear Multiphoton Process in the XUV Region and its Application to Autocorrelation Measurement K. Midorikawa, Y. Nabekawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

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Contents

11 Controlling Light Polarization for Attosecond Pulse Generation E. Constant, E. Mével . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12 Some Fundamental Concepts of Femtosecond Laser Filamentation S.L. Chin, W. Liu, F. Théberge, Q. Luo, S. A. Hosseini, V. P. Kandidov, O.G. Kosareva, N. Ak¨ ozbek, A. Becker, H. Schroeder . . . . . . 243 13 The Transport of Relativistic, Laser-Produced Electrons in Matter – Part 1 D. Batani, R.R. Freeman, S. Baton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 14 The Transport of Relativistic, Laser-Produced Electrons in Matter – Part 2 D. Batani, R.R. Freeman, S. Baton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 15 Ultrafast, Ultrahigh Intensity Lasers: Challenges and Perspectives K. Yamakawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

List of Contributors

Neset Ak¨ ozbek Time Domain Corporation Cummings Research Park 7057 Old Madison Pike Huntsville AL 35806, USA Andre D. Bandrauk Laboratoire de Chimie Théorique Faculté des Sciences Université de Sherbrooke Sherbrooke Quebec J1k 2R2, Canada and Canada Research Chair–Computational Chemistry and Photonics [email protected] Samira Barmaki Secteur Sciences Université de Moncton Campus de Shippagan 218, Boul. J.D. Gauthier Shippagan New Brunswick E8S 1P6, Canada Dimitri Batani Dipartimento di Fisica “G. Occhialini” Universit` a di Milano Bicocca Piazza della Scienza 3 20126 Milano, Italy

Sophie Baton Laboratoire LULI Ecole Polytechnique 91128 Palaiseau Cedex, France Andreas Becker Max-Planck Institut f¨ ur Physik komplexer Systeme Nötnitzer Str. 38 01187 Dresden, Germany Szczepan Chelkowski Laboratoire de Chimie Théorique Faculté des Sciences Université de Sherbrooke Sherbrooke, Quebec J1k 2R2, Canada See Leang Chin Départment de Physique de Génie Physique et d’Optique and Centre d’Optique, Photonique et Laser (COPL) Université Laval Québec, Québec G1V 0A6, Canada Eric Constant Centre Laser Intense et Applications (CELIA) Université Bordeaux 1 – CEA-CNRS 351 cours de la libération 33 405 Talence Cedex, France [email protected]

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Anthony DiChiara Physics and Astronomy Department University of Delaware Newark DE 19716, USA Adam Falkowski Physics and Astronomy Department University of Delaware Newark DE 19716, USA Richard R. Freeman Mathematical and Physical Sciences 425 Stillman Hall The Ohio State University Columbus OH 43210-1123, USA Yuichi Fujimura Department of Chemistry Graduate School of Science Tohoku University Sendai 980-8578, Japan fujimurayuichi@ mail.tains.tohoku.ac.jp Ionut¸ Georgescu Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38 01187 Dresden, Germany Isaac Ghebregziabher Physics and Astronomy Department University of Delaware Newark DE 19716, USA Ingolf Volker Hertel Max Born Institute Berlin-Adlershof and Physics Department Freie Universit¨ at Berlin Germany

Kunihito Hoki Department of Chemistry Graduate School of Science Tohoku University Sendai 980-8578, Japan

Seyed A. Hosseini Départment de Physique de Génie Physique and Centre d’Optique Photonique et Laser Université Laval Québec G1V 0A6, Canada

Emily L. Huskins Physics and Astronomy Department University of Delaware Newark DE 19716, USA

Gerard Lagmago Kamta Laboratoire de Chimie Théorique Faculté des Sciences Université de Sherbrooke Sherbrooke, Quebec J1k 2R2, Canada

Kensuke S. Ikeda Faculty of Science and Engineering Ritsumeikan University Noji-cho 1916 Kusatsu 525-0055, Japan

Valery P. Kandidov International Laser Center Physics Department Moscow State University Moscow 119992, Russia


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Fanao Kong The Institute of Chemistry Chinese Academy of Sciences 2, 1st Street N. Zhongguan Cun Beijing 100080 P.R. China [email protected]

Eric M´ evel Centre Laser Intense et Applications (CELIA) Université Bordeaux 1 – CEA-CNRS 351 cours de la libération 33 405 Talence Cedex, France

Olga G. Kosareva International Laser Center Physics Department Moscow State University Moscow 119992, Russia

Katsumi Midorikawa Laser Technology Laboratory RIKEN 2-1 Hirosawa Wako Saitama 351-0198, Japan [email protected]

Manchikanti Krishnamurthy Tata Institute of Fundamental Research 1 Homi Bhabha Road Mumbai 400 005, India [email protected] Tim Laarmann Max Born Institute Berlin-Adlershof Germany Weiwei Liu Institute of Modern Optics Nankai University Tianjin, 300071 P. R. China Qi Luo Départment de Physique de Génie Physique and Centre d’Optique Photonique et Laser Université Laval Québec G1V 0A6, Canada Deepak Mathur Tata Institute of Fundamental Research 1 Homi Bhabha Road Mumbai 400 005, India [email protected]

Yauso Nabekawa Laser Technology Laboratory RIKEN 2-1 Hirosawa Wako Saitama 351-0198, Japan

Takaaki Onishi Department of Physics Tokyo Metropolitan University Minami-Ohsawa Hachioji 192-0397, Japan

Dan Pajerowski Physics and Astronomy Department University of Delaware Newark DE 19716, USA

Sasi Palaniyappan Physics and Astronomy Department University of Delaware Newark DE 19716, USA

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Howard R. Reiss Max Born Institute 12489 Berlin, Germany and American University Washington DC 20016-8058, USA Jan M. Rost Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38 01187 Dresden, Germany [email protected] Ulf Saalmann Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38 01187 Dresden, Germany [email protected] Hartmut Schroeder Max Planck Institut f¨ ur Quantenoptik Laserchemie Hans-Kopfermann-Str. 1 85748 Garching, Germany Claus Peter Schulz Max Born Institute Berlin-Adlershof Germany Akira Shudo Department of Physics Tokyo Metropolitan University Minami-Ohsawa Hachioji 192-0397, Japan

Christian Siedschlag FOM Institute for Atomic and Molecular Physics Kruislaan 407 1098 SJ Amsterdam The Netherlands Francis Th´ eberge Départment de Physique de Génie Physique and Centre d’Optique Photonique et Laser Université Laval Québec G1V 0A6, Canada Barry C. Walker Physics and Astronomy Department University of Delaware Newark DE 19716, USA [email protected] www.physics.udel.edu/∼bcwalker Koichi Yamakawa Japan Atomic Energy Agency (JAEA) 8-1 Umemidai Kizu Kyoto 615-0215, Japan [email protected] Masahiro Yamaki Department of Chemistry Graduate School of Science Tohoku University Sendai 980-8578, Japan

1 Foundations of the Strong-Field Approximation Howard R. Reiss Max Born Institute, 12489 Berlin, Germany American University, Washington, DC 20016-8058, USA

Summary. The nonperturbative analytical approximation method known as the SFA (Strong-Field Approximation) is traced to its roots in the early days of strongfield physics. Although the SFA is now the primary analytical method used in practice, several departures from the appropriate rules of application have arisen that needlessly restrict the broad power of the SFA. These include matters such as the use of intensity parameters and rules of scaling, and the often-inappropriate use of tunneling and multiphoton concepts. Extensions to momentum distributions and to the relativistic SFA are discussed briefly.

1.1 Introduction Strong-field physics began more than forty years ago with a small group of theoreticians. As laser development progressed to the point where strong fields could be explored in the laboratory, the number of people involved grew slowly at first, but has now mushroomed to a size that was hard to imagine forty years ago, and continues to increase. One effect of this pattern of growth is that most of the people presently in the field are nearly unacquainted with the early history. Fortunately, almost all of the originators of strong-field physics are still alive and active, and so an oral history remains possible. As is often the case, this oral history is much richer than what can be gleaned from the published literature. The early history of strong-field research is given in Sect. 1.2, followed by a definition of the SFA in Sect. 1.3. Section 1.4 gives a critique of a few current practices that are unfortunate, and can serve only to impede progress. Among these impediments are included the current failure to distinguish clearly between tunneling theories and S-matrix theories. Theories based on an S-matrix formulation are not confined to tunneling. The strong-field approximation (SFA) in the title is the simplest S-matrix approach to strong fields. The identity of a reliable index of when a field must be considered strong is a source of confusion. As will be shown in Sect. 1.4.3, the strength of the electric field of the laser in the focal region is not such an index. (The expression “laser field” is often used in place of the more accurate expression “strong field”. This should not cause a problem.) It is far more useful to use

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H.R. Reiss

intensity parameters that are ratios of energies. Furthermore, as discussed in Sect. 1.4.4, more than one intensity parameter is necessary for scaling. Another target for scrutiny is the commonly-made distinction between the “tunneling domain” and the “multiphoton domain”. This terminology is flawed, since it lacks internal consistency. It also involves the use of the Keldysh parameter as the sole determiner of which domain exists. This is not sufficient to characterize strong-field phenomena. A new, general approach to the definition of transition amplitudes is developed in Sect. 1.5 as prelude to the discussion in Sect. 1.6 of how the SFA is applied in practice to rates, spectra, and momentum distributions. Section 1.7 has brief remarks about the relativistic SFA. This subject is important since it is possible to do a complete Dirac treatment in three dimensions within the domain of SFA applicability. Space limitations do not allow more than basic remarks about this topic. There are also discussions of gauges and gauge transformations, high vs. low frequency phenomena, and the general subject of stabilization.

1.2 Early History 1.2.1 Prehistory Certain basic tools existed in the literature on quantum mechanics that proved useful for the pioneers in strong-field physics. Three will be mentioned here: tunneling, the Volkov solution, and S-matrix methods. Tunneling created no conceptual problems. A brief description will be sufficient. The way in which the Volkov solution is used was profoundly misunderstood by some investigators for many years, although that particular problem has largely dissipated. The use of S-matrix techniques was initially rejected by most of the growing numbers of strong-field physicists because of its presumed limitation to scattering problems. The present situation is that it has become such a standard tool to those theorists who use analytical approximations, that the younger members of this community find it hardly credible that it was not always viewed as an obviously useful approach. On the other hand, some physicists continue to misconstrue the S-matrix technique. Tunneling The most familiar concept is the explicitly quantum property of tunneling, where the presence of a static or quasistatic electric field can make possible the passage of an electron through a potential-energy barrier that would be impervious in classical mechanics. The investigation of tunneling due to

1 Foundations of the Strong-Field Approximation

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an electric field dates to an early stage of quantum mechanics [1]. Tunneling transition rates have the characteristic that the electric field strength F enters into the problem by way of the exponential function exp (−C/F ) ,

(1.1)

where C depends on the physical parameters of the system being explored. If the electric field is not static, but oscillatory with low frequency, then F in (1.1) can be replaced by F (t). Volkov Solution The Volkov solution [2] is an exact solution of the Dirac equation of relativistic quantum mechanics for an electron in a plane wave electromagnetic field. The field may be monochromatic, or it may be a wave packet all of whose components propagate in the same direction. It has come to be standard to apply the name “Volkov solution” also to those problems where the spin of the electron is neglected, and even to a dipole-approximation limit. It would be more proper to call it a Gordon solution [3] in those cases, since the wave function used is more closely related to the exact solution of the Klein-Gordon equation for a spinless charged particle in a plane wave field, given by Gordon. Confusion about the way in which the Volkov solution is used came from a lack of awareness that, in any transition amplitude, there must be one interacting state and one non-interacting (or reference) state. Since the Volkov solution refers to a detached particle in a final state in the presence of the laser field, that means that the initial bound state must be free of the field. It is this last aspect that has caused misunderstandings. S-matrix Methods The S-matrix approach to the description of transitions was introduced in 1937 by Wheeler [4], and owed much of its early development to Heisenberg [5] and St¨ uckelberg [6] in the 1940s. Its original purpose was for the description of scattering events (the origin of the “S”), where it was used in formal quantum field theory as well as in practical applications in high-energy physics, nuclear physics, and ion-atom scattering. In the 1960s, there was a need for a general formulation of transition amplitudes that could be used in every type of quantum transition, including bound-bound, bound-free, free-bound, as well as free-free; and that could be used in a strong-field, nonperturbative context. The standard approach to writing a transition amplitude was the “Golden Rule”, associated with Fermi, but this is explicitly perturbative. A 1970 paper [7] was the first to show that the S-matrix technique has general applicability, and is confined neither to scattering theory nor to perturbative processes.

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1.2.2 The Sengupta Preliminary The first application of Volkov solutions to strong-field problems was by N.D. Sengupta [8] of the Tata Institute in Bombay (now Mumbai), India in 1952. Although nonperturbative tunneling and WKB methods were familiar at this time, methods based on the Volkov solution are far more versatile for the strong-field environment, and open the door to the treatment of a wide variety of strong-field problems. However, Sengupta published in the Bulletin of the Calcutta Mathematical Society, and his contribution remained largely unknown outside India for many years. He is, unfortunately, now deceased. 1.2.3 The 1962 Beginnings The next published contributions to strong-field physics were made in 1962 by the present author. Two 1962 papers [9, 10] were the outgrowth of work begun in 1955. John S. Toll, then at the University of Maryland, knew of the Volkov solution and recommended it as a tool in dissertation research on a formal exploration of the properties of perturbation theory in quantum electrodynamics. It has turned into much more than that. The 1962 papers marked the beginning of continuous development of strong field physics. As related by Nikishov and Ritus [11] in 1990, and more recently by Narozhny [12], these 1962 papers “started strong-field physics”. The articles came to the attention of Soviet Academician and now Nobel Laureate V.L. Ginzburg in Moscow. He recommended the papers to the physicists of the Lebedev Institute in Moscow, and that triggered the flow of strongfield physics papers from the Soviet Union beginning in 1964. The first of this stream of papers is by Nikishov and Ritus [13]. Also, one of my coworkers (Zoltan Fried) in the 1960s, brought Volkov techniques to the attention of Brown and Kibble, whose 1964 paper [14] in Physical Review became widely known. A further description of the 1962 papers is appropriate at this point. The first of the 1962 articles [9] was a calculation of electron pair production by the collision of an intense low-frequency field (frequency ω) with an energetic photon of frequency ω . The threshold for pair production occurs when 2 ω ) > mc2 , (1.2) (n0 ω) ( where n0 is the lowest photon order that can satisfy this inequality. This process provides a practical means to observe the creation of matter from pure energy. This differs from the pair production that has been long familiar in the laboratory, which occurs through scattering off particles with mass. The process where no mass at all is present in the initial state, published in 1962, was first observed in 1997 [15]. Within the context of relativistic quantum mechanics (RQM), the 1962 calculation is free of approximation. To apply it,


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however, one must explore possible experimental parameters. The suggestion and the necessary estimates to perform experiments at the Stanford Linear Accelerator Center (SLAC) were published in 1971 [16]. The purposes of the 1962 paper were broader than the calculation of a particular physical process. A primary aim was to investigate the properties of perturbation theory. In quantum electrodynamics (QED), it has long been known [17] that the radius of convergence of perturbation theory is zero. Although QED has given the most precisely correct predictions known in physics, QED is an “asymptotic” perturbation expansion. It will give accurate results up to a certain order, and then begin to diverge. This was first remarked upon in 1952 [17], and has not been refuted to date. The purpose of the second 1962 paper [10] was to show that strong-field RQM has a nonzero radius of convergence. A side effect of the demonstration that strong-field physics does not suffer from a “singularity at the origin” as does QED, was that upper as well as lower bounds were found on the radius of convergence for a perturbation expansion in RQM. The intensity-dependent upper limits on RQM perturbation theory are easily surpassed even with the lasers of twenty years ago. For low intensity with n0 = 1 in (1.2), the 1962 results reduce to the lowest order perturbation theory expression first attained by Breit and Wheeler [18]. For high intensities with very low frequencies such that n0 1, a tunneling limit can be found. This is not surprising. On an energy diagram, the process corresponds to a tilting by the electric field of the relativistic mc2 and −mc2 thresholds so that it is possible to tunnel from a negative to a positive energy state. Correct attainment of both perturbative and tunneling limits show the power of the Volkov approach. An anecdote is appropriate here. The 1962 papers were originally submitted to Physical Review. They were returned with the referee’s comment that it was interesting work, but the necessary fields were too strong ever to be achieved. In his judgment, the papers amounted to a mathematical exercise, and were more suitable for Journal of Mathematical Physics. Thirty five years later, the “physically impossible” intensities were available in the laboratory [15]. 1.2.4 The Lebedev Developments From 1964 and continuing to the experimental era, the bulk of strong-field research was done by Soviet physicists in general, and in particular by physicists at the P.N. Lebedev Institute in Moscow. As already described, the 1962 papers [9, 10] were noticed by Nobelist Ginzburg, who brought the papers to the attention of the Lebedev physicists. Professor Ginzurg recommended the subject of strong-field physics, as well as the techniques of Refs. [9, 10]. Thus started the great outflow of papers by Nikishov and Ritus [13]; Gol’dman [19]; Keldysh [20]; Popov, Perelomov and Terent’ev [21]; Rapoport, Zon, Narozhny, Krainov, Delone, Smirnov, and many others.

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1.2.5 Strong-Field Beginnings in the West The 1962 papers were delayed in publication. They are based on a Ph.D. dissertation at the University of Maryland, completed in 1958 with Professor John S. Toll as advisor. He had gone directly from his Ph.D. work with Professor John Wheeler at Princeton to Chairman of the Physics Department at the University of Maryland. In a modest duplication of such a move, I went straight from the Ph.D. to Head of the Nuclear Physics Division at the Naval Ordnance Laboratory. In addition to the nuclear physics activities, a theoretical group in relativistic quantum field theory and high energy physics was formed. Early members of this group were Zoltan Fried from Boston University, William M. Frank from MIT, and Joseph H. Eberly from Stanford University as a National Research Council Postdoctoral Associate. The group was introduced to strong-field problems. Only the most fundamental results attained will be mentioned here. Fried and Eberly published a paper in 1964 [22] in which they employed a fully field-theoretical formulation of strong-field Compton scattering, with classes of Feynman diagrams exactly summed under the premise of large occupation numbers for a single mode of the photon field. They obtained almost the same result as calculations in relativistic quantum mechanics using the Volkov solution. The one feature seemingly absent was the so-called strongfield mass shift first discussed in one of the 1962 papers [9]. Eberly and I later showed [23] that the strong-field mass shift is actually present in the field-theoretical calculation, arising from the finite exact sum of a class of divergent diagrams omitted from the Fried and Eberly paper. Demonstration of the equivalence of a Volkov-method calculation with a quantum field-theoretic calculation is a landmark result in that it showed that strong fields render second quantization unnecessary. Another basic paper [24] found the exact relativistic Volkov Green’s function in a form corresponding to the Feynman Green’s function of QED, showing the location of all of the relativistic Floquet states. A well-known paper from this era was that of Brown and Kibble [14] on strong-field Compton scattering, the same process originally calculated by Sengupta [8]. The Brown and Kibble paper was almost coincident with the first of the many Nikishov and Ritus papers [13]. It can also be said to have the same provenance. Brown and Kibble [14] credit Zoltan Fried with having introduced them to strong-field physics. As already mentioned, Fried had his introduction to strong fields as part of my group.

1.3 Definition of the Strong-Field Approximation The SFA will be introduced here, since it is the central topic of this work. There is confusion in the strong-field community about the precise meaning of the SFA. A clear definition can be achieved in a single step from the


7

general expression for a transition amplitude. The defining features of the SFA are: the use of S-matrix methods; the time-reversed form of the transition amplitude; and the replacement of the final fully interacting state in the S-matrix by a Volkov solution. We shall return to the SFA in more detail in Sect. 1.6 after presenting in Sect. 1.5 the formal foundations on which it rests. 1.3.1 S-matrix As demonstrated rigorously in Sect. 1.5, it is possible to write a completely general transition amplitude for any interaction that occurs in a volume bounded in both space and time. This is not very restrictive, since most experiments are done this way. In particular, experiments with pulsed lasers brought to a focus obey these restrictions. The transition amplitude can be written in either of two equivalent forms. There is the “direct-time” amplitude ∞ (S − 1)f i = −i

dt (Φf , HI (t) Ψi ) ,

(1.3)

dt (Ψf , HI (t) Φi ) .

(1.4)

−∞

and the “time-reversed” amplitude ∞ (S − 1)f i = −i −∞

In these expressions, Sf i is the S-matrix element for a transition from an initial state identified by the subscript i, to a final state bearing the subscript f . From this is subtracted (1)f i or, equivalently δf i , the probability amplitude that the system ends up in the state from which it started. One could simply replace (S − 1)f i by Mf i and make no further reference to S-matrices. The states Φ and Ψ in (1.3) and ( 1.4) satisfy the non-interacting and interacting Schr¨ odinger equations i∂t Φ = H0 Φ , i∂t Ψ = [H0 + HI (t)]Ψ .

(1.5) (1.6)

Equations (1.3)–(1.6) are in atomic units, as are all equations hereafter. It can be presumed that the Φ states will be known exactly but, in the presence of very strong fields, the Ψ states are generally very difficult to approximate. This is especially true if one attempts to introduce an approximation for a bound initial state Ψi as in (1.3). No matter how strong the applied field may be, the Coulomb singularity at the center of the atom can never be fully dominated. It is far easier and inherently more accurate to introduce an approximation for Ψf in (1.4). For the ionized electron, one can say that

8

H.R. Reiss

a sufficiently strong laser field will dominate residual Coulomb effects on the free electron. With the ponderomotive energy Up representing the energy of interaction of the applied field with the detached electron, and with the binding energy EB as a measure of the residual Coulomb effects, the condition z1 1 ,

z1 ≡ 2Up /EB ,

(1.7)

provides the grounds for replacement of the exact Ψf function in (1.4) with the Volkov state ΨfV . This is the SFA: ∞ SFA

(S − 1)f i

= −i

dt ΨfV , HI (t) Φi .

(1.8)

−∞

The general term of a strong-field expansion of the matrix elements (1.3) or (1.4) can be found from a Green’s function approach. These results are in Appendix A of [25]. That is, all correction terms to the basic SFA transition amplitude are stated there. A derivation of perturbation theory also follows in an elementary way from (1.3) or (1.4). 1.3.2 Choice of Gauge Equation (1.8) is the starting point for implementation of the SFA. There remains, however, a choice of gauge. Unlike perturbation theory, that can be shown to be gauge invariant at any given order, there seems to be no strongfield analytical approximation that is gauge invariant. The SFA terminology was originally introduced to avoiod the confusion that followed from the fact that the SFA predicts stabilization. This not possible with a tunneling theory like that of Keldysh, but most practitioners were unaware of the fundamental differences between tunneling theory and SFA. The SFA was intended to be used with the velocity-gauge interaction 1 HI = A · p + A2 , (1.9) 2 although some authors have elected to use the length-gauge interaction HI = r · F .

(1.10)

This would be consistent only if it can be shown that the length gauge exhibits stabilization. Until such time as this is done, the use of (1.10) in the SFA expression (1.8) risks perpetuating the original confusion between tunneling theories and the properly-defined SFA. The subject of gauge will arise again, so it will be convenient to introduce the abbreviations L gauge and V gauge for the two commonly encountered gauges. The L-gauge analog of the SFA, because it uses only a scalar potential that can be viewed as altering the Coulomb attractive well for a bound electron, shares many of the qualitative features of tunneling methods. It seems to be


9

better suited to prediction of total ionization rates than is V-gauge SFA, at least in the lower range of nonperturbative intensities. In the limited context of using only a wave function corresponding to a delta-function potential, the resulting simplicity has made possible application to few-cycle pulses [26] and to the introduction of a Coulomb correction to the Volkov function as shown in [27]. An advantage of the V-gauge SFA is that it requires two components of the electromagnetic four-potential Aμ , as is necessarily true in any fully quantum treatment. Since this is a fundamental feature of a transverse field, such as a laser field, it is to be expected that velocity-gauge SFA will perform better in the higher ranges of intensity. It confirms this expectation in that it connects smoothly into the relativistic SFA [28–30]. Velocity-gauge SFA exhibits considerable structure in angular distributions of photoelectrons. It does an excellent job on spectra [31–33]. Velocity-gauge SFA is also well adapted to the treatment of specific initial bound-state properties. There are other properties of V-gauge SFA that have never appeared in the literature on the L-gauge analog of the SFA. In addition to the relativistic connection already mentioned, V-gauge SFA can be applied to the study of stabilization [34,35], and to high frequencies as well as to low frequencies [36]. There are some frequency limitations that are explicated in Sect. 1.4.2. Further discussions of gauge matters appear in Sects. 1.6.1 and 1.8.

1.4 Strong-Field Terminology and Concepts In any field of endeavor, a specialized terminology arises that serves as a universally acknowledged “shorthand”. It is very important, since it allows a simple word or phrase to replace a detailed explanation of some concept that is frequently encountered in that discipline. However, if the terminology is illchosen or ambiguous, it can lead to real harm in that different people can ascribe different meanings to a supposedly universal concept. Such a situation exists now with respect to some widespread terminology in strong-field physics. A few current problems with terminology that directly affect SFA matters will be mentioned. In addition, the concept of the dominance of the electric field in electromagnetic interactions will be examined critically. 1.4.1 Tunneling Domain and Multiphoton Domain Tunneling theories make no mention of a concept of individual photons. When the energy spectrum of electrons ionized from an atom or molecule by a strong field appears to be a smooth, continuous spectrum, such a spectrum is taken to define what has come to be called the tunneling domain.

10

H.R. Reiss

A photoelectron energy spectrum in which individual peaks can be discerned, with a separation of ω between the peaks, is sometimes called an ATI (above-threshold ionization) spectrum. The name comes from the fact that perturbation theory normally predicts that only the lowest, or threshold order, would be observable. The additional peaks, first observed in 1979 [37], were thus referred to as above-threshold peaks. It is this 1979 work that started the experimental phase of strong-field physics that continues to expand to the present. When individual peaks separated by the energy of a single photon can be distinguished, the intensity domain in which this is observed is called the multiphoton domain. The strong-field parameter that is customarily employed to distinguish between the multiphoton domain and the tunneling domain is the dimensionless Keldysh parameter γ [20], that can be defined as (1.11) γ = EB /2Up , ω = 2EB , (1.12) F where EB is the binding energy of the ionized electron in the field-free environment before the laser pulse arises, and Up is the ponderomotive energy of the free electron in the field that causes the ionization. Equation (1.12) replaces Up by its equivalent in terms of the field frequency ω and the electric field amplitude F . In terms of γ, the exponential tunneling factor given in (1.1) is C 4 EB exp − = exp − γ . (1.13) F 3 ω Theories of tunneling ionization [20, 21, 38] give the domain of applicability for a tunneling theory as γ1.

(1.14)

The parameter γ is related to the intensity parameter z1 of (1.7) by the simple expression z1 = 1/γ 2 .

(1.15)

The physical understanding of the concept of tunneling comes from expressing a laser field in the L gauge in the dipole approximation, in which the field is described by the scalar potential V = −qr · E (t) = r · E (t) ,

(1.16)

where the electric charge q is replaced with the electron value −1 in a.u. There is no vector potential at all in the dipole-approximation L-gauge. The sum of the Coulomb potential and the length-gauge laser potential produces a barrier


11

that is elevated on one side of the atom and depressed on the other. On the depressed side, the summed potentials produce a peak of a height limited to negative energies. For a sufficiently strong field, a bound state of the atom can ionize by tunneling through the barrier. For example, consider groundstate hydrogen subjected to laser radiation with a wavelength of 800nm. In the direction in which the Coulomb barrier is maximally depressed by the laser field, the top of the barrier will be lowered to the energy of the ground state at an intensity such that γ ≈ 1. From (1.11) and (1.14), we see that the tunneling domain (γ < 1) occurs for higher intensities. When the top of the barrier is depressed below the level of the ground state, ionization occurs over the barrier. It is for lower intensities (γ > 1) that the bound electron must tunnel through the barrier in order to escape the atom. This is the reverse of the standard naming convention. That is, the socalled multiphoton domain, where individual ATI peaks are observed, is at γ > 1 where tunneling must be involved. The so-called tunneling domain, where a continuous spectrum is observed, is at γ < 1 where ionization occurs over the barrier, and no tunneling is necessary. To avoid this utterly confused situation, it would be better to refer to multi-peaked (or ATI) spectra and continuous spectra, and make no mention of tunneling at all. In the V-gauge (or Coulomb gauge), where the laser field is described by a vector potential, the concept of tunneling through a depressed barrier does not exist. In the V-gauge, multi-peaked spectra occur in the lower ranges of strong-field phenomena. There, the intensity distribution in the laser focus will produce the peaks in the spectra from the most intense part of the focal intensity distribution. As the maximum laser intensity is increased, more and more of the focal intensity distribution will contribute to the observed spectrum. Different regions of the focal intensity distribution will have different ponderomotive energies, so that the final kinetic energy acquired by the photoelectron will be different for the same photon order in accordance with the energy conservation condition p2 = nω − Up − EB . 2

(1.17)

The result of this superposition of focal regions with different Up is to broaden each ATI peak to the degree that the spectrum appears to be continuous. It is the fusing of many high-order ATI peaks that produces the continuous appearance of strong-field spectra, not the tunneling process. 1.4.2 KFR Theory The initials K, F, R represent theories by Keldysh [20], Faisal [39], and Reiss [25]. They are all quite different. K is a tunneling approximation requiring Up EB and EB ω; F is a high-frequency approximation requiring Up EB and ω EB ; and R is a strong-field approximation requiring only

12

H.R. Reiss

Up EB . When the KFR association was first suggested by Bucksbaum et al. in 1988 [40], it seemed to be a convenient shorthand for nonperturbative theories of atomic ionization. With further progress in understanding strongfield ionization, the KFR designation has come to be a source of confusion rather than enlightenment. To explain why this is so, the evolution of atomic ionization analytical approximations will be sketched. In the 1960s, after the initial spate of relativistic strong-field papers, the Lebedev strong-field physicists focused their attention on atomic ionization. There were, in rather quick succession, tunneling-theory papers by Keldysh [20], by Nikishov and Ritus [41], and by Perelomov, Popov, and Terent’ev [21]. Until the experimental era of strong-field physics arrived, strongfield analytical approximations in atomic physics were referred to generically as Keldysh-like approximations. Even after strong-field experimental work was published starting in 1979 [37], no immediate connections between theory and experiment occurred because the experimental spectra were all of the multi-peak variety, outside the scope of tunneling theories. Thus, there was no need to refine the terminology. The original SFA paper, although published [25] in 1980, was actually written before the 1979 ATI results were published [37]. The SFA paper was delayed by lengthy debates with referees. The 1980 SFA paper predicted and explained most of the strong-field effects found in the laboratory: the ATI phenomenon; channel closings; the qualitative differences between linear and circular polarization spectra; the failure of perturbation theory; etc. The 1980 SFA paper did not attract notice until a talk at Cortona [42] in Italy in which it was shown that the theory predicted very accurately the results of new experiments by Bucksbaum et al. [43] that yielded a multipeaked circular polarization spectrum of a type qualitatively distinct from linear polarization spectra. The Cortona results later appeared in a journal article [31]. Bucksbaum made a detailed comparison [32] that is reproduced here as Fig. 1.1. The lack of gauge invariance in strong-field analytical theories quickly came into evidence in that the most accurate predictions of total ionization rates came from the ADK tunneling theory [38, 44], whereas the SFA theory [32, 33] was most successful at predicting spectra. There remains a tendency to make no distinction between the Keldysh approximation and the SFA. This is unfortunate because the SFA is capable of making predictions that are entirely out of the question with the theory of Keldysh. This has had the consequence that some important SFA results have been discounted because “everybody knows that kind of theory cannot give such results”. Some distinguishing features will be listed here, and some of the most striking differences will be enlarged upon. A summary is given in Table 1.1. “Stabilization” is that strong-field property wherein an intensity can be found beyond which further increases cause a decline in transition probabil-


13

Fig. 1.1. A figure from [32] showing the first detailed agreement between experiment and the velocity-gauge SFA. Note that this is a multi-peak spectrum, impossible to model with a tunneling theory

ity instead of the usual increase. It seems to be a general strong-field property, not confined to ionization. For example, see the non-ionization problems shown to exhibit stabilization [45, 46]. It is impossible for a tunneling theory to predict stabilization. The intensity dependence is dominated by the tunneling exponential and, as (1.13) shows, the exponential function increases monotonically from a value of zero when F → 0 to a value of unity as F → ∞. By its very definition, a tunneling theory applies only to low frequencies. The first two entries in the Keldysh (or tunneling theory) column of Table 1.1 have now been explained.

14

H.R. Reiss

Table 1.1. Comparison between the properties of the Keldysh approximation and the velocity-gauge SFA Property

Keldysh theory

Velocity-gauge SFA

frequencies

ω EB

no limit

stabilization

impossible

good agreement with other theories

angular distribution

fwd peak only

many peaks poss., min or max fwd

spectrum

only continuous

continuous or multi-peak

spectrum limit as E → 0

max of spectrum

→ 0 for lin pol, followed by peak

parallel mom. distrib.

peaks at p = 0

can have min or max at p = 0

perpend. mom. distrib.

can have a cusp at p⊥ = 0

The SFA requires only high intensity, defined as z1 1, as in (1.7). Alternatively, z1 =

I 2ω 2 EB

1,

(1.18)

where I is the field intensity in a.u. As shown in (1.15), the intensity parameter z1 is related to the Keldysh parameter as z1 = 1/γ 2 , so that z1 increases linearly with intensity, whereas γ declines as intensity increases. As will be discussed more fully below, when the SFA transition amplitude is derived, the only condition required for validity of the SFA is z1 1. This condition does not directly place limits on the frequency. There is, however, an indirect limit on frequency. When examined for stabilization behavior, the SFA predicts [47] that the stabilization intensity for low frequencies behaves as low Istab =

4 3 E . 9 B

(1.19)

For high frequencies, the stabilization intensity behaves as high Istab = 4ω 3 .

(1.20)

The numerical coefficients in (1.19) and (1.20) depend on the atom. Those shown refer to ground-state hydrogen. If the intensity for validity of the SFA is higher than the stabilization intensity, then the SFA is of little use under such circumstances. If we take the SFA to be valid when z1 ≥ 10, and find the domains in which this is lower than the stabilization intensities in (1.19) and (1.20), then we obtain an upper limit for low frequency applicability and a lower limit for high frequency applicability. Figure 1.2 shows how the SFA prediction for stabilization in


15

Fig. 1.2. Comparison of SFA predictions for the location of the stabilization intensity with those found by numerical solution of the Schr¨ odinger equation [48]. The model atom had EB = .0276 a.u. Curves labeled “SFA low freq” and “SFA high freq” correspond to (1.19) and (1.20) here. (Reproduced from [35])

the transition region between high and low frequencies compares with stabilization results from numerical solution of the Schrödinger equation [48]. As noted, tunneling theories cannot describe stabilization, nor has it been demonstrated for the SFA in L gauge. The remaining rows in Table 1.1 will be commented upon below after transition probabilities have been exhibited explicitly. 1.4.3 Electric Field Dominance In perturbative AMO (Atomic, Molecular, and Optical) physics, and in strong-field physics using the L-gauge, the magnetic field component of a plane wave electromagnetic field seems to be of no consequence. The Lorentz force on a charge q,

v F Lorentz = q F + × B , (1.21) c diminishes the role of the magnetic field B because of the generally small magnitude of v/c. Furthermore, since the magnetic component of the Lorentz force is perpendicular to the direction of motion v, it can do no work on the charge q. That is, the B field does not cause changes in the energy of the

16

H.R. Reiss

charged particle. The notion that effects of the B field are unimportant is reinforced by the success of tunneling theories, where the tunneling factor, (1.1) or (1.20), depends only on the electric field, with no mention at all of the magnetic component of a laser field. Even though the magnetic field does not insert field energy into the charged particle, it couples to the electric field in such a fashion as to distort the linear oscillatory motion (in the coordinate frame in which the particle is at rest on average) into a “figure-8” motion [49, 50]. Figure 1.3 shows the shape of such a pattern for very strong fields. The amplitude α0 is the well-known free-electron amplitude of motion: α0 =

2 1/2 F0 Up = 2 , ω ω

(1.22)

where F0 is the electric field amplitude. The combination of electric and magnetic fields in a propagating plane wave induces a component of motion in the direction of the propagation vector k. This less familiar amplitude β0 is β0 =

Up F2 = 03 . 2ωc 8ω c

(1.23)

Equations (1.22) and (1.23) are nonrelativistic limits of known relativistic results. As will be seen in what follows, both α0 and β0 arise directly in the V-gauge SFA. The thickness β0 of the figure-8 pattern gives an index of when the dipole approximation breaks down. When β0 becomes large enough to be significant on the scale of the atom, then magnetic field effects will make a difference in the behavior of the system. We can set this limit at 1/10 of a Bohr radius, so that the validity of the dipole approximation is confined to intensities such that z < 0.1 2c z < 0.2c ≈ 27 .

β0 =

(1.24) (1.25)

Fig. 1.3. Figure-8 motion induced by the coupling of the electric and magnetic fields in a plane wave. The amplitude α0 is in the direction of the electric field, and β0 is in the direction of the propagation vector k perpendicular to the electric and magnetic fields


17

The quantity z in these equations is the dimensionless intensity parameter z ≡ Up /ω

(1.26)

that was introduced in Ref. [25], and (1.25) follows from the fact that, in a.u., the velocity of light has the value c = 1/α ≈ 137.036 .

(1.27)

Normally, one thinks of the validity of the dipole approximation as being a function only of the wavelength of a field as compared to the size of the system with which it interacts. Equation (1.25) involves the field intensity as well, so that it is explicitly an intensity limit. We can also point to the fact that the laser field will induce relativistic effects when the ponderomotive energy of the electron approaches the mass energy of the electron. In terms of the long-familiar [9, 51] free-electron intensity parameter zf ≡ 2Up /c2 ,

(1.28)

it can be said that intensity-caused relativistic effects will occur when zf > 0.1 .

(1.29)

When written entirely as a function of intensity I and frequency ω, (1.25) and (1.29) are I 100 ω 3 , I 4000 ω 2 .

(1.30) (1.31)

Equations (1.30) and (1.31) are plotted in Fig. 1.4, where (1.30) is the lower slanted line and (1.31) is the upper slanted line. The different ω dependences give the lines different slopes. Figure (1.4) illustrates that the electric field strength is a poor indicator of intense field effects. Although such an interpretation might be supportable in the vicinity of the often-used Ti-Sapph frequency, the figure shows that, at 100eV, F0 = 1 is merely perturbative, whereas F0 = 1 is strongly relativistic for the CO2 laser. Another indicator of the poor performance of the electric field as an index of intensity can be seen from the location in the lower left of Fig. 1.4 of experiments done with a CO2 laser [52]. Experimental conditions were very intense. The parameters z1 and z, defined in (1.7) and (1.26), have the values z1 = 154 and z = 7530. The ponderomotive energy of the ionized electron is 77 times as large as the binding energy of the valence electron. Nevertheless, the electric field amplitude is only F0 = 0.05 a.u.

18

H.R. Reiss

Fig. 1.4. Intensity limits on the dipole approximation and for relativistic behavior are shown as a function of frequency. Some representative strong-field wavelengths are shown on the upper x axis. An electric field of one a.u. is seen to have no special significance. The 1989 CO2 laser experiments of Corkum [52], shown by the dot at the lower left, have all the features of a very intense-field environment that is almost relativistic, but the electric field is only 0.05 a.u.

1.4.4 Scaling It is immediately evident from Fig. 1.4 that a single intensity parameter cannot suffice for a scaling comparison of one set of data with another. The boundary between validity and failure of the dipole approximation behaves as I/ω 3 [ (1.30)], whereas the onset of relativistic behavior depends upon the ratio I/ω 2 [ (1.31)]. The limits of perturbation theory in atomic ionization scale as I/ω 3 (Ref. [25]), whereas perturbative limits for free electrons in strong fields scale as I/ω 2 (Ref. [10]). It is impossible to maintain both ratios at the same values when altering experimental conditions.

1.5 General Transition Amplitude The word “general” in the section heading refers to the fact that the transition amplitude about to be derived has only one limitation: it applies to any experimental configuration where the measuring instruments lie outside the space and time domain in which the transition-causing interaction takes place. This


19

is certainly true of all experiments with pulsed lasers. The derivation to be given differs from the standard approach in that there is no recourse to “adiabatic decoupling”; the assumption that is conventionally made to describe how the interaction of the atom or molecule with the field is turned on or off. Furthermore, all ambiguity is removed from how a gauge transformation is applied to transition amplitudes. 1.5.1 Strategy to Relate Transition Amplitudes to Experiments We start with the idea that the quantum system whose transitions we wish to describe is enclosed in a finite spatial volume (as in a laser focal volume), and is subjected to a transition-causing interaction (the laser field) for a finite time (the laser pulse length). The laboratory instruments that determine what has happened in the interaction space-time region are located outside the interaction region. The measuring instruments never experience the field. It is necessary to deduce what has happened to the quantum system in the interaction zone by examining “readings” performed outside the interaction zone. Everything is done within the context of an S-matrix theory. “S-matrix” is used here in the broad context of application to any quantum process whose description requires nothing more than a statement of how the system is prepared, and how the final outcome is analyzed. The fact that the system obeys the Schr¨ odinger equation is implicit in the formulation, but nothing that occurs during the interaction needs to be examined. Analogous procedures can be followed with the Pauli, Dirac, and Klein-Gordon equations. 1.5.2 Quantum Equations and Asymptotic Correspondences The measuring instruments obey the Schrödinger equation that contains information about atoms or molecules (and the electrons they contain) without laser interaction. The measurement system “knows” nothing about laser fields directly. We write the Schr¨ odinger equation simply as i∂t Φ = H0 Φ .

(1.32)

The solutions of (1.32) form a complete set {Φn }. The atoms or molecules (or the electrons thereof) subjected to the laser field experience the additional interaction arising from the field, so their description is in terms of the more extensive Schrödinger equation i∂t Ψ = (H0 + HI (t)) Ψ .

(1.33)

The solutions of (1.33) form a complete set {Ψn }. The hypothesis is made that lim HI (t) = 0 ,

t→±∞

(1.34)

20

H.R. Reiss

but nothing needs to be stated about the way in which (1.34) is satisfied. The conventional adiabatic-decoupling demand that an exponential fall-off in HI occurs, is not needed here. We organize the {Φn } states by the statement that one element of this set, Φi , defines the initial state before an experiment is performed. The {Ψn } set is then correlated with the {Φn } states in terms of the initial conditions by making the correspondence lim Ψn (t) = Φn (t) .

t→−∞

(1.35)

The limits referred to in (1.34) signify nothing more than times before and after the laser pulse passes over the target atom. The limit t → −∞ and the limit t → +∞ may refer to times separated only by femtoseconds in a short-pulse experiment. 1.5.3 Transition Amplitude and Transition Probability In the laboratory, the outcome of an experiment is analyzed by instruments not themselves subjected to the laser pulse. This action can be expressed by saying that the initial state Ψi (−∞) has evolved to Ψi (+∞). The physical content of this evolution is defined by forming overlaps with all the possible states of the set {Φn } that can occur. That is, the amplitudes Sf i = lim (Φf (t) , Ψi (t)) t→+∞

(1.36)

are a complete description of what has happened during the experiment. Normally, any residual amplitude that remains in the initial state is not counted, so the transition amplitude is (S − 1)f i = lim (Φf (t) , Ψi (t)) − lim (Φf (t) , Ψi (t)) , t→+∞ ∞

=

dt −∞

t→−∞

(1.37)

∂ (Φf (t) , Ψi (t)) , ∂t

(1.38)

dt (Φf (t) , HI Ψi (t)) .

(1.39)

∞

= −i −∞

The last term in (1.37) comes from the orthogonality of the {Φn } states and from (1.35), so that lim (Φf (t) , Ψi (t)) = lim (Φf (t) , Φi (t)) = δf i = (1)f i .

t→−∞

t→−∞

(1.40)

The step from (1.37) to (1.38) is a simple identity. Equation (1.39) is a result of carrying out the time derivatives in (1.38) and inserting the Schr¨ odinger equations (1.32) and (1.33).


21

The transition probability per unit time follows from the square of the transition amplitude divided by the time, and integrated over the phase space constituting the possible final states of the system. That is, the transition probability per unit time is 2 1 (1.41) w = lim (S − 1)f i , τ →∞ τ wV d3 p W = (1.42) 3 , (2π) 3

3

where V is a normalization volume, and the divisor (2π) → (2π) is the size of a single cell in phase space. See Refs. [25] and [29] for more details. 1.5.4 Time-Reversed Transition Amplitude An alternative procedure, equivalent to the above, is to associate a particular final state Ψf with a member of the set {Ψn } , and to make the correspondence lim Ψf (t) = Φf (t) .

t→+∞

(1.43)

Then overlaps of Ψf (t) with the elements of {Φn } are used to obtain the probability amplitudes for Ψf (t) to have arisen from each of the possible Φi states. A procedure completely analogous to the preceding section leads to ∞ (S − 1)f i = −i

dt (Ψf (t) , HI Φi (t)) .

(1.44)

−∞

1.5.5 Fundamental Principles Established Equations (1.39) and (1.44) are quite general. We could have added a terminology relating to “in-states” and “out-states”, as is usually done in scattering theory, but we can overlook this for present purposes. One important matter is clear: Each transition amplitude must contain one fully interacting state (Ψ ) and one non-interacting or “reference” state (Φ). This must be true for any quantum process in which measurements (using Φ states) are made outside the space-time region in which transitioncausing interactions (described by Ψ states) are taking place. This cannot be emphasized too strongly. For many years after the introduction of the SFA, it was dismissed by many people in the strong-field community as “crude” because (1.44) contains no interaction in the initial state. The reminder that the appearance of Φi for the initial state is a necessity and not an approximation fell upon deaf ears for many years. This fundamental failure of understanding seems finally to be fading. The benefits of a structure where the initial state is a Φ state will be explicated in the following section.

22

H.R. Reiss

Another sometimes-contentious issue that is clarified by the derivation of (1.39) and (1.44) is the method by which gauge transformations are introduced into transition amplitudes in quantum mechanics. Generally, discussions about gauge transformations in quantum mechanics textbooks are confined to the demonstration that the Schr¨ odinger equation remains forminvariant under a change in gauge, as long as the wave functions in the two gauges differ by a unitary factor containing the generating function of the gauge transformation. However, such discussions do not normally extend to the matter of how transition amplitudes in two different gauges are related. Disagreements have occurred repeatedly because one group may prefer L gauge, and then find that their means of transformation to V gauge produces results in conflict with those found by people who begin in V gauge; and vice-versa. The resolution of this matter is simple. Equations (1.39) and (1.44) contain Φ states that are independent of gauge because the Φ states relate only to the world outside the interaction region, where no laser field exists. The Ψ states and the HI interaction Hamiltonian take on values appropriate to their own separate gauges. If we designate two gauges by superscripts (1) and (2), we will find generally that ∞

∞

(1) (1) (2) (2) dt Ψf (t) , HI Φi (t) = −i dt Ψf (t) , HI Φi (t) . (1.45) −i −∞

−∞

This is as it should be. In some fields of physics, the enforcement of gauge invariance often supplies a missing step to make a derivation complete. The assumption frequently found in the AMO literature that gauge-transformed transition amplitudes must be identically the same would obviate such a step. All that is demanded of a gauge transformation is that physically observable quantities should be gauge invariant.

1.6 The SFA In application of the above formalism to strong-field ionization, a choice must be made between the direct-time (1.39) and the time-reversed (1.44). It can be assumed that noninteracting states Φ are known exactly, or can be represented by well-established accurate approximations. No strong field is associated with any of the Φ states, so it is possible to draw on the long history of atomic and molecular physics. To the contrary, we begin with very little information about the Ψ states, since they are strongly affected by an intense field. The treatment of Ψ requires that some approximation must be made that is more general than perturbation theory. For simplicity, we use the language of single-independent-electron ionization of an atom. The initial Ψi (t) required for (1.39) means that we must be able to describe an electron bound to an atom and subjected to a strong


23

field. This is fundamentally difficult, because no matter how strong the laser field is, the Coulomb potential always has a singularity at the origin. Neither potential can be said to be the dominant one. By contrast, the final Ψf (t) in (1.44) refers to an unbound electron in a strong field. For this case, it is possible to say that, when Up for the detached electron in the field is very large (interpreted as Up EB , or as z1 1), then the effects of the laser field will dominate residual effects of the Coulomb force on the free electron. The Volkov solution is an exact solution of the quantum equations of motion for a free electron, so when Up EB it is a good approximation to simply replace Ψf (t) by the Volkov solution ΨfV (t) . This is the Strong-Field Approximation, already stated in (1.8): ∞ SFA (S − 1)f i = −i dt ΨfV (t) , HI Φi (t) . (1.46) −∞

The only requirement that needs to be stated is that z1 1, with no mention of frequency. The SFA agrees [36] with other analytical approximations at both high and low frequencies. Even the requirement on z1 can be relaxed if the physical problem depends only on high-lying parts of the spectrum, where a photoelectron has enough kinetic energy that residual Coulomb effects are unimportant. 1.6.1 Choice of Gauge in the SFA As remarked upon in Section 1.3.2, the SFA was intended to be used with the V-gauge to avoid confusion with the narrower applicability of tunneling theories. Specifically, because it has never been shown to date that the SFA formalism with an L-gauge substitution will exhibit stabilization, the L-gauge version does not possess the defining property of the SFA. It is necessary to note that gauge invariance does not hold with any known non-pertubative analytical approximation. Some basic remarks are made here about the properties of the V-gauge and the L-gauge when compared directly. A plane-wave electromagnetic field is a relativistic object since it propagates in free space at the velocity of light c. It is described by the four-vector potential Aμ . Not all components can be independent because, like all fourvectors, one can form the scalar invariant Aμ Aμ that must have the same value in every Lorentz frame of reference. This means that there are really only three independent components of Aμ . The electromagnetic field also satisfies the condition kμ Aμ = 0 , where kμ is the propagation four-vector

ω kμ : ,k . c

(1.47)

(1.48)

24

H.R. Reiss

The fact that Aμ satisfies both conditions (1.47) and (1.48) means that it has only two independent components, generally taken to be specified as two independent polarization states. The notation in (1.48) has the significance that k0 , the timelike component of kμ is given by the scalar quantity ω/c, and the three-vector part is k. The magnitude of k is |k| = ω/c, and so kμ kμ =

ω 2 c

− k2 = 0 .

(1.49)

The above remarks are intended to emphasize the fact that any full description of the electromagnetic potentials must include a vector potential. It is only when the dipole approximation is also imposed that it is possible to find a gauge transformation that can remove the vector potential and replace it with a scalar potential, constituting a reduction to only a single component of Aμ . This means that, with no vector potential employed, the L-gauge is limited to the region below the lower slanted line in Fig. 1.4. This is adequate for a wide variety of problems. The V-gauge, however, not only allows a continuation across the lower slanted line in Fig. 1.4, but there is a natural continuation across the upper slanted line as well. A fully Dirac-relativistic SFA [28–30] reduces smoothly to the velocity-gauge, dipole-approximation SFA when the appropriate limits are taken. This article stresses the V-gauge SFA. The flexibility of the V-gauge permits a broad discussion of strong-field physics. 1.6.2 Differential Transition Probability Explicit evaluation of the total transition probability of (1.42), using the time-reversed S matrix of (1.44) can be found in Refs. [25, 29]. We state the results here, but in a different and more revealing form than employed in those older references. The differential transition probability per unit time is, for linear polarization, 2 2 ∞ 2 dW 1 p 2 = p + E φ (p) [Jn (α0 p cos θ, −β0 c)] ,(1.50) B i 2 dΩ 2 (2π) n0 p2 = nω − Up − EB . 2

(1.51)

The φi (p) function in (1.50) is the momentum-space wave function for the spatial part of the initial wave function, lim Ψi (r, t) = Φi (r, t) = φi (r) exp (−iEi t) , φi (p) = d3 r exp (−ip · r) φi (r) .

t→−∞

(1.52) (1.53)


25

The function of two arguments in (1.50), Jn (u, v), is the generalized Bessel function that was originally encountered long ago [9, 25], and seems always to occur in strong-field problems, especially when done relativistically. For the definition and a collection of basic properties, see Appendices B to D in Ref. [25] or Appendix J in Ref. [53]. The α0 and β0 in the arguments of Jn (u, v) in (1.50) are the same quantities illustrated in Fig. 1.3. Equation (1.50) shows that all dependence on the angle of observation is contained in the generalized Bessel function. This function has a complicated structure that allows for non-trivial angular distributions. Tunneling theories always peak in the forward direction. This fact is listed in the third row of properties contained in Table 1.1. In application, the momentum-space wave function in (1.50) is represented by an analytical Hartree-Fock wave function [54]. Such wave functions are superpositions of hydrogenic wave functions and, in turn, the momentum space wave functions associated with hydrogenic functions are known [55] exactly. This means that the initial state Φi (t) can be regarded as completely accurate in the SFA. The only approximation made is the replacement of Ψf (t) by ΨfV (t) in the S matrix. The point just made, that the SFA requires no approximation whatever in the initial state, cannot be stressed too strongly. For a surprisingly long time, much of the strong-field community labeled the SFA as inaccurate, because the initial state was free of field interaction. Various “corrections” were suggested involving alterations to Φi (t). Such alterations only introduce error. A powerful advantage of the SFA, stemming from the use of a timereversed S matrix, is the fully accurate treatment of the initial state. (Some authors have been tempted to compensate for the substitution Ψf → ΨfV by introducing “corrections” to Φi . This can be done only if it can be demonstrated that the resultant loss of a true Volkov final state has improved, and not damaged the SFA approximation. I know of no successful attempt of this type.) Much qualitative information can be gleaned from (1.50). The multiplicative factor p is a remnant from the integration over phase space, which contributes p2 dp. When energy conservation is introduced from a delta function in the energy, this serves to eliminate only p dp. The p that remains in (1.50) means that every predicted spectrum will tend towards zero at zero energy. This is an essential deviation from tunneling theories, where the energy spectrum of photoelectrons always has a maximum at zero energy. Furthermore, for all hydrogenic bound-state wave functions, φi (p) behaves [55] as pl for small momentum, where l is the orbital angular momentum quantum number. This will now be discussed further. 1.6.3 Momentum Distributions Equation (1.50) is written in spherical coordinates. When the formalism is expressed in cylindrical coordinates, the distributions of momenta in parallel

26

H.R. Reiss

and perpendicular components can be found directly. For the distribution of photoelectron momentum components parallel to the polarization direction of the electric field, the result is 2 ∞ 2 1 p2 dW 2 = + EB φi (p) [Jn (α0 p , −β0 c)] , dp 2π n 2

(1.54)

0

where p in cylindrical coordinates is just p cos θ in spherical coordinates. For the distribution of photoelectron momentum components perpendicular to the polarization vector, one finds 2 ∞ 2 dW 1 1 p2 2 = + EB φi (p) [Jn (α0 p , −β0 c)] , (1.55) p⊥ dp⊥ 2π n p 2 0 p = 2 (nω − Up − EB ) − p2⊥ . (1.56) Equation (1.55) is written in terms of p even though p has been removed by integration over a delta function in p . What is meant by p in ( 1.55) is the expression stated in (1.56). The dependence of φi (p) on p already noted, means that dW/ dp will always have a minimum near p = 0 whenever l = 0. It can be seen from (1.56) that, in general, p = 0 can occur only on a “set of measure zero”. The usual situation is that the minimum of the momentum distribution near the origin is a local minimum that is quite smooth and decidedly nonzero. We note here that early experimental observation of that minimum led to speculation that it was a rescattering effect, largely on the basis that tunneling theories always show a maximum there. Instead, it means that the simplicity of tunneling theories is such that some essential details of the physics are missing. The transverse momentum distribution in (1.55) has the property that p occurs in the denominator. It also occurs in the first argument of the generalized Bessel function. In Appendix B of Ref. [25], giving the general properties of the generalized Bessel function Jn (u, v), (1.8) is Jn/2 (v) , n even Jn (0, v) = . (1.57) 0, n odd That is, Jn (0, v) is Jn/2 (v) whenever n is even, but it goes to zero when n is odd. Equation (B2) of that paper, Jn (u, v) =

∞

Jn−2k (u) Jk (v) ,

(1.58)

n=−∞

shows that the approach to zero as u → 0 is proportional to u, which means that, for odd n values, the p = 0 in the denominator of (1.55) is canceled. This is not the case for even n. As pointed out above, p = 0 can occur only


27

on a set of measure zero, which means that one does not expect a singularity there. However, the cusp that has been observed in laboratory measurements can be explained this way, meaning that the tentative explanation that the cusp is a Coulomb effect is not supported by the theory.

1.7 Relativistic SFA Velocity-gauge SFA not only extends continuously to the relativistic domain, one could say that the SFA is naturally a relativistic theory that possesses a simple nonrelativistic limit [28,29,56]. It is this limit that has been discussed above. A full discussion of the relativistic SFA would take far too much space, so only a brief summary of the main points will be mentioned. There are two very important points about the RSFA (Relativistic Strong Field Approximation) [28–30,56–58]. One is that a relativistically strong field in the laboratory can be achieved only with very short pulses on the order of tens of femtoseconds. This means that the rise time through intensities too low for the SFA to be valid are so rapid that very little ionization occurs in that preliminary period. That is, essentially all of the actual ionization takes place within the domain of applicability of the SFA and RSFA. The RSFA reduces algebraically to the SFA for sufficiently low intensities (under the upper slanted line in Fig. 1.4), so that the lower bound on applicability of the RSFA is the same as for the SFA. When all the factors involved in practical application are taken into account, this means that the RSFA is essentially an exact theory. By “factors involved in practical application” is meant: depletion of the un-ionized atoms, temporal and spatial shape of the pulse, and realistic representation of the initial bound state. In the nonrelativistic velocity-gauge SFA, such calculations [33] are done by using the solution of a rate equation (to incorporate depletion) integrated over the space-time profile of the focused laser beam, in which the instantaneous SFA rate is used within the rate equation for the field intensity existing at each point within the space-time profile of the focus. The SFA rate itself is calculated with the initial state given by the analytical Hartree-Fock momentum-space wave function, as described above. The second important point about the RSFA is that it exists in both Dirac and Klein-Gordon versions [30] with all three spatial dimensions retained. Numerical solutions of the Dirac equation that are now possible retain only one or two spatial dimensions. Relativistic conditions, wherein the magnetic field is fully as important as the electric field, and with electron spin incorporated as well, cannot be treated properly in less than three spatial dimensions. The RSFA transition amplitudes will simply be stated here. For the Dirac case, the RSFA amplitude is RSFA = −i d4 xΨfV (x) qAμ (x) γμ Φi (x) , (1.59) (S − 1)f i

28

H.R. Reiss

where the “natural” = 1, c = 1 units are used, the relativistic conventions are in the standard form such as that given in the Bjorken and Drell textbook [59], and the bar over the ΨfV signifies the Dirac adjoint Ψ = Ψ †γ0 .

(1.60)

The equivalent Klein-Gordon (spinless) transition amplitude is in Refs. [29, 56]. The full formalism, including explicit derivation of the final Dirac transition amplitude is in Ref. [28] for circular polarization. The analogous forms for Dirac linear polarization and for the Klein-Gordon cases have not been fully published. They are given in the 1994 Ph.D. dissertation of Crawford [30]. The full correction terms to the basic RSFA amplitude are stated in Ref. [29].

1.8 Differences Between Velocity Gauge and Length Gauge The lack of gauge invariance in analytical strong-field approximations has already been remarked upon. To date, no systematic study has been made of differences between the two commonly employed gauges, although differences can sometimes be substantial. Some general remarks can be made that might illuminate aspects of the long-running debate between those who favor L gauge and those who favor V gauge. A central fact in the debate over gauges, not always recognized by the debaters, is that physical interpretations are gauge dependent. This is true in classical physics and in quantum physics; both perturbative and non-perturbative. Even when gauge invariance holds true, the qualitative explanation (i.e., the “physical” explanation) of what happens between the initiation of an experiment and the observation of the final results may be completely different in different gauges. A famous example comes from second-order perturbation theory. In the calculation of the line shape in an electric-dipole interaction, Lamb [60] found the correct answer in L gauge, but an incorrect answer in V gauge even though gauge invariance should be satisfied. Fried solved [61] this problem in 1973. Lamb, accustomed to L gauge, knew that of the infinite set of intermediate states required in second-order perturbation theory, only states close in energy to the initial and final states contribute significantly, and other states can be neglected. However, this is an L-gauge perception. When Lamb applied this same assumption in V gauge, the answer was clearly wrong. Fried showed that, in V gauge, all intermediate states must be considered, with the usually neglected continuum states supplying a substantial part of the transition amplitude. The correct answer then follows also in V gauge. The physical intuition that only nearby states are important in the transition is a gauge-dependent interpretation. Gauge-dependent differences can be strong in a nonperturbative environment. For example, the L-gauge anolog of the SFA seems not to contain


29

a possibility for stabilization, whereas V-gauge SFA not only predicts stabilization, but shows the excellent agreement with TDSE (numerical solution of the time-dependent Schrödinger equation) [48] exhibited in Fig. 1.2. The V-gauge SFA can be applied to high frequencies, but the L-gauge analog of the SFA cannot. There seem to be systematic differences that make one or the other of the gauges suited to a particular aspect of strong-field physics. A limited investigation was done in Ref. [62] for a short-range binding potential, with the conclusion that the L-gauge analog of the SFA is closer to dipole-approximation TDSE results than is the V-gauge SFA. One general judgment seems to be possible. The V-gauge should be the better choice as intensities increase into the higher ranges of intensity, whereas the L gauge seems to perform better in the lower ranges. The reasons for this conclusion are simple and direct. For the relativistic environment, only the Coulomb gauge would be used. The Coulomb gauge is generally defined [63] as that gauge in which longitudinal fields (such as the Coulomb field) are represented by a scalar potential, whereas transverse fields (such as a laser field) are represented by a vector potential. In a relativistic context, the Coulomb gauge is used exclusively. As intensities decline from the relativistic domain, the relativistic SFA connects smoothly to the nonrelativistic V-gauge SFA [28]. (The L gauge can be extended to the relativistic situation, but it is not really practical to do so, since the relativistic extension requires both a scalar and a vector potential [64].) The L-gauge analog of the SFA represents the laser field as if it were a slowly varying quasistatic electric field. Said another way, if one has an almost-constant electric field, but one which can vary slowly in time, then the appropriate formulation would be identical to the L gauge. The conclusion is that the L-gauge formalism could be a better performer than the V-gauge SFA for the lower range of nonperturbative intensities, but V-gauge SFA models the action of a strong photon field more completely than does the L-gauge.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

J.R. Oppenheimer: Phys. Rev. 31, 66 (1928) D.M. Volkov: Z. Phys. 94, 250 (1935) W. Gordon: Z. Phys. 40, 117 (1926) J.A. Wheeler: Phys. Rev. 52, 1107 (1937) W. Heisenberg: Z. Phys. 120, 513, 673 (1943) E.C.G. St¨ uckelberg: Helv. Phys. Acta 17, 3 (1943); 18, 3 (1945); 19, 242 (1946) H.R. Reiss: Phys. Rev. A 1, 803 (1970) N.D. Sengupta: Bull. Calcutta Math. Soc. 44, 175 (1952) H.R. Reiss: J. Math. Phys. 3, 59 (1962) H.R. Reiss: J. Math. Phys. 3, 387 (1962) A.I. Nikishov, V.I. Ritus: private communication (1990)

30

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12. N.B. Narozhny: private communication (2005) 13. A.I. Nikishov, V.I. Ritus: Zh. Eksp. Teor. Fiz. 46, 776 (1964) [Sov. Phys. JETP 19, 529 (1964)] 14. L.S. Brown, T.W.B. Kibble: Phys. Rev. 133, A705 (1964) 15. D.L. Burke, R.C. Field, G. Horton-Smith et al: Phys. Rev. Lett. 79 1626 (1997) 16. H.R. Reiss: Phys. Rev. Lett. 26, 1072 (1971) 17. F.J. Dyson: Phys. Rev. 52, 631 (1952) 18. G. Breit, J.A. Wheeler: Phys. Rev. 46, 1087 (1934) 19. I.I. Gol’dman: Phys. Lett. 8, 103 (1964); Zh. Eksp. Teor. Fiz. 46, 1412 (1964) [Sov. Phys. JETP 19, 954 (1964)] 20. L.V. Keldysh: Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)] 21. A.M. Perelomov, V.S. Popov, M.V. Terent’ev: Zh. Eksp. Teor. Fiz. 50, 1393 (1966) [Sov. Phys. JETP 23, 924 (1966)] 22. Z. Fried, J.H. Eberly: Phys. Rev. 136, B871 (1964) 23. J.H. Eberly, H.R. Reiss: Phys. Rev. 145, 1035 (1966) 24. H.R. Reiss, J.H. Eberly: Phys. Rev. 151, 1058 (1966) 25. H.R. Reiss: Phys. Rev. A 22, 1786 (1980) 26. D.B. Milosevic, G.G. Paulus, W. Becker: Phys. Rev. Lett. 89, 153001 (2002) 27. W. Becker, A. Lohr, M. Kleber: J. Phys. B 27, L325 (1994) 28. H.R. Reiss: J. Opt. Soc. Am. B 7, 574 (1990) 29. H.R. Reiss: Prog. Quantum Electron. 16, 1 (1992) 30. D.P. Crawford: Relativistic ionization with intense linearly polarized light. PhD Thesis, American University, Washington, DC (1994) 31. H.R. Reiss: J. Phys. B 20, L79 (1987) 32. P.H. Bucksbaum: In: Atoms in Strong Fields, ed by C.A. Nicolaides, C.W. Clark, M. Nayfeh (Plenum, New York, 1990) p. 381 33. H.R. Reiss: Phys. Rev. A 54, R1765 (1996) 34. H.R. Reiss: Phys. Rev. A 46, 391 (1992) 35. H.R. Reiss: Opt. Express 8, 99 (2000) 36. H.R. Reiss: J. Opt. Soc. Am. B 13, 355 (1996) 37. P. Agostini, M. Fabre, G. Mainfray et al: Phys. Rev. Lett. 42, 1127 (1979) 38. M.V. Ammosov, N.B. Delone, V.P. Krainov: Zh. Eksp. Teor. Fiz. 91, 12008 (1986) [Sov. Phys. JETP 64, 1191 (1986)] 39. F.H.M. Faisal: J. Phys. B 6, L312 (1973) 40. P.H. Bucksbaum, M. Bashkansky, D.W. Schumacher: Phys. Rev. A 37, 3615 (1988) 41. A.I. Nikishov, V.I. Ritus: Zh. Eksp. Teor. Fiz. 50, 255 (1966) [Sov. Phys. JETP 23, 168 (1966)] 42. H.R. Reiss: Non-perturbative theory of above-threshold ionization. In: Photons and Continuum States of Atoms and Molecules, edited by N.K. Rahman, C. Guidotti, M. Allegrini (Springer, Berlin, 1987) pp 98 – 103 43. P.H. Bucksbaum, M. Bashkansky, R.R. Freeman et al: Phys. Rev. Lett. 56, 2590 (1986) 44. N.B. Delone, V.P. Krainov: Multiphoton Processes in Atoms (Springer, Berlin, 1994) 45. H.R. Reiss: Phys. Rev. Lett. 25, 1149 (1970) 46. H.R. Reiss: Phys. Rev. C 27, 1229 (1983) 47. H.R. Reiss: Laser Phys. 7, 543 (1997)


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48. A.M. Popov, O.V. Tikhonova, E.A. Volkova: J. Phys. B 32, 3331 (1999) 49. L.D. Landau, E.M. Lifshitz: The Classical Theory of Fields (Pergamon, Oxford, 1975), p. 118 50. E.S. Sarachik, G.T. Schappert: Phys. Rev. D 1, 2738 (1970) 51. J.H. Eberly: In: Progress in Optics, vol 7, edited by E. Wolf (North-Holland, Amsterdam, 1969), p. 361 52. P.B. Corkum, N.H. Burnett, F. Brunel: Phys. Rev. Lett. 62, 1259 (1989) 53. V.P. Krainov, H.R. Reiss, B.M. Smirnov: Radiative Processes in Atomic Physics (Wiley, New York, 1997), p. 273 54. A.A. Radzig, B.M. Smirnov: Reference Data on Atoms, Molecules, and Ions (Springer, Berlin, 1985) 55. H.A. Bethe, E.E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957) pp 40 – 47 56. H.R. Reiss: Phys. Rev. A 42, 1476 (1990) 57. D.P. Crawford, H.R. Reiss: Phys. Rev. A 50, 1844 (1994) 58. D.P. Crawford, H.R. Reiss: Opt. Express 2, 289 (1998) 59. J.D. Bjorken, S.D. Drell: Relatrivisitic Quantum Mechanics (McGraw-Hill, New York, 1964) 60. W.E. Lamb: Phys. Rev. 83, 259 (1952) p. 268 61. Z. Fried, Phys. Rev. A 8, 2835 (1973). 62. D. Bauer, D.B. Milosevic, W. Becker: Phys. Rev. A 72, 023415 (2005) 63. J.D. Jackson: Classical Electrodynamics, 2nd edn (Wiley, New York, 1975) 64. H.R. Reiss: Phys. Rev. A 19, 1140 (1979)

2 Trajectory Description of Ionization Processes in Strong Optical Fields Takaaki Onishi1 , Akira Shudo1 , and Kensuke S. Ikeda2 1

2

Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa, Hachioji 192-0397, Japan Faculty of Science and Engineering, Ritsumeikan University, Noji-cho 1916, Kusatsu 525-0055, Japan

Summary. We discuss the recent progress in the theoretical study on ionization processes of rare-gas atoms in strong optical fields, in atomic physics and in nonlinear dynamics. Special emphasis is placed on using the notion of trajectories in the study, which are particular Feynman paths in quantum mechanics. In terms of trajectories, both mechanisms of tunnel ionization and chaotic ionization are explained, and a clear difference is made between them. A possible experimental setting is discussed to observe the crossover of the two ionizations.

2.1 Introduction Rapid progress of optics and laser technology in the last decade has prompted the theoretical study on ionization of rare-gas atoms, ions and molecules by laser light in the field of atomic and molecular physics (Becker et al. 2002, Delone and Krainov 1998, Dorner et al. 2002, Faisal 1986) on one hand. On the other hand, detailed theoretical studies have been developed for the ionization processes using chaotic transportation as the basic mechanism, which was first proposed by (Leopold and Percival 1978) to explain unexpectedly large ionization rates of highly excited Rydberg atoms in microwave fields (Bayfield and Koch 1974, Bayfield et al. 1977). The intensity of laser fields in infrared experiments now becomes as strong as that of the Coulomb fields of hydrogen atoms at the Bohr radius. In microwave experiments, intensities of microwave fields and Coulomb fields can be made comparable by exciting valence electrons in Rydberg states. Characteristics of ionization processes have been established in the two schemes of experiments: tunnel ionization in infrared experiments, and chaotic ionization in microwave experiments. Tunnel ionization is characterized by the direct excitation of valence electrons from the initial bound states to the continuum by strong laser fields, which vary in time much slower than classical motion of the electrons. With the high intensity infrared laser, valence electrons in the ground states of hydrogen and noble-gas atoms are ionized in a few cycles of laser fields.

34

T. Onishi, A. Shudo, K. S. Ikeda

Choatic ionization in microwave experiments is characterzied by the chaotic excitation of electrons absorbing a large number of photons via many Rydberg states and by the classical chaotic mechanism of ionization processes. Valence electrons in Rydberg states of hydrogen atoms are ionized in hundreds of cycles of microwave fields. In either of the two ionization processes, ordinary perturbation theories cannot be applied because of the high intensity of optical fields relative to the Coulomb fields, and of the essential roles of extremely high order effects in the perturbation theories. Alternative theories to them have been proposed and developed. Unfortunately, it seems that investigations to the two ionization processes have been developed without tight connection in the two different disciplines, i.e., atomic and molecular physics and nonlinear dynamics. The main reason will be because of the considerable difference in the methods and conditions for the two schemes of experiments; choice of sample atoms, preparation of the initial states, and in particular, difference in light sources (laser and microwave generator). Even in the theoretical descriptions for the two ionization processes there seems to be a serious gap. The gap will be attributed to the difference in theoretical interests in both areas of physics. The manifestation of classical chaos in quantum mechanics is the main concern in nonlinear dynamics (Gutzwiller 1990), but it is not always in atomic physics. In fact, in the ionization processes dominated by classical chaos, the manifestation of chaos is observable quantum mechanically by the distribution of ionization widths and the time dependency of survival probability for the initial quantum wave packets of electrons, for instance. To our knowledge, however, it is not known what is the manifestation of chaos in tunnel ionization, even when the ionization is described in terms of electron trajectories and amplitudes associated with them. Rather, it seems that a question should be addressed in which situation the notion of chaos is meaningful in the semiclassical or semiquantal interpretation of tunnel ionization (Becker et al. 2002). In order to answer the question, it is desirable to understand tunnel and chaotic ionizations on the same theoretical footing, and it must be a preliminary to the construction of the footing to review the progress in theoretical studies on the ionizations in atomic physics and in nonlinear dynamics. In this article, we would like to review two major theories which describe tunnel and chaotic ionizations: one is a quantum and semiquantal theory of tunnel ionization which has been developed in the field of atomic physics and is often called the Keldysh-type theory (Ammosov et al. 1986, Faisal 1973, Keldysh 1964, Perelomov et al. 1966, 1968, Reiss 1980), and the other is classical and semiclassical theory of tunneling and ionization induced by chaos which has been developed in the field of nonlinear dynamics (Bohigas et al. 1990, 1993; Delande and Zakrzewski 2003, Leopold and Percival 1978, Shudo and Ikeda 1995, 1998; Tomsovic et al. 1994, Zakrzewski et al. 1998). We put emphasis not only on specifying motivations, theoretical frameworks,

2 Trajectory Description of Ionization Processes in Strong Optical Fields

35

and main results in the two approaches, but also on clarifying the relationship between them. This article is organized as follows. In Sect. 2.2, the Keldysh-type theory and its recent developments are overviewed, and the mechanism of tunnel ionization is explained within the framework of the Keldysh-type theory. In Sect. 2.3, the classical and semiclassical theories of chaotic ionization are overviewed, and the physical mechanism together with the mathematical background of chaotic ionization is discussed. In Sect. 2.4, the relationship between the two ionization processes is discussed. In particular, the regimes of two ionization processes are located in a parameter space, by taking a system of a hydrogen-like atom in an optical field as an example. Finally summary and conclusion follow.

2.2 Tunnel Ionization Tunnel ionization was first predicted theoretically by Keldysh (1964), and his theoretical framework has been further developed by many researchers (Ammosov et al. 1986, Faisal 1973, Perelomov et al. 1966, 1968; Reiss 1980). We briefly overview his theory according to the formulation by (Reiss 1980). Throughout this article, we will restrict ourselves to the problem of single electron ionization in a time-periodic monochromatic optical field, and skip the effects associated with multi electrons, multi poles, relativity, and the polarization of the optical field etc, because the aim of our article is to discuss the relationship between the two theories of ionization processes; the strong-field theory founded by Keldysh and nonlinear dynamical theory justified in the semiclassical limit, i.e., the zero limit of the effective Planck constant [see (2.22b)]. For many-electron problems, readers are recommended to refer to (Walker et al. 1994) for experiments and (Faisal 1996) for a theory. We discuss atomic systems given by the Hamiltonian H = H0 + VA + VB , where H0 is the kinetic energy of an electron, VA and VB are an optical field and a binding potential, respectively. The theory starts from a pair of the recursive relations for time-dependent wavefunctions: 1 |ψ(t) = |ψB (t) − i

+∞

ˆB (t, t )VÂ (t ) |ψ(t ) , dt U

(2.1a)

Â (t, t )VˆB (t ) |ψ(t ) , dt U

(2.1b)

t

1 |ψ(t) = |ψA (t) − i

+∞

t

which are respectively the perturbation expansions taking H0 + VB and Â(B) (t, t ) are H0 + VA as the unperturbed Hamiltonians. ψA(B) (t) and U

36


an eigenfunction and a unitary operator of time evolution due to the unperturbed Hamiltonian. Here we consider the limit that the optical field force is stronger than the atomic binding force, and it is reasonable to take H0 + VA as the unperturbed part. Then one obtains an expansion for the S-matrix for an emitted electron: 1 (S − 1)fi = i

+

+

+∞ dt ψAf (t)|VÂ (t)|ψBi (t) −∞ +∞

1 (i)

t dt

2 −∞

−∞

+∞

1

t dt

3

(i)

−∞

Â (t, t )VÂ (t )|ψBi (t ) dt ψAf (t)|VˆB (t)U

dt

−∞

t

dt ψAf (t)|

−∞

Â (t, t )VˆB (t )U Â (t , t )VÂ (t )|ψBi (t ) + . . . × VˆB (t)U

(2.2)

The subscripts i and f indicate the initial bound state and the final state in the continuum, respectively. The first term in the r.h.s. represents the amplitude associated with an electron which is first excited from the initial bound state to the continuum by a single interaction with the optical field, and then is emitted to a detector. Higher-order terms represent the amplitudes associated with an electron which is scattered by a parent ion many times after the excitation to the continuum. The theoretical framework of ionization processes based on the above type of perturbation expansion including its variants such as the ones obtained by integration by parts for individual terms (Kopold et al. 2000; Lohr et al. 1997) or replacement of final states by more improved ones (Reiss and Krainov 1994), is called the Keldysh-type theory. Trajectory description of ionization is achieved by the saddle point approximation of integrals for individual terms. The condition of this evaluation for the first term is N≡

Eion 1, ω

(2.3)

where Eion is the minimal ionization energy, ω is a single “photon” energy of the optical field and N represents the minimal “photon” number necessary for ionization. Hence this condition means that an electron must absorb many optical field “photons” to ionize. To evaluate the scattering amplitude, we implement the saddle point approximation for the integration over times and intermediate variables (momenta in this case). Then the resulting semiquantal amplitude (Becker et al. 2002), up to the second order of the expansion, is given by


⎡ (S − 1)fi ≈

⎢ ⎣ s

×e

iS

37

⎤1/2 5

(2πi)

det

(s) (s) ∂ 2 Spf /∂qj ∂qk

(t ,t ,ps )/ f 1s 0s

⎥ ⎦

j,k=1,... ,5

mpf (t1s , t0s , ps ) ,

(2.4)

(s)

where qi (i = 1, . . . , 5) runs over the five variables, t0s , t1s , and the three components of ps . t0s is the time when the electron is excited from the initial bound state to the continuum, t1s is the time when the electron is rescattered by the parent ion, and ps is the drift momentum of the electron before the rescattering. The sum is taken over all the trajectories connecting the above two times and satisfying the following “classical” equations of motion: [ps − eA(t0 )]2 = −2mEion , t0 (t1 − t0 )ps = dτ eA(τ ) ,

(2.5a) (2.5b)

t1

[ps − eA(t1 )]2 = [pf − eA(t1 )]2 ,

(2.5c)

t where m and e are the mass and charge of an electron, A(t) = − dτ E(τ ) is the vector potential of the applied electric field E(t), and pf is the drift momentum finally detected. The first equation decides the ionization time t0 , and the second equation requires the electron to revisit the parent ion at t = t1 , and the final equation is the energy matching condition for the rescattered electron. The five unknown parameters are decided by solving five equations of (2.5) simultaneously. Spf (t1 , t0 , ps ) is an action and mpf (t1s , t0s , ps ) is the transition matrix element between |ψps and |ψpf , which are the eigenstates of the system H0 + VA called the Volkov states, and represent the ionized states driven by the optical field with the drift momentum ps and pf , respectively. Particularly in the case of t0 = t1 , the three equations reduces to a single one [pf − eA(t0 )]2 = −2mEion ,

(2.6)

and it describes the direct ionization process that the electron is excited from the initial state at t = t0 and then moves only under the influence of the optical field to reach the detector. It is immediately seen that the solution t0 is always a complex number, which means that the action also has an imaginary part. Such a feature is inherited to the second order (and higher order) process of t1 = t0 , where a single rescattering by the atomic potential is taken into account. This is the reason why the Keldysh-type processes

38


can be interpreted as ionization processes via tunneling. More precisely, an imaginary time implies non-classical properties in scattering processes. Thus the above saddle point approximation results in the non-classical evolution of the electron under the influence of the optical field being scattered by the atomic potential several times. The itinerary of such an evolution is often called a quantum trajectory. Let Equiv be the average kinetic energy of an electron quivering in an optical field: 2

|eE| , (2.7) 4mω 2 where ω is the frequency of the field, and the electronic motion has the same frequency. Then a simple classical analysis reveals that on one hand, the energy of an electron emitted by the direct ionization has energy in the range 0 < E < 2Equiv , in which the ionization spectrum decays at a considerable rate with increase in E. On the other hand, the electron ionized and rescattered once forms a remarkable plateau in the ionization spectrum in the range 2Equiv < E < 10Equiv (Paulus et al. 1994). We introduce the dimensionless parameter γ, which is called the adiabaticity parameter. It plays an important role in the following arguments, and is defined by the square root of the ratio of the ionization energy to the quivering energy: Eion . (2.8) γ= 2Equiv Equiv =

Equation (2.6) leads to [pf − ep(t0 )]2 /(2mEquiv ) = −2γ 2 . If γ 1, there can exist a solution with Im eA(t0 ) sufficiently small (Becker et al. 2002). When one adopts this solution and integrates over the final momentum, the ionization rate is obtained (Ammosov et al. 1986): Eion Γ (t) ∝

2n∗ −|m |−1 3 3 4 2mEion 4 2mEion exp − , |eE(t)| 3 |eE(t)|

(2.9)

where m is the magnetic quantum number and n∗ is the effective principal quantum number. The exponent of the above formula is the imaginary action of the tunneling motion of an electron through the quasi-static potential barrier made up by the optical field and the binding potential: under the application of the optical field E(t), the exit of the tunneling barrier is qtun = Eion / |eE|, and the electron moves through the potential barrier at the imaginary momentum √ ptun ≈ i 2mEion and thus the tunneling action is roughly estimated by ptun qtun , which explains the exponent of the r.h.s. of (2.9) (divided by ). Equation (2.9) tells us that the maximal rate is reached at the peak hight of the optical field.


39

In terms of a tunneling time through the potential barrier, ttun ≡ 2mqtun / |ptun |, the adiabaticity parameter can be represented by γ = ωttun . Hence the condition γ 1 means the adiabatic temporal variation of the optical field with respect to the tunnel frequency of the electron. This parameter defines two regimes of ionization processes, tunnel (or adiabatic) regime for γ 1 and multi-photon (or non-adiabatic) regime for γ 1. In the latter 2N regime, the electron emission rate is proportional to |E| , which indicates that emission processes are due to the high order perturbation effects of the applied field and thus is extremely small. The series expansion for the S-matrix (2.2) is given by using (2.1a) once, and then (2.1b) recursively (Reiss 1980). Therefore in order for the series to be meaningful, it is sufficient that the second term is much smaller than the first one in the r.h.s. of (2.2), which is satisfied when γ 1.1 Each term in the r.h.s. of (2.2) describes that an atom is first excited from the initial Â (t1 , t2 ) and state to a Volkov state propagating with the evolution operator U then rescattered multiple times by the atomic potential. In order that such an expansion is meaningful, the intermediate bound states other than the initial state, whose influence may result in resonant behaviors in excitation processes and in rescattering processes as well, play no significant role. A typical case is that ω is large enough such that all the energy levels of the intermediate bound states are above the ionization level minus ω as shown in Fig. 2.1; an idealized situation will be that there is no bound state other than the initial state. As a result, the conditions sufficient for tunnel ionization are (C1) γ 1 , (C2) N 1 , (C3) intermediate bound states play no significant role. intermediate bound states |→ ionization levels ↓ ↓ |−−−−−−−−−−−−−−−−−−| | | | · · · · · · · · · · · · · · · · · · · · → E

ground state

−−−−−−−−−−−−−−−−−−−−−−−−− ω Fig. 2.1. A typical case that the expansion in (2.2) is meaningful. On the energy axis are the ground state level and intermediate bound state levels. Each angle bracket indicates a single photon energy of the optical field. All the intermediate bound state levels are placed within a single photon energy 1

The second factor is smaller than the first one by the factor dp ψ Apf |VB | ψAp / (p2f − p2 )/2m − n ω . The factor is evaluated as O (kR0 )−1 ≈ γ (R0 is the force range of the atomic potential). Reiss gives, however, a different estimation (Reiss 1980).

40


The conditions (C1) and (C2) were claimed by (Keldysh 1964), as pointed out by (Reiss 1980)2 . In (Becker et al. 2002), the structure of energy spectra of emitted electrons in the tunnel regime is well explained by a model having only a single bound state and satisfying the conditions (C1) and (C2), which supports the above claim. If the intermediate bound states play an essential role in excitation processes, the Keldysh-type theory cannot work well. A typical case is the microwave ionization of Rydberg atoms, where amplitudes for tunnel ionization are highly suppressed due to the large quantum numbers of the initial states. By a brief discussion presented below, one can see that in such a situation, the adiabaticity parameter loses usefulness. If we take hydrogen atom as an example, the bound states are quantized Kepler orbits. The classical Hamiltonian Hhyd , which is expressed by the action variable I, and the Kepler frequency ωK are given by 2 Eg , I2 3/2 dHhyd (I) 22 Eg 2Eion = = = , 1/2 dI I3 Eg

Hhyd (I) = −

(2.10a)

ωK

(2.10b)

where Eg = [me2 /(2πε0 )]2 /(2m), i.e., the energy of the ground state, and the action is quantized by I = n. Then the conditions (C1) and (C2) are rewritten by using the ratio ω/ωK , the quantum number of the initial state n and the ratio of the optical field and the Coulomb field at the Kepler orbit |E| / |E K |: γ ≈ (ω/ωK )/ (|E| / |E K |) 1 , Eion /ω = n/(ω/ωK ) 1 .

(2.11a) (2.11b)

If ω/ωK 1 and n is of order of unity, which matches typical situations of infrared experiments, the above two conditions reduce to γ 1. However, the tunnel action, which is proportional to n/ (|E| / |E K |), diverges in the limit n → +∞ with ω/ωK and |E| / |E K | being constant. Such a limit is realized in microwave experiments with Rydberg states rather than infrared experiments with the ground state. In this case, the tunneling ionization rate is vanishingly small because of the large quantum number of Rydberg states, without regard to the adiabaticity, which manifests that the picture of tunnel ionization loses its meaning. In this regime, the ionization may be described more successfully by classical dynamics than by quantum perturbation theory. This regime is called the chaotic ionization regime, in which perturbation theory based on unperturbed states used in the Keldysh-type theory (namely, unperturbed atomic states and/or laser-perturbed electron states) breaks down due to chaos. 2

By using the parameter z = Equiv /( ω), (C2) reads N = γz 2 1. In Reiss (1980), it is concluded that γ 1 and γz 1 are sufficient conditions for tunnel ionization, which are weaker ones than (C1) and (C2).


41

2.3 Chaotic Ionization For highly excited Rydberg atoms in microwave fields, the excitation of an electron to the continuum needs a huge number of photons. The Keldysh-type theory predicts vanishingly small tunnel ionization rate (2.9), even though the adiabaticity parameter γ is less than unity (Leopold and Percival 1978). For the Keldysh-type theory to be valid in the limit n → +∞ with γ 1, where n is the principal quantum number, the tunnel ionization must dominate over the multi-photon ionization, even if the former ionization rate is exponentially small. However, microwave experiments with Rydberg hydrogen atoms (Bayfield and Koch 1974, Bayfield et al. 1977) have shown that ionization probabilities are much larger than those expected from tunnel ionization rate, and thus the Keldysh-type theory cannot describe the ionization processes. This will be quite natural considering that perturbative approaches are not valid when an extremely large number of photons are necessary for ionization. Instead, when the underlying classical dynamics is highly chaotic and electrons are in sufficiently highly excited states, the experimental results are in excellent agreement with those of classical-dynamical simulation (Leopold and Percival 1978). In the above experiments, tunnel ionization rate was found to be negligible even when the experimental settings satisfied the conditions γ < 1 and N 1 for tunnel ionization to occur. Therefore one may doubt that the premise for the Keldysh-type expansion does not hold, in other words, intermediate bound states of the unperturbed system play an essential role in excitation processes. In fact, if the ionization continuum is classically accessible from the initial bound state, the intermediate states with energies between the initial energy level and ionization level should be strongly mixed by applying periodic perturbation, which means the breakdown of the perturbation expansion. Even if the continuum is classically inaccessible, hybrid ionization processes, which are composed of purely quantum (maybe semiclassical) processes from the initial state to the classical region followed by the classical ionization mentioned above, will be possible. Here we discuss why chaos dominates in the ionization processes in the Rydberg regime in the limit n → +∞. From (2.10) the Kepler frequency of 3/2 1/2 a classical orbit is ωK = 2Eion /(Eg ). As the microwave field of frequency ω is applied, the Kepler orbit exhibits nonlinear resonance with the applied field if the integer multiple of the period of the applied field coincides with the period of the Kepler orbit: lωK = ω (l = 1, 2, . . .) . Hence the resonance occurs at the energy 1/2 2/3 ωEg El = − (l = 1, 2, . . .) , l

(2.12)

(2.13)

42


Fig. 2.2. A universal feature in energy for any periodically perturbed system having an ionization level. Energy domains distinguish different dynamical structures in phase space. Ec is the threshold energy given by (2.14)

and the infinite series of resonances are accumulated close to the ionization level. At the beginning of the 20th century, Poincare, and afterward Birkoff developed a mathematical theory which claims that a very complex dynamical structure generically emerges close to every resonance (Poincare-Birkoff theorem). Nowadays it is known that such a complicated structure is the very origin of chaos, which is localized around every resonance if the amplitude of microwave field is small enough. As the field amplitude increases, the localized chaos around each of the resonances, which are accumulated close to the ionization level, merges with another one to form a globally extended chaotic region above a threshold energy Ec ( 3). Such a way of changing f makes the tunnel action scaled by the Planck constant, 2γN , decrease as (ni /4)(n/ni )η−3 with n. The ionization probability is thereby enhanced in the TI regime to an observable level. On one hand, ionization processes occur purely quantum-mechanically for the parameters outside the CI region. On the other hand, chaos plays an important role for the parameters being not so deep in the TI region. In the CI region and around the CI border, we think that an important role is played by UPO’s produced in the nonlinear resonance between Kepler orbits and the optical field, because, as discussed in Sect. 2.3, our previous studies on chaotic tunneling in time domain suggest that tunneling orbits are guided


49

Fig. 2.4. A scheme to observe the crossover between TI and CI, presented in the (F, Ω) plane. Parameters are varied as the function of the principal quantum number n along the line a {(F, Ω, eff ) = [Fi (n/ni )4−η , Ωi (n/ni )3 , 1/n] with η = 3.5 and Fi = Ωi = 0.023/4 }. The line b (γ = 1) is the TI boundary, and the lines c (F = 0.04Ω 4/3 ) and d (F = 0.02Ω −1/3 ) give the CI boundary. A wavy line indicates the weak perturbation limit F = 1

Fig. 2.5. (a) Phase space of one dimensional hydrogen atom driven by an oscillatory electric field. Red points are trajectories of bounded orbits. Green and blue curves are stable and unstable manifolds of an unstable fixed point (solid circle), respectively. (b) enlarges the area around the origin in (a)

50


by the stable and unstable manifolds, W s ’s and W u ’s, of the UPO’s. In fact, Fig. 2.5 shows that the key of tunneling transfer is W s and W u of the UPO in the middle of the classical escape region. Tunnel orbits initially being, e.g., in an EBK torus are attracted by W s , repelled by the UPO, and emitted along W u . The wavefunction which has tunneled and is emitted along W u is stretched and folded in the phase space many times as the time elapses, and the tunneling tail has almost the same probability along W u (Onishi et al. 2001, 2003; Shudo and Ikeda 1995, 1998), which results into the formation of a remarkable plateau in the tunneling spectrum (Takahashi and Ikeda 2005). It is quite interesting to study the relationship between this structure and the plateaus in energy spectra of ionization well explained by the Keldysh-type theory in terms of quantum trajectories (Becker et al. 2002). It is not fully understood how the transition occurs between two excitation processes, TI and chaotic tunneling, associated with instanton trajectories, and W s and W u , respectively. Investigations on a simple model of barrier tunneling under periodic perturbation suggest a scenario of the transition (Takahashi and Ikeda 2003). If the perturbation strength is weak enough, the intersection of W s and the initial manifold is in a very deep imaginary domain of the complex phase space. As is elaborated in (Shudo et al. 2006), entrances to the dominant chaotic tunneling paths will be aggregated close to the intersection. Since the intersection is sited much deeper than entrances to the instanton path contributing to TI, the classical action along the chaotic tunneling path has much larger imaginary part than that along the instanton path. Thus the tunneling is dominated by the Keldysh’s mechanism which uses the instanton path. As the strength of perturbation increases, the intersection of W s and the initial manifold comes closer to the real phase space and destroy the instanton path. It is in such a regime that the “imaginary” tunneling particles are converted into real particles, which become real at the exits of the chaotic tunneling paths which approach the real phase space very rapidly. The ionization processes are then dominated by the chaotic tunneling.

2.5 Summary and Conclusion We have overviewed the progress in semiclassical and semiquantal approaches for ionization of rare-gas atoms in optical fields. In two regimes of ionization, tunnel and chaotic ionization regimes, excitation processes of an electron are well explained respectively by the Keldysh-type theory and by the classical and semiclassical theory of tunneling and ionization induced by chaos. These regimes are located in distinct limits in a parameter space. The adiabaticity parameter in the former theory has its useful meaning when the effective Planck constant in a scaled system is not so small that the chaos of classical dynamics manifests itself in quantum mechanics. When the adiabaticity parameter and the effective Planck constant are both much smaller than unity,


51

it is not trivial which ionization regime is realized. The crossover between the regimes may be observed in microwave experiments with hydrogen atoms and hydrogen-like ions initially prepared in Rydberg states. By the development of optics and laser technology, new light is being shed on the ionization of atoms and molecules, and revealing the linkage of atomic and nonlinear physics. Acknowledgement. One of the authors (T.O.) is grateful of K. Takahashi and A. Tanaka for stimulating discussions.

References 1. Ammosov MV, Delone NB, Krainov VP (1986) Zh. Eksp. Teor. Fiz. 91, 2008 [(1986) Sov. Phys. JETP 64, 1191] 2. Bayfield JE, Koch PM (1974) Phys. Rev. Lett. 33, 258 3. Bayfield JE, Gardner LD, Koch PM (1977) Phys. Rev. Lett. 39, 76. 4. Becker W, Grasbon F, Kopold R, Milosevic DB, Paulus GG, Walther H (2002) Adv. At. Mol. Opt. Phys. 48, 35 5. Bohigas O, Tomsovic S, Ullmo D (1990) Phys. Rev. Lett. 64, 1479 6. Bohigas O, Tomsovic S, Ullmo D (1993) Phys. Rep. 223, 45 7. Creagh SC, Whelan ND (2000) Phys. Rev. Lett. 84, 4084 8. Davis MJ, Heller EJ (1981) J. Chem. Phys. 75, 246 9. Delande D, Zakrzewski J (2003) Phys. Rev. A 68, 062110 10. Delone NB, Krainov VP (1998) Physics Uspekhi 41, 469 11. Delone NB, Krainov VP, Shepelyansky DL (1981) Sov. Phys. Usp. 26, 551 12. Dorner R, Weber Th, Weckenbrock M, Staudte A, Hattass M, Moshammer R, Ullrich J, Schmidt-Bocking H (2002) Adv. At. Mol. Opt. Phys. 48, 1 13. Faisal FHM (1973) J. Phys. B: Atom. Molec. Phys. 6, L89 14. Faisal FHM (1986) Theory of Multiphoton Processes (Plenum Press, New York, 1986) 15. Faisal FHM, Becker A (1996) Selected Topics on Electron Physics (Ed. by Campbell and Kleinpoppen, Plenum Press, New York), pp 397 16. Frischat SD, Doron E (1998) Phys. Rev. E 57, 1421 17. Gutzwiller MC (1990) Chaos and Quantum Physics (Springer, New York) 18. Keldysh LV (1964) Zh. Eksp. Teor. Fiz. 47, 1945 [(1965) Sov. Phys. JETP 20, 1307] 19. Kopold R, Milosevic DB, Becker W (2000) Phys. Rev. Lett. 84, 3831 20. Leopold JG, Percival IC (1978) Phys. Rev. Lett. 41, 944 21. Lohr A, Kleber M, Kopold R (1997) Phys. Rev. A 55, R4003 22. Onishi T, Shudo A, Ikeda KS, Takahashi K (2001) Phys. Rev. E 64, 025201 23. Onishi T, Shudo A, Ikeda KS, Takahashi K (2003) Phys. Rev. E 68, 056211 24. Paulus GG, Nicklich W, Xu H, Lambropoulos P, Walther H (1994) Phys. Rev. Lett. 72, 2851 25. Perelomov AM, Popov Vs, Terent’ev MV [(1966) Sov. Phys. JETP 23, 924] 26. Perelomov AM, Popov VS, Kuznetsov VP [(1968) Sov. Phys. JETP 27, 451] 27. Reiss HR (1980) Phys. Rev. A 22, 1786

52 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

T. Onishi, A. Shudo, K. S. Ikeda Reiss HR, Krainov VP (1994) Phys. Rev. A 50, R910 Shudo A, Ikeda KS (1995) Phys. Rev. Lett. 74, 682 Shudo A, Ikeda KS (1998) Physica D 115, 234 Shudo A, Ikeda KS (2000) Prog. Theor. Phys. Suppl. 139, 246 Shudo A, Ishii Y, Ikeda KS (2002) J. Phys. A; Math. Gen. 35, L225 Shudo A, Ishii Y, Ikeda KS (2007), preprint “Julia set and chaotic tunneling I, II”, to be submitted to J. Phys. A Takahashi K, Ikeda KS (2001) Found. Phys. 31, 177 Takahashi K, Ikeda KS (2003) J. Phys. A: Math. Gen. 36, 7953 Takahashi K, Ikeda KS (2005) Euro. Phys. Lett. 71, 193 Takahashi K, Yoshimoto A, Ikeda KS (2002) Phys. Lett. A 297, 370 Tomsovic S, Ullmo D (1994) Phys. Rev. E 50, 145 Walker B, Sheehy B, DiMauro LF, Agostini P, Schafer KJ, Kulander KC (1994) Phys. Rev. Lett. 73 1227 Zakrzewski J, Delande D, Buchleitner A (1998) Phys. Rev. E 57, 1458

3 Many Electron Ionization Processes in Strong and Ultrastrong Fields Anthony DiChiara, Isaac Ghebregziabher, Sasi Palaniyappan, Emily L. Huskins, Adam Falkowski, Dan Pajerowski, and Barry C. Walker Physics and Astronomy Department, University of Delaware, Newark, DE 19716, USA Summary. Total ionization yields are reported for the noble gases in ultrastrong fields. The intensity range studied is from 1015 W/cm2 yy to 1018 W/cm2 with ionization charge states from one to twelve. Sequential ionization processes are modeled by tunneling ionization and shown to be accurate within a factor of two near saturation. Nonsequential, multielectron ionization is observed for most species and involves the correlation of up to four electrons; furthermore, correlated ionization is observed to occur for electrons initially in different atomic shells. A semiclassical, 3D relativistic rescattering model is compared to the data with limited success. The model shows the qualitative behavior of nonsequential ionization but fails to accurately predict the yields for many higher charge states. In neon the model accounts for 15% of the observed nonsequential ionization but in xenon only 1% is accounted for. The laser magnetic field does not play a role in many nonsequential ionization processes and a rescattering deflection parameter is presented to predict when magnetic field effects will impact rescattering processes such as high harmonic generation and nonsequential ionization.

3.1 Introduction The interaction of light with matter has advanced beyond the excitation of valence shell electrons in molecules and atoms with binding energies of order 10eV [1]. Modern lasers [2] create intensities sufficient to ionize inner shell electrons bound by 100eV to 1000eV. These new higher fields are the progression of the field-matter interaction beyond nonperturbative multiphoton [3,4] and strong field [5] interactions into a new ultrastrong field regime [6, 7]. The increase in the Coulomb and external laser field strengths, with the commensurate rise in the value of v/c for the photoelectron, brings into play the possibility of new dynamics involving the magnetic component of the external field. It has been shown the Lorentz deflection [8–15] from the laser magnetic field will affect the total ionization rate by suppressing the rescattering and nonsequential ionization mechanism (NSI) [16]. This is significant since multielectron nonsequential ionization can account for >99% of the observed charge state when the intensity is a factor of two below one-electron ionization saturation intensity. At sufficiently high field, one must also question the role of the laser magnetic field (Blaser ) and how it may affect the sequential ionization (SI) process and the semiclassical, one-electron hydro-

54

A. DiChiara et al.

gen wave function tunneling ionization rate given by Ammosov, Delone and Krainov (ADK) [17]. One of the most descriptive measurements of the ultrastrong field interaction is the intensity dependent total ionization rate. We present here current measurements on the intensity dependent ionization yields in strong and ultrastrong fields. The data is compared to the results of a relativistic, 3D semiclassical model and an analytical rescattering deflection parameter that describes how the wave function dynamics are changed due to the Lorentz force in ultrastrong fields. The total ionization rates presented here allow us to answer many of the questions about ionization in strong to ultrastrong fields. Finally, we discuss a rescattering deflection parameter that provides an easy and accurate calculation of the division between strong and ultrastrong fields.

3.2 Experiment The experimental apparatus used to measure the ionization of atoms in ultrastrong fields [18, 19] consists of a chirped pulse regenerative and terawatt multi-pass amplifier to generate the ultrastrong field. A time-of-flight ion spectrometer analyzed the ionization. The regenerative amplifier pulse energy was 3.5mJ with 1.6% energy fluctuations at a repetition rate from 10Hz to 1000Hz. It created intensities up to 3 × 1016 W/cm2 . The terawatt multi-pass amplifier (10Hz) generated intensities up to 1018 W/cm2 and produced 250mJ pulses with 4.5% energy fluctuations. The laser pulse duration was 40 ± 5fs, had a center wavelength of 780nm and a TEM00 spatial mode. The spectrometer (Fig. 3.1) analyzed the ions every shot with full isotope resolution (Fig. 3.1), which prevented unwanted counts from oxygen and carbon in the vacuum background. The data is averaged into 10% intensity bins; typically each data point represents five independent collections, each of 105 shots. The laser intensity is calibrated to a factor of two with linearity better than 20% over one decade of intensity. The ion signal error is estimated to be a factor of two. The ion signal errors come primarily from day to day differences in the setting of the peak intensity, which is magnified by the nonlinearity of the signal. The intensity calibration was done with He+ and He2+ and the lower three charge states of neon, which are known to agree well with the ADK ionization rate with 40fs pulses at 800nm. These calibrations are also in agreement with intensity expected from the focal spot and pulse duration measurements. The linearity of the data was ensured by using a waveplate and polarizer to attenuate the intensity. The contrast limit of the attenuation setup was five times lower than the minimum attenuation used in the experiments, e.g. when the leakage from the attenuator was 1mJ, the minimum energy used in the experiment was 5mJ.

3 Many Electron Ionization Processes in Strong and Ultrastrong Fields

55

Fig. 3.1. Scale drawing of experimental chamber with the focused laser light and trajectory paths for the ions superimposed. The chamber is 1.1 m from left to right. Also shown (lower section) is a time of flight ion spectrum showing Xe11+ to Xe7+ at 6 × 1016 W/cm2 . 128 Xe8+ is obscured by an oxygen contaminant

A f /2 off-axis parabola in vacuum focused the laser into an effusive gas beam. The primary difficulties with creating the ultrastrong field are the low quality of the final focusing optic along with chromatic and temporal distortions of the final pulse in the focus due to small misalignments in the compressor gratings or uncompensated phase distortions across the spectrum of the laser pulse. In the experiments, these errors led to peak intensities two to three times less than expected for a perfect focus and transform limited spectrum.

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Signal to noise difficulties in the experiment come from the large numbers of multiply charged ions created before the ion state of interest, which occurs at the peak of the field. Including the spatial averaging across the focus, literally millions of sample and background atoms are singly or multiply ionized for every sample ion that interacts with the ultrahigh peak intensity. In our experiments, saturation of the micro-channel plate detector by the lower ionization charge states has been avoided by including a mass filter (Fig. 3.1). The mass filter consists of three grids that are perpendicular to the ion flight path. The outer grids are grounded and the inner grid provides a window function. The voltage on the center grid is grounded for 1μs to allow the ion species of interest to pass by and raised to 1kV at other times to prevent hydrogen and less ionized charge states from saturating the detector. The scheme works well but has a disadvantage that the window grid voltage must be higher than the ion repeller voltage, which is typically kilovolts. The gas jet has been optimized to also limit space charge and background ions. The effusive gas jet has an orifice of 250μm and is then collimated by 380μm skimmers several millimeters from the jet. The skimmers isolate the gas jet source chamber from the UHV interaction chamber with differential pumping giving a pressure factor of 40 times between the two chambers.

3.3 Experimental Measurements In this section we discuss the intensity-dependent ionization yields of high charge states in Ne (Fig. 3.2), Ar (Fig. 3.3), Kr (Fig. 3.4), and Xe (Fig. 3.5). These figures plot the pressure corrected data points for the number of detected ions in units of ions/(laser shot-torr) at different intensities. Ionization saturates in the collections near 106 ions/(shot-torr). At this point the ionization probability for the species at the center of the focus is exp(−1) after the laser pulse. Above the saturation intensity the signal increases due to focal averaging, i.e. atoms away from the peak intensity at the center of the focus being ionized. Near saturation spatial averaging across the laser focus plays a large role in the observed yield. Slight distortions in the laser focus, variations in the geometric collection efficiency across the interaction region, and a higher MCP detection efficiency for higher charge states can lead to a “pinching” of the experimental ion yields with a lower charge state having a weaker intensity dependence than higher charge states. These effects are within the stated factor of two for the ion yield accuracy. Future, studies with an ion signal accuracy better than a factor of two will be needed to determine the exact physical mechanism. Below saturation the ionization decreases as the atom is less likely to ionize in weaker fields. At yields of 102 ions/(shot-torr) to 104 ions/(shot-torr) another ionization mechanism, the multielectron NSI process, is apparent in many of the collections. The measurements in Figs. 3.2–3.5 all show the


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Fig. 3.2. Experimental Ne+ to Ne8+ (left to right) yields with ADK (dash) and our calculation (line)

signature “knee” structure [20] in the ion yield curves indicating the clear presence of NSI in addition to SI. For Ar, Kr, Xe, the valence shell is stripped and inner shell electrons ionize. The ionization yields show NSI for the 9+ charge state in these species. Since this NSI in the next shell (i.e. 9+) is correlated with ionization in the previous, valence shell we call this ionization “cross-shell” NSI. Perhaps the biggest surprise with cross-shell NSI is how much of the ion signal is created by this mechanism. For cross-shell NSI in Ar, Kr, Xe, the NSI and SI yields are equal at between 0.1% and 1% of the ion yield at saturation, i.e. ion yields of 103 ions/(shot-torr) to 104 ions/(shot-torr). In argon and krypton the presence of NSI is clear even though the intensities where NSI is measured falls between 1017 W/cm2 and 1018 W/cm2 , well above the intensity where one might expect v/c and Lorentz effects to become apparent. When looking at ion yields, it is important to remember the data represents a spatially averaged signal across many intensities. For NSI this is

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Fig. 3.3. Experimental Ar6+ to Ar9+ (left to right) yields with ADK (dash) and our calculation (line). Shaded regions indicate extremes of the calculation from the cross sections and pulse duration/intensity

important because the process is a correlated event. If any one of the steps involved in the correlated ionization process saturates, the corresponding NSI also saturates. When the focal averaging is taken into account, a large NSI yield reported at 1017 W/cm2 may come from an electron ionization process that saturated at a lower intensity and whose yield increased due to focal averaging by I 1.5 . For example, if NSI in a 9+ charged ion at 1017 W/cm2 comes from an (e,4e) process it will saturate with the ionization of 6+ charge state, which can occur nearer to 2 × 1016 W/cm2 . Additional observations can be made from the data for each gas and we will now comment on the collections, starting with neon. The yields of Ne+ to Ne8+ are presented in Fig. 3.2 along with the calculated SI and NSI yields to be discussed later. Small fluctuations in the curves above ionization saturation, i.e. Ne3+ and Ne8+ yields >106 ions/(shot torr), are within the measurement uncertainty. Ionization yields for Ne5+ are measured with 22 Ne5+ to avoid 16 O4+ from the water vacuum background. Prominent NSI for Ne2+ , Ne3+ , Ne4+ and Ne5+ is apparent in Fig. 3.2 by the deviation from the SI yield for ion yields 103 ions/(shottorr)) at an intensity of 1017 W/cm2 or greater. However, NSI in Kr9+ is a process that is correlated with lower charge states that ionize at or below 2 × 1016 W/cm2 . The lack of quantitative agreement between the model and the data can be attributed to one of several factors that may be important in ultrastrong fields. These are (1) chain NSI rescattering, (2) NSI from excited states, either Rydberg population in the atom or inner shell electron holes, and (3) unaccounted for field assisted impact processes or other unaccounted for errors in collision cross sections in ultrastrong fields. Determination of NSI from excited states (2) will first require measurements of the excited populations in atoms. Following this, calculations or experiments will need to be done to determine the collision cross sections for excited states. The possibility of creating electron holes in atoms has been suggested for some time [51]. Whether these holes are created in ultrastrong fields and what impact they have on

Fig. 3.10. Ne+ to Ne7+ rescattering fluence (a) and Ne2+ to Ne8+ yields (b) calculated with (line)/without (dash) Blaser


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NSI when they decay is an area of ongoing investigation at this time. It is very possible that some of the unaccounted for NSI yield is from the decay of electron holes ejecting four or five electrons in the strong field. Further measurements of recoil ion momentum spectroscopy and electron momentum [52, 53] could corroborate the observations made here and shed insight into the physical NSI mechanism in highly charged Xe. Higher order ionization processes, i.e. Auger and Coster-Kronig transitions and subsequent rearrangement of inner-shell holes, are known to lead to multiple ionization of higher Z target atoms [54]. The possibility of creating inner-shell holes in atoms from high intensity laser matter interactions is a long-standing question that may be part of the mechanism behind NSI in ultrastrong fields. The theory results from Becker and Faisal [55] show larger NSI yields for Xe7+ and Xe8+ than those from our model. At this time, the physical source of this disagreement is not known but it may be due to the large variance in cross sections reported for Xe with the two models using different impact cross sections. The comparison of the models across the multiple charge states for Ar, Kr, and Xe should reveal any anomalies due to cross sections. First, the error in the cross sections decreases as one proceeds from Xe to Ar and, secondly, the number of charge states measured in this study reduces the chance that any erroneous cross section will overly impact the interpretation of the physics. If we shift our NSI curves, increasing the NSI from the model, the agreement between the model and the experiment is greatly improved. Figure 3.11 is a plot of Xe much like that of Fig. 3.5 except the theory curves are adjusted in the following manner. The NSI yields (only) are increased by a factor of order 102 : Xe7+ = (100 × NSI + SI), Xe8+ = (200 × NSI + SI), Xe9+ = (70 × NSI + SI), Xe10+ = (20 × NSI + SI), Xe11+ = (40 × NSI + SI). Such a factor is equivalent in the model to either increasing the rescattering fluence or impact cross sections and could be explained by an enhanced rescattering process such as collision ionization of an atom in an excited state. Because the three-step model used here has significant predictive power (e.g. cross shell ionization) and follows significant trends in the data, the authors do not believe it should be interpreted as an inconsequential physical component of the ionization mechanism. However, it is clear additional theoretical studies will be required to fully explain the observations and physics behind NSI in ultrastrong fields.

3.5 Wave Function Deflection Parameter To better understand the Lorentz deflection and any role it may play in NSI we have taken the wave function deflection shown in Fig. 3.7 and described it with a simple Lorentz deflection rescattering parameter, ΓR = Z02 /2χ2 where Z0 is the distance from the parent ion and wave packet center measured along the laser wave vector (which is also the direction of deflection due to Blaser and the Lorentz force) and the width of the returning wave packet (mea-

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Fig. 3.11. Ion yields of Xe7+ ( ), Xe8+ ( ), Xe9+ ( ), Xe10+ ( ), Xe11+ ( ), and Xe12+ ( ) with our modified (see text) calculated yield (solid line)

sured perpendicular to Elaser ) is given by χ. The parameter ΓR is a natural extension of strong/ultrastrong field studies where the transition point in the dynamics needs to be known. For the parameter ΓR , a value much less than one is consistent with traditional strong field approximations. However as v/c, Blaser , and the Lorentz deflection of the electron increase, ΓR also increases. When ΓR = 1, the rescattering electron fluence will decrease by exp(−1), marking the beginning of the ultra-strong field regime. Finally when the Lorentz deflection is much greater than the spread in the rescattering wave function transverse width (ΓR 1) the interaction is well into the ultrastrong field regime. Following the dynamics of Z0 , and χ analytically as a function of the scaled time (τ = ωt) we are able to derive a simple form for ΓR . Additional approximations that allow a concise analytical form include: treating the electron motion in the continuum nonrelativistically, the initial (i.e. birth) displacement and momentum [56] of the photoelectron relative the parent


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ion are zero, and the transverse width of the rescattering electron probability wave packet is much larger than the initial width. As we will show, the non-relativistic treatment of electron motion for the ΓR derivation is a reasonable approximation until the interaction is well within the ultra-strong field regime. With these approximations, Z0 and χ may be expressed in terms of the laser frequency, ionization potential of the species under study, and the peak field of the laser pulse as, ΓR =

η 2 q4E04 82 m4ω 6 c2

2

2 (τ − τ0 ) 2ω 2 m2 χ20

−1 ,

(3.1)

where c is the speed of light, χ0 the initial width of the photoelectron probability wave packet, and the dynamics from the integration of the equations of motion are primarily contained in η = 2 (τ − τ0 ) cos (2τ0 ) − sin 2τ + sin 2τ0 − 8 sin τ0 (cos τ0 − cos τ − (τ − τ0 ) sin τ0 ) .

(3.2)

For rescattering, i.e. the return of the photoelectron to the parent ion, and the additional simplification that relevant phases for rescattering are at small τ0 and τ corresponding to final kinetic energies between Up and 3.17Up we have, 8UP2 χ20 ΓR ∼ = 92 c2

(3.3)

The initial width, χ0 , of the quantum electron wave packet from the uncertainty principle when it tunnels through the laser electric field suppressed Coulomb barrier is given by √ VIP χ20 = √ e 2m E0 cos τ0 [27], where VIP is the ionization potential of the charge state of the species under study. Again constraining τ0 to represent rescattering phases near 3 Up , this further simplifies ΓR to its final form, UP3 VIP 1 ∼ ΓR = . (3.4) m 3c2 ω In this derivation we neglected the effect of Coulomb field from the parent ion core and the relativistic suppression of wave packet spreading [57], which comes into play for highly relativistic dynamics where ΓR is 1. Figure 3.12 gives the intensity for the threshold between the strong and ultrastrong field regime, i.e. the ΓR = 1 intensity, as a function of the laser wavelength for a species with a typical ionization wave packet width, χ0 , of

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Fig. 3.12. ΓR = 1 intensity as a function of wavelength for χ0 = 2 au. The ultrastrong field region is shaded gray and the strong field in dark gray

2 atomic units. Above ΓR = 1, shown with a gray fill in the figure, v/c = 0, Blaser = 0 and both must be considered in the dynamics. Below ΓR = 1, shown with a dark gray fill in the figure, v/c = Blaser = 0 approximations are adequate to describe strong field dynamics. From Fig. 3.11, one can see the ultra-strong effect threshold intensity approaches 1018 W/cm2 for UV wavelengths such as the 3rd and 4th harmonic of Ti:sapphire, but can be less than 1015 Wcm2 for mid-IR wavelengths now used for high harmonic radiation (HHG) studies [58]. The authors note the derivation of ΓR corroborates the derivation of the nonrelativistic dipole/multipole moment in the context of HHG radiation in [59] where |dipole moment/multipole moment| ≈ exp(αz Up /4aBohr mc2 ) where αz = eE0 /mω 2 is the electron excursion along the electric field and aBohr is the Bohr radius. Applying ΓR to the total ionization measurements allows a separation of strong field NSI and the ultrastrong field where NSI will be suppressed due to Blaser and v/c for the rescattering electron. The calculated intensity dependent ionization yields for Ne are shown in Fig. 3.13 for three different wavelengths, 500nm, 2μm, and 5μm. The calculations are done in the nonrelativistic strong field limit (solid line in figure) as well as the ultra-strong field case (bold dashed line). In the figures, the two regimes ΓR < 1 and ΓR > 1 are shown by shaded regions. For 500nm, 800nm, 2μm, and 5μm, the intensities at which the NSI yield drops by exp(−1) due to the ultra-strong field are 1.2 × 1017 W/cm2 for Ne9+ , 3.4 × 1016 W/cm2 for Ne8+ , 4.7 × 1015 W/cm2 for Ne5+ , and 1.2 × 1015 W/cm2 for Ne3+ , respectively. The ion yield units are referenced to saturation, i.e. an ion yield of exp(−1) corresponds to an ionization probability of exp(−1) for an atom at the focus. As can be seen from the figure, both calculations give the same result for 500nm light with the valence shell ionization of Ne up to 1017 W/cm2 . The ultra-strong field regime (shown with light gray in Fig. 3.13) is reached by the ionization of Ne9+ by 500nm light with an order of magnitude reduction in the calculated Ne9+ NSI yield. For 2μm light shown in Fig. 3.5(b), the break between the strong (gray fill) and ultra-strong (light gray fill) response


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Fig. 3.13. Figure 3.13. Ionization yields for Ne2+ to Ne9+ including both SI and NSI in a 500 nm (a), 2 μm (b), and 5 μm (c) laser field. The calculated yields are shown in a normal, strong field dipole approximation (solid, thin lines) and with relativistic and Lorentz effects (dashed, bold lines). The region of ΓR > 1 is light gray while ΓR < 1 is gray

occurs near 6 × 1015 W/cm2 and the atomic response differs by an order of magnitude in the NSI yield for Ne5+ . For even longer wavelengths, such as 5μm shown in Fig. 3.5(c), the break between the regimes occurs as the intensity changes from 10 × 1014 W/cm2 to 1015 W/cm2 . For Ne3+ the NSI ultra-strong field response is only 1% of the NSI yield expected from a strong field nonrelativistic, dipole approximation. In this case, the modest intensity and Ne3+ ionization state might normally be interpreted as well within the

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strong field. Our results however indicate this interaction is well within the ultra-strong field.

3.6 Conclusion Ionization yields are reported for Ne, Ar, Kr, and Xe in ultrastrong fields. As an aspect of the ultrastrong field – atom interaction, NSI is shown to be a robust and general feature. NSI yields that are observed in ultrastrong fields, are coming from (e,2e), (e,3e) and (e,4e) processes that saturates at lower field strengths and hence are less sensitive to Blaser and v/c and more likely to be accurately modeled in the dipole approximation. A Lorentz deflection parameter is consistent with the NSI measurements and suggest v/c and Blaser effects may occur at intensities from 1015 W/cm2 to 1016 W/cm2 at longer wavelengths, i.e. mid-IR wavelengths from 2μm to 5μm. Lorentz deflection does not play a significant role in the 800nm ionization of noble gas atom states with binding energies less than 400eV. NSI yields measured are consistent with the trends predicted by a rescattering model, but as one proceeds to higher Z atoms more NSI is observed than predicted theoretically and the quantitative agreement is poor. In lower Z atoms, such as Ne, the agreement is better than in heavier atoms; for Xe the simple rescattering model accounts for only 1% of the observed NSI in some charge states. Additional mechanisms that may need to be considered in future theories of ultrastrong field – atom interactions include “chain” NSI, NSI from excited states of the atom (e.g. Rydberg states or inner shell holes), and the possibility of ultrastrong field enhanced recollision/impact processes, whereby NSI electrons also rescatter and set up a “chain” of (e,2e) excitation or ionization rescattering events in the field (Xe+4 + nhν → Xe+5 + e− + n hν → Xe+6 +2e− +n hν → Xe+7 +3e− +n hν → Xe+8 +4e− ) or a mechanism whereby an anomalously high number of excited states or electron holes (denoted by [Xen+ ]∗ ) are populated and subsequently ionize or Auger decay, e.g. Xen+ + mhν → [Xe(n+1) ]∗ + e− + m hν → Xe(n+4)+ + 4e− ). These mechanisms will require experimental probing of the excited states during and following the laser field. Such experiments have not been completed at this time and the population of excited states in ultrastrong fields is not known. Acknowledgement. This material is based upon work supported by the National Science Foundation under Grant No. 0729785.

References 1. L.F. DiMauro and P. Agostini, in Ionization Dynamics in Strong Laser Fields, ed. B. Bederson and H. Walther, Advances in Atomic, Molecular, and Optical Physics Vol. 35 (Academic Press, San Diego, CA., 1995)

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4 Molecular Rearrangements in Intense Laser Fields Manchikanti Krishnamurthya and Deepak Mathurb Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India a [email protected]; b [email protected] Summary. Irradiation of molecules by strong optical fields readily leads to rupture of one or more molecular bonds. Such rupture is a consequence of field-induced multiple electron ejection from the molecule, leaving behind two or more ionic cores that experience strong Coulombic repulsion. The resulting Coulomb explosion occurs on ultrafast time scales. It now appears to be the case that bond formation processes can also occur on similar, ultrafast time scales. Ultrafast intramolecular rearrangements leading to bond formation upon intense field irradiation of a series of linear alcohols has been experimentally observed. The rearrangement process gives rise to an unusual ionic fragment, the hydrogen molecular ion, which is formed with substantial amount of kinetic energy. Results of polarization dependence measurements confirm that such rearrangement, leading to bond formation, occurs within the duration of a 100 fs-long laser pulse and that the unimolecular process is coaxed by the strong optical field.

4.1 Introduction The study of chemical reactions at the microscopic level, single collision by single collision, has been one of the prominent developments in the field of chemical physics in the last quarter century. The possibility of understanding the dynamics of the motion of atoms in a molecule in microscopic detail and, concomitantly, the feasibility of controlling such dynamics, has been the major driving force for research in chemical physics. Gas-phase ion chemistry [1], which paved the way for directly accessing information on single collision events as against information on ensemble averages in the traditional wet chemistry, was an important first step in the modern study of chemical reaction dynamics. In broad terms, the archetypal chemicalreaction was perceived as a two-step process. The first step involved a gain of internal energy in the course of a collisional event: AB + M → AB ∗ .

(4.1)

The temporal evolution of the molecule in the excited state, AB∗ , depends on the nature of the potential energy surface that is accessed. For example, the excitation could be to a repulsive state which would lead to the second

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Fig. 4.1. Schematic of the dissociation of a diatomic molecule

step, molecular dissociation: AB ∗ → A + B .

(4.2)

Figure 4.1 shows the schematic of the electronic transition that occurs in the first step and the quantum mechanical evolution of the molecular reaction as the molecule “rolls down” the repulsive potential energy state. A major draw back in collisional studies is the lack of control on the amount of internal energy that is gained by the system. The centre-of-mass collisional energy can be as large as a few hundred eV, even for moderate collision energies of a few keV [2–4]. Thus, it is not just one but a number of potential energy states in the excitation manifold of the molecular system that are accessed in most collisional interactions. Though the interaction time that gives rise to molecular excitation is very short (of the order of fs for keV collisions, and attoseconds for MeV collisions) [5], there is no control over the time at which the energy is actually gained by the system.

4.2 Historical overview The advent of lasers provided a major breakthrough that enabled this shortcoming to be neutralized to some extent and made the study of laser-assisted chemical reaction dynamics prominent. The first step of laser-induced reaction, now termed a half collision, is brought about by photoexcitation of the reactants: AB + hν → AB ∗ .

(4.3)

4 Intense-Field Rearrangements

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The use of dye lasers brought about the possibility of using the entire visible, VUV and infrared regions of the electromagnetic spectrum, and expanded the range over which energy could be transferred so as to excite molecules. Technological development in controlling the modes within the laser cavity, as in ring dye-lasers [6], have provided some measure of control on the excitation energy and have enabled high resolution spectroscopic investigations to yield information on the energy flow within the molecule and on how energy disbursement leads to molecular motion. Precise measurement of the energetics of the reactions can also be used to derive information on the temporal dynamics of molecular motion [7]. Dye laser technology also paved the way to ultrashort laser pulses [8–10]. The possibility of precisely timing the excitation of molecules allowed one to “follow” molecular motion and opened the regime of real-time chemical dynamics. Starting from the initial studies of dissociation of simple diatomic molecules [11], the technique has been used to study dynamics of unimolecular reactions even in big biological molecules [12], and to probe molecular rearrangements [13] and bimolecular reactions [14]. Even the role of the solvent and the associated multi-particle interactions has been studied by suitably chosen clusters that have the reactant embedded in a limited number of solvent molecules [15]. The need and desire to not only observe the molecular motion but to control the motion of molecules [16] provided a major thrust for further research. Many techniques, both experimental [17] and theoretical [18], have been developed to implement control over chemical reaction. The shaping of the laser pulse, either by an acousto-optical modulator [19] or by masking techniques like the liquid crystal modulator [20], have become popular tools in the hands of chemical physicists. The experimental implementation of these tools mandatorily requires precise mapping of the molecular potential energy surfaces. Consequently, these methodologies proved to be of utility for only a limited set of simple reactions. The subsequent development of feedback algorithms and their implementation with computer-controlled pulse shapers helped overcome some of these drawbacks [21]. The yield of a desired fragment in laser-induced molecular fragmentation could, in principle, be selectively improved by a learning algorithm which optimized the pulse shape so as to channel the dissociation process into specific potential energy curves that gave the desired products [22]. Thus control of chemical reactions began to be implemented even for relatively complex reaction systems [23]. Another development which led to rampant expansion of activity in realtime dynamics of molecular motion was the Kerr lens mode locking technique [24] applied to Ti:Sapphire laser systems. Pulse widths as short as 100fs became routinely available, facilitating the precise clocking of the vibrational motion of molecules. The ultrashort pulses that used the solid state lasing technology also opened the possibility of achieving higher intensities than were hitherto available. Amplification by means of Chirp Pulse Ampli-

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fication (CPA) [25] brought to the fore laser intensities that had only been possible with large, expensive and user-unfriendly systems. Laser system that can fit in an “average” laboratory are now readily accessible off the shelf; these are capable of delivering Terrawatt power levels and intensities of up to 1018 W cm−2 [26]. Non-resonant, multiphoton excitation now became an intrinsic part of studies of reaction dynamics. While multiphoton ionization has found interest in the overall study of non-perturbative physics [27], the dynamics of molecular motion in intense laser fields has begun to attract interest as new methods to control chemical reactions in the strong field regime are sought [28].

4.3 Rearrangement Reactions in Optical Fields In a chemical reaction the most important parameters that govern the rate and course of the reaction are [29] a) the potential energy surface though which a reaction occurs, b) the bond energy or strength of the bond that needs to be broken, c) the orientation of the molecule in the chemical reaction, and d) the energy flow in the vibrational manifold of the molecule that leads to a desire product channel. Might intense laser fields offer the possibility of “tuning” each of these aspects? One of the earliest aspects to attract attention with regard to the use of intense fields in molecular reaction dynamics was the possibility of controlling the orientation or alignment of molecules in the presence of strong, linearly polarized optical radiation [30]. It was observed that the fragmentation of molecules was not isotropic when ionization was achieved using strong, linearly polarised laser light [31]. The propensity for observing a fragment ion is much larger along the polarization vector than at the perpendicular direction. Large electric fields that are associated with the intense laser fields polarize molecules and induce a dipole moment. The torque exerted on the molecule due to the induced dipole moment results in spatial alignment of the most-polarizable bond in a molecule along the polarization vector of the laser. Since the discovery of such spatial alignment a large number of studies have been extended not only to understand the dynamics of alignment, but also to control the alignment [32] and use the aligned molecular ensemble to induce chemical reactions [33]. The potential energy surfaces over which the reactions occur in field free reactions are very different to those that exist in the presence of the laser field. The field-dressed states can be dynamically tuned to provide ‘bond hardening’ or ‘bond softening’ [28]. The vibrational motion of the molecules during the laser pulse can lead to new concepts like ‘enhanced’ ionization [34]. Experiments to control the molecular motion by populating a desired field-dressed state that results in a desired product has been shown to be possible by optimal shaping of the incident laser pulse [35]. Though a number of chemical reactions have been probed, and methods to control those have been explored,


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the important class of chemical reactions that involve the rearrangement of atomic constituents seem to be relatively less studied. A rearrangement reaction is a reaction that happens between two different moieties of a polyatomic molecule as schematically illustrated in Fig. 4.2 The B and C moieties in the polyatomic molecule are, initially, not directly bonded to each other. In the course of a rearrangement reaction, the bond angles and bond lengths change in such manner so as to lead to the formation of a new bond between B and C. Rearrangement reactions, in general, are well known and well studied in chemistry. One of the most notable in gas-phase ion chemistry is the McLafferty rearrangement [36] that occurs as a single molecular ion event in the fragmentation of many organic molecules. Generally these reactions occur as a result of the motion of moieties within a particular excited-state potential energy surface. The time required for such motion is generally considered to be several vibrational periods. A priori, rearrangement reactions may be considered very uncommon in intense laser fields, especially in the super-intense regime of 1015 W cm−2 and larger. It is well established that for intensities beyond 1012 W cm−2 multiple ionization of molecules dominates the dynamics, and subsequent Coulomb explosion sets the time limit for the ensuing fragmentation. In such intensity regimes the possibility of molecules to rearrange their constituents such that bond formation can occur between different moieties so as to yield a rearrangement product may be considered very rare and unlikely: prevailing wisdom would have it that Coulomb explosion occurs on much shorter timescales. Rearrangement reactions in the lower-intensity multiphoton regime, where the Coulomb explosion is not dominant, has attracted some attention [37–39]. The possibility of using intense-field dressed states to control such reactions has been theoretically explored [40]. Experimentally, these reactions have been observed in a few organic molecules [41–44]. The rearrangement product ions were observed at intensities in the 1012 Wcm−2 range with femtosecond laser pulses being used to irradiate large organic

Fig. 4.2. Schematic of a rearrangement reaction

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molecules. Using feedback algorithms such relatively weak laser pulses could be shaped so as to increase the yield of certain products [38]. Among rearrangement reactions, hydrogen migration has attracted attention, including that of astrophysicists who wish to understand the origin of H+ 3 ion signals obtained from auroral emission from the outer atmospheres of Jupiter, Saturn and Uranus [45]. It was found that a number of organic molecules yielded H+ 3 ion when excited by UV light of wavelength in the region of 300nm. Might hydrogen migration be of any consequence in molecular dynamics that occur in intense, femtosecond-duration laser ionization? We address this issue in the following with reference to a series of linear alcohol molecules. As mentioned earlier, when the light intensities are as high as 1016 Wcm−2 , the irradiated alcohol molecules would inevitably be multiply ionized. With the tetrahedral geometry of the sp3 -hybridized carbon atoms in the aliphatic alcohols, the Coulomb explosion would lead to protons that have their individual velocity vectors that are aligned to each other at about 109.8◦ . In such a scenario the possibility of two or more of such protons interacting with each other is clearly not easily feasible. Proton migration might be expected to occur in such cases only if the molecule were suitably ro-vibrationally excited in such manner that the migration occurred prior to the Coulomb explosion. Might there be a possibility of intense laser fields “coaxing” such rearrangement, on ultrafast timescales that match Coulomb explosion times? The first evidence for such an event was reported recently [46] in connection with rearrangements involving two H-atoms in linear alcohol molecules irradiated by 100-fs long infrared light pulses of peak intensity as high as 1016 W cm−2 . Subsequent experiments have further revealed that migration of more than two protons may also occur in such alcohols [47, 48], leading to + the formation of H+ 3 products. However, the H3 formation has been shown to occur long after the laser pulse, so the reaction occurs in the expected manner: the alcohol molecule is excited to the ro-vibrational manifold in the appropriate electronically excited state, and the rearrangement reaction occurs as the system rolls down this potential energy surface. Let us, therefore, focus attention only on the unexpected rearrangement reaction, the one that leads to H+ 2 formation in the course of less than 100fs in an intense optical field generated by light intensities in the 1016 W cm−2 range.

4.4 Linear Alcohols in Intense Fields: Fragmentation Before specifically considering ultrafast rearrangements it is necessary to lay the proper experimental background by first discussing how ionization and fragmentation is induced in the linear alcohols by the intense optical fields they are subjected to. Figure 4.3 shows typical fragment ion yields that are obtained upon irradiation of some linear alcohol molecules by linearly- and circularly-polarized, 806nm, light at intensities in the 1016 W cm−2 range.


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These spectra were measured using intensity values that ensured that the optical fields that were experienced by the molecules were the same irrespective of the polarization state. We note that circular polarization results in a distinct suppression of fragment ion yields in all the molecules. This observation also holds for the doubly-charged molecular ions like, for instance, CH3 OH2+ . The observation of such dications in time-of-flight spectra indicates that their lifetimes against unimolecular dissociation certainly lie in the microsecond range, which covers the flight times for such ions through the spectrometer that was used for making these measurements. Figure 4.3 also shows Cq+ (q=1–3), O2+ atomic fragments as well as CH+ and CH+ 3 molecular fragment ions obtained from molecules like C2 H5 OH. As in the case of methanol, suppression of fragment ion yields with circular polarization is clearly observed. The spectra for hexanol and dodecanol show similar suppression in ionization with circular polarization. The fragmentation of each of the alcohol molecules that were probed is, as discussed above, induced by population of electronic excited states of the molecular ion whose potential energy surfaces are repulsive, at least in the Franck-Condon region that is vertically accessible from the ground electronic state of the neutral precursor. The potential energy surface and its energy are distorted by the intense optical field. The fragment ions that are formed will depend on the nature of the field-distorted state and on the minimum energy path in the multidimensional potential energy surface. Gaining proper theoretical insight along these lines remains a challenge for theorists. Nevertheless, it is still possible to make some general observations on the basis of the fragmentation spectra that have been measured. Electron rescattering in the laser field is invoked to qualitatively rationalize the suppression in ion yields that is observed when circular polarization is used. Electron rescattering is a special feature of strong-field ionization dynamics. Electrons that are field-ionized from a molecule continue to “feel” the effect of the light even after the initial ionization event is over. The wavepacket that describes the ejected electron initially moves away from the vicinity of the parent molecule. However, when the incident laser light is linearly polarized, the electronic wavepacket is pulled back towards the parent molecule half an optical cycle after its initial formation. The probability of recollision between the electron and the parent molecule depends on the laser phase as well as on the initial velocity and initial position of the electronic wave packet. Rescattering of the ionized electrons in the presence of the laser field strongly influences multiple ionization and electronic excitation in the molecular ions which, in turn, influence the fragmentation of the molecular ion. The following is the chronology of events that occurs. Upon initial irradiation, the alcohol precursor first tunnel ionizes when the applied field intensity is large enough. The ionized electron does not totally “leave” the alcohol molecule, but interacts with it under the influence of both the Coulomb force generated by the residual molecular ion and by the optical field generated by the

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Fig. 4.3. Fragment ion yield measured upon irradiation of alcohols (clockwise, from top left: methanol, ethanol, dodecanol, and hexanol) with 100 fs laser pulses of 1016 W cm−2 intensity. Solid bars: linear polarization, striped bars: circular polarization. The optical field was of the same magnitude for both polarization states

laser radiation. At low relatively values of laser field (corresponding to intensity ∼ 1014 W cm−2 ) the Coulomb field has a large influence in determining the motion of the wavepacket that describes the ejected electron. However, at large fields (in the 1016 W cm−2 intensity range and beyond), the electric field of the interacting laser becomes comparable in magnitude to the Coulomb field and, therefore, exerts a much larger influence on the electron trajectories. A decrease in the energy of the rescattered electron or in the


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probability of rescattering would adversely influence the multiple ionization of the molecular ion and, in turn, the fragment ion yields. Thus the lowering of fragment ion yields is attributed to the lowered probability of the rescattered electrons inducing dissociative ionization when circularly polarized light is used for irradiating the alcohols. Effectively, circular polarization switches electron rescattering “off”: the effect of this is clearly discernible in the fragmentation spectra. Detailed studies of the role of electron rescattering have recently been carried out [49, 50] and data that have been obtained appear to establish that molecular structure effects are also important in determining the degree of suppression that can be achieved by changing the polarization state of the incident laser radiation from linear to circular. Those fragmentation channels that require the largest transfer of energy from the optical field to the molecule are suppressed most markedly by using circularly polarized light; the suppression is less marked for those channels that require smaller amounts of energy transfer. Moreover, the kinetic energy release values that are measured for different fragmentation channels indicate that bond breakage occurs at distances that are larger than the equilibrium bond length. This is indicative of the enhanced ionization mechanism, and there is now experimental evidence that this mechanism also holds in the circular polarization regime.

4.5 Proton Migration in Strong-Field-Irradiated Alcohols As already noted, it might a priori be thought legitimate to suppose that rearrangement reactions are likely to be of little or no concern to the dynamics that are initiated when molecules are irradiated by ultrashort laser light whose intensity is large enough so that the associated electric fields start to match intramolecular Coulombic fields. Light intensities of 1016 W cm−2 achieve this situation. Even at light intensities of 1013 W cm−2 and above, optical field-induced multiple ionization becomes inevitable in most polyatomic molecules. The deposition, on femtosecond timescales, of energy from the optical field into molecular potential energy leads to ejection of one or more molecular electrons and, also, to molecular fragmentation. Most often, the molecular ion peak that is measured in a mass spectrum in such instances is negligibly small compared to the yield of fragments ions [51]. This is even more so in measurements that are made in “intensity-selected” mode [52] wherein only the central (most intense) part of the laser focal volume is sampled and contribution to the mass spectrum from the lower-intensity regions of the focal volume is avoided. In such intense fields, conventional wisdom has it that it is very improbable that the multiply charged molecular ion that is formed will undergo re-orientation so as to lead to rearrangement products being formed. Note also that multiple ionization in the Franck-Condon region

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would lead to fragment ions being driven by the Coulomb repulsion energy between the charged molecular fragments, thereby resulting in the release of potential energy in the form of translational energy of the fragment ions [2–4]. We focus attention in the following on results of recent intense field experiments that have probed intramolecular reactions involving proton migration in linear alcohols: methanol, ethanol, hexan-1-ol, and dodecan-1-ol. The peak laser intensities employed in these studies were as high as 8 × 1015 W cm−2 . Quite unexpectedly, it was discovered that proton migration occurs within the multiply charged parent molecular ion that is formed by optical field induced tunnel ionization. Moreover, such proton migration occurs much before the Coulomb repulsion makes the fragment ions fly apart. The fast rearrangement leads to formation of unusual products of dissociative ionization of alcohols, namely H+ 2 ions that are formed with substantial kinetic energy in the centerof-mass (cm) of the molecule. This ultrafast re-arrangement reaction is seen to be a feature that is common to the fragmentation dynamics of all the linear chain alcohols under similar fields. Experiments have also been conducted in which the laser polarization direction is changed and results establish clearly that the rearrangement apparently occurs within the 100fs pulse duration that is used. The short intense light field actually coaxes molecular rearrangement in addition to causing molecules to undergo fragmentation into ions. Figure 4.4 a) shows part of a typical time-of-flight (TOF) spectrum obtained when ethanol molecules were irradiated with pulses of peak intensity 8 × 1015 W cm−2 . Focusing only on the proton migration reaction, note that along with the multiply charged molecular ion peaks the TOF spectrum offers evidence for multiply charged fragment ion peaks, like C2+ . In the case of multiply charged atomic fragments, the large kinetic energy release (KER) in the fragmentation event manifests itself in TOF spectra as splitting in the arrival times of ions. The splitting indicates that fragments that are initially scattered in a direction towards the detector in the TOF spectrometer reach early, and are labeled as forward scattered ions (marked f in the figure), while those that are initially formed in the opposite direction reach the detector later, and are denoted backward scattered ions (marked b in the figure). The time difference between the forward and backward ions provides a measure of the kinetic energy release (in the centre-of-mass frame) that accompanies a particular ion formation channel [2–4]: δT =

2(2mUo )1/2 , qEs

(4.4)

where δT is the time separation between the forward and backward scattered peaks, Uo is the energy release in the center-of-mass frame, m is the mass of the ion, q is the ionic charge, and Es is the ion extraction field used in the TOF spectrometer. We shall make use of the ability to deduce KER values in the following, but first note the not-insubstantial signal that is observed in Fig. 4.4 a) at


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Fig. 4.4. A portion of the TOF spectra observed when ethanol is subjected to linearly-polarized optical fields of intensity 8 × 1015 W cm−2 . Left panel : bar format. Right panels: a) A typical raw TOF spectrum. The spectrum obtained with same laser intensity and number of laser shots at background pressure (≤1×10−9 Torr) is also shown. b) Spectra observed with the plane of polarization parallel and perpendicular to the spectrometer axis. Note the disappearance of forward and backward components in the latter case

flight times of 0.75μs that corresponds to H+ 2 molecular ions being formed; Fig. 4.4 b) depicts this somewhat more clearly. The H+ 2 peak shows clear features on either side of the central peak due to the forward and backward scattered ions; these ions are certainly formed with substantial kinetic energy. This is an unexpected feature since in TOF spectra, one does not expect to find molecular ions with large energy in the cm frame. Figure 4.4 a) also shows a typical spectrum of ion signals obtained at base pressure, in the total absence of alcohol molecules in the experimental chamber. At the lower end of the 10−9 Torr range, trace H+ ions that are desorbed from stainless steel are observed but we note that there is no evidence for H+ 2 ions in the background after the vacuum chamber was thoroughly degassed by hightemperature baking for prolonged periods of time. H+ 2 ions are only observed once alcohol molecules are introduced into the laser-molecule interaction zone. An important question that arises is whether + these H+ 2 ions result from an intermolecular reaction. Is H2 formed when one alcohol molecule interacts with an hydrogen-atom fragment from an-

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Fig. 4.5. Measured H+ 2 ion yield as a function of the pressure of the alcohol vapors introduced in the TOF spectrometer

other molecule within the laser focal volume? Or is their formation due to intramolecular reactions? One way to experimentally decipher these questions would be to measure the H+ 2 signal as a function of alcohol number density, and some typical results are depicted in Fig. 4.5 (see also [47, 48]). At pressures below 6 × 10−8 Torr the ion signal that was obtained showed, very clearly, a linear dependence. The H+ 2 yield deviated from linear dependence at higher pressures, indicating the onset of bimolecular collisions. In all measurements care was taken to use very large extraction fields so as to ensure that ion collection efficiency was close to unity, even for energetic ions. Although the measurements that have been described so far seem to indicate that H+ 2 ions are formed by a unimolecular process following an intramolecular rearrangement, unambiguous confirmation comes from polarization-dependent measurements. Figure 4.4(b) shows a typical spectrum obtained when the plane polarized laser light was perpendicular to the TOF axis as compared to when the plane of polarization was made parallel to the TOF axis. As is obvious from the data, the forward- and back-scattered peaks are suppressed with the use of perpendicularly polarized light. This brings to the fore three facets of what has been observed. Firstly, the polarization dependence enables deduction of the fact that the energetic H+ 2 ions are formed by an intramolecular rearrangement. If they were formed from a bimolecular reaction involving the interaction of a proton


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with another alcohol molecule, the process would certainly not exhibit any dependence on the polarization of the ionizing laser pulse. Moreover, it would exhibit a quadratic dependence on gas pressure, contrary to observations. Secondly, the data shown in Fig. 4.4 indicate that the rearrangement reaction occurs within the duration of a single laser pulse, and is driven by the strong optical field. If the rearrangement occurred long after the laser pulse, one would not expect the polarization state to affect the angular distribution of the ion emission. Thirdly, if H+ 2 formation were simply a consequence of rearrangement in the chemical bonds of the molecule, possibly in an excited ionic state that evolved from the Franck-Condon region of the neutral ground state, the H+ 2 yield would, again, not be expected to show a dependence on the incident laser polarization. The TOF spectrum indicates that H+ 2 ions are formed with substantial kinetic energy in the center-of-mass frame. Figure 4.6 shows the kinetic energy spectrum that is deduced from the forward and backward scattered peaks in TOF spectra measured for methanol precursors. In addition to a thermal energy component, H+ 2 is produced with an energy component that extends up to ∼12eV. The hollow spheres in Fig. 4.6 show the measured data and the lines shows the thermal energy component and the high energy component that constitute the measured kinetic energy spectrum. The two energy components arise from a simple double-Gaussian fit to the measured energy spectrum. The H+ 2 kinetic energy arises from the Coulomb repulsion from the charge on the rest of the multiply-charged alcohol Since a most probable value as high as 5eV is observed in our measurements (and such values are typical of those expected upon Coulomb explosion of multiply charged molecules [3]), the clear implication is that the rearrangement of the chemical bonds occurs well before the Coulomb explosion happens. In the absence of a proper theoretical framework within which strongfield molecular dynamics issues can be handled, simple quantum chemistry has been invoked in order to gain some further insight: minimum energy path calculations have been carried out on the multiply charged methanol using an ab initio method [53] to determine the fragmentation path for the linear chain alcohols. Self-consistent field Hartree-Fock computations were carried out using a 3-21G basis. It is clear that such calculations are not rigorous enough for deriving the detailed energetics of the fragmentation reaction. Nevertheless, they do provide qualitative insight into the main features of the reaction pathway. Using the steepest-descent method it is found that a doubly- and triplycharged methanol molecule reorients to form the H2 moiety that separates from the rest of the charged molecule. In Fig. 4.7 are shown snap-shot pictures of the minimum energy path that illustrate this. This feature was found to be apparent in all linear chain alcohols. Note how the two hydrogen atoms that are denoted HA and HB take part in the rearrangement so as to give rise

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to formation of a HA –HB bond in panel f. The novel feature of this rearrangement reaction is that it involves a molecular ion that is multiply charged. The multiple charging has been accomplished by the action of a laser field whose intensity is large enough to lead one to intuitively postulate that Coulomb explosion is the only possible pathway that is open. The rearrangement reaction that is depicted in Fig. 4.7 is induced by the strong optical field.

Fig. 4.6. Kinetic energy release imparted to the H+ 2 ion in the center-of-mass frame computed from the observed forward and backward scattered ion in the TOF spectrum. Note the low-energy and high-energy components (see text)

Fig. 4.7. Snapshot picture of the reaction pathway taken by a doubly-charged methanol molecular ion, as computed by the steepest descent method using a HFSCF method (see text)


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The sequence of events that occur are the following: i) at the rising edge of the laser pulse, one of the three C–H axes of the alcohol molecule becomes aligned with respect to the laser light’s electric field vector, ii) the aligned molecule undergoes multiple ionization, and iii) also undergoes rearrangement so as to form the H+ 2 moiety that Coulomb explodes from the remaining part of the charged molecule. So, what has been observed is ultrafast intramolecular rearrangement upon intense field irradiation of linear alcohol molecules in the gas phase. The rearrangement process gives rise to an unusual ionic fragment, H+ 2 , that is formed with a substantial amount of kinetic energy (a most probable value of about 5eV in the center-of-mass frame). The H+ 2 fragment is formed by a unimolecular reaction under intense light fields. The results of polarization dependence measurements indicate that the rearrangement occurs within the duration of the laser pulse and is driven by the strong light field. Strong field rearrangement reactions are an unexpected facet of lasermolecule interactions in the high intensity regime. Subsequent work by Yamanouchi and coworkers [43, 47, 48] has also established the discovery and has extended it by observations of rearrangements involving not just two but of three protons from alcohol precursors. Very recently, ultrafast proton rearrangements have been rediscovered in North America [54].

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5 Optical Control of Chiral Molecular Motors Kunihito Hoki, Masahiro Yamaki, and Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan, e-mail: [email protected]

Summary. Results of theoretical treatments of optical control of chiral molecular motors driven by linearly polarized laser pulses are presented. Both quantum and classical simulations of time-dependent quantum mechanical expectation values were performed to identify the rotational direction of real molecular motors. The rotational direction was found to be toward the gentle slope of the asymmetric potential energy surface of a chiral molecule of interest, which is the intuitive direction of rotation. The mechanism of its unidirectional motion is a non-resonant multi-photon forced-rotation induced by linearly polarized intense laser pulses. Relaxation effects of randomly oriented molecular motors were investigated using the Lindblad-type quantum master equation. Quantum control of unidirectional rotation of a chiral molecular motor is presented. Counter-intuitive rotation as well as intuitive rotation is generated by using a quantum control theory. Time- and frequency-resolved spectra of the electric field of the optimal laser were evaluated to analyze the origin of both intuitive and counter-intuitive rotations. A femtosecond pump-dump control method via an electronic excited state is shown to be one of the effective methods for avoiding effects of couplings between motors and solvents.

5.1 Introduction Recent advances in laser science and technology have led to considerable progress in optical control of physical and chemical processes in molcular systems [1–4]. In optical control, atoms or molecular groups are guided to a target state for a dynamic process by using tailored laser pulses. Quantum control has been applied to simple diatomic molecules and medium-sized molecules such as aromatic bio-molecules [5]. Recently, we have paid attention to optical control of quantum dynamics of chiral molecules for creation of functioning. There is an increased interest in control of molecular motors that are potentially important as functional molecular devices [6–21]. Chirality is essential for unidirectional rotation of a functional group as an engine in a molecule. Molecular motors can be more effectively controlled by lasers than by thermal or electrical methods. In this review, we present recent results of optical control of quantum dynamics of chiral molecules from a theoretical viewpoint. Our attention is focused on mechanisms of unidirectional rotation of a molecular functional

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K. Hoki, M. Yamaki, Y. Fujimura

group driven by nonhelical, linearly polarized laser pulses since the direction is not determined by laser fields but by the characteristic features of chiral molecules themselves. We call such a molecule in which a functional group rotates in a controlled direction a molecular motor. In Sect. 5.2, we first review unidirectional rotation of chiral molecular motors driven by a linearly polarized laser from the viewpoint of a symmetric consideration. We then present quantum and classical simulations to identify rotational direction of molecular motors. Here, we chose a real chiral molecule as a molecular motor. We evaluated the time evolution of quantum mechanical expectation values of an angular momentum of randomly oriented molecular motors at finite temperatures under various laser field conditions with changing central frequency and amplitude of laser pulses. We show that the rotational direction is toward the gentle slope of the asymmetric potential of a chiral molecule of interest, which is the intuitive direction of rotation. Generally, an asymmetric potential V (ξ) with respect to the rotation has asymmetric gradient dV (ξ))/ dξ. Therefore, the slope of the uphill for a mass point goes to the positive direction of ξ and that goes to the negative direction of ξ are different. An intuitive direction has a gentle slope to climb, and the other one has a steep slope. The results of the simulations showed that the mechanism of the molecular motor, that is, a unidirectional motion is induced by linearly polarized intense laser pulses through nonresonant multi-photon absorption processes. Relaxation effects of molecular motors investigated using the Lindblad-type quantum master equation are also described. In Sect. 5.3, we present optical control of unidirectional rotation of a chiral molecular motor driven by linearly polarized laser pulses. We show that counter-intuitive rotation as well as intuitive rotation is generated by using quantum control theory. The time- and frequenc-resolved spectrum of the electric field of the optimal laser was evaluated to analyze the mechanisms of both intuitive and counter-intuitive rotations. A femtosecond pumpdump control method was introduced in Sect. 5.3. This is one of the effective methods for conserving the energy of the motor rotation by avoiding the energy dissipation into the other motions such as overall molecular rotation and energy flow into solvent molecules. Concluding remarks are presented in Sect. 5.4.

5.2 Characteristics of Chiral Molecular Motors Driven by Linearly Polarized Lasers 5.2.1 Role of Molecular Chirality in Unidirectional Rotation Consider a molecule that consists of n electrons and N nuclei in an external linearly polarized electric field E(t) at time t. The Hamiltonian of the system is expressed as

5 Optical Control of Chiral Molecular Motors

95

2 2 Δi − Δj + Vcoulomb (r1 , r 2 , . . . , R1 , R2 , . . .) 2m i 2Mj j ⎫ ⎧ ⎨ ⎬

(5.1) + −e rZi + qj RZj E(t) , ⎭ ⎩ n

N

H (t, E) = −

i

j

where m is the mass of an electron, Mj is the mass of the jth nucleus, Vcoulomb (. . . ) is the Coulomb interaction between the electrons and the nuclei, ri is the position of the ith electron, Rj is the position of the jth nucleus, and qj is the charge of the jth nucleus. Here, the wave length of the electric field is assumed to be much longer than the size of the molecule, and the Z-axis is set to the direction of the electric field vector. Let us assume that the molecule has chirality. In this section, we call a pair of chiral molecules L and R molecules. The initial density operators ρL 0 and ρR 0 of L and R molecules are transformed by each other by the inversion operator P . That is, P ri P −1 = −r i , P Rj P P H (t, E) P

(5.2a)

−1

= −Rj ,

(5.2b)

−1

= H (t, −E) ,

(5.2c)

−1 P ρL 0P

=

ρR 0

.

(5.2d)

Under the electric field E(t), time propagation of the molecular density operator ρΓ0 (t, E), where Γ = L or R, is formally written as ρΓ (t, E) = U (t, E) ρΓ0 U † (t, E) ,

(5.3)

i t U (t, E) = T exp − H (τ, E) dτ . 0

(5.4)

where

Here, T is the time-ordering operator. From Eqs. (5.2)–(5.4), the density operator satisfies P ρL (t, E) P −1 = ρR (t, −E) .

(5.5)

We now introduce the angle of the internal rotation ξ and the corresponding angular momentum l. Both have anti-symmetric properties with respect to P as P ξP −1 = −ξ , P lP

−1

= −l .

(5.6a) (5.6b)

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The expectation value of the angular momentum at time t, lΓ (t, E), which is called the instantaneous angular momentum in this paper, is written as lΓ (t, E) = TrlρΓ (t, E) .

(5.7)

From Eqs. (5.5), (5.6b) and (5.7), we obtain the symmetric property of the instantaneous angular momentum for the internal rotation ξ as lL (t, E) = −lR (t, −E) .

(5.8)

We note that −1 R⊥ (π) H(t, E)R⊥ (π) = H(t, −E) , −1 R⊥ (π) lR⊥

(π) = l ,

(5.9a) (5.9b)

where R⊥ (π) is the operator representing the π rotation around an axis perpendicular to the Z-axis. If the initial state ρΓ0 is symmetric with respect to R⊥ (π), Eq. (5.8) is rewritten as lL (t, E) = −lR (t, E) .

(5.10)

Equation (5.10) shows the relation between molecular chirality and direction of temporal angular momentum under linearly polarized electric fields. As a specific example, we consider an idealized case where the internal rotation is well characterized as torsion of two rigid groups. Here, the system Hamiltonian in the electronic ground state within the Born-Oppenheimer approximation can be written as H Γ (ξ) = −

2 ∂ 2 1 Γ + V Γ (ξ) − μΓZ (ξ) E(t) − αZZ (ξ) E(t)2 + O E 3 , 2 2I ∂ξ 2 (5.11)

where I is the moment of inertia, ξ is the coordinate of the internal rotation, μΓ (ξ) is the dipole moment vector, and αΓ (ξ) is the polarizability matrix. Note that the nuclear configuration in the rigid groups determines molecular chirality Γ . We now assume that the rotating rigid group is much lighter than the entire molecule to omit the effects of overall molecular rotation. The Euler angle-averaged expectation value of a vector component projected on the rotation axis of the angular momentum operator introduced in Eq. (5.7) can be represented as 1 ∂δ (ξ − ξ) Γ Γ (t, E) = 2 dΩ dξ dξ −i ρ (ξ, ξ , t, E) , (5.12) 8π ∂ξ where Ω is the Euler angle of the molecular frame, δ(ξ) is the Dirac delta function, and ρΓ (ξ, ξ , t, E) is the density matrix in the ξ representation. Here, the expectation value is averaged over the ensemble of molecules whose molecular frame is isotropically distributed.


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5.2.2 Unidirectional Rotation of Molecular Motors In this section, first we show that the unidirectional rotation of chiral molecular motors is induced by nonhelical, linearly polarized lasers. We next now show by using both quantum and classical dynamics simulations that the rotational direction is toward the gentle slope of the asymmetric potential energy of a chiral molecule (i.e., intuitive rotation) [22, 23]. From the simulations, the mechanism of the unidirectional rotation was clarified [24]. On the other hand, it is well known that circularly polarized laser pulses drive molecular motors, where the direction of rotation is determined in the laboratory fixed frame. We adopted (R)-2-chloro-5-methyl-cyclopenta-2,4-dienecarbaldehyde as a model of a chiral molecular motor as shown in Fig. 5.1(a). Here, chirality of the molecule is labeled R or S, which is a conventional notation in chemistry when a molecule has asymmetric carbons. The internal rotation of the CHO group around the C2 −C3 bond is driven by laser pulses. The dihedral angle ξ between the O1 −C2 −C3 and C2 −C3 −H planes is the variable in the motor. The molecule has several properties that are essential for

Fig. 5.1. (a) (R)-2-chloro-5-methyl-cyclopenta-2,4-dienecarbaldehyde as a molecular motor. The aldehyde group is an engine driven by a laser field. The dihedral angle ξ is a variable. (b) Potential energy surface of (R)- and (S)-motor. The rotation toward the negative direction of ξ has a gentle slope to climb. (c) Cartesian components of the dipole moment vector in the molecular frame

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a molecular motor driven by laser pulses: (1) the potential energy surface is asymmetric, that is, V R (ξ) = V L (ξ) and V R (ξ) = V L (−ξ), (2) the components of the dipole moment vector vary to a fairly extent along ξ because of the large electro-negativity of the oxygen atom of the CHO group, and (3) rotational constants are of a few GHz, so that we can safely ignore the effects of overall rotation of the molecule within a few hundred picoseconds [22]. Figure 5.1(b) shows the three components of the dipole moment vector in the molecular fixed Cartesian coordinates. Here the z-axis is on the C2 −C3 axis, heading from C3 to C2 , and the x-axis is placed on the C2 −C3 −H4 plane, heading to H4 and at right angles to the z-axis. The potential energy and the dipole moment were calculated by using the Gaussian 98 package of programs [25]. The potential energy shown in Fig. 5.1(c) was calculated at every dihedral angle O1 −C2 −C3 −H4 at which all of the other structural parameters were optimized. The moment of inertia of the internal rotation was assumed to be constant, 17.6amu ˚ A2 , which was estimated at the most stable geometrical configuration of the molecular motor. The molecular fixed coordinate system is fixed in space because the rotational constants are 1.97, 1.16 and 0.79GHz, which means the time scale of the overall rotation is a few hundred picoseconds and longer than the time scale of a motor rotation within several ten picoseconds. Figure 5.2(a) shows a linearly polarized laser field E(t) used in the simulation: E(t) = f (t) cos(ωt) for 0 ≤ t ≤ tf E(t) = 0 for tf < t .

and

(5.13a) (5.13b)

Here, tf , was 30ps, the central frequency of the pulse ω was 3.72 × 1012 Hz, which corresponds to the frequency 124cm−1 , and the envelope function of the laser pulse f (t) was set to be πt f (t) = E0 sin2 , (5.14) tf where the maximum amplitude of the pulse E0 was set to be 3.4GV/m. The resulting magnitude of motor-laser interaction energy 2μE0 | sin θ| could become larger than that of the hight Vmax of the potential barrier of the internal rotation, 2μE0 > Vmax . μZ is replaced by μ thereafter. This relation ensures that the induced rotation overcomes the potential barrier even if the frequency of laser is non-resonant. Figure 5.2(b) shows the time evolution of the angular momentum of randomly oriented molecular motors, Γ (t, E), where initial density is set to the Boltzmann distribution at T = 300K [22]. The axis on the left-hand side represents angular momentum in units of , and on the right-hand side the angular frequency in units of Hz. In Fig. 5.2(b), we obtain R (t, E) ≈ −1.3 for (R)-motors and S (t, E) ≈ 1.3 for (S)-motors after application of a laser pulse. Those angular motions are maintained even if the pulse is turned off.


99

Fig. 5.2. (a) Linearly polarized electric field used in the simulations. (b) Expectation values of the internal rotational angular momentum of randomly oriented molecular motors [22]. Reproduced by permission of American Institute of Physics

This means that the internal motion is not a pendulum motion but a unidirectional rotational motion across the potential barrier. We note that the rotational directions of the randomly oriented motors are opposite in (R)- and (S)-motors, and that the angular momentum being zero for achiral molecules as indicated by the dashed line in Fig. 5.2(b). Here, an achiral motor is obtained by substituting a methyl group in the chiral motor with a chlorine atom so that its potential is symmetric as V R (ξ) = V L (ξ). To clarify the mechanism of the unidirectional rotational motion of chiral molecular motors driven by linearly polarized laser pulses, we examined the dependence of ω and E0 on the instantaneous angular momentum Γ (t, E). Figure 5.3 shows contours of randomly oriented (R)-motors R (t, E) at t = 30ps and T = 0K [24]. In Fig. 5.3, absolute values of angular momentum larger than about 0.1 are plotted. The directions of internal rotations are persistent because these can be seen at a time longer than 30ps. The counter map can roughly be divided into two regions. Its dividing line is denoted by a broken line in Fig. 5.3. The critical value of E0 is nearly 2.4GV/m, which corresponds to the magnitude of the laser-motor interaction energy, 1650 cm−1 . In the region above the dividing line, the sign of the angular momentum is negative, which means that (R)-motors rotate toward the gentle potential energy side. In this region, the magnitude of

100


Fig. 5.3. Contour plots of the angular momentum of randomly oriented (R)-motors at t = 30 ps and T = 0 K [24]. Reproduced by permission of American Institute of Physics

the laser-motor interaction is larger than that of the internal rotational potential barrier, i.e., 2μE0 > Vmax , which ensures that the induced rotation overcomes the potential barrier with nonresonant condition of the laser frequency. In the region below the dividing line, on the other hand, counterintuitive rotations toward the steep slope of the potential energy curve occur. Figure 5.4 shows temporal behaviors of (R)-motors in a regime of medium laser intensity, 2μE0 < Vmax . Contrast to the case in Fig. 5.3, it can be seen in Fig. 5.4 that counter-intuitive rotation occurs just after the transient intuitive rotations around the time t = 15ps at which the electric field E(t) takes the maximum value. This type of counter-intuitive rotational motion is due to a quantum effect. To confirm this, counter plots of the angular momentum calculated by using classical mechanics are shown in Fig. 5.5 as well. Here, R (t, E) was obtained by solving the classical equation of motion. The initial distribution was set to be a canonical distribution at T = 150K. The results of the classical treatment show that neither intuitive nor counter-intuitive rotation was induced in the regime of medium laser intensity indicated by the dividing line as that in Fig. 5.3. When the laser intensity is larger than the potential barrier, the intuitive rotational motion is induced by non-resonant linearly polarized lasers. In this case, the direction of unidirectional rotational motions can qualitatively be understood based on a dynamical motion of the rotational wave packet on the effective potential energy of a chiral molecular motor as shown in Fig. 5.6. Figure 5.6 shows the effective potential and rotational wave packets on the (R)-motor at four Euler spatial configurations. The Euler angle θ was set


101

Fig. 5.4. Temporal behaviors of (R)-motors in a regime of medium intensity in which 2μE0 < Vmax is satisfied. Here E0 = 2.2 GV/m and ω = 114.0 cm−1 were adopted. The solid line denotes R (t, E) calculated using the quantum master equation, and the dotted line denotes that calculated using classical mechanics [24]. Reproduced by permission of American Institute of Physics

Fig. 5.5. Contour plots of the angular momentum of randomly oriented (R)-motors at t = 30 ps and T = 150 K, which were calculated using the classical equation of motion [24]. Reproduced by permission of American Institute of Physics

to be 0.5π, which gives the maximum laser-motor interaction. The amplitude of the electric field was set to be the same as that in Fig. 5.2. The center of the rotational wave packet was initially localized at ξ = 0. It can be seen in Fig. 5.6(a) that the initial wave packet at the configuration χ = π at ωt = 0 or χ = 0 at ωt = π moves toward the left-hand side with a gentle slope when the wave packet is shaken by an intense pulsed laser. On the other hand, Figs. 5.6(c) and 5.6(d) show that the wave packet shaken by a pulse laser at the configurations χ = ±0.5π cannot obtain sufficient angular momentum to cross over the potential barrier after the laser pulse is turned off. This shows that the direction of the unidirectional motion is determined by the shape of the asymmetric potential of the internal rotation of the chiral molecule. The laser acts as an accelerator of the molecular motor.

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Fig. 5.6. Schematic view of a localized rotational wave packet on the effective potential energy functions of the (R)-motor at specified Euler angles χ and times ωt [24]. Reproduced by permission of American Institute of Physics

We now discuss relaxation effects that we omitted so far [24]. We used the Lindblad-type quantum master equation as the equation of motion of the reduced density operator ρ(t) as [26] i

( ∂ i ' ρ(t) = [H(t), ρ(t)] + [An ρ(t), A†n ] + [An , ρ(t)A†n ] . ∂t 2

(5.15)

Here, H(t) is the Hamiltonian of the molecular motor, and [ , ] denotes a commutator. The second term of the right-hand side in the above equation denotes the relaxation effects due to the system–bath coupling, where the bath is assumed as a Markovian bath. Here, the bath represents all other degrees of freedom, including both of intra-molecular vibration modes and surrounding other molecules. Furthermore, by assuming that the motor system was a weakly fluctuating system, we ignored ultrafast inertial effects. Such a system can be seen for motors surrounded by solvent cage molecules or imbedded in a rigid solvent under low temperature conditions. The interaction operator, An was set to be An = |n + 1 an n| + |n bn n + 1| with n = 0, 1, . . . . Here, an and bn are the interaction parameters that were assumed to be expressed as

an bn

2

En+1 − En = exp − kT

and a2n + b2n =

En+1 − En , τ0 E1 − E0

(5.16)


103

in which En is the nth quantum state of the molecular motor, and τ0 is an input parameter of the model, which specifies the relaxation time from the first excited quantum state to the ground state. The quantum master equation was solved by means of the split operator and a finite difference method with a fast Fourier transform algorithm using 256 grids for the internal rotational coordinate. Figure 5.7 shows relaxation effects on temporal behaviors of the instantaneous angular momentum of randomly oriented (R)-motors at T = 150K. Here a linearly polarized laser pulse with the same envelope function and parameter set as those in Fig. 5.2 was adopted. In Fig. 5.7, strong τ0 -dependence can be seen. This is mainly because effective relaxation times were taken to be proportional to the excited energy of the internal rotation in the relaxation model. The magnitude of the effective relaxation time in the 54th quantum state just above the rotational potential barrier height is shorter than that of τ0 by about two orders of magnitude, for example, 5ps for τ0 = 100ps. One of the methods for maintaining a constant rotational motion under such a relaxation condition is to sequentially apply intense laser pulses to the motor to recover the rotational energy loss and to accelerate the rotation.

Fig. 5.7. Temporal behaviors of the instantaneous angular momentum of randomly oriented (R)-motors denoted by a solid line, and those of (S)-motors denoted by a dotted line at T = 150 K. τ0 denotes the relaxation time from the first excited state to the lowest state [24]. Reproduced by permission of American Institute of Physics

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5.3 Optical Control of a Chiral Molecular Motor 5.3.1 Quantum Control of Unidirectional Rotation of a Chiral Molecular Motor In this section, we present a scenario for controlling the rotational direction of a chiral molecular motor using quantum control theory and clarify the control mechanism by analyzing the time- and frequency-resolved spectrum of the motor and the time-dependent population in a non-stationary, angular momentum state representation [27, 28]. A molecular motor can be driven in the intuitive or counter-intuitive direction by using techniques of quantum control methods. There are two types of theoretical methods for designing optimal laser pulses: one is a global optimization method and the other is a local control method [3, 29]. We adopted a local control method [30, 31]. In a local control method, the electric field of laser pulse E(t) is expressed as E(t) = −2A Im Ψ (t)| W μ |Ψ (t) .

(5.17)

Here, Ψ (t) is a rotational wave packet at time t, A is) a regulation parameter of the laser intensity, and W , which is expressed as wn |n n|, is a target operator of a multi-level quantum system, whose quantum-mechanical expectation value gives the maximum value at a final time. Weighting factors {wn } satisfy the condition wn > wn−1 > · · · > w1 > w0 > 0 for securing the population transfer from the ground state to the target state at the final time. Let us consider a molecular motor that is driven from the ground state |0 to a final state by laser pulses [28]. The target state is expressed in terms of a linear combination )of the eigenstates of a rotational angular momentum, |m, as |T = m0 Cm |m for a counter-intuitive rotation. Here, the optimization procedure is first to determine the electric field, E(t0 ), at the initial time t0 after substitution of Ψ (t0 ) into the expression of the electric field above and then to obtain a motor wave function after an infinitesimally increased propagation time by solving the time-dependent Schr¨ odinger equation with the initial condition of Ψ (t0 ). The procedure described above is repeated until the electric field at the final time is determined. In applying the locally optimized control method to dynamic systems such as molecular motors in which the final state is non-stationary, the target state should be set to the initial state by reversing the time order of the control procedure [28]: the target operator has to commute with the motor Hamiltonian to guarantee a monotonic increase in the observable in the target state [30, 31]. Figure 5.8 shows the results obtained when the quantum control method is applied to (R)-2-chloro-5-methyl-cyclopenta-2,4-dienecarbaldehyde as a model molecular motor. Figure 5.8(a) and 8(b) show the quantum mechanical expectation values of instantaneous angular momentum (t) of


105

Fig. 5.8. Instantaneous angular momentum, (t), of the molecular motor driven by controlled laser pulses: (a) rotation in the intuitive direction and (b) rotation in the counter-intuitive direction. [28] Reproduced by permission of the PCCP Owner Societies

the molecular motor in intuitive and counter-intuitive directions, respectively [28]. The instantaneous angular momentum, (t), is defined as 2π ∂ ∗ dξΨ (t) −i (t) = Ψ (t) . (5.18) ∂ξ 0 where it is assumed that the molecule is fixed at the laboratory fixed axis XYZ. The target states for√ intuitive rotation and counter-intuitive rotation were √ set to be (|66 − i |65) / 2 and (|66 + i |65) / 2 at the final time t = 300ps, respectively. Here, |65 and |66 are the eigenstates whose frequencies are equal to each other within 0.001cm−1 . That is, these eigenstates are practically degenerate in the observation time. The target states correspond to those of the angular momentum quantum numbers m = −33 and m = 33 in a free-rotation model. Figure 5.8 indicates that rotational directions of the motor are well controlled, that is, the instantaneous angular momentum becomes constant values of about −23 and 23 for the intuitive and counter-intuitive directions, respectively. Figure 5.9 shows time- and frequency-resolved spectra of the electric fields of the designed laser pulses that drive the molecular motor in the intuitive (counter-intuitive) rotational direction as shown in Fig. 5.8. The time- and frequency-resolved spectrum is defined as S(ω, t) = dτ E(τ )g(t − τ ) exp(iωt) . (5.19)

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Fig. 5.9. Time- and frequency-resolved spectra of the electric fields of laser pulses obtained by using the locally optimized control procedure: (a) for the intuitive direction of internal rotation and (b) for the counter-intuitive direction of internal rotation [28]. Reproduced by permission of the PCCP Owner Societies

Here g(t) is a window function. It can be seen in Fig. 5.9 that the designed electric fields consist of four components (ε1 , ε2 , ε3 and ε4 ) in a good approximation. The first two components (ε1 , ε2 ) simultaneously operate at the initial stage of motor initiation (0–160ps). The third component, ε3 , with its low slope reflects the low-frequency regime of the highly anharmonic, torsional potential of the motor. This component bridges between the initial state and the final stage (180–250ps) at which unidirectional rotation starts. The fourth component, ε4 , accelerates the rotational motion. The frequency of ε4 is around 60cm−1 , which is nearly equal to the frequency difference between two quasi-degenerate eigenstate pairs n = 63, 64 and n = 65, 66. Figure 5.10 shows the time-dependent population of the controlled molecular motor in the angular momentum state representation. In Fig. 5.10, the intuitive (counter-intuitive) rotation case is characterized by negative (positive) values of m at the final time. It can be seen that about 90 of angular momentum states {|m} make a significant contribution. This is mainly due to the effect of the potential of the chiral motor. It is notable that intuitive rotation begins at 240ps, while counter-intuitive rotation begins at 260ps. The difference between these two rotations corresponds to about 1.06cm−1 , which is close to the frequency difference between the two eigenstates n = 57 and n = 58 from numerical analysis of the eigenstates.


107

Fig. 5.10. Time propagation of the population of the controlled molecular motor in the angular momentum state representation: (a) for the intuitive direction of rotation and (b) for the counter-intuitive direction of rotation [28]. Reproduced by permission of the PCCP Owner Societies

This clearly indicates that the direction of rotation is determined by the phase of a coherent superposition of these eigenstates. The direction of rotation can be controlled by timing of the laser pulses. It is also notable that intuitive rotation is realized earlier than counter-intuitive rotation. It should be noted that in addition to m = −33 and m = 33 in Fig. 5.10, several angular momentum states make a dominant contribution to unidirectional rotation at the final time of 300ps. This is simply because the target states expressed in terms of a linear combination of the eigenstates of the chiral motor Hamiltonian are not the eigenstates of the potential free Hamiltonian. 5.3.2 Motor Control by a Femtosecond Pump-Dump Method So far, we have presented a chiral motor control in the ground electronic state by using IR or FIR pulses. In this section, we show another method of a chiral motor control via an electronic excited state. Contrast to the previous subsection, a chiral motor is controlled by UV pulses in a femtosecond time scale. Under such an ultra-short time condition, it is possible to avoid competing relaxation dynamics due to intra-molecular and intermolecular interaction.

108


As a first example, we present a pump and dump method for the chiral motor control that is one of the simplest control scenarios. Since the pumpdump laser pulses have a simple shape of function E(t), the resultant rotor dynamics can easily be analyzed. It is also possible to employ the quantum control technique such as the local control method shown in Eq. (5.17) to obtain high quantum yield. Figure 5.11 shows a schematic illustration of a femtosecond pump-dump laser ignition method. Here V0 and V1 denote the potential energies in the ground and electronic excited states, respectively. Solid lines with an arrow denote a pump process (1) and a dump process (2). A femtosecond visible or UV pump pulse transfers a rotational wave packet from the ground state to the excited state. When the wave packet moves along V1 and reaches the most preferred position where kinetic energy is maximum in terms of the FranckCondon principle, it is transferred back to V0 by applying the dump pulse (2). The direction of the rotational motion is determined by the gradient of

Fig. 5.11. A schematic illustration of the femtosecond pump-dump laser ignition method. α denotes the national coordinate. Reprinted with permission from [32]. Copyright (2004) American Chemical Society


109

the potential energy surface V1 around its Franck-Condon region. If the motor has a symmetric plane as in the case of an achiral molecule, its gradient should be zero from symmetry consideration, which means that an achiral molecular motor cannot produce unidirectional motion by using linearly polarized laser pulses. On the other hand, (S) and (R) molecular motors rotate in opposite directions. The rotational directions under the pump-dump laser pulses are characteristic of the chirality of the motor, and the gradient of field-free potential energy surface on V1 around the Franck-Condon region can be estimated using molecular orbital methods. For demonstrating the applicability of the ignition method, a chiral molecular motor, (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde, was adopted instead of (R)-2-chloro-5-methyl-cyclopenta-2,4-dienecarbaldehyde shown Sect. 5.2.2 [32]. The former molecule has large potential energy displacements between the ground and electronic excited states and therefore has a large kinetic energy to be transferred to rotational motions, compared with the latter. In Fig. 5.12, temporal behaviors of the molecular motor driven by

Fig. 5.12. Upper figures: contour plots of the rotational wave packet created on the electronic ground state as a result of pump-dump pulses. Lower figure: expectation value of the angular momentum. Time regimes indicated by a, b, c correspond to those in the upper figures, respectively. Reprinted with permission from [32]. Copyright (2004) American Chemical Society

110


a femto-second pump and dump method are shown. Here the electric fields of the laser pulses E(t) were assumed to be given as E(t) = Ap f (t, tp , tp + Tp ) cos(ωp t) + Ad f (t, td , td + Td ) cos(ωd t) , where

* sin2 (t − t1 )π/(t2 − t1 ) (t1 ≤ t ≤ t2 ) f (t, t1 , t2 ) = 0 (t < t1 , t2 < t)

(5.20a)

(5.20b)

with Ap = Ad = 1010 V/m, Tp = Td = 100fs, tp = 50fs, td = 180fs, ωp = 34 300cm−1 and ωd = 32 800cm−1 . The initial state was set on the lowest rotational state in the ground state. The upper three panels in Fig. 5.12 show the rotational wave packets created on the electronic ground state. Here wave packets trapped in the potential well in the ground state were omitted. In the early stage, it takes about 550 fs for one turn of internal rotation, which corresponds to the angular frequency of ω = 1.1 × 1013 s−1 . In this case, the expectation value of the angular momentum of the motor is estimated to be ≈ ωI = 31. The lower panel in Fig. 5.12 shows the quantum mechanical expectation value of the angular momentum of the motor at time t. The time evolution of the expectation value can be divided into three regions, a, b and c, within the rephrasing time 2πI/ = 1700fs. The expectation value of the angular momentum is about 30, which agrees with the value estimated above. Revival structures of the angular momentum are shown in Fig. 5.12. These are called fractional revivals that originate from the nonlinearity of the system Hamiltonian [33, 34]. Another quantum ignition method by means of ultra short IR + UV laser pulses has been designed to obtain a high angular momentum about 110 [35]. Such a high angular momentum excitation is due to additive effects between rotational mode excitations in the ground state by IR pulses and those in an optically active electronic excited state by UV pulse.

5.4 Conclusions Results of theoretical treatments of optical control of chiral molecules driven by non-helical, linearly polarized laser pulses are presented. The origin of the unidirectional motions of molecular motors, which is due to chirality of molecules, has been clarified by performing both quantum and classical simulations of time-dependent quantum mechanical expectation values. The rotational direction of molecular motors is toward the gentle slope of the asymmetric potential of a chiral molecule, which is called the intuitive direction of rotation. The mechanism of the unidirectional motion is nonresonant multi-photon forced rotation induced by linearly polarized intense


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laser pulses. Relaxation effects of molecular motors were also investigated using the Lindblad-type quantum master equation. Counter-intuitive rotation as well as intuitive rotation is generated by using quantum control theory. Time and frequency-resolved spectra of the optimal electric field with terahertz frequency regimes were obtained to analyze the origin of both intuitive and counter-intuitive rotations. A femto-second pump-dump control via an electronic excited state is shown to be an effective method for avoiding competing dynamics of motors such as intra-molecular interactions or solvent interactions. Acknowledgement. This research was partly supported by a Grant-in-Aid for Scientific Research (No. 1555002) and a Grant-in-Aid for Scientific Research on priority areas, “Control of Molecules in Intense Laser Fields” (Area No. 419) from the Ministry of Education, Science, Sports, Culture and Technology, Japan.

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19. G. S. Kottas, L. I. Clarke, D. Horinek, and J. Michl, Chem. Rev. 105, 1281, 2005 and references therein. 20. R. A. van Delden, M. K. J. ter Wiel, M. M. Pollard, J. Vicario, N. Koumura, and B. L. Feringa, Nature 437, 1337, 2005 21. J.-P. Collin and J.-P. Sauvage, Chem. Lett. 34, 742, 2005 22. K. Hoki, M. Yamaki, S. Koseki, and Y. Fujimura, J. Chem. Phys. 118, 497, 2003 23. K. Hoki, M. Yamaki, and Y. Fujimura, Angew. Chem. Int. Ed. 42, 2975, 2003. 24. K. Hoki, M. Yamaki, S. Koseki, and Y. Fujimura, J. Chem. Phys. 119, 12393, 2003 25. M. J. Frisch et al., Gaussian 98, Revision A. 9, Gaussian, Inc., Pittsburgh, PA, 1998 26. G. Lindblad, Commun. Math. Phys. 48, 119, 1976 27. M. Yamaki, K. Hoki, Y. Ohtsuki, H. Kono, and Y. Fujimura, J. Am. Chem. Soc. Commun. 127, 7300, 2005 28. M. Yamaki, K. Hoki, Y. Ohtsuki, H. Kono, and Y. Fujimura, Phys. Chem. Chem. Phys. 7, 900, 2005 29. Y. Ohtsuki, M. Sugawara, H. Kono, and Y. Fujimura, Bull. Chem. Soc. Jpn. 74, 1167, 2001 30. M. Sugawara and Y. Fujimura, J. Chem. Phys. 100, 5646, 1994 31. Y. Ohtsuki, H. Kono, and Y. Fujimura, J. Chem. Phys. 109, 9318, 1998 32. K. Hoki, M. Sato, M. Yamaki, R. Sahnoun, L. Gonz´ alez, S. Koseki, and Y. Fujimura, J. Phys. Chem. 108, 4916, 2004 33. J. A. Yeazell and C. R. Stroud, Phys. Rev. A 43, 5153, 1991 34. T. Seideman, Phys. Rev. Lett. 83, 4971, 1999 35. Y. Fujimura, L. Gonz´ alez, D. Kr¨ oner, J. Manz, I. Mehdaoui, B. Schmidt, Chem. Phys. Lett. 386, 248, 2004

6 Non-Coulomb Explosions of Molecules in Intense Laser Fields Fanao Kong1 and See Leang Chin2 1

2

The Institute of Chemistry, Chinese Academy of Sciences, 2, 1st Street N., Zhongguan Cun, Beijing, 100080 P.R. China e-mail: [email protected] Department of Physics, Engineering Physics and Optics & Center for Optics, Photonics and Laser, Laval University, Quebec City, G1V 0A6 Canada

Summary. Polyatomic molecules undergo neutral dissociation in moderate intense laser fields. As a typical molecule species, methane has been studied in the laser intensity of 1012 – 1013 W/cm2 of a Ti-sapphire laser (λ = 800 nm). The fragments of electronically excited CH (A 2 Δ, B 2 Σ, C 2 Σ+ ) and H (n = 3), where n is the principal quantum number, are observed in the fluorescence spectrum. It is interesting that the same products were found in the synchrotron radiation previously, where the XUV photons of 10 – 22.5 eV were used. The above phenomena show that the fragmentation in the intense laser field through multiphoton absorption and in the synchrotron radiation through single XUV photon absorption are due to the excitation and dissociation of super-excited states, which have more energy than the ground state of molecular ion.

6.1 Introduction There is a growing interest in the study of molecular behavior in intense laser fields. Being a relatively new field in molecular science, strong laser fieldmolecule interaction, especially dissociation of molecules, is very promising from many points of view including controlling chemical reaction [1]. In this paper, the intense femtosecond laser we refer to is the most popular femtosecond Ti-sapphire laser whose wavelength is around 800nm. When a molecule is placed in a strong laser field, the field applies a force to both the electrons and the atomic cores of the molecule, all the internal degrees of freedom of a molecule can be influenced [2–5]. Electrons of the molecule are affected by the intense laser pulse immediately. Some abnormal phenomena thus appear. In the “moderate” laser intensity of 1013 – 1014 W/cm2 , molecules are usually first tunnel ionized [6–8]. The swinging of the ionized electron in the alternative laser field may cause a re-scattering effect. The effect would result in high harmonic generation (HHG) [9], abovethreshold ionization (ATI) [10], and double ionization (DI) of molecules [11]. Atomic cores of a molecule may also be affected by such intense laser pulse. Immediately after the ionization, the molecule may dissociate in a strong laser field. The phenomenon is usually explained by Coulomb explosion (CE) [12]. Beyond the laser intensity of 3 × 1014 W/cm2 , doubly or multiply charged

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molecular ions can be generated. Driven by the intra-molecular Coulomb repulsive force, the ionic skeleton undergoes CE, yielding ionic products. The product species are detected by mass spectroscopic techniques in most cases [13–15]. However, neutral fragments of molecules yielded in intense laser fields have also been found recently, even at the relatively “low” laser intensity of 1013 W/cm2 [16–18]. Fluorescence spectroscopy was used to detect neutral fragments, which could not be detected by mass spectroscopy. Since “low” intensity laser could not produce multiple charged ions, the appearance of the neutral products cannot be explained by the CE of multiply charged ions. Therefore, a new mechanism is needed to explain the neutral fragmentation of molecules at relatively low laser intensity. It is interesting to compare the fluorescence spectra obtained in the intense laser fields to those appearing in synchrotron radiation (SR). The spectra recorded in both cases are analogous to each other. This implies that the final products are yielded from the parent molecules with similar mechanisms. The single photon energy of about 15eV used in the SR is almost equal to the total energy of ten laser photons which a typical molecule absorbs in the multiphoton excitation regime. It is therefore strongly suggested that the neutral dissociation of molecules in intense laser fields and that in SR have a common mechanism, that is, the dissociation of the super-excited molecule, whose energy is higher than the first ionization potential of the molecule. López-Martens et al. have studied the dissociation of NO2 under the intensities between 3 and 20TW cm−2 by femtosecond two-color fluorescence depletion spectroscopy [19,20]. The dependence of the NO (A 2 Σ+ → X 2 Π+ ) fluorescence intensity on the laser intensity suggests that a parent molecule NO2 is excited to an electronically excited state by absorbing three 3.1eV (400nm) photons, and dissociates into NO (A 2 Σ+ ) and O (3 P). Moreover, the authors suggest that this state has a “3sσ 2 Σ+ Rydberg state character” and is located “0.5eV below the first ionization threshold.” So the dissociative state discussed in these two papers cannot be considered as a super-excited state, which we focus on in this paper. The purposes of this paper are: (1) to introduce a new type of fragmentation induced by relatively moderate laser intensity (1012 –1013 W/cm2 ), and (2) to propose a mechanism to explain the neutral dissociation of superexcited molecule after the laser excitation.

6.2 Fluorescence Detections The high power femtosecond laser system consists of a Ti:Sapphire oscillator (Spectra Physics, Maitai), a regenerative amplifier (Spectra Physics, Spitfire) and a two-pass Ti:Sapphire amplifier. The laser beam is focused, using a 100cm focal length lens, into a cell with a CaF2 input window containing gases with a variable pressure of 3–50torrs. The fluorescence emitting

6 Non-Coulomb Explosions of Molecules in Intense Laser Fields

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from the focal volume along the direction perpendicular to the laser propagation direction is collected after it passes through a fused silica window with a diameter of 8cm, and is imaged by two 50.8mm fused-silica lenses onto the entrance slit of an imaging spectrometer (Acton Research Corp., Spectrapro-150). The dispersed fluorescence is detected by a liquid nitrogen cooled CCD camera (Princeton Instruments 770E). The slit is parallel to the laser propagating direction. A broadband reflector (band width 100nm) with the center wavelength of 800nm is placed in front of the slit to block the scattered laser light. The spectral range is from 300 to 1000nm. The spectral resolution is about 0.4nm using a grating of 1200gr/mm (blazed wavelength at 500nm) with 100μm entrance slit width. The detector with 1152 × 298pixels has a dynamic range of 16bits. The fluorescence is collected within 100ns after the laser shines. All the spectra presented here are accumulated for 200shots. The sample methane gas is of ultra high purity (Praxair Inc.).

6.3 Fragment Emission Spectrum For the sake of detecting the neutral fragments generated by femtosecond intense laser at the moderate intensity of 1012 –1013 W/cm2 , it is suitable to use fluorescence spectroscopy, instead of mass spectroscopy. Very recently, Kong and co-workers reported the neutral dissociation of methane molecule in femtosecond intense laser fields [18]. Methane was irradiated by strong femtosecond laser (1013 W/cm2 ). The emitting products were identified as electronically excited CH (A 2 Δ, B 2 Σ, C 2 Σ+ ) and H (n = 3, 4, 5), where n is the principal quantum number (Fig. 6.1a). Three other polyatomic molecules, ethylene (C2 H4 ), n-butane (C4 H10 ) and 1-butene (1-C4 H8 ), were also investigated with the same procedures [18]. Small fragments CH, C2 and H were identified by their fluorescence spectra (Fig. 6.1b, c, d). Emissions of CH (A 2 Δ, B 2 Σ− , C 2 Σ+ → X) radicals showed that the C–C single bonds in these molecules had been cleaved. The Swan bands of C2 (d 3 Πg → a 3 Πu ) were observed also in the fragmentation of ketene derivatives, indicating that the hydrogen atoms had been stripped off from the carbon chain of the molecules. In addition, the emission from H (n = 3) has also been observed. The fluorescence spectra of the different kinds of molecules indicate that the formation of neutral fragments in their electronically excited states is a common process occurring in an intense laser field.

6.4 Electronic Excitation of the Molecules in the Intense Fields In a very strong laser field, it is feasible to excite a molecule to the highlying excited states from the ground state through two possible regimes. The

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Fig. 6.1. The fluorescence spectra of (a) C2 H4 , (b) n-C4 H10 , and (c) 1-C4 H8 respectively. The gases are irradiated by intense laser (810 nm) at the laser intensity of 1 × 1015 W/cm2 . The emission bands at 314.5 nm, 388.9 nm and 431.4 nm are assigned to the 0–0 bands of C 2 Σ+ → X 2 Π, B 2 Σ → X 2 Π, A 2 Δ → X 2 Π transitions of CH radical, respectively. Balmer Hα line at 660 nm is recorded in (c) also. Five Swan bands (d 3 Πg → a 3 Πu ) of C2 are observed. The bands near 437 nm, 473 nm, 516 nm, 563 nm and 610 nm are assigned to the Δv = 2, 1, 0, −1 and −2 progressions, respectively. The pulse duration at the output of the compressor is 42 fs and the central wavelength is 810 nm with a band width of 23 nm (FWHM). The maximum output energy is about 20 mJ per pulse with a repetition rate of 10 Hz


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re-scattering process of an ionized electron may transfer the energy from the laser fields to the parent molecule. A molecule may also be excited by absorbing multiple photons from the laser fields. 6.4.1 Re-scattering of Electrons The re-scattering model, first proposed by Kuchiev [21] and developed by Corkum [9], is employed to explain the highly excitation of electrons in atom. The model has been widely used to interpret the atomic ionization process and its related phenomena, high-harmonic generation (HHG), abovethreshold ionization (ATI) and double ionization (DI) [9–11]. The mechanism consists of two steps. Firstly, when the ponderomotive potential Up = E 2 /4ω 2 of an electron in the laser field exceeds the ionization potential of an atom (ω and E are the angular frequencies of the laser pulse and the electric field amplitude at the time of ionization), tunneling ionization takes place. The ionization rate can be calculated using a tunneling ionization model. In the ionization process, the outmost electron is ejected from an atom. If the laser intensity is ‘moderate’, on the order of 1014 –1015 W/cm2 , the ejecting electron will swing in the alternating laser field. Secondly, after a half period of the laser oscillation, the swinging electron revisits the ionic core if the electron was ejected at a proper moment (phase) of the laser field oscillation. The revisiting electron with a kinetic energy of up to a maximum value of 3.17Up excites or ionizes the atom further, inducing the atoms to generate HHG, ATI or DI. Some groups have attempted to interpret the molecular fragmentation in the intense laser field using this re-scattering model. The mechanism they suggested is that the re-scattering of the electron causes DI of the molecule, followed by CE of the doubly charged parent ion [22–28]. Unfortunately, again, the mechanism cannot explain the generation of neutral products we observed. However, re-scattering may be a possible excitation mechanism, which explains how an alternating laser field stimulates molecules to its superexcited states. Some other groups have considered that the single-electron re-collision event is not sufficient to excite the molecules because the energy transfer through the re-collision is a low efficiency process [18, 29]. Furthermore, we have conducted an experiment to compare the fluorescence yield using both linearly and circularly polarized fs laser pulses in the case of ethylene. It was found that the same fluorescence spectra were observed in both polarization cases. According to the re-scattering theory, circular polarization should reduce the probabilities of HHG, ATI and DI to nearly zero; thus no scattering should occur. Hence, no fluorescence is expected as long as the re-scattering is the only process depositing the energy to molecules. Figure 6.2 shows the laser intensity dependence of the C2 Swan band of ethylene at 3torrs. It clearly shows that the fluorescence still occurs even using circular polarization. This immediately confirms that the re-scattering is not the process responsible for the fluorescence emission.

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Fig. 6.2. Intensity dependence of C2 swan band from ethylene at 3 torr. The circles and the squares represent linear and circular polarizations, respectively

Furthermore, as shown in Fig. 6.2, the fluorescence yield at the circular polarization with respect to that at the linear polarization increases as the intensity increases, the common super-excited states acting as the gateway states embedded in the ionization continuum, and approaches unity at a higher intensity before saturation sets in [30]. This may indicate that the ionization continuum is prepared in both polarization cases. The underlying physics that argues against re-scattering is the following. The re-scattering model begins with the assumption that the electron is first tunnel ionized. In order that the tunnel ionization proceeds, the Keldysh parameter γ = [E0 /2Φ0 ]1/2 should satisfy the condition γ ≤ 1/2 [7], where E0 is the ionization potential of the molecule and Φ0 is the ponderomotive potential of the laser field. With the ionization potential E0 = 12eV of methane, γ = 3.2 ∼ 1 is obtained when the laser field intensity is 1013 –1014 W/cm2 at λ = 800nm. Considering that the laser field intensity is less than or at best around 1014 W/cm2 , the tunnel ionization would not occur. This in turn means that multiphoton absorption into the continuum (γ > 1) is more probable than the re-scattering. 6.4.2 Multiphoton Excitation of Molecules In the multiphoton regime, molecules can be excited to the highly excited electronic states after absorbing several photons within a laser pulse. For


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example, Castleman and co-workers investigated acetone molecules and acetone clusters using a femtosecond pump-probe technique [31–33]. They explained that the molecules were excited by multiphoton absorption. Absorbing two 624nm photons (1.99eV/photon), an acetone monomer can be excited to lower regions of the {S1 , T1 } mixed state, whereas absorbing four photons it can be excited to 3d Rydberg state. The molecules prepared in the excited states (CH3 )2 CO∗ result in a complete dissociation into CH3 and an excited acetyl intermediate, a portion of which further dissociates into CH3 and CO [31]. In another experiment, they found that absorption of three 585nm photons can excite acetone to the 3s Rydberg state [32], and three photons of 390nm to the 3s Rydberg state near the ionization continuum [33]. In an intense laser pulse, many Rydberg states can be in resonance through dynamical Stark shift, which enhances the excitation probabilities [34]. After these excitations, the molecules may undergo dissociation. In the case of methane, we have measured the fluorescence yield of the CH (A–X) band as a function of the laser intensity in a log-log plot (Fig. 6.3). The fluorescence intensity increases with increasing the laser power with a steep slope of 10 ± 1 before it saturates. This implies that methane molecules are excited through the absorption of ten photons into the electronically super-excited states before they dissociate into CH [18].

Fig. 6.3. The laser intensity dependence of the CH (A → X) fluorescence intensity is shown in the insertion of the plot. The rising slope of the curve is 10 ± 1

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6.5 Features of the Fragmentation in Intense Laser Fields 6.5.1 Multiple Dissociation Channels The fluorescence spectra of the different molecular species indicate that the fragmentation proceeds widely at the neutral stage in intense laser fields on the order of 1013 W/cm2 . The variety of the fragment species appearing in the spectra indicates that a number of dissociation channels coexist in the decomposition processes. For example, methane at least has four paralell dissociation channels. The number of dissociation channels can be more, since some fragment species can be prepared in their ground states, from which no visible emission is expected. These fragment species cannot be generated via sequential absorption of two stages because (i) the temporal width (43fs) of the laser pulses, which is as short as the time required for severing one C–H bond [35], greatly lowers the probability of a second photoabsorption taking place after the bond has been broken, and (ii) the absorption bands of the CH(X), CH(A), and CH(B) fragment species do not match the laser wavelength of 800nm and thus cannot absorb the second photon for further fragmentation. 6.5.2 Different from Conventional Photolysis The fragments produced in intense laser fields are completely different from those in weak UV laser photolysis. The conventional UV photolysis usually “cleaves” one chemical bond only, taking place on the ground or low-excited surfaces. Such photolysis is a moderate process, releasing a small amount of energy to the products. The dissociation products are thus in the ground state or low-lying excited states. For example, some details in the dissociation mechanism of the conventional UV photolysis of methane have been studied by quantum chemical calculations [36, 37] and experimental investigations [38–41]. The results are summarized and illustrated in Fig. 6.4. Either the parent molecule of CH4 or the products of CH3 , CH2 and H are prepared in their ground states or in the low-lying excited states. In contrast, the spectra in Fig. 6.1 show that the fragments generated in intense laser fields are also prepared in higher-lying excited states. Some fragments might also be produced in non-emitting dark states in the laser beam, but they could not be identified by the present measurements. 6.5.3 Analogy to Synchrotron Radiation It is interesting to compare the observed fluorescence spectra in the intense fs laser field with those obtained in XUV photolysis. Since the 1990’s, fragmentation processes of many molecules in the XUV region have been investigated using synchrotron radiation (SR) combined with the fluorescence spectroscopy [42]. Similar to the observation of the emission from CH (A, B, C)


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Fig. 6.4. Correlation diagram between CH4 and the possible dissociation products. The dissociation pathways of the low-lying excited states are shown by the linking lines in the lower portion of the diagram. The energy data are the ab initio calculated results [36, 37]. The upper portion of the diagram shows the dissociation pathways of the super-excited states and the fluorescent transitions of the bright fragments initiated by intense laser or XUV excitations

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Fig. 6.5. Fluorescence spectra produced by photoexcitation of CH4 at the excitation of 14.9 eV [43]

Fig. 6.6. 3D-representation of the dispersed visible fluorescence spectra of C2 H4 as a function of synchrotron excitation energy, obtained at 0.5 eV energy intervals, in the 10 – 22 eV energy region [46]


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Table 6.1. Fluorescence of the neutral fragments produced by synchrotron radiation and intense lasers Molecule Synchrotron Radiation Products

Intense Laser Excitation Energy

CH4

CH (A, B → X) 14.5 – 29 eV H (Balmer-α, β, γ, δ)

C2 H4

CH (A, B → X) C2 H∗

N2

N (3s 4 P, 4 So )

C2 H5 OH CH H OH (A → X)

Ref. Products [44]

12.4 – 41.3 eV [45]

6.2 – 27 eV

Ref.

CH (A, B → X) H (Balmer-α)

[54]

CH (A → X) C2 (d → a) H (Balmer-α)

[18]

[55]

N2 (C 3 Πu → B 3 Πg ) [56]

[57]

CH (A, B, C → X) C2 (d → a)

[58]

and H (n = 3) in the intense laser fields (Fig. 6.1a), Lee et al. in 1990 obtained the fluorescence spectra of CH (A 2 Δ, B 2 Σ− → X) emission after exciting methane by the SR source at the photon energies of 14.9eV and 16.2eV (Fig. 6.5). They also observed the Balmer Hα and Hβ lines by exciting at 22.5eV [43]. In 2002, Hatano’s group obtained similar results as Lee et al. [44]. For ethylene molecules, Kameta et al. measured the fluorescence excitation spectra using SR in the photon energy between 12.4eV and 41.3eV. The emission of CH (A → X and B → X) and C2 H was observed (see Fig. 6.6) [45]. Almost the same type of measurements were performed by Alnama et al. [46], who observed, in addition to the CH and C2 H emission, the C2 swan band and Balmer H (3 → 2) [46] emission. Almost the same emission spectrum as recorded in these SR studies has been identified in the emission spectra we observed in the intense laser fields (Fig. 6.1b). Similar comparisons were made for other molecules, as summarized in Table 6.1. The fragment species generated by the XUV irradiation are similar to those generated by intense fs laser excitations with only small differences. The similarity between the SR excitation and multiphoton excitation may not be accidental. In fact, one could use either one XUV photon with appropriate energy or use an intense laser pulse to provide multiple photons to couple the ground electronic state of the molecule with super-excited states embedded in the low lying ionization continuum. Once the super-excited state is reached, the consequence may be the same.

6.6 Neutral Dissociation of Super-Excited Molecules The super-excited state of neutral molecules possesses energy higher than its first ionization potential. It is known that the fragmentation of molecules

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induced by XUV radiation is a neutral dissociation (ND) process from superexcited states [47]. The super-excited molecule AB∗ can undergo dissociation spontaneously through the ND, i.e., AB∗ → A + B, and one or both of the two neutral fragments A and B may be in the electronically excited states, emitting UV or visible lights. For example, by scanning the SR light in the photon energy range 12– 41eV, Kato et al. [48] recorded an excitation spectrum of methane and found five super-excited states, which can undergo either ND or dissociative ionization. The emission spectra of the CH products obtained in the intense laser fields were found to be similar to those of the lowest super-excited (1t2 )−1 state located at around 14.5eV. The possible dissociation pathways of the excited methane molecules around this energy region are shown in the correlation diagram (Fig. 6.4). The diagram is drawn based on the previous experimental observations and the ab initio calculations in the literature [36, 37, 48–50]. An H2 elimination reaction of methane, CH4 → CH2 + H2 , undergoes also via the super-excited CH4 molecule. Some of the CH2 products are in their electronically excited states, and a sequential dissociation reaction, CH2 → CH + H, may take place further. The excited states of CH2 (2 3 B1 ) and CH2 (1 3 B2 ) are correlated to CH (A 2 Δ) + H [48, 49]. The fluorescence from CH (B 2 Σ− ) or CH (C 2 Σ+ ) can thus be observed [50]. It can be concluded that, similar to the case of excitation by XUV photons, molecules such as CH4 are prepared in one or more of the super-excited states in an intense fs laser field, and the superexcited states become the “gateway states” from which further reactions proceed.

6.7 Conclusions Femtosecond intense laser at the moderate intensity of 1012 –1013 W/cm2 is capable of inducing fragmentation of neutral molecules. Some products are neutral species. To detect the neutral fragments, fluorescence spectroscopy instead of mass spectroscopy, are used in the investigation. Methane is irradiated by strong femtosecond Ti-sapphire laser (1013 W/cm2 ). The emitting products have been identified as the electronically excited CH (A 2 Δ, B 2 Σ, C 2 Σ+ ) and H (n = 3), where n is the principal quantum number. The fluorescence spectra of the different kinds of molecules indicate that the fragmentation of neutral polyatomic molecules is a universal phenomenon, existing widely in the intense laser fields. The fragmentation is a thorough multi-channel disintegration of the neutral molecules, yielding small energetic products. For methane molecules, the emissions of CH (A 2 Δ, B 2 Σ− , C 2 Σ+ → X) and H (n = 3, 4, 5 → 2) are found in the XUV single photon excitation also. The spectra, and thus the products, are analogous to those obtained in intense laser fields. The fluores-


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cence intensity increases with increasing the laser power with a steep slope of 10 ± 1. This implies that the methane molecule may absorb ten photons and become a super-excited state before it dissociates into the CH species. The above observation leads us to propose that the two types of excitation, namely by synchrotron radiation and by intense laser pulse, could reach the same super-excited states which decay into almost identical fragments. Acknowledgement. We would like to acknowledge the help of Di Song and Kai Liu for preparing this manuscript. The experiments were carried out by Qi Luo, Huailiang Xu and M. Sharifi (see [18]). This work was supported in part by NSERC, DRDC-Valcartier, Canada Research Chairs, CIPI, FQRNT and China Ministry of Science and Technology, China Natural Science Foundation.

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20. Schmidt, T.W., L´ opez-Martens , R.B., Roberts, G.: J. Chem. Phys. 121 (2004) 4133 21. Kuchiev, M. Yu.: JETP. Lett. 45 (1987) 404 22. Rajgara, F.A., Krishnamurthy, M., Mathur, D.: Phys. Rev. A 68 (2003) 023407 23. Rajgara, F.A., Krishnamurthy, M., Mathur, D.: J. Chem. Phys. 119 (2003) 12224 24. Bhardwaj, V.R., Rayner, D.M., Villenueve, D.M., Corkum, P.B.: Phys. Rev. Lett. 87 (2001) 253003 25. Alnaser, A.S., Voss, S., Tong, X.M., Maharjan, C.M., Ranitovic, P., Ulrich, B., Osipov, T., Shan, Chang, B.Z., Cocke, C.L.: Phys. Rev. Lett. 93 (2004) 113003 26. Tong, X.M., Zhao, Z.X., Lin, C.D.: Phys. Rev. Lett. 91 (2003) 233203 27. Voss, S., Alnaser, A.S., Tong, X.M., Maharjan, C., Ranitovic, P., Ulrich, B., Shan, B., Chang, Z., Lin, C.D., Cocke, C.L.: J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 4239 28. Bhardwaj, V.R., Corkum, P.B., Rayner, D.M.: Phys. Rev. Lett. 93 (2004) 043001 29. Yatsuhashi, T., Nakashima, N.: J. Phys. Chem. A 109 (2005) 9414 30. Ammosov, M.V., Lazaresku, S., Augst, S., Chin, S. L.: Laser Physics. 7 (1997) 706 31. Buzza, S.A., Snyder, E.M., Card, D.A., Folmer, D.E., Castleman, Jr., A.W.: J. Chem. Phys. 105 (1996) 7425 32. Buzza, S.A., Snyder, E.M., Castleman, A.W.: J. Chem. Phys. 104 (1996) 5040 33. Zhang, Q., Poth, L., Castleman, Jr., A.W.: J. Chem. Phys. 110 (1999) 192 34. Levis, R.J., Menkir, G.M., Rabitz, H.: Science 292 (2001) 709 35. Wang, C., Song, D., Liu, Y.Y., Kong, F.N.: Chin. Sci. Bull. 51 (2006) 1269 36. Mebel, A.M., Lin, S.H., Chang, C.H.: J. Chem. Phys. 106 (1997) 2612 37. Mebel, A.M., Lin, S.H.: Chem. Phys. 215 (1997) 329 38. Heck, A.J.R., Zare, R.N., Chandler, D.W.: J. Chem. Phys. 104 (1996) 4019 39. Ye, C., Sato, M., Lee, L.C.: J. Chem. Phys. 89 (1988) 2797 40. Berkowitz, J., Greene, J.P., Cho, H., Ruscic B.: J. Chem. Phys. 86 (1987) 674 41. Welch, A.R., Judge, D.L.: J. Chem. Phys. 57 (1972) 286 42. Whitehead, J.C.: Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 94 (1998) 293 43. Ma, G., Suto, M., Lee, L.C.: J. Quant. Spectros. Radiat. Transfer. 44 (1990) 379 44. Kato, M., Kameta, K., Odagiri, T., Kouchi, N., Hatano, Y.: J. Phys. B 35 (2002) 4383 45. Kameta, K., Seta, T., Watanabe, M., Kouchi, N., Hatano, Y.: J. Elect. Spect. Rela. Phen. 101–103 (1999) 65 46. Alnama, K., Boye , S., Douin, S., Innocenti, F., O’Reilly, J., Roche, A.-L., Shafizadeh, N., Zuinb, L., Gauyacq, D..: Phys. Chem. Chem. Phys. 6 (2006) 2093 47. Hatano, Y.: Bull. Chem. Soc. Jpn. 76 (2003) 853 48. Bohn, B., Stuhl, F.: J. Chem. Phys. 102 (1995) 8842 49. Kassner, C., Stuhl, F., Luo, M., Lehner, M., Fink, R., Jungen, M.: J. Chem. Phys. 105 (1996) 4605 50. Bearda, R.A., van Hemert, M.C., van Dishoeck, E.F.: J. Chem. Phys. 97 (1992) 8240 51. Han, J.C., Ye, C., Suto, M., Lee, L.C.: J. Chem. Phys. 90 (1989) 4000 52. Urban, B., Bondybey, V.E.: Mole. Phys. 100 (2002) 1925

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7 Excitation, Fragmentation and Control of Large Finite Systems: C60 in Moderately Strong Laser Fields Tim Laarmann, Claus Peter Schulz, and Ingolf Volker Hertel Max Born Institute, Berlin-Adlershof, Germany Summary. Recent progress in the understanding of the primary excitation mechanisms of the C60 fullerene in intense laser pulses is reported. By analyzing mass spectra as a function of pulse duration, laser intensity and time delay between pump- and probe pulse insight into fundamental photoinduced processes such as ionization and fragmentation is obtained. Using ultrashort sub-10 fs pulses excitation times are addressed which lie well below the characteristic time scales for electron–electron and electron–phonon coupling. The measured saturation intensities of multiply charged parent ions indicate that for higher charge states the well known C60 giant plasmon resonance is involved in creating ions and a significant amount of large fragments through a non-adiabatic multi-electron dynamics. To enhance the formation of large fragments femtosecond laser pulses tailored with closed-loop, optimal control feedback were used. A characteristic pulse sequence excites oscillations in C60 with large amplitude by coherent heating of nuclear motion. Again, the experimental findings can be understood by a laser-induced multi-electron excitation via the electronically excited resonance followed by efficient coupling to the radial symmetric breathing vibration of C60 .

7.1 Introduction The discovery of fullerenes by Curl, Kroto, and Smalley in the mid 80th of the last century [1] has opened a flourishing field of interdisciplinary research (see e.g., [2] and references therein). An ever increasing wealth of experimental and theoretical studies on structure and dynamics has emerged since a convenient method to produce C60 in macroscopic quantities was established [3] and continues to reveal often surprising properties and facets of the interaction of C60 with photons, electrons, and ions. With its highly symmetric structure, 174 nuclear degrees of freedom, 60 essentially equivalent delocalized π- and 180 structure defining, localized σ-electrons C60 may be considered as a model case of a large finite molecular system. Structural and dynamical studies in the gas phase offer a direct access to isolated C60 molecules free from interaction with a bulk environment. A wide variety of processes has been studied, leading to a detailed understanding of the mechanisms involved in energy deposition, redistribution, ionization, fragmentation and finally cooling of C60 . Experimental and theoretical investigations have covered phenomena such as thermal heating [4], single-photon [5–7] or multi-photon absorption [8], elec-

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tron impact [9], collisions with neutral particles [10], atomic ions, including highly charged ions [11–16] as well as molecular ions [17,18], cluster ions [19], and surface collisions [20–22] – to mention just a few early and some recent examples. These studies have shown that C60 is very resilient and can absorb a surprisingly high amount of energy without complete disintegration on a ns or even μs time scale, e.g., much more than 100eV in intense, pulsed laser fields or fast collision processes. Many of these interesting properties of C60 originate from its special geometric structure, a truncated icosahedron belonging to the Ih symmetry point group. Its 12 pentagons and 20 hexagons make C60 the most stable one of the fullerene family and generally one of the most strongly bound molecules with respect to dissociation. The investigation of photon-induced energetics and dynamics has revealed that C60 shows atomic properties such as above-threshold ionization (ATI), as well as bulk properties such as thermionic electron emission (delayed ionization) [23]. In this sense, photo physical studies of fullerenes using strong laser fields cover the whole range from atomic through molecular to solid state physics. In the present contribution we report on recent progress in the understanding of primary excitation mechanisms in an intense laser pulse. Although the interaction of intense laser radiation (1011 −1017 W cm−2 ) with C60 and polyatomic molecules in general has a history of more than 10 years, many fundamental questions still remain unanswered: How do the electrons, of which quite a few may be excited in a strong laser field, interact in a large molecular system? Which states are most strongly affected and how do they contribute to the overall response of the system? Or more specifically: When does the single active electron (SAE) dynamics which dominates the strong field response of atoms, [24–26] give way to the richer multi active electron (MAE) response [27–30] expected in large finite systems? One specific idea to disentangle the rich dynamics is to use extremely short excitation pulses, which may allow to separate the time scales of energy deposition and rearrangement. In the first experimental Sect. 7.2 of the present progress report we will discuss recent results from exploring this concept with ultrafast sub-10fs pulses for excitation – allowing for an interaction time below the characteristic timescales of electron–electron and electron–phonon coupling [31]. In Sect. 7.3 we focus on another fascinating and presently extremely active field of research, quantum or optimal control of the molecular response using temporally shaped laser pulses. Fundamental questions can be addressed, such as to the role of intermediate electronic excited states in ionization and/or dissociation processes [32, 33]. In particular, the efficient coupling to nuclear degrees of freedom is of high current interest [34, 35]. We show that adaptive closed-loop feedback allows to control the energy deposition process in photoexcited C60 molecules and to obtain detailed insight into ultrafast electronic and nuclear dynamics on the molecular scale [36].

7 Excitation, Fragmentation and Control of C60

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7.2 Ultrafast Excitation of C60 on a Time Scale Below Coupling to Electronic and Nuclear Degrees of Freedom One of the big challenges in ultrafast laser spectroscopy of isolated molecules and clusters is to follow the motion of electrons and atomic particles in real-time during the formation and rearrangement of electronic orbitals, and molecular bonds [37]. In particular, the interaction of larger molecular systems with intense laser fields is a subject of high current interest [33, 38–41]. However, while continuous efforts to decrease laser pulse durations have pushed the limit of time resolution to a few femtoseconds [42] or even to the attosecond regime [43], up to now only very few experiments have exploited these new possibilities for studying photoinduced electronic and nuclear dynamics in the gas phase, and none so far for large molecular systems at high laser intensities (>1013 W cm−2 ). Long term stability and pulse to pulse reproducibility needed for gas phase experiments combined with high intensities are still a major experimental challenge. In this context, the model system C60 is particularly well suited for prototypical studies since one may hope that its photo physics may still be tackled by high quality theoretical approaches. The present state of knowledge has been reviewed in [44] and some important facts and facets are given below. One of the surprising early observations was the delayed ionization of neutral C60 on a μs time scale upon irradiation with ns laser pulses [45]. This has been explained by statistical, thermionic electron emission from vibrationally excited molecules. During photo excitation the strong electron– phonon coupling leads to extensive heating of the nuclear motion accompanied by ionization of the molecule [46, 47]. Recently, it was found that the ionization behavior sensitively depends on the duration of the excitation, i.e., the laser pulse duration Δτ [23]. The spectacular difference observed in the mass spectra when changing the pulse duration from 25fs to 5ps is illustrated in Fig. 7.1. These mass spectra were obtained for nearly equal laser pulse energies (fluences) of about 20J cm−2 , the corresponding intensities being 1×1015 W cm−2 and 3.2×1012 W cm−2 , respectively. A strong contribution of multiply-charged Cq+ 60 ions together with their large fragments (created by sequential C2 losses) is very clearly seen in the 25fs spectrum. However, extremely little fragmentation is detected for singly charged C+ 60 – as illustrated by the enlarged inset – and only a few small fragments if any. In contrast, only singly charged ions and massive fragmentation are observed with 5ps pulses. The large fragment ions in both case are highly vibrationally excited up to an effective temperature of 4000K and undergo metastable fragmentation on a μs–ms time scale after the initial energy deposition has occurred [48]. The corresponding mass peaks are marked by asterisks. Figure 7.1 also shows for 5ps pulses the typical delayed ionization tail on the C+ 60 mass peak which is not present in the 25fs spectrum. On first sight, the large finite molecular system appears to behave as one might intuitively expect: For short pulses of 25fs length one (active) electron

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Fig. 7.1. Typical mass spectra taken from [44] obtained from C60 by ionizing with Ti:Sa laser pulses of 5 ps (top) and 25 fs duration (bottom) at equal laser fluence. The C60 signal on expanded mass scale shown on the right illustrates the high mass resolution allowing for isotope separation as well as – for the 5 ps case – delayed ionization. For details, see the text

is ionized by absorption of many photons and carries most of the absorbed energy. The ionization occurs promptly with no asymmetric broadening of the C60 ion peak due to delayed ionization [23], as can be clearly seen from the insets in Fig. 7.1. The absence of large singly charged fragments indicates again, that only a small amount of the absorbed energy remains in the C+ 60 ion. In contrast, a large amount of energy can be transferred into vibrational heating during a laser pulse of 5ps duration [46]. However, the very different behavior of singly and doubly charged ions remain unexplained in this simplistic picture: Why are multiply charged fragments so dramatically more abundant after ionization with ultrashort laser pulses than singly charged ones – a prominent phenomenon observed for all pulse durations below a few 100fs and for a wide range of intensities? Interesting parallels between ultrafast photo excitation and collisions with fast atomic ions can be found looking at the electronic and nuclear response of C60 (see e.g. [49,50]). Ultrashort pulses as well as fast collisions deposit energy predominantly into the electronic system. Thus, it would be highly desirable to disentangle the mechanisms of exciting the electronic system, redistributing the energy within the electronic degrees of freedom, and coupling to the nuclear backbone, the latter two being thought to occur on time scales 0). This may be due to resonant intermediate excitation steps and the “correct” slopes at 2+ 2+ 3+ much too low intensities observed for C+ 60 → C60 and C60 → C60 could be caused by significant preexcitation to high lying electronic states close to ionization threshold (possibly the Rydberg states reported in our earlier work [56, 57]). Simulations using time-dependent density functional theory (TDDFT) [27] and the many body density matrix formalism [28, 34] predict multi electron excitation at the leading edge of the intense, ultrashort light pulses even at intensities far below the presently observed saturation values. Consequently, many electrons are excited prior to ionization and can possibly support simultaneous multi electron ejection. An additional or alternative interpretation of the observations in Fig. 7.2a arises from noting that the largest slope of N 11 observed in the present experiment (even for higher q) corresponds to a total photon energy of N hν = 17.8eV with the 9fs pulse at 765nm. With this energy one may in principle already excite the first maximum of the σ-part of the C60 giant plasmon resonance [54] to which many electrons collectively contribute (see


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e.g., [58–61]). Figure 7.2b confronts this energetics with the known single photon ion yield. We see that even with N = 8 photons (q = 2) there is already substantial oscillator strengths in the ionization continuum. Thus, one can easily imagine that once the plasmon resonance is sufficiently excited, further excitation due to subsequent absorption of photons by the hot electron-hole plasma is very efficient. The transition thus saturates and no further increase in the slope is observed while at the same time multiple ionization occurs. This might indeed explain (i) the observed slopes of N = 8 (q−1)+ for q = 2 and 11 for q = 3 at low intensities, where the precursor ion C60 is not yet saturated, and (ii) the fact that also for higher charge states the slopes do not increase. Once this region of highly correlated electron excitation is accessed the system is bound to undergo nonadiabatic multi electron dynamics (NMED) as described by Levis and collaborators [62]. Sequential or non sequential nonadiabatic excitation is then driven by the strong laser field which mixes the dense manyfold of electronic states in this energy range. Ionization follows by electron–electron interaction and on a longer time scale energy will be exchanged between the electronic and nuclear system leading to the massive fragmentation observed for higher charge states. However, we have to note that the recent SAE S-Matrix work by Jaro´ nBecker et al. [63] appears to describe certain aspects of multiple C60 ionization in strong laser fields quite well, particularly the behavior at longer wavelengths [55] as well as the overall trend of the ion yield at 800nm as a function of pulse duration [31]. They even showed, that for C+ 60 the saturation intensity decreases significantly for 400nm as observed experimentally – although this may seem somewhat surprising after the above discussion. 7.2.2 Memory Effect in the Photo Ionization Process Finally, we turn to time-resolved mass spectroscopy with the general goal to directly observe fingerprints of multi-electron effects (MAE/NMED) in the initial excitation steps of C60 irradiated with ultrashort 9fs pulses. More specifically, one aims at finding indications for excitation in the electronic system prior to ionization and to probe this by ionizing the system with a second laser pulse. The question is whether the system has such a memory possibly filled through a doorway state and how quickly the energy deposited relaxes. In general, the coupling of excited electrons to atomic motion leads to nuclear rearrangement in the ionic or in the neutral molecular system. According to recent theoretical work [28, 34], this results in characteristic oscillations which one may aim to observe with time-resolved mass spectroscopy. Both, the density of excited electrons and the nuclear geometry are expected to affect the photo ionization yield of C60 in a time-dependent study. For pump-probe spectroscopy, the radiation is divided into two beam paths without lengthening the light pulses. The ultrashort pump pulse with an intensity of 7.9 × 1013 W cm−2 solely deposits the energy into the electronic systems during the interaction. The energy redistribution within the

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electronic and nuclear degrees of freedom is then probed by a delayed, slightly less intense probe pulse (6.8 × 1013 W cm−2 ). Figure 7.3a shows the measured time dependence of the normalized C+ 60 ion signal. Since both pulses, pump and probe, act individually on the molecule and already cause some ionization at these intensities, the sum of single pulse signals has been subtracted from the pump-probe ion yield. The limitations of this procedure due to the highly nonlinear excitation steps involved are obvious, however, the graph gives first insight into the time evolution of the photoinduced processes. Particularly, the comparison with the simultaneously measured Xe+ signal included in Fig. 7.3a is instructive because it constitutes a genuine direct MPI process with probably only one active electron determining the systems response, and, thus, can be taken as an auto-correlation measurement. The C+ 60 ion signal is clearly broadened at the bottom of the spectrum. As shown in Fig. 7.3b, the deconvolution of the total ion yield results in

Fig. 7.3. (a) C+ 60 ion yield (open triangles) as a function of the time delay between pump (7.9 × 1013 W cm−2 ) and probe pulse (6.8 × 1013 W cm−2 ), normalized to the maximum signal. t = 0 is defined by the auto-correlation function (ACF, dotted line) derived from a fit to the simultaneously measured Xe+ signal (closed circles). (b) Contributions from direct SAE/MPI (dark gray shaded) and MAE/NMED (light gray shaded ) refer to the tentative deconvolution of the C+ 60 photo ion yield [31], for details see the text


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two main contributions: direct MPI of C60 from the neutral ground state to the continuums state (dark gray shaded), which essentially follows the Xe auto-correlation plus a significant contribution exhibiting dynamics on a sub100fs time scale (light gray shaded) which is slightly shifted towards positive time delays, when the stronger pump pulse leads the weaker probe pulse. This deviation of the C60 ion pump-probe signal from the auto-correlation function is interpreted as a clear indication of multi-electron excitation in a sub-ensemble of C60 during the laser interaction [31]. Supported by recent theoretical work [27, 28, 34], this is tentatively attributed to a finite probability to initially excite two or more electrons into the t1g resonant state, which in turn acts as a doorway (bottleneck) to ionization. The observed dynamics (τ = 43 ± 4fs) is comparable to the characteristic time for thermalization within the electronic system due to electron–electron scattering as previously concluded from single pulse experiments [23, 46]. As intuitively expected the density of the hot electron cloud depends on the laser intensity, and its time evolution on the electron–electron scattering time constant [28]. The excited electron density in the doorway state determines the transition probability into the ionic continuum. Since pump and probe pulse have slightly different intensities (7.9:6.8) the ion distribution due to doorway state excitation is slightly shifted to positive time-delays, as shown in Fig. 7.3b. Based on a rough fit with two response functions in each direction for the undelayed, direct SAE/MPI process (proportional to the ACF signal) and the MAE/NMED with its memory effect (taken as exponential decay) an estimate of 65% and 35% for the contribution of SAE and NMED processes to the signal is obtained, respectively. In this study with ultrashort but non-resonant excitation at 765nm, a direct indication of an oscillation on the sub-100fs time scale postulated in theoretical work (e.g. [28, 34]) is not observed. We believe that this is due to efficient coupling among different vibrational modes which for reasons of computational simplicity is not accounted for in the theoretical treatments. However, in the following section we present experimental data where temporally shaped, fs laser pulses were used to excite oscillations in C60 with large amplitude. Complementary twocolor pump-probe data and TDDFT calculations give direct information on the laser-induced multi electron excitation via the t1g resonance.

7.3 Ultrafast Control of Energy Dissipation Processes in C60 Using Optimally Tailored Femtosecond Laser Pulses The previous section has focused on the analysis of photo physical processes in C60 by comparing mass spectra as a function of characteristic laser parameters, such as intensity, pulse duration, photon energy or pump-probe delay. In the following, recent results are presented that go one step further,

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such as controlling the molecular response with judiciously tailored fs laser pulses [36]. The control of photo physical processes with temporally shaped laser pulses is a cutting edge topic in modern laser science and might pave the way to optically controlled organic chemistry [64]. Many pulse shaping experiments have been performed since the concept of closed-loop, adaptive feedback control was proposed [65]. Making use of self-learning optimization algorithms, the method enables one to determine the optimum pulse shape driving the system towards a specific, pre-defined target state. Thus, it became possible to obtain control over very different photoinduced processes such as selective fragmentation [66], bond dissociation and rearrangement [41], laser-induced fluorescence [67], stimulated Raman emission [68], high harmonic generation [69] or even energy transfer in biological photosynthetic antenna complexes [70]. The motivation behind this powerful method is twofold: (i) to maximize a specific, selected reaction and (ii) to learn about the thus induced (optimized) process itself. The latter is usually very difficult to achieve for large finite systems with many electronic and nuclear degrees of freedom [70], in particular so, when strong field laser pulses are used which – on the other hand – may be very efficient in reaching the former purpose [71]. In the present progress report we discuss selective enhancement of C2 evaporation, a typical energy loss channel of vibrationally excited fullerenes. Combining the results of such an optimal control experiment with complementary two-color pump-probe experiments and time-dependent density functional theory (TDDFT) allows to pinpoint the “microscopic” process discussed in detail in the following subsections. 7.3.1 Coherent Heating of Nuclear Motion A typical learning curve for maximizing the C+ 50 fragment ion yield is plotted in Fig. 7.4a. The resulting optimal pulse shape for this pre-defined target is characterized by cross-correlation frequency-resolved optical gating (SHXFROG) shown in the inset. Finding the optimal pulse shape is done in analogy to similar optimization processes occurring in nature by a feedback learning-loop. From the different approaches possible [72] an evolutionary strategy is used in this work by which robust optimal solutions with good convergence were obtained. One starts with 20 randomly chosen phase masks (1st generation). The response of the molecular system is then evaluated for each mask by recording a mass spectrum averaged over 1000 laser pulses. The two pulse shapes with the best fitness value, e.g., the C+ 50 ion counts, are carried over to the next generation without changes. The remaining 18 members of the new generation are created by crossover events, where the phase pattern from two randomly chosen masks are partially exchanged. In a second step the newly created members are modified in a mutation event, where with a 20% probability the pixels of the phase mask are replaced by a random value. Convergence is achieved after typically 30–40


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Fig. 7.4. (a) C+ 50 signal as a function of generation of the evolutionary algorithm. The inset shows the XFROG trace of the optimal solution. On the right, mass spectra recorded with optimal (b), stretched to 340 fs (c), and unshaped pulses of + 31 fs FWHM are plotted. The derived ratios R = C+ 50 /C60 are given

generations. During the learning process the C+ 50 ion signal increased by a factor of 2.4 compared to the signal recorded with unshaped pulses (0th generation). The C+ 50 ion abundance was chosen as optimization target because it is a measure for the temperature of the nuclear backbone. It is well-known that cooling of highly vibrationally excited C60 occurs mainly through sequential evaporation of C2 units in a statistical process [44]. Thus, a strong correlation of the optimization curve with those of the neighbor+ ing fragments C+ 48 and C52 is observed (not shown here). It is important to note that this control scheme does not optimize selective bond breaking. Rather, it maximizes the process of energy flow into vibrational modes of C60 . For comparison the C+ 50 were measured with a stretched pulse of the same overall energy and duration (Fig. 7.4c) as the optimal pulse (Fig. 7.4b). As a matter of fact the signal for the optimal pulse was more than two + times stronger. The ratios R = C+ 50 /C60 even increased by factors of 7 and 32 compared to those of the stretched and the unshaped pulse, respectively. These observations thus go far beyond the common wisdom according to which the response of fullerenes to intense laser fields is found to be determined by the pulse duration as recently reviewed [44]: the XFROG trace in Fig. 7.4a clearly shows that a sequence of pulses is best suited for most efficient energy coupling into nuclear motion. The wave packet generated by this pulse sequence prevails apparently for at least 6 cycles, surprisingly long considering the large number of electronic and nuclear degrees of freedom into which energy can finally flow. Thus, the process may be called coherent heating of nuclear motion due to the strong specifically shaped pulsed field.

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7.3.2 Multi Electron Excitation of Giant Breathing Motion To corroborate these findings and to glean further insight into the underlying many body dynamics complementary 2-color pump-probe experiments were performed with a relatively weak frequency-doubled 400nm pump pulse, resonant with the dipole-allowed HOMO(hu ) → LUMO + 1(t1g ) transition, assumed to mimic essentially the excitation effect of the leading edge of the strong, shaped 800nm pulse in the previous experiment. The dynamics of the energy redistribution is then probed by a time-delayed 800nm probe pulse. Fig. 7.5a shows the metastable C3+ 48 ion signal as a function of the time-delay between 400nm pump (1.7 × 1013 W cm−2 ) and 800nm probepulse (7.3 × 1013 W cm−2 ) with pulse durations of 25fs and 27fs, respectively. Triply charged, large fragments are most abundant using fs pulses as visible in Fig. 7.4d and metastable fragmentation (on a μs-ms timescale) is a particularly sensitive probe of the temperature of Cq+ 60 generated in the initial photo absorption process [44]. At negative time delay, when the red pulse leads, almost no signal from C3+ 48 is observed. Once pump and probe pulse overlap, the ion yield increases strongly and a maximum fragment signal is found at a delay of 50fs. It can be inferred from this observation that the resonant pre-excitation of the LUMO + 1(t1g ) state by the blue laser pulse significantly enhances multiple ionization and massive fragmentation both induced by the subsequent red pulse. The t1g state has already been identified as the doorway state for the population of Rydberg states in C60 [56] and it seems to act generally as a bottleneck for photo physical energy deposition into C60 . Closer inspection of the pump-probe transient reveals a weak modulation on top of the C3+ 48 ion signal. By fitting the data to the laser cross correlation function convoluted with a single exponential function decay function and subtracting this from the measured signal, this modulation – albeit small – is found in all pump-probe data from multiply charged large fragments with a periodicity of 80 ± 6fs. This is shown in Fig. 7.5b. Obviously, nuclear rearrangement upon electronic excitation via the t1g state occurs. The oscillation is then probed by the 800nm probe pulse, assuming (by virtue of FranckCondon arguments) that the absorption cross section for further energy deposition depends on the C60 oscillation. The comparison of the pump-probe modulation with results from pulseshaping experiments (Fig. 7.4a) gives evidence that these different spectroscopic techniques probe very similar dynamics. Figures 7.5c and d show for comparison the optimal temporal shapes for excitation with 220μJ and 280μJ pulses, respectively, derived by projecting the XFROG-traces onto the time-axis. The key findings reproduced in several optimization runs are as follows: (i) Each pulse shape consists of two distinct regimes with periodicity T1 and T2 . (ii) The periodicity is smaller on the leading edge of the pulse than on the trailing edge (T1 < T2 ). (iii) It increases with increasing pulse energy. The observed values range from T1 = 84 ± 9 fs at 220μJ up to T2 = 127 ± 12fs at


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Fig. 7.5. (a) Metastable C3+ 48 ion signal as a function of time delay between blue pump and red probe pulse. An exponential decay is fitted to the data. (b) A modulation is found for all large fragments C3+ 60−2m by subtracting the fits from the measured transients. (c) Optimized temporal shape with 220 μJ laser pulses and (d) 280μJ pulses (for details, see the text)

280μJ. All observed times including the pump-probe result are much larger than the well known radially symmetric breathing mode ag (1) of neutral C60 molecule, which has an experimentally determined period of 67fs [73]. On the other hand, the observed periods are in general shorter than the lowest prolate–oblate mode hg (1) (122fs [73]) recently suggested as the dominantly excited mode of C60 due to the strong laser-induced dipole forces acting in intense 1500nm pulses [55]. To glean information on the nuclear motion excited in strong 400nm and 800nm laser fields calculations were performed using the so-called nonadiabatic quantum molecular dynamics (NA-QMD) [74], developed recently. In this approach, electronic and vibrational degrees of freedom are treated simultaneously and self-consistently by combining time-dependent density functional theory (TDDFT) in basis expansion with classical molecular dynamics. The NA-QMD theory has already been successfully applied to excitation and fragmentation mechanisms in ion-fullerene collisions [49] and laserinduced molecular dynamics [75–77]. We note that the calculations are limited due to the computational effort by using the frozen core approximation and only a minimal basis set (i.e., 2s, 2px , 2py , 2pz orbitals) and, thus, describing the ionization mechanism not very realistically. However, in principle it

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is possible to include the full ionization process into NA-QMD, demanding however many more basis functions and a reliable absorber potential [78]. In Fig. 7.6 results for exciting C60 by an intense laser field (λ = 370 nm, τ = 27fs) at different intensities are presented. The wavelength of 370nm is close to the experimentally used 400nm pump pulse and matches the first optical resonance calculated in the LDA approximation. The calculations shown in Fig. 7.6a predict an efficient excitation of many electrons by the laser field. At the highest laser intensity (5 × 1013 W cm−2 ) nearly 31 valence electrons are strongly excited resulting in an impulsive force that expands the molecule dramatically up to 9.4 ˚ A which is 130% of the C60 diameter, orders of magnitude larger than expected for any standard harmonic oscillation. At high laser intensities most of the kinetic energy of the nuclei is stored in the radially symmetric breathing mode ag (1) in contrast to the rather small contribution of the other vibrational modes as shown in Fig. 7.6b in agreement with other theoretical work [34]. The new equilibrium position as well as the oscillation period of the ag (1) depend on the excited electronic configuration, and, thus, on the absorbed energy. Fig. 7.6a shows a strong increase of the oscillation period with increasing absorbed energy as observed experimentally. The calculated oscillation period of highly excited C60 is in good agreement with the results of the pump-probe experiment in Fig. 7.5b as well as with the first time regime of the optimally shaped laser pulse given in Fig. 7.5c, d. The longer period seen experimentally in the second time regime are not

Fig. 7.6. (a) Period of the ag (1) breathing mode (circles) and number of excited electrons (diamonds) as a function of deposited energy derived from NA-QMD simulations (laser parameters: λ = 370 nm, τ = 27 fs). (b) Vibrational energy of normal modes after laser excitation (I = 3.3× 1013 W cm−2 ). The kinetic energy is mainly stored in the ag (1) mode


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reproduced by the theory, possibly due to the lack of absorbing boundary conditions [78], since the breathing mode period of ionized C60 is increased in comparison with the neutral molecule. This is shown in Fig. 7.7 where the oscillation period of the ag (1) breathing mode in the neutral species as

Fig. 7.7. Period of the ag (1) breathing mode as a function of the number of excited electrons (full circles) derived from NA-QMD simulations by Schmidt et al. [private communication] (laser parameters: λ = 370 nm, τ = 27 fs, I = 3.4 × 1012 – 5 × 1013 W cm−2 ) and as a function of charge state (open circles) derived from hybrid B3LYP level of the DFT method by Kono et al. [79, private communication]

Fig. 7.8. (a) Absorbed energy (circles) and number of excited electrons (squares) of C60 after excitation with pump (λ = 370 nm, I = 3.3 × 1013 W cm−2 , τ = 27 fs) and probe pulses (λ = 800 nm, I = 7.3 × 1013 W cm−2 , τ = 27 fs) as a function of time delay. The horizontal line indicates the absorbed energy after the pump pulse alone (292 eV). (b) C60 radius at the maximum of the probe pulse as a function of delay time

144


a function of excited electrons calculated by Schmidt et al. [private communication] is compared to the period of this mode in different charge states q of Cq+ 60 which Kono et al. [79, private communcation] recently derived. The pump-probe experiment has also been simulated by calculating the total absorbed energy and the number of excited electrons as well as the radius of C60 oscillation in this giant breathing mode as a function of the pump-probe delay time of the laser pulses. Figure 7.8 shows a significant correlation between the actual radius, e.g., the ag (1) mode of the excited C60 with T = 75 fs, and the energy deposited after the probe pulse which is the key for understanding the experimental observation.

7.4 Conclusion and Outlook In this progress report we have covered some important aspects of the present state-of-the-art of research on the ultrafast laser interaction with C60 – an ideal model of a large finite system with many active electrons and vibrational degrees of freedom. The comparison of experimental results using laser pulses as short as 9fs for single-pulse and time-resolved mass spectroscopy with recent theoretical work gives a strong indication that complex non-adiabatic multi electron dynamics (NMED) plays a key role for the understanding of the molecular response to short-pulse laser radiation. We are, nevertheless, still far from fully understanding the intricacies of intense field interaction with such a relatively complex system. Rigorous theoretical efforts are needed to quantitatively explain the key aspects of the experimental observations presented here and those to emerge in the near future. On the other hand, further experimental work is desirable – preferably with even shorter pulses (and better tunability of the fs light sources) to perform sophisticated and direct multi-color pump-probe experiments. Experimental detection schemes need to become more sophisticated, e.g., dynamical imaging promises a whole new multi dimensional view into the dynamics discussed here. Electron and ion coincidence techniques will allow characterize the role of intermediate resonances, charge states and collective plasmon excitation upon energy deposition, as well as to follow fragmentation cascades directly. One other fascinating experimental aspect, which has been highlighted already in this progress report, will be surely further combining analysis and control of photo physical processes by means of fs laser pulses tailored with closed-loop, optimal control feedback. We presented work, where this technique was used to study specific aspects of C60 photo physics, namely efficient coupling to nuclear motion. As an example, a characteristic pulse sequence was found to excite large-amplitude oscillations by coherent heating of nuclear motion. Complementary 2-color pump-probe studies allowed to identify the t1g state to play the key role in the energy deposition process. With the help of TDDFT calculations one has been able to connect the experimentally observed periods and the calculated, laser-induced giant vibrational motion.


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Last but not least, the future is bright at short wavelength! New laser schemes for soft- and hard X-rays both, laboratory-based and large scale facilities, are being proposed and are presently further developed. High-harmonic generation, table-top plasma sources and Free-Electron Lasers are expected to open completely new horizons for fs studies with photons capable to excite and ionize matter directly. Acknowledgement. We wish to thank the groups of R¨ udiger Schmidt (Dresden, Germany) and Hirohiko Kono (Sendai, Japan) for creative collaboration and many stimulating discussions, as well as for their permission to communicate and compare their original, in this form yet unpublished data shown in Fig. 7.7. Financial support from the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 450 TP A2 is gratefully acknowledged.

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8 Theoretical Description of Rare-Gas Clusters Under Intense Laser Pulses Ionut¸ Georgescu1 , Ulf Saalmann1 , Christian Siedschlag2 , and Jan M. Rost1 1

2

Max Planck Institute for the Physics of Complex Systems · N¨ othnitzer Str. 38, 01187 Dresden, Germany e-mail: [email protected], [email protected] FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam · The Netherlands

Summary. We discuss the ionization and explosion dynamics of rare-gas clusters under intense femtosecond laser pulses. On one hand, we show how a microscopic treatment by means of a mixed quantum/classical approach provides detailed and time-resolved insight into the mechanisms of excitation and ionization of the irradiated clusters. Furthermore, we compare the cluster response to standard 780 nm light pulses with the response to 100 nm pulses, already obtained at an vuv-fel source, and with 3 nm light, which will be available from future xfel sources. On the other hand, we present a simple analytical model which idealizes the single-cluster dynamics but considers the real experimental scenario, i.e., a laser beam profile and a cluster size distribution. Only thereby one can achieve agreement with experimental data on a quantitative level as demonstrated for the kinetic energy distribution of the ionic fragments.

8.1 Introduction Clusters absorb light very efficiently which becomes clear from the phenomena observed, like the production of highly charged ions [1, 2], energetic electrons and ions [3–5], X-rays [6, 7], or neutrons [8–11]. A more exhaustive list of references, also to theoretical work, can be found in a recent review [12]. The efficient energy absorption is rooted in the nature of clusters: between the condensed and the gas phase, a cluster is much denser than a gas, thus absorbing more energy than isolated particles. Yet, it does not have as many dissipation channels, e.g., lattice vibrations, as a solid. A number of experiments have additionally revealed that an optimum pulse length Tcrit exists for maximum energy absorption, where Tcrit changes substantially under different conditions, such as the kind of cluster atoms, the size of the cluster and the peak intensity of the laser pulse [13–18]. Here, we will give a brief review over energy absorption mechanisms with particular emphasis on the conditions for optimum absorption. Under this aspect we will also briefly discuss the first results at 100nm wavelength and set the stage for light with even another order of magnitude shorter wavelength, namely 3nm. These new parameter regimes of light-matter interaction will become accessible with the advent of intense vuv and xuv pulses from free-

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I. Georgescu, U. Saalmann, C. Siedschlag, J. M. Rost

electron laser sources [19]. To quantify the different regimes we present in Table 8.1 the response of free electrons to lasers of various wavelengths λ and intensities I in terms of the quiver amplitude (i.e. the excursion length) √ xquiv := Iλ2 /c2 and the ponderomotive energy (i.e.the mean kinetic energy) Epond := Iλ2 /4c2 . We will later refer to the values shown in this table. Common to all clusters subjected to strong laser pulses of different wavelength is the three-step scenario sketched in Fig. 8.1. In a first phase (termed I), the light couples to the atoms as if they were isolated, the cluster environment does not have an effect. In phase II, the critical and interesting phase, the cluster expands due to the ions created in phase I leading to a decreasing density of ions. On the other hand, the density of so called “quasi-free” electrons in the cluster does not necessarily decrease. These electrons are said to be “inner-ionized” but not yet “outer-ionized”, i.e., they are still bound to the cluster but no longer to a specific ion. The net change of Table 8.1. Ponderomotive energies Epond and quiver amplitudes xquiv for long and short laser wavelengths λ at different intensities I intensity I wavelength

1014 W/cm2

1016 W/cm2

1018 W/cm2

λ = 780 nm

Epond xquiv

5.67 eV 8.28˚ A

567 eV 82.8 ˚ A

56.7 keV 828 ˚ A

λ = 100 nm

Epond xquiv

93 meV 0.136 ˚ A

9.3 eV 1.36 ˚ A

932 eV 13.6 ˚ A

λ = 3.5 nm

Epond xquiv

0.1 meV 0.0002 ˚ A

0.01 eV 0.002 ˚ A

1.1 eV 0.02 ˚ A

Fig. 8.1. Typical cluster dynamics under a strong laser pulse in terms of the time dependent cluster radius R(t). Atomic ionization (phase I), critical expansion (II) and relaxation (III), see text. The laser pulse is also indicated

8 Clusters under Intense Laser Pulses

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their density depends on the balance of inner and outer ionization at each instant of time. Finally, during phase III energy is redistributed within the cluster, e.g., through recombination. The cluster completely disintegrates and the final (measurable) distribution of ions and electrons is built up. This relatively simple yet quite universal scheme facilitates the understanding and assessment of the very different mechanisms of energy absorption we will discuss. Concerning the theoretical description we will restrict ourselves to the approach we have followed, namely a classical molecular dynamics simulation of the cluster explosion where the coupling of the bound electrons to the laser light and/or existing electric fields is described by quantum rates. This is to date the only approach which permits to reach relatively large clusters without additional approximations and has been developed by a number of authors [20–31].

8.2 Quasi-classical Microscopic Description The key idea [22] which has proven to provide physical insight and numerical efficiency is the division of the ionization process into inner and outer ionization3 . Here, inner ionization means energy absorption of bound electrons resulting in so-called quasi-free electrons. They are not bound anymore to a particular atom but still to the cluster as a whole, which can provide a sufficiently strong space charge to hold the electrons back. Eventually, quasi-free electrons may be further heated until they are ejected into the continuum, which we call outer ionization. The dynamics of bound electrons with typical oscillation periods of a few attoseconds are not treated explicitly. Rather one uses a statistical approach to describe it by means of the occupation number of bound levels which may change after each time step. The probability for a particular transition within a time step is calculated as the product of the corresponding rate Γ and the time step Δt. This probability p = Γ · Δt is compared to a random number ξ distributed uniformly in the interval [0,1]. A transition takes place if p > ξ. The rates Γ may crucially depend on the laser (intensity and frequency) and the current state of ion, see the following sections 8.2.1 to 8.2.3. For clusters, the Coulomb field of the neighboring ions and electrons has to be taken into account for a proper description of the inner-ionization process. An inner-ionization event “gives birth” to a quasi-free electron, which is subsequently propagated classically along with the ions and other quasifree electrons with all mutual Coulomb forces included. This propagation 3

In a strict sense this is only possible in rare-gas clusters. However, the delocalized valence electrons of metallic clusters should be of minor importance for the creation of the high charge states observed, since these electrons are emitted early in the pulse.

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accounts for electron–electron and electron–ion scattering which is important because of the high particle density in the cluster volume. Furthermore, for infrared/visible and vuv frequencies the laser may additionally heat these quasi-free electrons. 8.2.1 Inner Ionization in Low-Frequency Fields In low-frequency laser pulses, the inner ionization occurs from top to bottom, i.e., the most weakly bound electron is ionized with the highest probability. For sufficiently high fields the ionization of an electron with the binding energy Eip to an ion of charge q is due to barrier suppression (above-thebarrier ionization). For weaker fields the electron can still leave the ion by tunneling through the barrier. The tunneling probability may be obtained [24] from the adk formula [32]. Since the formula is derived for a homogenous electric field one should be careful for the case of clusters where additional contributions to the electric field from the other particles (electrons and ions) may be important. Therefore, it has been proposed [26] to calculate directly the tunnel integral [33] 1 s=

dτ

V (r τ ) − Eip ,

rτ = R + τ X

(8.1)

0

with the electric total field at the ionic position R pointing in the direction X, with |X| = 1. The potential V in (8.1) is composed of the laser and all the other particles not just the ion under consideration. The tunnel probability is finally P = exp(−2s). 8.2.2 Inner Ionization in VUV Fields Atomic ionization by 100-nm lasers is perturbative below a laser intensity of ∼1016 W/cm2 (see Table 8.1). Hence, calculating the atomic photoabsorption rates [34] for the respective outermost electron of each cluster atom forms the starting point for inner ionization. With the frequency of 12.7eV used in the first fel experiments at desy [35] one can only singly ionize an isolated xenon atom. In a cluster environment neighboring ions lower the effective threshold for inner ionization given by the effective binding energy Eeff = Ebar − Eb , where Ebar is the energy of the closest barrier to the atom or ion out of which the electron is to be ionized, and Eb the energy of the bound electron (taken to be the binding energy of an isolated atom/ion plus the additional potential energy due to the laser field and the surrounding charges). Whenever a photon is absorbed and ω > |Eeff |, the outermost electron of the ion is ionized and henceforth treated as a classical particle. This process is, in principle, repeated until all electrons are inner-ionized. In practice, however, it turns out that in almost all cases only the 5s and 5p electrons of xenon are ionized.


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8.2.3 Intra-Atomic Processes in High-Frequency Fields For high-frequency laser impact, the laser-atom interaction is of non-relativistic and perturbative nature, as in the previous case, yet the light acts fundamentally differently on the bound electrons: Ionization starts from the inside because photo-ionization cross sections at X-ray wavelengths are considerably higher for the inner shells than for the valence shells [36]. In firstorder perturbation theory cross sections scale as σ ∝ Eb 7/2 for ω > Eb . Typically, the inverse rates are 1 . . . 10 fs and thus much smaller than the pulse length of about 100 fs. Hence, multiple single-photon ionization is possible, in particular since the inner-shell holes created by photo-ionization are refilled by Auger-like processes. The Auger decay is only weakly dependent on the atomic charge state4 and occurs fast, typically in 0.2 . . . 5 fs [37]. Due to this almost instantaneous refilling of the inner shells they can be ionized many times during the pulse and the atoms can be “pumped dry” efficiently. The ionization occurs “inside-out” exactly opposite to the mechanism in the visible and vuv-wavelengths regime where the most weakly bound electrons are removed first. The ionization cascade may stop for highly charged ions where the increasing binding energy of the remaining electrons may prevent both, photo- and auto-ionization, for energetical reasons. Non-dipole effects in the interaction with single atoms/ions do not have any crucial influence apart from distortions of the angular distribution of the photoelectrons [38]. For the interaction with the clusters they are negligible because of the vanishing impact of high-frequency light on quasi-free electrons. 8.2.4 Outer Ionization The numerical (classical) propagation of the inner-ionized electrons is straight forward apart from two aspects: instability of classical particles and an unfortunate scaling with the particle number. To circumvent the first problem one may introduce a smoothed Coulomb interaction [21] qi qj Wij (r i , r j ) := rij 2 + α

(8.2)

for two particles with charge qi and qj separated by the distance rij 2 = (ri − rj )2 . The smoothing parameter α “regularizes” the Coulomb potential and prevents the collapse of ions and electrons. The same effect can be obtained by a short-range repulsive part which additionally accounts for elastic scattering [22]. Furthermore, (8.2) simplifies the numerical integration by avoiding strong gradients for close collisions. 4

For the idealized case of hydrogenic wave functions it can be shown that the matrix element for Auger decay according to Fermi’s golden rule is completely independent of the nuclear charge.

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Due to the long-range nature of the Coulomb interaction the calculation of the forces on all N particles of the systems scales as N 2 . In order to handle clusters with more than ∼103 atoms (including the electrons N ∼ 104 ) one is forced to use special algorithms which take advantage of the longrange interactions between a large number N of particles, e.g., hierarchical tree codes [39]. Originally developed for gravitational N -body problems in cosmology [40], such hierarchical tree codes allow one to follow the dynamics of all charged particles over a few hundred femtoseconds with typical time steps of attoseconds [27, 31]. In the long-wavelength and the vuv regime the laser is coupled to the electrons as a classical field. The quasi-free electrons are driven over distances of the respective quiver amplitudes (see Table 8.1), or experience substantial inverse bremsstrahlung (ibs) heating due to repeated forced collisions with the ions. In the case of high-frequency radiation, the field is oscillating so fast that an electron cannot gain substantial velocity and one can completely neglect the laser field in the classical equations of motion. Note, that despite the short wavelength the dipole approximation remains valid, since the quiver amplitude is much larger than the wavelength, even for high intensities (see Table 8.1). Furthermore, Compton scattering effects are not taken into account [28] due to their small cross section.

8.3 Dynamical Phenomena and Their Dependence on Laser Wavelength and Cluster Size 8.3.1 Cooperative Behavior in Small Clusters (λ = 780 nm) Simulation of the charging in small rare-gas clusters along the lines described in the last section have revealed the existence of an optimum pulse length for maximum charging of the cluster. This optimum pulse length was traced back to an optimum mutual ionic separation Rei , where “ei” stands for enhanced ionization (ei). First discovered for diatomic molecules [41, 42] it applies for small clusters in full analogy [26,43]. It is characterized by an independence of Rei from the laser frequency (as long as its period is adiabatically slow compared to the orbital times of the electrons). A slightly different expansion speed of the cluster for different laser frequencies leads to a small frequency dependence of the experimentally accessible optimum laser pulse length although the mechanism of ei is operative. In contrast to diatomic molecules ei occurs in clusters also for circularly polarized light [26]. This is easy to understand, since in a (spherical) cluster the rotating polarization vector always finds two ions in a line, which is required for ei. The “simplest” way to proof ei experimentally would be to use a laser with a frequency which is higher than the plasmon frequency of the unperturbed cluster. Then resonant absorption (see next section) cannot occur and ei could be clearly attributed to the mechanism of ei.


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ei is by no ways a collective behavior, it is rather the cooperative effect of two ions in line (there are a few of these pairs in a cluster) with the instant polarization vector which helps to outer-ionize the electron. Clearly, it does not work if there is another shell of ions beyond the outer ion which prevent outer ionization. Hence, ei is limited to small clusters. However, the number of atoms of the cluster is not the only limiting factor. Another one is the number of quasi-free electrons generated during the pulse, i.e., those electrons, which are inner-ionized, but not immediately outer ionized. These quasi-free electrons may absorb (possibly collectively) energy from the laser pulse, a mechanism which we will discuss next. 8.3.2 Collective Behavior in Medium Sized Clusters (λ = 780 nm) The collective behavior emerges from the possibility to match an internal frequency of the cluster, namely that of the center-of-mass (cm) motion of the quasi-free electrons Ω, with the external driving frequency ω of the laser. In general, considering typical electron densities in the cluster, this frequency is much too high. However, when the cluster expands, the density decreases, and in turn also the eigenfrequency of the quasi-free electrons, (8.3) Ω(Q, R) = Q/R3 . The usual picture, from which this quantitative relation is deduced, starts with two spheres of constant but opposite charge density which are shifted with respect to each other along a line out of the force free equilibrium through the external force (in our case the dipole coupling to the laser field). As a consequence a restoring harmonic force is generated whose force constant Ω provides the eigenfrequency of the collectively excited system of electrons and ions. This excitation is identical to a surface plasma excitation5 of a confined electron plasma. Since the density is a function of the cluster radius, the frequency is also directly a function of the cluster radius, and the resonance condition Ω(Q, Rra ) = ω

(8.4)

leads, as in the case of enhanced ionization, to a critical cluster radius Rra , where “ra” stands for resonant absorption. However, there are striking differences. Quantitatively, the relation R0 < Rei < Rra holds, i.e., the radius for enhanced ionization Rei is smaller than that for resonance absorption Rra but larger than the equilibrium radius R0 . This has been revealed clearly in [45] where both mechanisms, enhanced ionization and resonance absorption, could 5

The eigenfrequency can be written in terms of the charge density ρion (assumed to /3. For a neutral system with ρion = ρquasi = ρ be homogeneous) as Ω = 4πρion √ it reads Ω = 4πρ/3 = ωpl / 3 with ωpl the plasma frequency. This is the classical surface plasmon frequency of a spherical cluster [44].

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be identified in a single cluster expansion. Secondly, collective excitation is strongly frequency dependent in contrast to enhanced ionization. Resonance absorption (ra) occurs also in metal clusters for the valence electrons. This requires of course laser fields which must not be so strong that the loosely bound valence electrons are lost immediately by field ionization, typical intensities are up to 1012 W/cm2 [46, 47]. Even small clusters exhibit ra if the number of quasi-free electrons is large enough, see the Platinum cluster experiment [13] or more recent experiments with silver clusters [18]. Surprisingly, the entire cm electron dynamics of the cluster can be described as the solution X(t) = At cos(ωt − φt ) to a driven damped oscillator with the equation of motion [27] ¨ + 2Γt X(t) ˙ X(t) + Ωt2 X(t) = F0 (t) cos(ωt) .

(8.5)

The amplitude At , phase φt , damping Γt , and eigenfrequency Ωt are quasistationary variables whose change in time, indicated by the index t, is much slower than the laser period 2π/ω. The four variables are not independent, one can express Γt and Ωt in terms of At and φt , Ωt2 = ω 2 + F0 /At cos φt ,

Γt = F0 /(2At ω) sin φt .

(8.6)

This allows us to extract the eigenfrequency and damping from the cmvelocity, provided it really obeys the dynamics X(t) of a driven damped harmonic oscillator. The result, along with the determination of the eigenfrequency directly from the density of ions in the expanding cluster, is shown in Fig. 8.2. One sees that the damping reaches its maximum with Ω ≈ Γ at resonance which is in the framework of (8.5) a direct consequence of a roughly constant amplitude A. This holds also true for clusters which contain an order of magnitude more atoms, i. e. 104 instead of 103 [31]. The strong change in (negative) slope of the lower bright trace in the electron energy spectrum at t = −300 fs in Fig. 8.3 is due to a sudden increase of positive background charge which indicates increased outer ionization. The reason is efficient energy absorption since the resonance condition is met as the phase lag of π/2 between the driving laser field and the electron cm motion shows (inset of Fig. 8.3). However, the laser pulse must be long enough so that the cluster can expand until the resonance condition at low enough electron density is met during the pulse. 8.3.3 Nonlinear Behavior in Large Clusters (λ = 780 nm) If the conditions are suitable, than ra is also for large clusters (with more than 104 atoms) the most efficient mechanism, simply because it includes the majority of the available electrons. This was already realized by Ditmire and coworkers who introduced the nano-plasma model [48]. It is the linear macroscopic equivalent to ra as described in the last section. Derived within a hydrodynamic approach it extends the mechanism of ra to very large clusters.


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Fig. 8.2. Parameters of the harmonic oscillator model (8.5) as calculated from the Xe923 dynamics. Shown are the eigenfrequency Ωt according to (8.3) as solid line and from (8.6) with circles, as well as the damping rate Γt from (8.6) with diamonds and the laser frequency (dotted line). Graph from [27]

Fig. 8.3. Time resolved energy spectrum of cluster electrons under a laser pulse of the form F (t) = exp(− log 2(2t/T )2 ) cos ωt with half width (pulse length) T = 400 fs (the laser pulse envelope is indicated in grey above the contour plot) for a cluster of 9093 xenon atoms. Bright color corresponds to large number of electrons having the corresponding energy at the respective time. The inset shows the phase lag φt of the electronic cm motion

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However, often it lasts longer than the pulse length before a large cluster has expanded sufficiently to reach the resonance condition (8.4). Hence for large clusters other less effective absorption mechanisms become relevant under laser pulses with a length of the order of 100fs. Since these mechanisms typically involve only a small number of electrons with properties which uniquely characterize the effect, these mechanisms are much harder to identify and to confirm experimentally. We only mention here a number of interesting ideas. They all have in common that they provide means to disrupt the harmonic quiver motion of electrons with the consequence of a phase difference and resulting energy absorption. Inhomogeneous fields in the cluster, including edge effects, can play a major role as argued in different ways [49–51]. Phase disrupting back-scattering in fair analogy to the Fermi-shuttle mechanism has been discussed [7]. 8.3.4 Efficient Inverse Bremsstrahlung for a Strongly Inner-Ionized Cluster in VUV Light (λ = 100 nm) The first (cluster) experiments at the free electron laser of desy with laser pulses of ∼1014 W/cm2 peak intensity, 50–100 fs duration and a wavelength corresponding to 12.7eV photon energy have opened a completely new regime of matter–light coupling [35, 52–54]. The concepts from 780nm light pulses were not suitable to understand the experimental results. While the photon frequency suggests inverse Bremsstrahlung (ibs) as the dominant absorption mechanism, the experimentally determined number of absorbed photons (about 50 photons per cluster atom) was much to high to be explained by standard ibs given the observed ionic charges (about 2 photons per atom according to simple model calculations [35]). Moreover, the average degree of ionization (for a cluster with roughly 80 xenon atoms, e.g., 1.5 fold) was significantly smaller than the highest charges observed (up to 6 fold charged xenon). The latter effect points to the built-up of charge inhomogeneities. This is indeed the case [29] and it is only possible in the absence of quiver motion (which is negligible here, cf. Table 8.1) while it wipes out all possible charge imbalances in distributions for small clusters under 780nm light. The high efficiency of energy absorption prompted the introduction of a model [55] with a modified atomic potential were quasi-free electrons in the cluster would “see” a higher ionic charge than the charge of a cluster ion suggested. However, a microscopic calculation along the principles as discussed in Sect. 8.2 revealed that standard ibs can explain the efficient energy absorption as well, considering that many more electrons are subjected to heating by ibs. These electrons are created by single photo ionization of xenon ions into the cluster due to a significant lowering of the barriers between neighboring ions. This is like a ladder-ionization, where the 12.7eV photon energy is always sufficient to ionize the least bound electron of an ion into the cluster. The quasi-free electrons cannot easily leave the cluster, due


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Fig. 8.4. Number of electrons as a function of time in an Xe147 cluster irradiated by a 90 nm laser pulse (indicated by the gray curve). We show quasi-free (dashed line) and free (dot-dashed line) electrons as well as the sum of both (solid line). Note the different time scales for t < 275 fs and t > 275 fs, respectively. (For the arrows see Fig. 8.5)

to the large background charge of the ions which has been built up which results in a relatively large number of quasi-free electrons compared to the much fewer ionized electrons, see Fig. 8.4. From the figure one also notes that, in contrast to infrared laser pulses, the ionization continues until long after the pulse is over. In fact, the quasi-free electrons form a nano-plasma for clusters as small as 100 atoms and the thermalized quasi-free electrons obey a Maxwellian velocity distribution (Fig. 8.5) whose high velocity tail supplies the ionizing electrons which slowly “boil off” the cluster even after the pulse is over. At times later than shown in Fig. 8.4, many of the quasi-free electrons will recombine, leading to the finally observed spectrum of charges. Hence, the transient average charge of the ions is much higher than concluded from the finally observed ion spectrum and the efficient energy absorption can be explained by standard ibs [29], which yields for Xe147 about 35 absorbed photons per atom. A similar conclusion was reached in [30], where, however, the inner-ionization and recombination processes in the cluster were interpreted differently. The recombined electrons have higher average energy than the quasi-free electrons and do not thermalize any more (light gray shaded region in the right panel of Fig. 8.5). Consequently, they have to be discarded when fitting a Maxwellian shape to the velocity distribution of inner-ionized electrons (dark grey shaded region in the right panel of Fig. 8.5).

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Fig. 8.5. Velocity distributions of the electrons at two times (marked by vertical arrows in Fig. 8.4): at t = 100 fs during the pulse and before cluster expansion (left panel ) and at t = 1000 fs after the pulse (right). For the later time we show distribution for all electrons in the cluster volume (light gray) and for the case when the recombined electrons are excluded (dark gray); for the earlier time both distribution coincide. Additionally, we show fits to a Maxwell distributions (solid lines)

8.3.5 Hollow Clusters Formed Under 3.5 nm Laser Pulses To illustrate the wealth of phenomena in laser-cluster interaction we discuss now laser pulses whose wavelength is roughly a factor 100 shorter than in the vuv light in the last section. Clearly, even with extremely high intensity, light with this frequency acts perturbatively on the electrons (Table 8.1). In fact, apart from single photo-ionization or excitation of bound electrons, there is no effect of the laser pulse so that the dynamical evolution is only due to the forces between charged particles of the cluster. The dynamics of clusters in strong short-wavelength X-ray laser pulses is much less studied and understood than the one of clusters exposed to pulses of longer wavelengths. The main reason is the lack of experimental data. Such data will be available only if the planned xfel machines [19] at desy in Hamburg [56], at the lcls in Stanford [57], or at bessy in Berlin [58] will start operating in the next years. The first theoretical studies concentrate on the ionization and fragmentation dynamics of various cluster types [28,59–61]. This is of crucial importance for the planned imaging investigations with xfel machines: On one hand the extreme brilliance of the xfel helps to get sufficient intensity in the diffraction pattern of a sample in the beam. On the other hand, however, this beam strongly ionizes the sample which will therefore undergo fragmentation. The key question is on which time scale compared to the laser pulse this “loss of structure” occurs. Atomic clusters will be among the first targets in strong X-ray beams to address this question experimentally. Hence, one needs to understand the basic ionization mechanism for these systems. Our first studies with small Argon clusters have concentrated on the electron dynamics at


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a photon energy of ω = 350eV which is only slightly larger than the binding energy of the L-shell of Argon. This permits insight into the importance of competing excitation/ionization processes. In other studies [28, 62] the laser frequency was chosen equal to the highest one available in the near future. Main emphasis was put on the question whether X-ray imaging will be possible or not when the atoms are stripped of their electrons and move due to their Coulomb repulsion. Figure 8.6 shows the final charge per atom of two different clusters at ω = 350 eVas a (measurable) indicator for the energy absorption. As an overall feature, we note lower final charges for the cluster atom compared to the isolated atom (where just the rate equations according to Sect. 8.2.3 have been solved), a difference which increases with the size of the cluster. This indicates a considerable influence of the cluster environment on the photoionization process with the result that clusters are less effectively ionized at high fields than atoms. The first reason for the reduced final charge of the cluster is the space charge. It suppresses outer ionization since the electrons have only gained the difference between the photon energy and the binding energy relative to the other cluster ions which is not enough to overcome the space charge. Secondly, the rapidly oscillating laser field cannot drive quasi-free electrons against the positive space charge out of the cluster which is evident from the small quiver amplitudes listed in Table 8.1. However, there is another, less obvious, reason for the reduced energy absorption due to the cluster environment. The inner-shell holes created by photo-ionization will decay by subsequent ionization processes like autoionization or shake-off processes. At a first glance one would expect that such intra-atomic processes are not affected by the cluster environment. However, the strong laser impact creates local charges in the cluster which deform weakly-bound electronic states and even delocalize electrons since the Coulomb barriers to neighboring ions are lowered. Hence valence electrons are “turned” into quasi-free electrons which screen local charges. The delocalization of the valence orbitals quenches the Auger decay rates because the overlap with the (localized) core holes becomes smaller. As a consequence, the inner shells to be ionized are no longer efficiently refilled by inter-atomic decay, i.e. the atoms in the cluster are temporarily hollow. In order to assess the relative importance of the screening effect compared to the suppression of ionization due to the cluster space charge one can artificially exclude tunneling of electrons to neighboring ions which keeps the electrons localized thereby increasing their ionization rates. Note that this applies to primary photo-ionization and secondary auto-ionization processes as well. The result is indicated in Fig. 8.6 where the difference with and without tunneling is marked by grey shading and accounts for the delocalization effect. The remaining difference between the restricted cluster calculation (the dash-dotted line in Fig. 8.6) and that for the atom reveals the space charge effect. For

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Fig. 8.6. Average charge per atom for two cluster sizes Ar13 and Ar55 produced by an xfel pulse ( ω = 350 eV, T = 100 fs) as a function of the field strength E (dashed line). Cluster calculation where intra-cluster screening was precluded (dot-dashed ) and the isolated atom (solid ) are shown for comparison. Graph from [59]

field strengths f ≥ 0.3 au, where differences between atoms and clusters appear, the space charge effect is initially weaker. This changes for stronger fields: Whereas for the smaller cluster Ar13 both are of the same magnitude at E = 10 au, for the larger cluster Ar55 the space charge effect dominates at this field strength. The reduced energy absorption of clusters compared to isolated atoms at high frequencies may have important implications for the possibilities to image structures with X-ray pulses: The damage threshold is higher than expected based on isolated atom data and therefore, a cluster or another large structure may sustain a higher photon flux than anticipated. If this also reduces the imaging signal by the same amount must be assessed by detailed calculations in the future.

8.4 Comparison of Theory and Experiment for Cluster Observables Beyond a Qualitative Level Theoretical considerations in strong field physics often take the laser simply as a spatially homogenous and temporally constant source with intensity I0 . Since the spatial and temporal variation of the laser parameters are usually adiabatic compared to electronic degrees of freedom, one can simply average the calculation for fixed I0 over the spatial and temporal laser beam profile. This holds in general true also for clusters. However, an additional complication arises from the fact that under experimental conditions clusters have not a unique size with N atoms, but rather a distribution g(N ) where the mean N0 depends on the way they have been generated.


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The kinetic energy distribution of the ions (kedi), e.g., differs substantially for a single cluster under a spatially homogenous laser beam (the theoretical standard) from the experimentally observed kedi [63]. If we start for simplicity with a homogenous charge distribution of ions in the cluster then three simple steps are sufficient to convert the single cluster kedi (Fig. 8.7a) to the realistic one, namely averaging over the spatial laser profile (Fig. 8.7b), averaging over the cluster size distribution (Fig. 8.7c), and taking into account saturation in the ionization (Fig. 8.7e). The basic mechanism underlying the kedi in clusters is their Coulomb explosion. It converts the potential energy Ecoul (r) of a (partially) ionized cluster atom at a distance r from the cluster center into kinetic energy E. The probability dP/ dr to find an atom at a distance r from the center of the cluster with a homogeneous atomic density is dP/ dr = 3r2 R−3 Θ(R − r), where R is the cluster radius. If the cluster is charged homogeneously by the laser pulse with charge q per ion and the ions have not moved yet, then the potential (Coulomb) energy of an ion at radius r ≤ R inside the cluster consisting of N atoms is given by Ecoul (r, q, N ) = N q 2 r2 /R3 .

(8.7)

The ions at the cluster edge R have the maximum energy, which sets the energy scale ER := Ecoul (R, q, N ) = q 2 N/R. Since the Coulomb explosion converts the entire potential energy Ecoul into kinetic energy E, combining dP/ dr with (8.7) and defining ε = E/ER gives directly the kedi [64, 65], dP 3√ = ε Θ(1 − ε) , dε 2

(8.8)

which is shown in Fig. 8.7a. The spatial profile of the laser pulse can be usually described by a Gaussian function with field amplitude F (ρ) = F0 exp(−ρ2 /2ξ 2 ), where ρ is the distance (radius) from the laser beam center in the plane perpendicular to the beam. Along the laser beam we assume a constant intensity since the experiments discussed later [66] are performed with a narrow cluster beam, i.e. a radius smaller than the Rayleigh length [67] of the laser beam. This does not hold for the experiments where the cluster beam is irradiated near the output of the gas-jet nozzle [68]. The charging of the cluster is assumed to be proportional to the field strength, q ∝ F . This applies for resonant charging of the cluster [27, 31]. Hence, we obtain the spatial distribution of charge q(ρ) by replacing F (ρ) with q(ρ) and F0 with q0 , the maximum charge per ion reached in the laser focus ρ = 0. After integration over ρ the laser profile averaged kedi reads in terms of the scaled energy ε = E/Ecoul (R, q0 , N ) dPlas πξ 2 N 1 − ε3/2 = Θ(1 − ε) , dε 2 ε

(8.9)

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Fig. 8.7. Kinetic energy distributions of ions (kedi) for Coulomb exploding clusters as function of scaled energy ε, see text. a: single clusters, see (8.8), b: convoluted with a Gaussian laser profile, see (8.9), c: convoluted with a log-normal cluster size distribution, see (8.11), d: effect of laser profile and cluster size distribution combined, see (8.12), e: including saturation of ionization in addition, see (8.14). Graph from [63]

as shown in Fig. 8.7b. What has changed compared to the original kedi from (8.8) is the qualitatively different behavior with ε−1 instead of ε1/2 for small ε. The formal divergence of (8.9) for ε → 0 can be cured at the expense of a more complicated expression by taking into account that beyond a maximum radius ρmax the laser intensity is too weak to ionize. The enhancement of small kinetic energies after averaging over the laser profile is easily understandable from the higher weight of laser intensities less than the peak intensity, which leads to less charging and, consequently, to more ions with smaller kinetic energy. In most experiments the laser beam interacts with clusters of different size N , which are log-normally distributed [69, 70] according to 1 ln2 (N/N0 ) exp − . g(N ) = √ 2ν 2 2πνN

(8.10)

Convoluting the single-cluster kedi from (8.8) with g(N ) yields in scaled units ε = E/Ecoul (R, q, N0 ) 3 √ dPsize = N0 ε erfc dε 4

3 ln ε √ 2 2ν

.

(8.11)

This size-averaged kedi, shown in Fig. 8.7c, reaches with its tail beyond the energy ε = 1, as more as larger the width parameter ν of the cluster size distribution (8.10) is. The fastest fragments are those from the large clusters in the long tail of this distribution. Note, that we have assumed the average charge q per fragment to be independent of the cluster size N . Of course, for a realistic experimental kedi one has to take into account both, the spatial profile of the laser beam and the cluster size distribution. This yields, in a similar manner as for the other distributions,


dPboth dε

2 2 2ν − 3 ln ε √ = ξ 4π Nε0 exp(ν 2 /2) 1 + erf 2 2ν 3 ln ε 3/2 √ −ε erfc . 2 2ν

165

(8.12)

Here, we have used ε = E/E0 , with the reference energy E0 = Ecoul (R, q0 , N0 ) defined as the maximum Coulomb energy of ions from clusters with the median size N0 6 at the laser focus (charge q0 ). The corresponding distribution is shown in Fig. 8.7d. Since the spatial laser profile modifies the low-energy part and the cluster size distribution the high-energy part of the ion distribution it is possible to gain information from a measured kedi on both effects separately. The final phenomenon which must be taken into account to understand an experimental kedi is saturation, i.e. the fact, that independent of the laser intensity provided, the charging cannot be higher than a certain maximum value qsat , either because the next atomic shell has a much higher ionization potential or because the atoms are completely ionized. We can model the situation by changing our spatial charging function q(ρ) to for ρ ≤ ρsat , qsat (8.13) q(ρ) = q0 exp(−ρ2 /2ξ 2 ) for ρ > ρsat , with qsat the maximum charge, which is realized for clusters close to the center of the laser focus with ρ < ρsat . The saturation can be characterized by the dimensionless quantity √ η := qsat /q0 ∈ [0, 1]. The radius of saturation in (8.13) is given by ρsat = ξ −2 ln η. The charging function (8.13) amounts to using the averaging over the spatial profile only for ρ > ρsat and suggests to define the energy scale as ε = E/Esat with the saturation energy Esat = Ecoul (R, qsat , N0 ). The result is the kedi dPsat (η) dPboth dPsize = − ln η , dε dε dε

(8.14)

which develops a characteristic hump before ε = 1, as can be seen in Fig. 8.7e. To illustrate how the presented expressions for kedi apply we fit them in Fig. 8.8 to experimental data of very different situations. Whereas xenon clusters do not show any noticeable saturation effect (η = 0.8, upper two graphs in Fig. 8.8), the large gap between the 1st and the 2nd shell of argon is responsible for the hump seen in the kedi (η = 0.35, lower left graph in Fig. 8.8). Finally, hydrogen clusters are extreme cases, since only one electron per atom is available (η = 0, lower right graph in Fig. 8.8). Of course, the kedi derived cannot only be used to interpret experimental spectra regarding mean size and saturation of the cluster. Much more they 6

The median size N0 separates the higher half of the distribution from the lower half. In a log-normal distribution it is different from the average size which is exp(ν 2 /2)N0 .

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Fig. 8.8. Ion energy spectra for Xe2500 [1], Xe9000 [71], Ar40000 [4] and (H2 )200000 [64] clusters from experiments (circles) and fits by our model, (8.14) (solid line). Graph from [63]

should make it possible to compare single cluster kedi obtained theoretically via the corresponding convolution to experimental results.

8.5 Summary Firstly, we have presented a mixed quantum/classical approach: Transitions, according to quantum transition rates, of bound electrons into quasi-free electrons are combined with a classical molecular dynamics for these quasi-free electrons and the created ions. If properly adapted, this approach allows for a microscopic description of laser-cluster interaction for a wide range of laser wavelengths λ, from the infra-red (λ = 780nm) over the vuv (λ = 100nm) to the X-ray (λ = 3nm) regime. The analysis of the, in principle straightforward, propagation showed that the excitation/ionization mechanisms are fundamentally different for the three regimes studied. At 780nm, where a wealth of experiments has been performed, the dependence of the ionization on the laser pulse length or on the delay in pump-probe measurements, could be traced back to a cooperative behavior of the electrons in small cluster (∼10atoms) and a collective dynamics in larger clusters (>100atoms). Going to 100nm, with the quiver amplitude about two orders of magnitude smaller, the dominant absorption mechanism seems to be inverse bremsstrahlung. At a even shorter wavelength of about 3nm, core-level ionization dominates. Surpris-


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ingly, this is strongly influenced by the cluster environment in such a way that the ionization of clusters is reduced compared to atoms. Secondly, we have presented a simple analytical model for the ion kinetic energy spectra of laser irradiated clusters. This model allows one to link quantitatively experimental spectra to typical theoretical single-cluster results. We have been able to fit a number of experimentally available size dependent kinetic energy distribution of the ions, which correspond to the experimental setup in terms of laser profile and cluster distribution we have assumed. We hope, that this link will allow the comparison of observables from cluster experiments with theory on a similar quantitative basis as it is done routinely for atomic or molecular observables.

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9 Molecular High Order Harmonic Generation Andre D. Bandrauk1,2 , Samira Barmaki3 , Szczepan Chelkowski1 , and Gerard Lagmago Kamta1 1

2

3

Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke. Sherbrooke, Quebec, J1k 2R2, Canada Canada Research Chair–Computational Chemistry and Photonics [email protected] Secteur Sciences, Université de Moncton - Campus de Shippagan, 218, Boul. J.D. Gauthier, Shippagan, New Brunswick, E8S 1P6, Canada

Summary. We present, in this chapter, numerical results of high order harmonic generation spectra (HOHG), in the Born-Oppenheimer approximation, of the di2+ under strong atomic molecular ion H+ 2 and the linear triatomic molecule H3 laser fields. We revisit, first, the standard electron classical re-collision model. The atomic limit of 3.17 Up for re-collision energy can be considerably exceeded in molecules. HOHG spectra results of H+ 2 at R = 2 a.u., under 10 cycles intense (I > 1014 W/cm2 ) 800 nm laser field, have shown to be strongly influenced by the orientation of the molecular axis with respect to the laser polarization axis. The interferences between contributions of each nucleus exhibit maximum at certain harmonic orders as a function of molecular orientation. A comparison between the acceleration and dipole formulations of the harmonic emission is presented. Finally, we present numerical results of harmonic spectra of the one electron linear H+ − −H+ 2 molecular system at large internuclear distances R = α , where α is the ponderomotive radius, using ultra-short 2 cycles intense and πα 2 (I = 4 × 1014 W/cm2 ) 800 nm laser pulse linearly polarized along the internuclear axis. The extended cut-off is shown to be related to the nature of electron transfer, whose direction is shown to depend critically on the absolute carrier envelope phase (CEP) of the ultrashort pulse. Constructive and destructive interferences in the HHG spectrum of coherent superpositions of electronic states in the H+ − H+ 2 system are interpreted in terms of multiple electron trajectories from a Gabor time analysis.

9.1 Introduction Advances in current laser technology allow for the shaping and focusing of laser pulses to light intensities exceeding the atomic unit of intensity I0 = cE 2 +16 W cm−2 [1] where c is the velocity of light (c = 137a.u.), 8π = 3.5 × 10 E the amplitude of the electric field. One obtains the a.u. of field strength E = ae2 = 5.14 × 10+11 V m−1 , where a0 = 5.29 × 10−11 m is the a.u. of 0 distance, the Bohr radius of the H atom in its 1s ground state [2].

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One of the most interesting phenomena which arises from the interaction of an atom with intense laser light is the production of odd-order harmonics of the incident radiation [3]. A previous attempt to develop a perturbation theory of high harmonic generation (HHG) quickly came to the conclusion that for intensities I > 1011 W cm−2 , harmonics of order n > 3 could not be reproduced perturbatively [4]. Thus HHG is a fundamental nonlinear process of strong-field physics. The high order harmonics are in fact sources of coherent soft-X-Ray radiation with energies beyond 700eV being achieved by focusing several-mJ, 10fs near infrared (λ = 800nm) pulses into a He gasjet [5]. The increasing interest in such coherent X-Ray sources is for the generation of “attosecond” (as) pulses [6,7]. High harmonic generation, HHG is currently the experimentally furthest advanced method that can supply coherent single as-pulses, which is leading to a new science “attosecond science”. While much previous work in application of ultrashort pulses to molecular systems was in the time regime of femtoseconds (1fs = 10−15 s) [8], attosecond (as) laser pulses are needed to observe the rapid electron motion (1a.u. of time = 24.2 as, 1as = 10−18 s). Hence one expects such pulses to allow for the temporal resolution of correlated electron processes, electronic dephasing processes in large molecules and solids or core electron processes (e.g., Auger decay). Furthermore HHG will allow for extension of nonlinear optics to the extreme ultraviolet (XUV) regime [9], where one expects new nonlinear effects due to the nondipolar radiation transitions expected at extremely short wavelengths [10, 11]. Krause et al. were the first to calculate the harmonic spectrum produced by atoms and ions exposed to intense laser radiation by numerically solving the 3-D time dependent Schroedinger equation (TDSE) for a one-electron 3-D effective atom [2]. For laser pulses longer than the period of the carrier frequency high harmonic emission can be qualitatively described by a simple semiclassical picture [13]. Under the influence of low frequency, near IR intense fields, the electron can be considered to ionize by tunnelling ionization [14]. When the field changes phase, i.e., when the field reverses, the ionized electron is made to rescatter and recombine by recollision with the parent ion, which produces harmonics up to a cut-off maximum energy called the maximum order Nm . From the energy conservation law, the value of Nm is Nm = (Ip + 3.17 Up )/ω. Ip is the ionization potential of the initial atom and Up is the ponderomotive energy of the electron acquired in the field, Up = 4ωI 2 . The atomic harmonic spectra consist therefore of a plateau with slowly decreasing intensity followed by a short cut-off around the energy Nm ω. For very short pulses, τ ∼ = 5fs, the highest harmonics eventually merge into a broad continuum band, which corresponds to a single attosecond X-Ray pulse in the time domain [15, 16]. It is theoretically predicted that combining such broad single as-pulses with intense ultrashort 800nm pulses leads to an even broader continuum and thus to amplification of as-pulses [17]. The study of the dynamics of HHG is thus

9 Molecular High Order Harmonic Generation

173

an important subject as it is currently the main source of attosecond pulse generation.

9.2 Theoretical Aspects of Short Laser Pulse-Molecule Interaction The interaction of molecules with intense laser pulses introduces new challenges due to the presence of nuclear motion [18, 19], i.e., extra degrees of freedom and thus additional timescales. Typical molecular ionization rates which can be estimated by tunnelling ionization models [13, 14], at intensities I ≈ 1014 Wcm−2 , at wavelength in the near IR regime, 1064–800nm are of order of 1014 s−1 [20], i.e., ionization times of 10fs. Laser cycle periods are 3.5fs (1064nm) and 2.7fs (800nm). The fastest nuclear motion is that of the proton (e.g. for C-H bonds, the vibrational frequency ωv ≈ 3000cm−1 corresponding to a vibrational period τ ≈ 10fs). This requires that the motion of this most elusive particle (important in many biological and chemical processes) needs to be considered on equal footing with radiative processes induced by current intense laser pulses. Furthermore, HHG in molecules is a new subject because of the multicenter Coulomb interference effects expected after recombination of the electron with the parent molecular ion or with neighboring ions [19, 23] with the result that LIED, laser induced Electron Diffraction [24, 25] can become a tool for tomographic imaging of molecular orbitals [26–28]. We present therefore in this chapter different theoretical aspects of the interaction of ultrashort intense laser pulses with molecules based on our previous and current nonperturbative simulations of TDSE’s for one and two electron molecules. We will address subjects as the qusiclassical recollision model [14], gauge issues [2,22], multicenter interference effects [21,24] and finally the difference between HHG in symmetric [22] and nonsymmetric molecules [29] using time series analysis tools for identifying relevant collision trajectories [30]. The HHG simulations presented here are for single molecule nonlinear responses in ultrashort intense laser pulses and do not include propagation effects. The latter include plasma effect which can shorten the coherence length for atomic harmonics [31]. Efforts are now being undertaken to extend propagation effects for molecular HHG [32], which effects are also important for atmospheric propagation of intense laser pulses leading to filamentation of pulses in the air medium [34]. 9.2.1 Laser Induced Collision and Recollision In the multiphoton ionization regime, for an atomic or molecular system under a strong laser field polarized along the z axis, an electron can tunnels the distance z0 and escape from the deformed potential barrier with zero

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velocity v0 = v(t0 ) = 0 at some t0 time. Within the first laser period after such tunnel ionization there is a significant probability that the electron will return to the vicinity of the ion with very high kinetic energy. High order harmonic generation (HHG) high energy above threshold ionization (ATI) and even two-electron ejection [35], are all consequences of this electron-ion interaction upon recollision of the electron with its parent ion. One important parameter which determines the strength of these effects is the rate at which the wavepacket spreads in the direction perpendicular to the laser electric field and/or laser polarization. In extended systems such as molecules and clusters, recollision can occur with neighboring ions [23, 36–38] thus producing longer plateaus or suppressing HHG by dephasing. We derive next from a simple classical model the energetic consequences of recollision and their dependencies on the relevant physical parameters, generalizing previous approaches by assuming the initial velocity is nonzero. This is achieved generally by preionizing the initial electron with a high frequency pulse in order to enhance HHG [17, 23]. We solve the classical equations of motion of an electron in a laser field E(t) = E0 cos(ωt), (we use atomic unit, e = m = = 1), d2 z dv = −E0 cos(ωt) , = dt2 dt

v(t0 ) = v0 ,

z(t0 ) = z0 = 0

(9.1)

The general solutions are at final time tf for the initial conditions in Eqs. (9.1) [39], E0 v(tf ) = v0 + [sin(φ0 ) − sin(φf )] , ω E0 (φf − φ0 ) E0 z(tf ) = v0 + sin(φ0 ) + 2 [cos(φf ) − cos(φ0 )] , ω ω ω

(9.2) (9.3)

where φ’s are laser phases at t = t0 and t = tf , φ0 = ωt0 and φf = ωtf . From (9.2) one obtains a maximum final velocity vf = v0 + 2 Eω0 when φ0 = π2 and φf = 3π 2 . If v0 = 0, the maximum energy Ef (a.u.) is Ef =

vf2 E2 = 2 02 = 8 Up , 2 ω

Up =

E02 I0 = 4ω 2 4ω 2

(9.4)

where Up is called the ponderomotive energy. This maximum energy can only be reached at z = 0, i.e., at |z(tf )| =

E0 (2n − 1)π , ω2

n = 1, 2, . . .

(9.5)

It is to be emphasized that unfortunately ionization is negligible at φ0 = π2 since E(t0 ) = 0 at such a phase, and therefore preionization is required to make such a HHG scheme efficient [17]. It was shown originally by


175

Corkum [14] that if the electron initial velocity is v0 = 0 at time t0 , then the maximum energy that the electron can acquire at its return to the parent v2

ion, i.e. when z(tf ) = 0 is Ef = 2f = 3.17 Up . We derive below the general result for initial nonzero velocity [39]. Imposing the condition of the electron return to the parent ion z(tf ) = 0 in (9.3), we can express v0 as a function of φ0 , φf or rather Δ = φ0 − φf , the initial and relative phase of the field at the final time tf : v0 /(E0 /ω) =

1 − cos(Δ) sin(Δ) − 1 cos(φ0 ) + sin(φ0 ) Δ Δ

(9.6)

Inserting (9.6) into (9.2) now gives for the final velocity vf vf /(E0 /ω) = f (Δ, φ0 ) = α1 cos(φ0 ) + α2 sin(φ0 ) ,

(9.7)

where α1 =

1 − cos(Δ) − sin(Δ), Δ

α2 =

sin(Δ) − cos(Δ) Δ

(9.8)

From (9.7) using (9.8), we obtain all possible return velocity with non-zero v0 . vf is then a simple analytical function of two independent variables φ0 and Δ. The extrema of the function f (Δ, φ0 ) are found by requiring its partial derivation to be zero: ∂f = α1 cos(φ0 ) + α2 sin(φ0 ) = 0 , ∂Δ ∂f = −α1 sin(φ0 ) + α2 cos(φ0 ) = 0 , ∂φ0

(9.9) (9.10)

where α1 and α2 are first derivatives of α1 (Δ) and α2 (Δ). After elimination of φ0 from (9.9) we get from (9.7) the following transcendental equation: F (Δ) = 2 − 2Δ sin(Δ) + (Δ2 − 2) cos Δ = 0

(9.11)

The function F (Δ) never exceeds 1, has zero’s at Δ’s. The corresponding energies En of the electron returning to its departure point z(tf ) = z0 = 0 are tabulated in Table 9.1. The first extremal energy E1 = 3.17 Up is the maximum return energy and corresponds to the maximum return velocity vf = 1.2586 (E0 /ω). For large Δ, from Eq. (9.7) the final velocity simplifies to: vf /(E0 /ω) = f (Δ, φ0 ) = − sin(φf )

(9.12)

Thus for large return excursions such that Δ >> 1, the final kinetic energy Ef = vf2 /2 for return at z(tf ) = 0 never exceeds 2 Up . In summary, the final energy of the electron return never exceeds 3.17 Up for all initial velocities v0 .

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Table 9.1. Zero’s of Δn and corresponding return energies En which are solutions of (9.11) for initial phases φ0 n

Δn /π

En /Up

φ0 /π

φf /π

1 2 3 4 5

1.301 2.427 3.435 4.458 5.461

3.173 1.542 2.404 1.734 2.246

0.098 0.037 0.032 0.021 0.019

1.398 2.463 3.467 4.478 5.480

We show next that v0 = 0 at any extremum of the function vf /(E0 /ω) defined by Eqs. (9.9) and (9.10). From (9.10), we find that tan(φ0 ) = α2 /α1 and inserting into Eq. (9.6) gives v0 /(E0 /ω) =

cos(φ0 )F (Δ) α1 Δ2

(9.13)

Thus the initial velocity v0 is proportional to F(Δ) defined in (9.11). The zeros of F (Δ) which occur at the Δn s in Table (9.1) correspond to the extrema of vf as a function of Δ. Clearly, these extrema correspond to v0 = 0. We have thus proven in general that the highest energies of electrons returning to the parent ion in the presence of linearly polarized light after ionization only occurs for initial velocity v0 = 0 with maximum energy E1 = 3.17 Up . We further note that Eqs. (9.6) and (9.7) can be rewritten as vf /(E0 /ω) = v0 /(E0 /ω) + sin(φ0 )(1 − cos(Δ)) − sin(Δ) cos(φ0 ) ,

(9.14)

so that if v0 /(E0 /ω) = 0, then vf /(E0 /ω) = sin(φ0 )(cos(Δ) − 1) + sin(Δ) cos(φ0 )

(9.15)

The maxima of vf are illustrated in Fig. 9.1 (1a), with the first occurring at φ0 = 0.1π, φf = 1.4π with energy E1 = 3.17 Up as noted above for vf (max) = 1.2586E0 /ω. For any value of v0 , φ0 , Δ (see Fig. 9.1 (1b)), |vf | ≤ 1.2586E0 /ω

(9.16)

For any value of vf > 1.2586E0 /ω, the electron will not return to the nucleus. Thus the energy E1 = 3.17 Up and |vf | = 1.2586E0 /ω corresponds to an absolute maxima in energy and velocity of recolliding electron in the presence of an intense linearly polarized field of amplitude E0 and frequency ω. If now we invert the role of vf and v0 and we solve Newton equation moving backwards in time that corresponds to changing the phase of the field by π, one could expect that electron initialized at φ0 = π − 1.4π = −0.4π with


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Fig. 9.1. (a) Return velocity vf of the electron as function of phase delay Δ = φf − φ0 and initial phase φ0 . (b) Initial velocity v0 corresponding to the final velocities vf shown in (a) (X = Eω0 )

v0 = 1.2586E0 /ω will return at the departure position at φf = π − 0.1π = 0.9π with the final velocity vf = 0. This implies that for any value of vf , |v0 | ≤ 1.2586E0 /ω. Instead of recollision with the parent ion, one can consider the electron collision with neighboring ions [23, 37, 38]. In this part, we consider that electron ionize from its parent ion at t0 = 0 with initial velocity v0 (t0 ) = 0 and it’s driven by the laser field to collide with neighboring ions. We describe

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A.D. Bandrauk, S. Barmaki, S. Chelkowski, G.L. Kamta 2

the classical electron equation as dd2zt = −E0 cos(ωt + φ), φ is an initial 2 arbitrary phase. One obtain dd2zt (max) at phase (ωt + φ) = π. This yields the velocity dz dt = E0 sin(φ)/ω with a maximum E0 /ω for φ = π/2. The resulting energy is E(max) = 2 Up . The corresponding minimum distance required by the electron to collide with a neighboring ion is then obtained from (9.3) as +π , − 1 = 0.57α0 , (9.17) z(tf ) = α0 [−1 + ωt] = α0 2 where we have defined α0 = ωE02 . Thus the maximum acceleration criterion yields a maximum harmonic order Nm = (Ip + 2 Up )/ω at internuclear distance R = 0.57α0 at phase ωt = π2 . The maximum energy E = 8 Up , with no constraint at z(tf ) = πα0 is gained during a half cycle, i.e. at (ωt + φ) = 3π 2 with φ = π/2 and ωt = π. Setting z(tf ) = πα0 /2 yields after simple calculation another cut-off law, Nm ≈ (Ip + 6Up )/ω with φ = π/2 and ωt = 3π/4. This was confirmed numerically from the TDSE’s of H+ 2 and H2+ 3 [23, 38]. To resume, the classical collision model predicts maximum kinetic energies 2 Up , 6 Up and 8 Up by collision with neighboring ions at the maximum 3π 0 distances z(tf ) = ( π2 −1) α0 at phase ωt = π2 , z(tf ) = πα 2 at ωt = 4 and πα0 at ωt = π. Thus it has been shown that atoms separated by a large distance R = πα0 generate harmonics of order up to Nm = (Ip + 8 Up )/ω [23, 37–39]. HHG results presented in [38] have shown that for two initially localized electrons of H2 only harmonics of order Nm = (Ip +8 Up )/ω are obtained, whereas for the two electrons completely delocalized in the symmetric H+ 3 harmonics of order up to Nm = (Ip + 12Up )/ω are observed. That was explained by the fact that, at the middle of the molecule, i.e., at R = πα0 /2, one electron exchanges 6 Up of kinetic energy by a collision with the other, thus giving a maximum of 12 Up as observed in the H+ 3. In summary one sees from the above simple classical model that recollision with the parent ion, i.e., z(tf ) = 0 and assuming initial zero velocity v(t0 ) = 0 gives the maximum harmonic order Nm = (Ip + 3.17 Up )/ω. Collision with neighboring ions can lead to a longer HHG spectrum with a cut-off up to the maximum harmonic order Nm given by Nm = (Ip + 8 Up )/ω. This larger extended cut-off law involving collisions with neighboring ions requires an initial phase φ = π/2. Thus in this more extended HHG spectrum, unfortunately little ionization occurs initially, which requires preionization for enhanced efficiency, i.e., with nonzero initial velocity [17, 23]. Assuming further that the initial velocity is nonzero, i.e., dz dt (0) = v0 = 0, one can in principle obtain harmonics beyond the 8 Up law [17, 23, 37, 38]. In all the above linear polarization schemes, Coulomb forces have been ignored. These have the effect of refocusing electron trajectories [41]. Combination of nonzero velocities, i.e., v0 = 0, and Coulomb forces can change even the recombination parameters giving rise to maximum ponderomotive energies beyond 3.17 Up [17].


179

9.3 Gauges/Representations The standard description of laser-particle interactions starts from the classical electric-field particle interaction. By successive unitary transformations on the molecular TDSE, one can obtain various representations where the particle-radiative couplings have different frequency dependencies [2, 18]. In general, one has to solve the following electronic molecular TDSE with rj=1,...,Nelec and Rα=1,...,Nnuc corresponding respectively to electron and nuclear position vectors (we use atomic units e = = me = 1): i

∂Ψ (rj , Rα , t) = HΨ (rj , Rα , t) = [Helec + Vext ] Ψ (rj , Rα , t) ∂t

(9.18)

where

Helec

⎤ ⎡

p2j

⎣ + = U (rj − rk ) + V (rj − Rα )⎦ 2 α j

(9.19)

k=j

Vext =

zj E(t)

(9.20)

j

U is the electron-electron repulsion and V is the attractive electron-nuclear potential. Equation (9.20) represents the electron-laser interaction in the dipole approximation (rj /λ 1) and long wavelength gauge [2]. Applying a first unitary transformation allows to derive new TDSE’s to

ˆ Ψ , i ∂Ψ = H Ψ , Ψ =U ∂t

ˆ† ˆ] ˆ †H U ˆ + i ∂U U H = [U ∂t

(9.21)

One can therefore re-express the electron-radiative coupling (9.20) as a fieldelectron momentum coupling, i.e. Ψ (rj , Rα , t) = exp(iA(t)

zj )Ψ (rj , Rα , t)

(9.22)

j

where A(t) = − E(t ) dt is the vector potential. It has been shown how, using ultrashort pulses, one can measure A(t) for a pulse E(t) from the asymmetry in the ionization [17]. The new TDSE for Ψ becomes: ⎡ ⎤ 2 2

p ∂ A (t) j i Ψ (rj , Rα , t) = ⎣ + A(t) · pzj + + U + V ⎦ Ψ (rj , Rα , t) ∂t 2 2 j j (9.23)

180


This is called the Coulomb gauge. For a field E(t) = E0 cos ωt, the radiative coupling depends on ω −1 since A(t) α ω −1 . A further unitary transformation originally used in QED [2, 18] allows to further remove field momentum coupling A·pzj terms thus resulting in an exponentiated radiative interaction, ⎡

Ψ (rj , Rα , t) = exp ⎣−

i 2

A2 (t ) dt + iα(t)

⎤

pzj ⎦ Ψ (rj , Rα , t)

(9.24)

j

and Ψ (rj , Rα , t) is the solution of the Pauli-Fierz or acceleration gauge TDSE, i

p2j ∂ Ψ (rj , Rα , t) = Ψ (rj , Rα , t) + eiα(t) j pzj (U + V ) ∂t 2 j

· e−iα(t)

j

pzj

Ψ (rj , Rα , t)

(9.25)

where α(t) =

A(t ) dt =

dt

E(t ) dt

(9.26)

One notes that for a field E(t) = E0 cos(ωt), α(t) depends on ω −2 , thus suggesting (9.25) as an appropriate representation for high frequency problems. One can transform (9.18) to an equivalent electron-translation frame. The space translation method is used in intense high frequency atom-laser interaction theories. To our knowledge it was first used in quantum mechanics by Husimi to solve the complete time dependant harmonic oscillator in an external time dependent force. In the following we transform (9.18) using appropriate unitary transformations. We consider the system under a spatially uniform field E(t). At first, we change from the r, t coordinates system to a new ξ, t coordinates system where ξ = r − α(t) from which it follows: ∂ξ = 1, ∂r

∂ξ = −α(t) ˙ ∂t

so ⏐ ⏐ ∂Ψ ∂Ψ ⏐ ∂Ψ ⏐ ⏐ ⏐ = −α(t) ˙ + ∂t ∂ξ ⏐t ∂t ⏐ξ Introducing now the transformation ˙ Ψ (ξ, Rα , t) = eiαξ Φ(ξ, Rα , t)


181

we obtain a new TDSE for Φ(ξ, t) i

2

pj ∂ Φ(ξ, Rα , t) = (ξ) + U (ξj − ξk ) + V (ξj + α(t) − Rα ) ∂t 2 α j k=j 2 α˙ +(¨ α − E(t))ξ − + E(t)α Φ(ξ, Rα , t) (9.27) 2

One sees that all electronic coordinates in U and V in Eq. (9.27) are displaced by α(t). In particular U = U [(rj − α(t) − (rk − α(t))] = U (rj − rk ) remains invariant to the presence of the field, the latter in contrast induce modification in the electron-nuclear attraction V . By setting α˙ 2 α˙ = E(s) ds, g(t) = + E(t)α 2 and performing another unitary transformation Φ(ξ, Rα , t) = exp i g(s) ds Ψ (ξ, Rα , t) we obtain the following TDSE equation ⎡

p2j

∂ ⎣ (ξ) + i Ψ (ξ, Rα , t) = U (ξj − ξk ) ∂t 2 j k=j

V (ξj + α(t) − Rα ) Ψ (ξ, Rα , t) +

(9.28)

α

In the Husimi representation, from (9.28), we conclude that the electron– electron repulsion remains invariant to the presence of the field E(t). All radiative interactions occur as displaced electron–nuclei distances in the attractive electron-nuclear potential V . Going beyond the dipole approximation introduces field corrections to U itself and further momentum–momentum couplings [44–47]. The original solution of (9.18) is given by ⎡ ⎤ 2

A (t ) dt Ψ (ξj , Rα , t) Ψ (rj , Rα , t) = exp ⎣iA (t) zj ⎦ exp −i 2 j (9.29) Equation (9.29) allows us to identify three important effects of the laser field E(t) on the electron wavefunction. The first effect is a) momentum displacement Δpz = A(t) through the phase factor exp[iA(t) zj ]. As an example of this is aδ-function pulse E0 δ(t) for which A(t) = E(t ) dt = E0 . The solution of this excitation for a H-atom can be obtained as Ψ (r, t = 0+ ) =

182


exp (iE0 z) Ψ (r, 0) [48]. Applying this solution to the ground 1σg and first excited electronic states 1σu of the H+ 2 molecular ion transform readily the nonperturbed electronic wave functions, 1 Ψ1σg(u) (r, R) = √ [φ1s (r − R/2) ± φ1s (r + R/2)] 2

(9.30)

where φ1s are hydrogen -1s orbitals, centered at ±R/2. The corresponding field induced momentum space functions f+ and f− of the two molecular orbitals (9.30), reflect the interference between the two atom centers obtained after Fourier transformation [24], ⎡

⎤

cos E0 + k · R/2

⎦ ϕ1S E0 + k (9.31) f± (k) = ⎣ sin (E0 + k · R/2 The fundamental difference between this result and previous weak field direct photoionization [49] or long pulse photoionization is the influence of the momentum transfer δpz = E0 on the interference in the momentum distribution (9.31) by an extra phase E0 R/2. This is an example of an extreme nonadiabatic excitation by a laser pulse. Clearly at high intensities, field-electron momentum transfer is non-negligible and has been measured recently [50]. The next phase factor exp[−i A2 (t) dt/2] represents the additional energy acquired by a free electron in the field and is called the ponderomotive energy Up which when averaged over many cycles for a field E0 cos ωt equals Up = I/4ω 2 where I is the intensity I = cE02 /8π. The final important effect of an intense ultrashort pulse is the coordinate displacement of the electron, i.e., ξj = rj − α(t). This has two important consequences: all electrons are displaced by the same amount α(t). This leaves first the electron–electron repulsion U (rj − rk ) unchanged but secondly will contribute strongly to further momentum changes via the forces ∂V /∂zj , etc. Thus expanding V (ξj + α(t) − Rα ) gives for the electron–nuclear attraction to second order, V = V (ξj − Rα ) + α(t)

∂V α2 ∂ 2 V (ξj − Rα ) + (t) 2 (ξj − Rα ) + . . . ∂ξj 2 ∂ ξj

(9.32)

The third term in (9.32) has been used previously to estimate Lamb-shifts in atoms as it is proportional to electron densities at the nucleus [2, 18]. The three gauges (Eqs. (9.18), (9.23) and (9.25)) are equivalent only if one uses complete basis sets in any calculation or simulation. Thus numerical grid simulations should give the same results in anyone gauge and this may be often used to check numerical accuracies. Thus only exact calculations are gauge invariant whereas perturbation expansions are not necessarily so. One can readily see this paradox by comparing the radiative matrix elements in the three gauges [18]. This also allows for assessing the convergence properties of any perturbation expansion for these gauges. For a classical oscillator


183

one has the relation between momentum p or velocity v, frequency ω and corresponding radius r, p/m = v = ωr. In the quantum picture this becomes an equation for matrix elements between |α > and |β > states,

p αβ =< α | [ˆ r, Hˆ0 ]|β >= ωαβ r αβ (9.33) m where ωαβ = Eα − Eβ is the energy difference between the two states and Hˆ0 is the field free Hamiltonian. Using the equivalence between the electro magnetic potential A and the electric fields E , A = E c/ω, gives the followi

ing relation between the momentum radiative matrix elements in the A · p (Coulomb) gauge, and the dipole radiative matrix elements (length gauge) in (9.20),

A · p αβ ωβα =i r αβ · E (9.34) mc ω Clearly, the radiative matrix elements are only equal in resonance where ω = ωβα . For nonresonant or virtual transitions, large matrix elements will

occur in the A · p gauge when ωβα /ω 1. In the case of multiphoton transitions, such nonresonant transitions are therefore important in calculating transition amplitudes in the Coulomb gauge. A comparison of the Coulomb and acceleration radiative transition matrix elements is obtained by expanding the radiatively shifted potential and using the commutation relation, ˆ 0 , pˆ], one obtains, i∇V ( r ) = [H

(∇V )αβ = iωβα p αβ

(9.35)

from which follows (assuming A(t) = A0 cosωt)

α(t) · (∇V )αβ

iA0 ωβα = p mc ω αβ

(9.36)

0 As in the earlier case, (9.34), the radiative couplings α(t) · (∇V )αβ and iA mc p αβ agree only in the resonant case. Clearly, perturbation expansions will converge most slowly in the acceleration gauge, followed by the Coulomb gauge. We note however, that at very high frequencies, i.e. ω ωβα , the acceleration gauge matrix elements will be smallest and is therefore called a high frequency regime [2, 18]. In the acceleration gauge, (9.25), the response of the molecular system is through gradients of the potentials, i.e. through the forces. A direct applications of this result is to the emitted radiation spectra, constituting the HHG spectrum in presence of an intense laser field. This can be obtained from the time dependent acceleration for an electron in a potential V (r) [40].

a(t) =

d2 < z >= ψ(t) | −∂V /∂z | ψ(t) dt2

(9.37)

184


The wavefunction ψ(t) is the exact solution of TDSE. The Fourier transform of a(t), a(ω) from which one obtains the power spectrum of HHG, |a(ω)|2 is given by [51] ˙ ) + iω eiωτ d(τ ) − ω 2 τ eiωt d(t) dt] (9.38) a(ω) = (2π)−1/2 [ e−iωτ d(τ 0 The final integral in (9.38) is the HHG power spectrum using the dipole form d(t) =< ψ(t)|r|ψ(t) >. Clearly, this is only valid if the transients ˙ ) = d ( d(t))t=τ and d(τ ) vanish at the end of the pulse. The accelerd(τ dt ation form emphasizes maximum acceleration as in the case of a recolliding electron in agreement with the classical model [40]. Since the energy E of such an electron is 3 Up ≤ E ≤ 10 Up where Up is the ponderomotive energy, at high intensities one has highly oscillatory electron wavefunctions so that the TDSE in any gauge requires highly accurate numerical procedures [33]. Optimal gauges for various frequencies and intensities have been examined by [52]. Finally, we emphasize that for an electromagnetic pulse propagating freely, the time integral of the electric field E must vanish, i.e. τf E(t) dt = A(τf ) − A(τi ) = 0

(9.39)

τi

This is a consequence of Maxwell’s equations and the fact that the pulse is limited in duration and spatial extent [1, 2]. It is thus nonphysical to have a freely propagating pulse with a non-vanishing time integral of the electric field E, i.e. for real physical pulses one should have A(τi ) = A(τf ) = 0. If this condition is not fulfilled then one obtains an artificially induced E − dc component of the pulse which will result in an nonphysical drift of the electrons [53].

9.4 High Order Harmonic Generation of H+ 2 In this section, we present results obtained by solving, in Born-Oppenheimer approximation, TDSE of H+ 2 under intense laser pulse having an arbitrary oriented linear polarization. V (r) the coulomb potential experienced by the electron in the field of the two identical nucleus of charge Z = 1, fixed at internuclear distance R, is V (r) = V (r1 )+V (r2 ) = − rZ1 − rZ2 where r1 = r+ R 2 and r2 = r − R are respectively the position vectors of the electron relative 2 to the nucleus 1 and relative to the nucleus 2. r is the electron position vector relative to the geometric center of the molecule. The harmonic spectrum S(ω) radiated by a system is proportional to the absolute square of the Fourier transform a(ω) of the dipole acceleration [51]: (9.40) a(ω) = eiωt Ψ (t) e · [∇V (r) + E(t)] Ψ (t) dt


185

Ψ (t)|∇V (r) + E(t)|Ψ (t) is the total dipole acceleration obtained via Ehrenfest’s theorem. E(t) is the electric field obtained from E(t) = - ∂A(t) and ∂t A(t) = A0 f (t) sin(ω0 t) e, e = sin(χ) ey + cos(χ) ez . Since we consider the molecule aligned along the z axis, then χ is the angle between the z axis and the laser polarization direction. We use laser frequency ω0 = 0.057a.u. (λ = 800nm), a trapezoidal f (t) pulse envelope, i.e., the pulse is turned on linearly over three laser periods, kept at constant intensity for four laser periods, and turned off linearly over the laser periods. The second term in (9.40) is the Fourier transform of the electric field with polarization vector e, whose contribution to the harmonic spectrum is essentially limited to the fundamental harmonic. It follows that the high order harmonic spectrum from the system is almost completely determined by |a(ω)|2 , where a(ω) =

eiωt Ψ (t) e · ∇V (r) Ψ (t) dt

(9.41)

9.4.1 Nuclear Contributions to Harmonic Generation Since V (r) = V1 (r) + V2 (r) for a two center system as H+ 2 , then we may separate the nuclear contributions a(ω) = a1 (ω) + a2 (ω)

(9.42)

where aj (ω) =

eiωt Ψ (t) e · ∇Vj (r) Ψ (t) dt

(9.43)

with j = 1, 2. Equations (9.41) and (9.43) indicate that aj (ω) is the analog of a(ω) for the nucleus j. This suggests the interpretation of |aj (ω)|2 as the harmonic spectrum originating from the nucleus j, in the presence of the other nucleus. Figure 9.2 shows the harmonic spectra originating from the nucleus 1 (i.e., |a1 (ω)|2 ) and from the nucleus 2 (i.e., |a2 (ω)|2 ) of H+ 2 , for various orientations χ of the molecule with respect to the laser polarization. These spectra are obtained from a Born-Oppeinhemer (static nuclei) solution of the 3-D H+ 2 one electron molecule [22]. In both cases, and for all orientations, features of these spectra strongly resembles those of harmonic spectra from atoms: a sharp decrease of the first few harmonics, followed by a “plateau”, and ending with a cutoff that determines the highest harmonic order achievable. The cutoff is independent of the molecular orientation and is located approximately at the 85-th harmonic for intensity I = 3 × 1014 Wcm−2 and wavelength λ = 800nm (ω0 = 0.057a.u.) , which is in agreement with the

186


energy cutoff formula given by Nm ω0 = Ip + 3.17 Up . However, except for the orientation χ = 90o , there is a feature in Fig. 9.2 that does not exist in the harmonic spectra from atoms. This feature is the presence of both odd and even harmonics. Even harmonics do not appear in the spectra of atoms due to the inversion symmetry that exists in these systems as a result of their spherical symmetry. In fact, with the other nucleus nearby, the potential experienced by the electron from one nucleus is not inversion symmetric (except for χ = 90o ), leading to both odd and even harmonics in the spectra originated from a single nucleus. In order to get an additional insight into the nuclear contributions to the harmonic spectrum, we use a Gabor analysis [54, 55], which provides the time profiles of the harmonic spectra originated from each nucleus. The time-profiles are obtained by taking the inverse Fourier transform of the product of aj (ω) by a Gaussian window function, which is centered at a selected reference harmonic, and which has a specified full width at half maximum (FWHM). The resulting time profile indicates the time at which the selected set of harmonics was emitted during the pulse. Fig. 9.3 shows the time profiles of harmonics emitted by the nuclei 1 and 2 of H+ 2 for various orientations of the molecule. The window function used to obtain these profiles is centered at the 85-th harmonic and has a FWHM of 5ω0 . This harmonic order is near the cut-off region where we anticipate the electron to return to the molecular core with the maximum energy 3.17 Up [14, 39]. In addition, we have solved the classical Newton equation x ¨ = −E(t) for the motion of a free electron driven by the electric field E(t) of the laser pulse. Assuming that at the initial time t0 , the electron is at the origin of the coordinates with zero velocity, we find the time of first return of the electron to the origin and the corresponding kinetic energy. These first returns are the so-called short electron trajectories [56]. By varying t0 throughout the laser pulse, we find and plot with dots in Fig. 9.3(e) the first return energies vs the return time. The resulting plots shows two peaks every laser period, corresponding to two classical returns with the maximum kinetic energy 3.17 Up [14, 57]. Throughout the 10 cycles long pulse used in this work, classical first returns with the maximum kinetic energy 3.17 Up occur at the times t1 = −1.3, t2 = −0.8, t3 = −0.3, t4 = 0.2, t5 = 0.7, t6 = 1.2, and t7 = 1.7 (in units of the laser period τ = 2.7fs at λ = 800nm). For all molecular orientations, the time profiles in Fig 9.3 show series of peaks separated by about half the laser period. These peaks, which indicate the instants (during the laser excitation) at which the 85-th harmonic is emitted by each nucleus, agree very well with peaks in the plot of the classical first return energy vs the return time in Fig. 9.3(e). This indicates that high order harmonics are indeed emitted every half-cycle by each nucleus when the electron wavepacket returns for a recollision with the molecular core.


187

Fig. 9.2. Harmonic spectra (in arb. units) originating from the nucleus 1 (i.e., |a1 (ω)|2 ) and from the nucleus 2 (i.e., |a2 (ω)|2 ) of H+ 2 , for various orientation angles: (a) and (e) for χ = 0o ; (b) and (f ) for χ = 30o ; (c) and (g) for χ = 40o ; (d) and (g) for χ = 90o . Plots (a), (b), (c) and (d) on the left correspond to |a1 (ω)|2 , while plots (e), (f ), (g) and (h) on the right correspond to |a2 (ω)|2 . For χ = 90o , |a1 (ω)|2 and |a2 (ω)|2 are identical. The equilibrium internuclear distance R = 2 a.u. is used. Laser frequency used: ω0 = 0.057 a.u. The pulse consists of a linear turn-on over 3 laser periods, followed by 4 laser periods at constant peak intensity, and linear turn-off over 3 laser periods for I = 3 × 1014 W cm−2 , λ = 800nm

188


Consider Fig. 9.3(a) for the parallel orientation of the molecule (χ = 0). On sees that at the first harmonic emission at time t1 , the intensity of the time profile for the nucleus 1 is larger than that of the nucleus 2. On the other hand the classical analysis indicates that the returning electron at time t1 encounters nucleus 1 first, and nucleus 2 next. At the next recollision time t2 , the returning electron encounters nucleus 2 first, then nucleus 1 next, and the intensity of the time profile at t2 is higher for the nucleus 2 than for the nucleus 1. This asymmetry repeats itself alternatively throughout the laser pulse, with the intensity profile of the nucleus 1 dominating that one recollision and the profile of the nucleus 2 dominating a half-cycle later at the next recollision, and so on. This suggests that at each return of the electron wavepacket to the molecular core, high order harmonics are emitted predominantly by the nucleus that experiences the first recollision. Indeed, after its recollision with the first nucleus, the electron wavepacket that reaches the second nucleus is diminished by its scattering and its recombination with the first nucleus. In other words, the nuclei screen each other from the returning wavepacket. This is further supported by the fact that as one tends from the parallel orientation (χ = 0o ) to the perpendicular orientation (χ = 90o ) in Fig. 9.3, the asymmetry in the time profile between the two nuclei decreases gradually and vanishes for the case χ = 90o where there is no such screening. As discussed previously, the full harmonic spectrum is almost entirely determined by |a(ω)|2 , while |aj (ω)|2 (j = 1, 2) is the harmonic spectrum originating from the nucleus j. Since a(ω) = a1 (ω) + a2 (ω), then the full harmonic spectrum is given by |a(ω)|2 = |a1 (ω)|2 + |a2 (ω)|2 + 2Re(a1 (ω)a∗2 (ω)) .

(9.44)

It is clear that ||a1 (ω)|2 + |a2 (ω)||2 is the harmonic spectrum without interferences, and 2Re(a1 (ω)a∗2 (ω)) is the interference term. Therefore, plotting |a(ω)|2 and |a1 (ω)|2 + |a2 (ω)|2 side by side provides a direct quantitative, unambiguous insight into influence of interferences in the harmonic spectra. Plots of |a(ω)|2 and |a1 (ω)|2 + |a2 (ω)|2 for various orientations of the molecule are displayed in Fig. 9.4 for the peak intensity I = 3 × 1014 W/cm2 , and in Fig. 9.5 for the peak intensity I = 5×1014 W/cm2 . Since the harmonic spectra |a1 (ω)|2 and |a2 (ω)|2 originating from each nucleus of H+ 2 exhibit both odd and even harmonics (except for χ = 90o ), it is not surprising that the harmonic spectrum of H+ 2 without interferences (see dotted lines in Fig. 9.4, and Figs. 9.5(f) – 9.5(i)) also contains both odd and even harmonics. However, it appears that only odd harmonic are present in the full harmonic spectra |a(ω)|2 (see solid lines in Fig. 9.4, and Figs. 9.5(a) – 9.5(e)) which includes interferences. In other words, interferences lead to a cancellation of even harmonics. In fact, the interference term restores the overall inversion symmetry of the system, leading to only odd harmonics. Another interesting feature from Figs. 9.4 and 9.5 is the fact that the interference term leads to a strong suppression of a relatively broad band of


189

Fig. 9.3. Time profiles (in arb. units) of the 85-th harmonic emitted by the nucleus 1 (solid lines) and by the nucleus 2 (dashed lines) of H+ 2 , for laser orientation angles χ shown. In the bottom plot ((e) χ = 90o ) where the time profiles for the two nuclei are identical, we also plot (with dots) the kinetic energy (in units of the ponderomotive energy Up ) of the returning classical electrons vs their first return times (see text for more details). Time is shown in units of the laser period

consecutive harmonics, leading to a minimum in the harmonic spectrum of H+ 2 . With increasing χ, the size of the band of suppressed harmonics increases, and the location of the minimum moves to higher harmonic orders. The location of these minima agrees with the interpretation in [21], where it was

190


shown that the occurrence of minima or maxima in the harmonic spectra depends on the interference term: k·R I(k) = eik·r1 + eik·r2 = 2 eik·(r1 +r2 )/2 cos (9.45) 2 where k = 2π/λ is the recolliding electron momentum, and λ its wave number. The electron momentum k is polarized along the electric field, so that k ·R = kR cos χ. Destructive interference (minimum in the harmonic spectra) occurs when I(k) = 0, i.e., when k · R = (2p + 1)π, p = 0, 1, 2, . . . .

(9.46)

Similar interferences were originally pointed out in angular distributions of ATI spectra of H+ 2 [20]. 9.4.2 Acceleration Versus Dipole Photon Emission It has been shown that the correct evaluation of the harmonic spectrum is to proceed via the Fourier transform a(ω) of the acceleration of the dipole or the equivalent (9.38) where d(t) = Ψ (t)|(−r)|Ψ (t) is the mean dipole moment of the electron, and Ψ (t) the exact time-dependent wave function at time t. For simplicity, the turn-on of the laser pulse is chosen at t = 0, and its turn-off at t = T . Exploiting Ehrenfest’s theorem, a(ω) can be expressed as in (9.40) in terms of the gradient of the Coulomb potential [51]. It is quite common [57] to evaluate the harmonic spectrum using the Fourier transform T of the dipole moment d(ω) = −ω 2 0 eiωt d(t) dt. Clearly, expressions of a(ω) ˙ ) and d(T ) vanish at the end of and d(ω) are equivalent if the transients d(T the pulse (see Equation 9.40). Using a(ω) (the acceleration form) emphasizes maximum acceleration at the nuclei, whereas using d(ω) (the dipole form) emphasizes the spatial extent of electron trajectories. Below, we analyze the two forms, in comparison with our exact numerical calculations. We start by approximating the time-dependent wave function by a superposition of the ground state wave function Ψ0 , and a continuum wavepacket Ψc , which describes the recombining electron [57]: Ψ (r, t) = β(t) e−iE0 t Ψ0 (r) + Ψc (r, t)

(9.47)

E0 = −Ip is the ground state energy, and β(t) is the ground state probability amplitude. We assume that there is little depletion of the initial state, so that β(t) ≈ 1. For a single recollision event, Ehrenfest’s theorem gives a(ω) = eiE0 t Ψ0 |e · ∇V (r)|Ψc (t) eiωt dt + c.c., (9.48)


191

Fig. 9.4. Full harmonic spectrum |a(ω)|2 (solid curve) of H+ 2 , and harmonic spectrum |a1 (ω)|2 + |a2 (ω)|2 of H+ 2 without interferences (dotted curve). The arrow covers the set of harmonics that are suppressed by at least one order of magnitude due to interferences. The internuclear distance and laser pulse duration are the same as in Fig. 9.2. The peak intensity of the laser is I = 3 × 1014 W/cm2

after neglecting terms involving continuum-continuum transitions, and after ignoring the term that involves the electric field E(t) as for (9.41). We approximate the initial state wave function by a linear combination of atomic orbitals (LCAO) [24, 58] Ψ0± = N0± e−α± r1 ± e−α± r2

(9.49)

192


o Fig. 9.5. Harmonic spectrum |a(ω)|2 from H+ 2 (left plots: (a) for χ = 0 ; (b) for o o o o χ = 30 ; (c) for χ = 40 ; (d) for χ = 50 ; and (e) for χ = 90 ), and harmonic spectrum |a1 (ω)|2 + |a2 (ω)|2 from H+ 2 without interferences included (right plots: (f ) for χ = 0o ; (g) for χ = 30o ; (h) for χ = 40o ; (i) for χ = 50o ; and (j) for χ = 90o ), for various angles χ shown. In each left plot, the arrow points to the approximate location of a minimum induced by interferences. For χ = 90o , |a(ω)|2 = 4|a1 (ω)|2 = 4|a2 (ω)|2 . The internuclear distance and laser pulse duration are the same as in Fig. 9.2. The peak intensity of the laser is I = 5 × 1014 W/cm2 and λ = 800nm

where N0± is a normalization constant and α’s are screening constants. We assume that the initial state could have a gerade (+) or ungerade (−) symmetry. The real parameter α± can be adjusted so that the energy associated with the wave function (9.49) agrees with the exact energy of ground or first


193

excited state of H+ 2 [2, 58]. The continuum wave function describing the electron returning to the molecular core with energy k2 /2 is approximated by the plane wave Ψc (t) = ei(k·r−E t) ,

Ek = k2 /2

(9.50)

This leads to a± (ω) = N0± Zδ(E0 − Ek + ω) −α± r1 e · r1 e · r2 −α± r2 · dr e ±e + 3 eik·r + c.c r13 r2

(9.51)

Consider the integral

e−αr1

e · r1 e · r2 + 3 r13 r2

eik·r dr

(9.52)

The quantity e−αr1 = e−α|r+R/2| is maximum at r1 = 0 (i.e., at r = −R/2), 1 2 and is negligible elsewhere. e·r is also maximum at r = −R/2, while e·r is r3 r3 1

2

maximum at r2 = 0 (i.e., at r = R/2) where e−αr1 = e−α|r+R/2| is negligible (provided that R is relatively large). Therefore, the second term in (9.52) is negligible compared to the first one, so that, e · r1 e · r1 e · r2 −αr1 ik·r e e + dr ≈ e−αr1 3 eik·r dr (9.53) 3 3 r1 r2 r1 Similarly, one can show that e · r1 e · r2 e · r2 ik·r e−αr2 + e dr ≈ e−αr2 3 eik·r dr r13 r23 r2

(9.54)

This gives a± (ω) = N0± Zδ(E0 − Ek + ω) −α± r1 e · r 1 ik·r −α± r2 e · r 2 ik·r · e e dr ± e e dr r13 r23 +c.c.

+ , e·r = N0± Zδ(E0 − Ek + ω) e−ik·R/2 ± eik·R/2 e−α± r 3 eik·r dr r +c.c. (9.55)

After making a change of variables r1 → r and r2 → r. The above equa tion clearly exhibits the more general interference term e−ik·R/2 ± eik·R/2 , where the + and − signs correspond the gerade (+) and ungerade (−) initial

194


states, respectively. For the initial ground state of H+ 2 (gerade) involved in this work, one retrieves the interference term I(k) given by Eq. (9.45). Such an interference was originally obtained in ATI spectra in H+ 2 [24]. Let Nmin be the harmonic order corresponding to a minimum in the harmonic spectrum of H+ 2 . The energy of the harmonic photon corresponding to this minimum is Nmin ω0 . If k is the electron momentum at the instant of recombination of the electron wavepacket with the ground state, then we may write 1 2 k = Nmin ω0 2

(9.56)

Note that we consider (9.56), instead of 12 k2 = Ip + Nmin ω0 which explicitly contains the ionization potential Ip of the molecule, and which naturally emerges from (9.55). The reason for this is that the recollision occurs near the nuclei, when the electron momentum k in (9.56) already encompasses the effect of the Coulomb potential. Obtaining k from the destructive interference condition given by Eq. (9.46), one can use (9.56) to derive the harmonic order Nmin at which a minimum in the harmonic spectrum is expected. For various orientation angles, we summarize in Table 9.2 the harmonic orders Nmin at which a minimum occurs in the harmonic spectra of H+ 2. The second column in Table 9.2 contains the harmonic orders obtained in this work, after solution of the TDSE (see arrows in Figs. 9.5 and 9.4). The harmonic order in the third column are obtained by using Eqs. (9.46) and (9.56) as described above (“Acceleration” values). It appears that results from the solution of the TDSE agree quite well with “Acceleration” values. This indicates that the simple emission model from a continuum to a bound state via an acceleration mechanism arising from electron recollision reproduces the interference pattern quite well. Note however, that the minima obtained in Figs. 9.5 and 9.4 from the full 3D time TDSE are rather broad, suggesting that many electron trajectories with different momenta actually contribute to each minimum. We examine next the transition matrix element TH which determines the intensity of the HHG spectrum. Consider the multiphoton transition from the initial electronic state |0 to a continuum state |k and then back to the same initial ground state by emission of a photon of energy E = N ω0 . The transition matrix element can be considered as a hyper-Raman process through the continuum state |k and is written as [60] 0|TH |0 =

dEk lim →0

= PP

dEk

0|TN |k k| d|0 E − Ek + i

(9.57)

0|TN |k k| d|0 − iπδ(E − Ek ) 0|TN |k k| d|0 E − Ek (9.58)


195

Table 9.2. Harmonic order at which an interference minimum occurs in the harmonic spectrum of H+ 2 , for various orientation angles χ. Results from the solution of the TDSE (second column) are compared with those obtained from the acceleration formula given by (9.45) using I(k) = 0 (third column), and with those derived from the dipole formula given by Eq. (9.64) using I(k) = 0 (fourth column) Angle

Harmonic order Nmin at the minimum

χ

TDSE (This work)

Acceleration + LCAO

Exact Dipole + LCAO

0 30 40 45 50 60 75 90

≈ 23 ≈ 27 ≈ 39 ≈ 43 ≈ 51 ≈ cutoff > cutoff None

22 29 37 43 52 87 323 ∞

42 55 68 79 95 153 553 ∞

where the integral sums over all intermediate continuum states |k. The total harmonic transition matrix element of TH is separated into a nonresonant principal part (PP) integral and a resonant transition E = Ek , where E is the initial (final) total energy of the laser-molecule system. TN is an (unknown) intermediate transition operator corresponding to the multiphoton transition from the initial bound state |0 to the continuum state |k from which photon emission occurs. In (9.58), we have emphasized the dipole form of the photon emission process. For continuum energies Ek much larger than the ionization potential Ip , which is the threshold energy for continuum excitation, the PP integral becomes negligible due to cancelation from fluctuations of the denominator: E − Ek > 0 and E − Ek < 0 in (9.57). Only in this limit can we assume that the total transition moment depends on the resonant recombination process, i.e., 0|TH |0 ∝ k| d|0

(9.59)

The absolute phase of (9.57) is an essential factor in attosecond pulse synthesis [7] and depends on both nonresonant and resonant contributions. Note however that instead of the acceleration formula of Eq. (9.48), one could use the traditional dipole formula d(ω) = eiE0 t Ψ0 |e · r|Ψc (t) eiωt dt + c.c. (9.60) which involves the transition matrix (9.59). Interestingly, using the wave functions (9.49) and (9.50), the above integral can be evaluated exactly, without

196


using the approximations described earlier in the evaluation of the integral (9.48). Indeed, ± i(E0 −E +ω)t d (ω) = e dt N0± e−α± r1 ± e−α± r2 e · r eik·r dr + c.c. −α± r1 = N0± δ(E0 − Ek + ω) e ± e−α± r2 e · (−i∇k ) eik·r dr + c.c. .

/ = −iN0± δ(E0 − Ek + ω)e · ∇k e−ik·R/2 ± eik·R/2 Φ± 1s (k) + c.c. (9.61) where Φ± 1s (k) =

8πα± e−α± r eik·r dr = 2 2 α± + k2

(9.62)

is the Fourier transform of the atomic orbital e−α± r . For the case of a gerade (+) ground state considered in this work, one obtains d+ (ω) = −2iN0 δ(ω − Ek ) Φ+ 1s (k) I(k)

(9.63)

where I(k) is given by I(k) =

e·R sin 2

k·R 2

+

4e · k 2 + k 2 cos α±

k·R 2

(9.64)

Imposing the destructive interference condition I(k) = 0, one arrives at discrete numerical solution that depends on an integer p as in (9.46). The resulting harmonics orders at which minima are predicted are given in the fourth column of Table 9.2. These results do not agree with our time-dependent calculations in the second column, as well as with the predictions of the interference term I(k) obtained from the acceleration form. This means that obtaining the harmonic spectrum via the dipole fails to predict the interference effects correctly. This agrees with previous studies suggesting that using the acceleration is the most appropriate way for calculating harmonic spectra [51]. The acceleration form emphasizes maximum acceleration at the nuclei, thus intuitively, the acceleration form more accurately reproduces the two-center character of the molecule, in contrast to the dipole form that emphasizes the spatial extent of electron trajectories. Furthermore, the acceleration and dipole matrix elements (9.48) and (9.60) are equivalent only if one uses exact wave functions, i.e., for the continuum state |k, one should use the exact Coulomb two-center wave functions of H+ 2 . We note that the free electron states eikr are not orthogonal to the initial wave functions (9.49), and thus can give spurious results [61]. The results above confirm the superiority of the acceleration form of the photo-emission matrix element in explaining interferences in molecular harmonic generation.


197

9.5 High Harmonic Generation at Large Internuclear Distances The recent theoretical works on two-center molecules has shown interesting effects specific to molecules as described in the last section. Extended systems E such as molecules at large internuclear distance R = α, πα 2 where α = ω2 is the ponderomotive radius offer new possibilities to extend the plateau of the emitted photon [23, 37, 38]. We present in this section, the results of high harmonic generation spectra (HHG) of the one electron linear triatomic + + + + H2+ 3 (H − −H2 ) where H and H2 are separated by large internuclear distances R submitted to an ultrashort (2 cycles) 800nm (ω0 = 0.057a.u.) laser pulse and intensity I = 4 × 1014 Wcm−2 (E0 = 0.1069a.u.) at CEP ϕ = 0, π. We considered the laser field polarized linearly along the internuclear z axis. We solved The TDSE describing the motion of the electron in the field of H+ and H++ 2 , in cylindrical coordinates, using split-operator method. CEP is the carrier envelope phase which introduces considerable asymmetries in ionization and ATI spectra for few-cycle pulses. These asymmetries provide tools for measuring CEP [62, 63]. The electric field we obtain from E(t) = −∂A(t) has at t = 0 a maximum positive value for ϕ = π and ∂t a minimum negative for ϕ = 0. For these laser parameters, the ponderomotive energy is Up = 0.879a.u. (24eV) and the ponderomotive radius is α = 33a.u. We thus set the distance R between H+ and the center of mass πα of H+ 2 , R = α = 33a.u. and R = 2 = 52a.u.. We consider the internuclear + distance of H2 fixed at RH + = 3a.u.. Since we fix the distance between H+ 2 πα and H+ 2 equal to R = α = 33a.u. and R = 2 = 52a.u., the maximum energy 2+ emitted by the H3 will be respectively E ≤ Ip + 4 Up and E ≤ Ip + 6 Up . For each R value, we have calculated the HHG spectra of H2+ in two different 3 coherent states which could be prepared by bifurcation of electronic states at avoided crossings of excited electronic states [64]. √1 First we consider H2+ 3 initially in the coherent state ψ0 = 2 (ψ1σ + ψ3σ ). We represent this latter state as a combination between the fundamental state 1σ which is essentially the bonding 1σg molecular orbital of H+ 2 and the third bound state 3σ of H2+ which is essentially the 1s atomic orbital of H. 3 A representation of this H2+ coherent initial state is presented in Fig. 9.6 3 (1a). We study also the HHG of H2+ initially in the coherent state ψ 1 = 3 √1 (ψ2σ +ψ3σ ), where 2σ is the second bound state of H2+ which is essentially 3 2 the antibonding molecular 1σu orbital of H+ 2 and the 3σ state is again the 1s H atom orbital. A representation of this coherent state is also illustrated in Fig. 9.6 (1d). 9.5.1 High Harmonic Generation Spectra at R = α = 33 a.u. We begin by presenting the results for the case H2+ 3 is initially in the coherent state ψ0 (H+ (1s)—H+ (1σ )). We show in Fig. 9.6 (1b) the HHG spectra of g 2

198


Fig. 9.6. Total harmonic generation spectra of H2+ for ϕ = 0, π for two coherent 3 initial states ψ0 = R = πα = 52 a.u. 2

1s(H)+1σg (H2+ ) √ , 2

ψ1 =

1s(H)+1σu (H2+ ) √ 2

and for R = α = 33 a.u.,

the system under a two cycle laser pulse of CEP ϕ = 0, π. Both spectra show a cutoff at about the harmonic 80th. To see by which species H+ or H+ 2 the 80th harmonic is emitted we calculate for both CEP the HHG originating from individual nuclei. We present in Figs. 9.7 (2a), (2b) the HHG of each nuclei for respectively ϕ = 0 and π. In Fig. 9.7 (2a), for ϕ = 0, the electron which is ionized from H+ travels the distance R = α to collide with H+ 2 and generate the harmonic 80th (80ω = Ip (H+ ) + 4U ) exceeding the atomic p 2 cutoff law Ip + 3 Up . In Fig. 9.7 (2a) the harmonic spectra of each nuclei + of H+ shows that the 2 exhibit maxima and minima. The spectrum of H + electron ionized from H recolliding with it emits harmonics of energy less than Ip (H+ ) + 2.5Up . In Fig. 9.7 (2b), for CEP ϕ = π, the harmonic 80th


199

Fig. 9.7. HHG originating from individual nuclei of H2+ initially in ψ0 = 3 1s(H)+1σg (H2+ ) √ 2

and ψ1 = is fixed to R = α = 33 a.u.

1s(H)+1σu (H2+ ) √ 2

for ϕ = 0, π. The internuclear distance

+ is generated by H+ . The electron ionized from H+ 2 travels to collide with H + emitting less intense harmonics of energy Ip (H ) + 4.5Up . The HHG of H+ 2 shown in Fig. 9.7 (2b) shows a cutoff at the harmonic 50th (2.3 Up ). This spectrum corresponds to the HHG yield of an electron originating from H+ 2 which after ionization recollides with its parent ion. In Fig. 9.6 (1b) we remark that at around the harmonic 80th the HHG spectrum for ϕ = 0 is six orders of magnitude more efficient than at ϕ = π. The first bound state 1σ of H2+ 3 which is essentially the 1σg (H+ 2 ) is strongly bound by the potential, so that 2+ ionization in H2+ 3 occurs mainly from H. When the H3 is initially in the ψ1 + coherent state (H+ (1s)–H2 (1σu )) submitted to the 2 cycle laser pulse, Fig. 9.6 (1e) shows that the HHG spectrum for both CEP ϕ = 0 and π has a cutoff at about the harmonic 77th, where the difference of efficiency of both spectra is + of one order of magnitude. The second bound state 2σ of H2+ 3 (1σu (H2 )) and the third bound state 3σ (1s H+ ) can be delocalized, so the ionization occurs then from both H+ and H+ 2 . We show in Figs. 9.7 (2c), (2d) the HHG spectra originating from individual protons of H2+ 3 for respectively ϕ = 0 and π. In Fig. 9.7 (2c), for CEP ϕ = 0 the 77th harmonic is emitted from H+ 2 (77ω =

200


Fig. 9.8. HG originating from individual nuclei of H2+ initially in ψ0 = 3 1s(H)+1σg (H2+ ) √ 2

1s(H)+1σu (H2+ ) √ 2

and ψ1 = = 52 a.u. is fixed to R = πα 2

for ϕ = 0, π. The internuclear distance

+ Ip (H+ 2 ) + 4 Up ) as a result of collision of the electron ionized from H . In + Fig. 9.7 (2d) for ϕ = π this harmonic is emitted from H (77ω = Ip (H+ ) + 4.3 Up ) after collision of the electron ionized from H+ 2 . The HHG spectra of non-symmetric systems under short laser pulse allows therefore for identifying the direction of electron transfer and thus allows for identifying the CEP via its influence on asymmetric ionization in asymmetric molecular systems [65]. Harmonic generation spectra for both ϕ = 0 and π and for H2+ 3 initially in ψ0 and ψ1 via short laser pulses show broad peaks and different minima. In the next subsection, where we present results of HHG for R = πα 2 , we explain the reason for the presence of these minima based on a harmonic time profile using Gabor time profile analysis [22]. We compare the HHG time profiles of some harmonics at the maxima and minima, thus identifying the classical electron trajectories responsible for the dynamics of HHG at large distances.

9.5.2 High Harmonic Generation Spectra at R =

πα 2

= 52 a.u.

We present in Figs. 9.6 (1c), (1f) the total HHG spectra of H2+ initially 3 in respectively the coherent states ψ0 and ψ1 for CEP ϕ = 0 and π. We


201

Fig. 9.9. Results for R = πα : (4a) Time profile of harmonics emitted by H+ 2 (1σg ) 2 at maximum N = 97 and minimum N = 88 for both inner and outer nucleus at ϕ = 0. (4c) Time profile of harmonics emitted from H+ 2 (1σu ) at maximum N = 95 and minimum N = 86 (outer nucleus), N = 83 (inner nucleus) at ϕ = 0. (4b) Plot initially in ψ0 around H+ of the electron density of H2+ 3 2 at three pulse time. (4d) 2+ the same as (4b) but for H3 initially in ψ1

see that for R = πα 2 , the cutoff region is extended as compared to R = α case. In Fig. 9.6 (1c) the cutoff occurs at the harmonic 97th. In Fig. 9.6 (1f) is initially in ψ1 the HHG spectra for ϕ = 0 show a cutoff where the H2+ 3 at about the harmonic 95th and a cutoff at about the 90th harmonic for ϕ = π. We calculate the HHG originating from individual protons to see from which nuclei the harmonics in the cutoff region is emitted. In Figs. 9.8 (3a), (3b) we show the results of the nuclear HHG spectra of H2+ initially 3 in ψ0 for respectively ϕ = 0 and π. In Fig. 9.8 (3a) where ϕ = 0 the 97th + harmonic (97ω = Ip (H+ 2 ) + 5.4 Up ) is emitted from H2 , in contrast with ϕ = π (9.8 (3b)) where it is less intense and emitted from H+ (Ip (H+ ) + 5.5 Up ). Figures 9.8 (3c) (3d) show the HHG originating from each nuclei for H2+ 3 initially in ψ1 for respectively ϕ = 0 and π. In Fig. 9.8 (3c) for ϕ = 0, + the harmonic 95th (95ω = Ip (H+ 2 ) + 5.3 Up ) is emitted from H2 (1sσu ). In Fig. 9.8 (3d) where ϕ = π, the harmonic 90th (90ω = Ip + 5.2 Up ) is emitted from H+ (1s). To explain the presence of the minima observed in all harmonic spectra, we calculate the time profile of some harmonics at some maxima and minima

202


of the HHG spectra. Since the mechanism of suppression of harmonics at the minima is the same for ϕ = 0 and π, we restrict our study to the case of the laser pulse CEP ϕ = 0. In Fig. 9.9 (4a), we show the harmonic time profile around the H+ 2 (1σg ) HHG maximum N = 97 and minimum at N = 88 for ϕ = 0. In Fig. 9.9 (4c) we illustrate the harmonic time profile at the maximum N = 95 and minimum N = 86 (outer nucleus) and N = 83 (inner) of the H+ 2 (1σu ) HHG. We note that the harmonic at order N = 97 is created by a single trajectory at t2 ≈ 0.2 cycle. The electron responsible of this harmonic collides with H+ 2 when the field is approaching zero, in agreement with the semiclassical recombination model of atomic HHG [57]. The electron density at this combination time of single trajectories is nearly symmetric (density 2 in Figs. 9.9 (4b), (4d)). At minima of the HHG spectra, we see now the contribution of always two trajectories, one electron combination at t1 ≈ 0.1 cycle and another possible combination at t3 ≈ 0.3 cycle (Figs. 9.9 (4a), (4c)). These can be readily correlated to asymmetric electron densities in H+ 2 since the electric field at t1 is negative and at t3 it is positive. The electron density consequently localizes on each nucleus (densities 1 and 3 in Figs. 9.9 (4b), (4d)). Of special note is that at t1 and t3 the corresponding H+ 2 electron densities are out of phase when comparing the electron combination into 1σg (Figs. 9.9 (4a), (4b)) and 1σu (Figs. 9.9 (4c), (4d)) molecular orbitals of H+ 2 . In Fig. 9.9 (4a), electron combination at t3 ≈ 0.3 cycle where the field is positive occurs mainly on the left (inner) nucleus of H+ 2 in agreement with localization of the electron on that nucleus (density 3 in Fig. 9.9 (4b)). In contrast when the H2+ 3 is initially in ψ1 , the electron combination at t1 ≈ 0.1 cycle, where the field is negative, occurs mainly on the left (inner nucleus) ( Fig. 9.9 (4c)), in agreement with localization of the electron in H+ 2 on that nucleus (density 1 in Fig. 9.9 (4d)). Similarly, for the outer nucleus of H+ 2 (1σg ) (Fig. 9.9 (4a)), the electron collides at t1 ≈ 0.1 cycle where the electron density is localized on that nucleus (density 1 in Figs. 9.9 (4b)). When H+ 2 is initially in 1σu , the electron from H+ collides with the outer nucleus at t3 ≈ 0.3 cycle (density 3 in Fig. 9.9 (4d)). We arrive therefore at the main conclusions: i) single electron combination trajectories produce maxima in HHG when the electric field is approaching zero in agreement with the atomic model [57], ii) double combination trajectories occur at minima of HHG, due to combination at nonzero fields and consequently localization of electrons on one or the other nucleus. The localization of the nonionized electron is out of phase for the 1σg orbitals as compared to the 1σu orbital. This reverses the combination times on each nucleus for the 1σg when compared to the 1σu orbital. The main reason for this reversal of combination times, which depends on the symmetry of the orbital is the change of sign in polarizability of the excited 1σu state as compared to the 1σg orbital. The frequency dependence of polarizabilities is 2 (ωexc −ω 2 )−1 , where ωexc is the excitation energy and ω is the laser frequency. For ground states, ωexc > ω whereas for excited states ωexc < ω generally. This changes the sign of polarizabilities and has been used as a tool previously


203

to focus or defocus atom beams [66] and control vibrational wavepacket [67]. In the present calculation, the 1σu electron is displaced in opposite direction to the 1σg electron (Fig. 9.9 (4b), Fig. 9.9 (4d)), thus creating reversal of combination times at each nucleus in H+ 2 for the 1σu state as compared to the 1σg state. Surprisingly, this difference of electron displacement of the nonionized electron in the 1σu and 1σg orbitals with respect to the combining electron does not seem to influence the dynamics, the net overall effect being a suppression of the harmonics, i.e., the creation of minima. We offer a tentative explanation of this suppression of harmonic intensities. Localization of electrons on the left nucleus into the left atomic orbital 1σ +1σ 1sL is obtained from the combination of molecular orbitals: u√2 g ≈ 1sL , whereas localization into 1sR , the right atomic orbital is obtained by the 1σ −1σ out of phase combination, u√2 g ≈ −1sR . Assuming a direct transition from the continuum state | c, the total transition moment from the two trajectories which produce low intensity harmonics such as the 88th (Fig. 9.9 (4a)) and 86th (Fig. 9.9 (4c)), is c |r| 1sL − c |r| 1sR 0, thus leading to suppression and minima in molecular HHG. Clearly it is the CEP which determines the direction of bound state localisation to which recombination of the ionized electron occurs.

9.6 Conclusion A most successful model of intense laser pulse interactions with atoms has been the recollision model of the ionized electron with its parent ion [14]. This model was generalized to collision with neighboring ions [23] thus allowing for extension of HHG plateaus beyond the atomic cut-off law N ω = Ip + 3.17 Up . HHG in molecules involves interferences due to the multicenter nature of the electron–nuclei recombination process. Current interest in molecular HHG is increasing due to its potential for molecular orbital tomography [26–28] and even possible monitoring of nuclear wavepacket motion on near fs time scales [68]. Furthermore, HHG being the only current source of coherent attosecond X-Ray pulses [7], molecular HHG is offering new avenues for creating high orders of harmonics. The dynamics of molecular HHG is thus of current interest due to the inherent interference process arising from the multicenter nature of electron-nuclei interaction in molecules. In the present chapter, we have reviewed some of the issues arising in calculating single-molecule HHG. Extension of the recollision model to nonzero initial electron velocity, different gauges for calculating and interpreting angular distributions of harmonics and finally HHG at large internuclear distances have been addressed in detail in order to lay a solid foundation for the new emerging imaging science: Laser Induced Electron Diffraction (LIED) [24,25]. In view of the general time scale of electron recombination with its parent ion or neighboring ions as such sub-cycle time periods, then HHG will become

204


a special tool for measurement of ultrafast imaging from few femtosecond to attosecond times in the case of coherently prepared electron wavepackets in molecules [69] or at large intermolecular distances where electron transfer can be controlled by the CEP of few cycle ultrashort laser pulses [70].

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10 Nonlinear Multiphoton Process in the XUV Region and its Application to Autocorrelation Measurement Katsumi Midorikawa and Yauso Nabekawa Laser Technology Laboratory RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan, e-mail: [email protected] Summary. Phase-matched high-harmonics produce a high focused intensity sufficient to induce nonsequential multiphoton processes such as two-photon double ionization and above threshold ionization of rare gases in the XUV region. Using two-photon double ionization in He, the pulse width of the 27th (42 eV) harmonic was measured by an autocorrelation technique, and found it to be 8 ns. A train of attosecond pulses was also visualized directly by the energy-resolved autocorrelation with the above threshold ionized electrons.

10.1 Introduction The progress of chirped pulse amplification and femtosecond Ti:sapphire laser enables the study of the interaction of strong optical fields with atoms and molecules. A variety of interesting phenomena, including high-harmonic generation, high energy radiation/particles emission, and Coulomb explosion of molecules, have been investigated intensively. The underlying physics of these nonlinear optical phenomena have been emerging through the interaction with low-frequency optical radiations such as infrared or visible light; however, little has been understood concerning the interaction of intense high-frequency radiation such as XUV or soft x-ray. Great interest has been aroused recently in the interaction of intense high-frequency radiation with matters. Yet no one could observe it because of the lack of an intense coherent light source in this spectral region. The induced phenomena are expected to be much different from those caused by low frequency radiation because the electron quiver energy-the cycle averaged electron energy in the optical field-is proportional to a square of the wavelength. Photon energy higher than ionization potentials of atoms and molecules also would differentiate multiphoton ionization process from those observed with infrared or visible photons. This research explores the generation of intense soft x-ray pulses and its application to nonlinear multiphoton processes. Using such nonlinear processes, the temporal width of the 42-eV soft x-ray pulse was measured directly by an autocorrelation technique. The observation of nonlinear optical process in the soft x-ray region has been a very attractive and challenging area of research in quantum electronics since the first observation of second-harmonic

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generation and two-photon excitation in the visible region in 1961 [1]. A train of attosecond pulses was also characterized by energy-resolved autocorrelation with photoelectrons produced through two-photon above threshold ionization (ATI). Although characterization of attosecond pluses is of great importance for the progress of ultrafast optical science, most research owe its success to the indirect methods using two-color ATI [2,3]. An autocorrelation signal given by the nonlinear interaction between optical fields and matter, on the other hand, directly reflects the shape of the pulse in the primary data, whether it is trainlike or isolated. Our method using two-photon ATI is the natural extention of the well-developed method in the infrared or visible region. The high-intensity coherent XUV radiation produced by phase-matched high-order harmonics explores these nonlinear optical processes in the XUV region and its applications, which are beyond the reach of conventional light sources.

10.2 High-Power High Harmonic Generation For application of high harmonics to nonlinear optics in the XUV region, high focused intensity is crucial because the cross section of nonlinear processes in atoms or molecules tends to decrease rapidly with decrease of the pump wavelength [4]. To increase high harmonic energy by improving conversion efficiency from the pump energy, phase matching is essential. It is, however, not easy to satisfy the phase matching condition along the interaction length, because in contrast to low-order harmonic generation in the perturbative regime, the atomic dipole phase is dependent on the driving laser intensity for high-order harmonic generation [5]. This means that the atomic phase varies rapidly around the focus when the pump pulse is tightly focused in the nonlinear medium. Furthermore, in addition to the medium’s dispersion, nonlinear phenomena such as self-focusing and plasma defocusing accompanying high-intensity pump pulse propagation also make the phase matching quite difficult to fulfill. Overcoming such difficulties, the energy scaling of high-order harmonics under the phase-matched condition has been achieved using a long interaction length and a loosely focused pumping geometry [6]. This method shows a linear increase in harmonic energy with respect to the geometrical focusing area of the pump pulse, while keeping an almost perfect spatial profile of the harmonic output [7]. Figure 10.1 shows the experimental setup for high-order harmonic generation using a long focusing pumping geometry. Details are described elsewhere [6]. Experiments were carried out with a 10Hz Ti:sapphire laser producing an output of 200mJ with a pulse width of 35fs and a beam diameter of 50mm. The wavelength was centered at 800nm. The pump pulse was loosely focused with an f = 5m long fused silica lens, and delivered into the target chamber through a CaF2 window. CaF2 was selected as a window material to avoid self-phase modulation at the input window. The input energy and

10 Nonlinear Multiphoton Process in the XUV Region

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Fig. 10.1. Schematic of experimental setup for high harmonic generation using a loosely focusing geometry

diameter were adjusted with an aperture placed in front of the focusing lense, depending on nonlinear media of rare gases. Diameters of the aperture were set at 15mm and 20mm for Xe and Ar, respectively. We set the focus around the entrance pinhole of the interaction cell. The interaction cell had two pinholes on each end surface of the bellow arms. These 1-mm diameter pinholes isolated the vacuum and gas-filled regions. The interaction length was variable from 0 to 150mm in the interaction cell and set typically at 100mm. Rare gases were statically filled in the interaction cell. With this experimental setup, peak powers of 130MW at 62.3nm in 0.6-torr Xe [8], 10MW at 29.6nm in 1.8-torr Ar [6], and 1MW at 13.5nm in 10-torr Ne [9] were obtained. The focusing property of the 27th harmonic wave was investigated by observing a visible image with Ce:YAG scintillator and an ablation pattern produced on a gold-coated mirror placed at the focal position. The diameter of the circular hole produced on a gold mirror by single-shot irradiation was approximately 2μm, which agreed well with the scintillator experiment. The focused intensity of the 27th harmonic wave was estimated from those experimental results of the energy, spot size, and pulse width. Assuming that the pulse width of the 27th harmonic wave was the same as the fundamental pulse, the maximum focused intensity of 1014 W/cm2 was obtained [10].

10.3 Two-Photon Double lonization in He at 42 eV As the first step in investigating the interaction of intense high-frequency radiation with matter, two-photon double ionization of the He atom has been selected as a target. At the very high frequency and low intensity limit, double ionization by single photon absorption has been investigated in detail using synchrotron radiation; this process has been understood as “shake off”. On the other hand, in the low-frequency and high-intensity regime realized by femtosecond high intensity Ti:sapphire lasers, the occurrence of double ionization has also been reported and the process was explained by a rescattering model. Between these two extreme experimental conditions, the occurrence of two-photon double ionization by intense XUV lights was theoretically pre-

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dicted about 15 years ago [11–13]. Particularly, two-photon double ionization in He has attracted much interest and has been studied theoretically in a number of papers because it provides insight into the electron–electron interaction (that is, electron correlation) and paves the way for unexplored aspects of the three-body problem. In the experiment, the generated harmonics (200nJ/pulse at the 27th harmonic) and the fundamental pulses were separated by a SiC beam separator [14]. Then, the harmonics were introduced into an interaction chamber with a time-of-flight (TOF) mass spectrometer. The harmonics were focused on 3 He gas with a SiC/Mg multilayer mirror with a focal length of f = 50mm. The SiC/Mg mirror has a reflectivity of 24% at the 27th harmonic. We truncated the undesired component of the harmonic beam by placing an aperture with a diameter of 3mm between the beam separator and the interaction chamber, while the central part of the harmonic beam, including approximately 70% of the harmonic energy, can pass through this aperture due to the small divergence (∼0.7m rad) of the phase-matched harmonics. This aperture also eliminates ∼94% of the energy of the remaining fundamental pulse reflected by the beam separator, with the result that the energy of the fundamental pulse at the focal point is reduced to ∼100nJ from 20mJ at the entrance of the f = 5m focal lens. Consequently, the intensity of the residual fundamental is estimated to be 2 × 1012 W/cm2 at the most. The polarization of the laser and harmonics is parallel to the TOF axis. The energy of the 27th harmonic was estimated to be 24nJ/pulse in the interaction region. The spot size was estimated to be ω0 = 3.1μm in a separate experiment [15]. Thus, the intensity of the 27th harmonic was estimated to be 2 × 1013 W/cm2 . Here, we regard the pulse width of the 27th harmonic to be 8fs, based on our autocorrelation measurement described below. The confocal parameter was also determined to be 250μm from the same experiment [15]. The sample gas of He atom uses the isotope helium-3 (3 He) because the signal of 4 He2+ ions appears very close to the signal of molecular ions that originate from the residual water molecules. The inset of Fig. 10.2 shows the ion spectrum produced by the interaction between the focused 27th harmonic pulses and 3 He. The production of 3 He+ results dominantly from the one-photon absorption of 3 He. The strongest peak of 1.8μs can be assigned to 3 He2+ . The signal of 3 He2+ clearly appears between the H+ signal and H+ 2 signal. The generation of doubly charged He2+ confirms the observation of a nonlinear optical process (two-photon absorption) in the soft x-ray region [16,17]. To further confirm that the He2+ ions are produced via the two-photon process, the 27th harmonic intensity dependence of the He2+ yield was investigated. In order to simultaneously measure the relative intensities of the 27th-harmonic pulse and the corresponding yields of doubly charged helium ions at those intensities, we utilize the yields of singly charged helium ions as indicators of the intensities. We can assume that the yield of the singly charged ions is proportional to the intensity of the 27th harmonic, because there is no saturation in their production. The


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Fig. 10.2. Doubly charge 3 He ion yields as a function of the 27th harmonic intensity. Inset is the time of flight ion spectrum

result of this experiment is shown in Fig. 10.2. The result clearly shows the slope of two, giving more evidence of the occurrence of the two-photon double ionization. Theoretical calculations of the cross section of two-photon double ionization in He have been reported many groups [18–23]. We have estimated the cross section from the experimenally oberved He2+ yield and measured a pulse width of 8fs (as described below). Our value of 4 × 10−53 cm4 s agrees with Feng’s [21], Piraux’s [22], and Ishikawa’s [23] data. The cross sections of the other groups [18–20] are approximately one order larger than our value. The difference may result from the difference in the photon energy. They calcualted the cross section of the two-photon double ionization at a photon energy of 45eV, which corresponds to the 29th harmonic, while we used the 27th hamonic (41.8eV). The calculation of the dependence of the cross sec-

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tion on photon energy by Feng et al. can qualitatively explain the difference, because their results for the photon energy dependence indicate that the cross section of the two-photon double ionization increases with an increase in the photon energy [21, 22].

10.4 Autocorrelation Measurement of 42-eVPulse The XUV nonlinear optical process is not only interesting in the field of atomic or molecular physics, it is also of importance for the direct measurement of the pulse width by autocorrelation method in the ultrafast optics field. With the two-photon double ionization in He, the pulse duration of the 42-eV soft x-ray pulse was measured by means of autocorrelation [17]. For the autocorrelation measurement, an autocorrelator splits a measured pulse equally into two pulses and spatially overlaps the separated delayed pulses at the focus point. The construction of the autocorrelator for the XUV light is, however, not straightforward because no beam splitter or high reflectance-/transmittance optics are available. Therefore, a novel autocorrelator using a split beam separator was designed [24]. The delayed pair of harmonic pulses is produced by spatially dividing the harmonic beam with two beam separators of SiC substrates that are aligned with each other with a separation of approximately 100μm (the photograph of the split beam separator is shown in Fig. 10.3(a)). This spatial division of the harmonic beam is confirmed with a MCP and a CCD camera attached to the spectrometer behind the TOF chamber. The spatial and spectral distributions are shown in Fig. 10.3(b). We observe the semicircular profiles of the divided harmonic beams. One of the divided pulses is delayed or advanced to another pulse by moving one of the separators placed on a translation stage with a piezo-actuator. The measurement principle using this split beam separator is the same as an all-reflective interferometric autocorrelator [25].

Fig. 10.3. (a) Photograph of the split beam separator. (b) The specral and spatial (27th) intensity distribution of the harmonics afier the beam separator


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Fig. 10.4. Autocorrelation trace (solid circles with dotted line) of the 27th harrnonic pulse obtained by utilizing the two-photon absorption for the yield of a doubly charged ion of 3 He. The full width at half maximum (FWHM) of the autocorrelation is determined by fitting a Gaussian shape (solid curve) to the trace

The yield of doubly charged helium ions from each delay is shown in Fig. 10.4. We can see a typical autocorrelation trace in this figure, while the pedestals, which are inherited from those of the fundamental pulse, appear at a delay of about 40fs. The ratio of the peak height to the background in the autocorrelation signal integrated to a certain volume of the interaction region should be approximately three, which agrees well with the ratio in the trace in Fig. 10.4. A curve fitted to the central region of the trace assuming a Gaussian temporal profile, indicated as a solid √ curve in Fig. 10.4, results in a pulse duration of 8fs corresponding to 1/ 2 of the full width at half maximum of the fitted curve.

10.5 Direct Temporal Characterization of XUV Attosecond Pulse Train Although the temporal width of the soft X-ray pulse was measured directly by an intensity autocorrelation technique, the pulse width is limited by the bandwidth of the 27th harmonic wave. The narrow bandwidth of a SiC/Mg multilayer mirror has high reflectivity only for the 27th harmonic wavelength. If several harmonics are simultaneously focused with a broadband mirror, a train of extremely short pulses would be observed. As is well known, Fourier synthesis of harmonic wave fields exhibits a train of ultrashort pulses. The pulse width decreases in inverse proportion to the number of harmonics. The optical field in the XUV region generated as high order harmonics of a femtosecond laser pulse is one of the most significant examples of the Fourier synthesis because it can serve an attosecond pulse train. For characterization of an attosecond pulse train, several harmonics should be simultaneously measured and the information of individual harmonic phase should be determined. But the phase information can hardly

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be obtained from ion signals. Therefore, a new method to reconstruct an attosecond pulse train was proposed [26]. This method uses photoelectrons produced by the two-photon above threshold ionization process. In the ATI process, an ejected electron absorbs the photons in excess of the minimum required for ionization. The ejected electrons carry the information of the phase of the XUV pulse used for ionization. Paul et al. first reported the measurement of an attosecond pulse train and determined that the pulse width in the train is 250 as by analyzing the electron energy spectra produced with the two-color ATI [3]. This method, resolution of attosecond beating by intereference of two-photon transitions (RABBITT), gives us an information of relative phase between adjacent harmonics, but can not directly visualize the pulse shape from the experimental data. We can not straightforwardly imagine the reconstructed attosecond pulse train form the sideband spectra of the two-color ionized electrons. In order to determine the relative phase precisely, the intrinsic atomic phase originated from the two-color two-photon transition also must be calculated theoretically [3, 27]. Autocorrelation signal produced by two-photon ATI electrons, on the other hand, directly reflects the shape of the pulse in the primary data, although focused intensity higher than ∼1012 W/cm2 is required to induce two-photon transition in the XUV region. It is widely used for characterizing femtosecond pulses of visible lasers. In this two-photon ATI experiment, a fundamental laser pulse with a duration of 40fs and an energy of 13mJ is delivered from a chirped pulse amplification system of a Ti:sapphire laser, and focused into a static gas cell filled with xenon gas in a vacuum chamber. The detailed parameters are fixed according to the results reported in [8]. The pulse energies of the 11th-, 13th-, and 15th-order harmonic fields just behind the gas cell are all estimated to be higher than 1μJ [8, 15], and these high pulse energies are critical for observing the nonlinear interaction of the harmonic field without the help of another intense laser field. Two replicas of the harmonic fields after a Si beam separator are introduced in a magnetic bottle photoelectron spectrometer and focused with a concave mirror made of silicon carbide with a radius of curvature of 100mm. Argon gas, as target atoms of two-photon ATI, is supplied with a capillary tube attached to a pulsed gas valve. Approximately half of the ejected electrons by photoionizations are guided with a magnetic field, and then go through a small bore with a radius of 700μm. The kinetic energies of the electrons are resolved by measuring the time of flight of the electrons traveling into a flight tube behind the bore with the guidance of the magnetic field. To simplify the analysis of the ATI electron spectra, only three harmonics from 11th to 15th were selected by passing through a Sn thin-foil filter. When these three harmonics are simultaneously focused to Ar gas, five peaks from the lowest-order mode at 22nd (11th + 11th) to the highest order mode at


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30th (15th + 15th) are expected to appear in the photo-electron spectrum, corresponding to the equivalent modes at the 22nd, 24th, 26th, 28th, and 30th orders. But, we could not observe the 22nd order, because it would be lower than that at the 24th order, and the large signals of one-photon absorption at the 21st and 23rd order harmonic fields degraded the resolution at the 22nd order. The ATI electron spectral peaks exhibited correlated signals by changing delay between two replica of harmonic fields, as shown in Fig. 10.5. The attosecond beats that originated from Fourier synthesis of three incident harmonics were clearly observed at the 24th, 26th, and 28th orders. The period of this beat was −1.33fs, which corresponds to the half period of the optical field of the fundamental laser pulse. According to the analysis for frequencyresolved optical gating technique (FROG), the mode-resolved autocorrelation trace at the 26th order S26 , to delay τ , is approximately given by . /2 S26 (τ ) ∝ I13 + 2 I11 I15 cos (2ωf τ )

− 8I13 I11 I15 sin2 2ωf 2φ¨ cos(2ωf τ ) (10.1) where In2+1 (n = 5, 6, 7) is the intensity of the harmonic field at 2n + 1-th order, ωf is the angler frequency of the fundamental laser field, and φ¨ is the chirp defined at the optical frequency of the 13th order harmonic field. We concluded that the duration of the pulse in this train was 450 attoseconds and φ¨ is 1.3 × 10−32 f s2 by fitting S26 (τ ) to the experimental data.

Fig. 10.5. Energy-resolved autocorrelation trace of two-photon above threshold ionization electrons in Ar and reconstracted train of attosecond pulses

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We expect that the attosecond pulse train can be fully characterized with the more accumulation of the data and the fine adjustment in the all delay because the obtained spectrogram of the electrons corresponding to all the even odder harmonics can be regarded as the result of a FROG of the attosecond pulse train. This photoelectron analysis of nonresonant two-photon ionization for harmonic electric-field reconstruction (PANTHER) [26] will be an effective method for characterizing the attosecond pulse train of the harmonic field.

10.6 Conclusion Generation of intense XUV pulses by high-order harmonic and its application to nonlinear multiphoton processes are described. Phase-matched high harmonics by a loosely focusing geometry produce high focused intensity sufficient to induce nonlinear optical phenomena in the XUV region. Using two-photon double ionization in He, the temporal width of the 42eV soft x-ray pulse was measured directly by an autocorrelation technique. A train of attosecond pulses was also visualized directly by mode-resolved autocorrelation with two-photon ATI around the 20eV region. When we apply this two-photon double ionization to molecules, we can expect the occurrence of Coulomb explosion with attosecond temporal precision [28]. The advantages of XUV lights on ultrafast Coulomb explosion imaging have been reported by Chelkowski and Bandrauk [29]. For the characterization of an attosecond pulse train, there are still some remaining issues. The most crucial issue for fully characterizing an attosecond pulse train is the spectral broadening of the kinetic energy of electrons due to the local electric field induced by a large number of ions left with one-photon absorption, the so-called space-charge effect. This effect would be relaxed by decreasing the density of the target atoms. The reduced number of ATI electrons per laser shot would be compensated with a higher repetition rate if we could use a terawatt-class laser system at a 1kHz repetition rate [30]. Acknowledgement. We wish to thank Drs. K. Yamanouchi, H. Hasegawa, T. Shimizu, K. Furusawa, K. Ishikawa, E. Takahashi, and T. Okino for their experimental contribution and helpful discussions.

References 1. P.A. Franken, A.E. Hill, C.W. Peters, G. Weinreich: Phy. Rev. Lett. 7, 118 (1961) 2. P. Agostini: Atoms, Solids, and Plasmas in Super-Intense Laser Fields, ed by D. Batani, C.J. Joachin, S. Martellucci, A.N. Chester (Kluwer Academic/Plenum Publishers, New York, 2001) pp. 59- 81


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3. P.M. Pual, E.S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H.G. Muller, P. Agositini: Science 292, 1689 (2001) 4. S. Klarsfeld, A. Maquet: J. Phys. B 12, L533 (1979) 5. P. Saliéres, A. L’Huillier, M. Lewenstein: Phys. Rev. Lett. 74, 3776 (1995) 6. E. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, K. Midorikawa: Phys. Rev. A 66, 021802(R) (2002) 7. E. Takahashi, Y. Nabekawa, M. Nurhuda K. Midorikawa: J. Opt. Soc. Am. B 20, 158 (2003) 8. E. Takahashi, Y. Nabekawa, K. Midorikawa: Opt. Lett. 27, 1920 (2002) 9. E.J. Takahashi, Y. NabekawA, K. Midorikawa: Appl. Phys. Lett. 84, 4 (2002) 10. H. Mashiko, A. Suda. K. Midorikawa: Opt. Lett. 29, 1927 (2004) 11. H. Bachau and P. Lambropoulos, Phys. Rev. A 44, R9 (1991) 12. M.S. Pindzola, F. Robicheaux: J. Phys. B 31, L823 (1998) 13. P. Lambropoulos, L.A.A. Nikolopoulos, M.G. Makris: Phys. Rev. A 72, 013410 (2005) 14. E.J. Takahashi, H. Hasegawa, Y. Nabekawa, K. Midorikawa: Opt. Lett. 29, 507 (2004) 15. E. J. Takahashi. Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, K. Midorikawa: IEEE Sel. Top. Quantum Electron. 10, 1315 (2004) 16. H. Hasegawa, E.J. Takahashi, Y. Nabekawa, K.L. Ishikawa, K. Midorikawa: Phys. Rev. A 71, 023407 (2005) 17. Y. Nabekawa, H. Hasegawa, E.J. Takahashi, K. Midorikawa: Phys. Rev. Lett. 94, 043001(2005) 18. T. Nakajima, L.A.A. Lambropoulos: Phys. Rev. A 66, 041420(R) (2002) 19. J. Colgan, M.S. Pindzola: Phys. Rev. Lett. 88, 173002 (2002) 20. S. Laulan, H. Bachau: Phys. Rev. A 68, 013409 (2003) 21. Feng, H.W. van der Hart: J. Phys B 36, L1 (2003) 22. Piraux, J. Bauer, S. Laulan, H. Bachau: Eur. Phys. J. D 26, 1 (2003) 23. K. Ishikawa: unpublished data 24. H. Hasegawa, E.J. Takahashi, Y. Nabekawa, K. Midorikawa: Laser Phys. 15, 812 (2005) 25. H. Mashiko, A. Suda, K. Midorikawa: Appl. Phys. B 76, 525 (2003) 26. Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, K. Midorikawa: Phys. Rev. Lett. 96, 083901 (2006) 27. E.S. Toma, H.G. Muller: J. Phys. B 35, 3435 (2002) 28. T. Okino, K. Yamanouchi, T. Shimizu, K. Furusawa, H. Hasegawa, Y. Nabekawa, K. Midorikawa: Chem. Phys. Lett. 432, 68 (2006) 29. S. Chelkowski and A.D. Bandrauk: Appl. Phys. B 74 S113 (2002) 30. Y. Nabekawa, Y. Kuramoto, T. Togashi, T. Sekikawa, S. Wanabe: Opt. Lett 23, 1384 (1998)

11 Controlling Light Polarization for Attosecond Pulse Generation Eric Constant and Eric Mével Centre Laser Intense et Applications (CELIA), Université Bordeaux 1 – CEA-CNRS, 351 cours de la libération 33 405 Talence Cedex, France, e-mail: [email protected] Summary. We developed a technique for controlling the duration of XUV pulses by modulating the polarization of an intense short pulse and using this pulse for high order harmonic generation. This approach allows confining the XUV emission down to the isolated attosecond pulse level leading to the generation of broadband and tunable isolated attosecond pulses.

11.1 Introduction The generation of ultrashort pulses allows the study of ultrafast processes by using pump-probe techniques where a pump pulse starts a process and, after a well known evolution time, a probe pulse gives an image of the evolving process. By controlling this pump-probe delay, it is possible to follow the temporal evolution of a transient state and thereby to access its dynamics. The temporal resolution accessible with such a technique is essentially limited by the duration of both the pump and probe pulses. Femtosecond pulses allow the study of nuclear motion in molecules [72] since their evolution typically occurs in the femtosecond (1fs = 10−15 s) to picosecond (1ps = 10−12 s) range. Electronic re-arrangement typically occur on a much shorter time scale and attosecond (1as = 10−18 s) pulses are required to study these processes [1, 53]. Generating these pulses requires a phase locked broadband light source. High order harmonic generation (HHG) in gases [23, 44] fulfils these requirements as will be presented here. In this chapter, we will describe the process of high order harmonic generation in gas that is at the basis of attosecond pulse generation [22, 27] and consider how this process allows generating train of XUV attosecond pulses. We will then focus on the possibility to confine the XUV emission especially by controlling the polarization of a short pulse [18]. After introducing the technique that we developed to modulate light polarization [67], we will describe experimental studies of this confinement and show that it gives access to a continuous confinement of the XUV emission down to the isolated attosecond pulse regime [60]. We will then conclude and highlight the perspectives offered by this confinement technique.

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11.2 High Order Harmonic Generation and Attosecond Pulse Trains High order harmonic generation occurs when a short intense laser pulse (later called the fundamental pulse and/or fundamental beam) is focused in a gas sample and partially ionises this gas [17]. At the intensities required to ionise it, the gas response is strongly non-linear and leads to the generation of new frequencies that can extend very far in the VUV to XUV domain. The main characteristics of HHG can be understood via the so called three steps model [17] described below. 11.2.1 The Three Steps Model In this simple model, the atom (i.e. gas medium) is considered as an ionic core (that creates a potential well) and a single electron, trapped in the fundamental state of the potential (characterised by Ip , its ionisation potential). The fundamental pulse is considered as a slowly evolving strong electric field. As the atom is irradiated by the strong field, the potential well is modified and a potential barrier is created. When the field amplitude is strong enough, the electron can escape the ionic core attraction by tunnelling through this barrier. This tunnel ionisation is the first step of HHG. Once the electron has tunnelled through the barrier, it is classically accelerated by the strong laser field and can quickly gain some kinetic energy. This laser acceleration is the second step of the model. As the laser field is oscillating, the electron can be driven back to the ionic core and it can collide with the ionic core with some kinetic energy. As this collision occurs, the electron can recombine with the ionic core provided it releases its extra energy (i.e. the kinetic energy of the re-colliding electron plus the ionisation potential of the atom) by emitting an XUV photon with the corresponding energy. This radiative recombination is the third step of this model. As ionisation (first step) is a probabilistic process, this full sequence can occur every time the field amplitude is strong enough to trigger it. For a weak ionisation probability and with long fundamental pulses, this leads to a periodicity of the process equals to half the optical period of the fundamental field (T0 /2). The accelerated electron is a quantum object (a wave-packet) and this electronic wave-packet has an accumulated phase that is linked to both the laser field and the phase of the initial fundamental wave-function. As recombination occurs into the initial state, the influence of the phase of the initial wave-function vanishes and the phase of the emitted light is imposed by the strong field characteristics. This implies that the XUV emission is coherent. The coherence of the emission and the T0 /2 anti-periodicity [16] of this coherent process (the emitted XUV field changes its sign every T0 /2 and the

11 Controlling Light Polarization for Attosecond Pulse Generation

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real periodicity is therefore T0 ) associated with the centro-symmetry of the gas medium implies that only odd harmonics of the fundamental frequency can be emitted as observed experimentally. This simple model also predicts that the maximum electron kinetic energy at recollision is 3.17 Up (Up = q 2 E 2 /4mω02 is the ponderomotive energy which corresponds to the average kinetic energy of an electron of mass m and charge q accelerated in a field having an amplitude of E and an angular frequency of ω0 ). Therefore the maximum photon energy that can be emitted is hνmax = Ip + 3.17 Up which is known as the cutoff law. This model allows to reproduce the harmonic spectra that present a broad plateau where the harmonic efficiency remains roughly constant (the efficiency is the ratio between the total energy in a given harmonics and the energy of the fundamental pulse), for moderate order harmonics, followed by an abrupt cutoff after which the emission efficiency quickly decreases. 11.2.2 Attosecond Pulse Train This semi-classical approach is also very convenient for understanding why the XUV light is emitted as a train of attosecond pulse [18]. The electron acceleration and motion can be simulated by considering the acceleration due to the electric field and a zero initial electron velocity at the time of ionisation, ti . Under these assumptions, the electron kinetic energies at the times of recollision, tr , (and therefore the energy of the emitted photon) are uniquely defined by the times of ionisation. The energies of the emitted photon (Fig. 11.1) change significantly within an optical period of the fundamental and therefore a spectral selection of the XUV light ensures that the

Fig. 11.1. Energy of the recolliding electron as a function of the phase of the fundamental field (top). Since this energy changes significantly during an optical period of the fundamental, photons of a given energy can only be emitted in a narrow temporal window. The bottom figure shows the fundamental field (dot) and the XUV bursts emitted at well defined phases (spectral selection of the cutoff photons)

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corresponding photons can only be emitted in a narrow temporal window. This window can easily be sub-femtosecond when the optical period is on the order of few fs (T0 = 2.66fs for a fundamental wavelength centred at 800nm). For instance, the cutoff photons having an energy hν > Ip + 3 Up can only be emitted over a 34◦ range [16] of the fundamental phase (i.e. 235◦ < ω0 tr < 269◦ for a field having its maximum at t = 0) which correspond to an emission duration of ∼250as for T0 = 2.66fs. After spectral selection of these photons, the XUV light consists in a train of attosecond pulses, each of them being separated by T0 /2. This model also shows that for photons of smaller energy (plateau harmonics), two possible times of emission exist within an half optical cycle. A spectral selection of these photon energy (for instance Ip + Up < hν < Ip + 2 Up ) would also lead in attosecond XUV bursts with a T0 /2 periodicity but there would be two bursts during each half optical cycle of the fundamental (each of these bursts corresponding to the so called “short” and “long” trajectories since the acceleration of the electron occurs for a short time in one case and a longer time in the other case). In this case, two intertwined attosecond pulse trains are emitted. This fast evolution of the photon energy also implies that the XUV pulses present a chirp when plateau harmonics are selected [37, 39]. Note that the emission times of the attosecond XUV bursts are defined as compared to the fundamental electric field. This semi-classical approach gives an intuitive understanding of the single atom response and shows that after spectral selection of a given XUV spectral range, attosecond pulse train are naturally emitted via HHG (the uncertainty principle has not been considered here and if the selected spectrum is not wide enough, this principle would of course increase the XUV pulse duration). This XUV emission has also been studied quantum mechanically [35] and these calculations have shown that the semi-classical model gives a good vision of the HHG process and that attosecond pulse trains are indeed extracted [4] by a spectral selection of a given spectral range. Collective effects are also very important in HHG since many atoms contribute to the emitted XUV field. However, the coherent build up of the radiation is possible only under restrictive phase matching conditions that ensure that the temporal profile of the XUV field is close to the field calculated for the single atom response when good phase-matching is achieved [18]. Considering these collective effects coupled with propagation, it has been shown that the attosecond structure of the pulses remains [4]. Furthermore, specific phase matching conditions [26, 27, 57] (controlled via the geometry of HHG) ensure that for the plateau harmonics it is possible to selectively enhance one of the quantum path (short or long) and reduce the emission due to the other. Even for the plateau harmonics, it is therefore possible to extract an attosecond pulse train consisting only in one attosecond pulse per half optical cycle [4]. These attosecond pulse train have now been observed experimentally in several places [38, 46, 51, 69].


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Extracting a single attosecond pulse out of this train requires breaking the periodicity of the process in order to prevent XUV emission everywhere except inside a single half optical period of the fundamental. Breaking this periodicity is possible by controlling the key parameters that define the HHG efficiency. 11.2.3 The Critical Parameters of HHG The efficiency of HHG depends on several crucial parameters. As mentioned before, collective effects are very important to optimize this emission and allow for instance to select one quantum path while destroying the other (a selection of the short quantum path is for instance possible by locating the gas medium after the laser focus). This optimization is usually not highly wavelength selective and it is therefore possible to optimize the emission over a broad wavelength range by using collective effects. For a given XUV wavelength, the intensity of the fundamental pulse is also a crucial parameter since an efficient emission can only be obtained when the considered wavelength is in the plateau or in the cutoff. If the intensity is so weak that the desired wavelength is above the cutoff, the emission yield will be very low. This effect can be used to confine the XUV emission by using a short fundamental pulse, if one chooses an harmonic that is in the cutoff at the peak intensity of the fundamental pulse. This harmonic will be above the cutoff in the edges of the fundamental pulse and the emission efficiency will be significant only around the peak of the fundamental. This naturally restricts the possible emission times and therefore the length of the attosecond pulse train. HHG is also very sensitive to the polarisation state of the fundamental pulse [8] and is usually maximum for a linear polarisation. For a relatively

Fig. 11.2. Critical ellipticity measured for different harmonic order and generating gas (open circle: Argon, full square: Neon). This critical ellipticity is the constant ellipticity that decreases the harmonic signal by a factor two as compared to the maximum signal obtained with a linear polarization (ε = 0)

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weak ellipticity, ε, of the fundamental (defined as ε = min(|Ex /Ey |, |Ey /Ex )|) with a laser field E = Ex cos(ω0 t) + Ey sin(ω0 t), the efficiency quickly drops. We define the critical ellipticity, εc , as the constant ellipticity that decreases the efficiency by a factor two as compared to the maximum efficiency (usually obtained for ε = 0). This critical ellipticity is around 10 to 15% and changes only slowly with the harmonic order (Fig. 11.2). To maximise the XUV emission, the ellipticity is usually set to zero but it can be modulated (see next paragraph) to confine the XUV emission.

11.3 Confinement of the XUV Emission It is possible to confine the XUV emission by rapidly changing the parameters that define the emission of HHG. When the confinement is very strong, an isolated attosecond pulse can be extracted after spectral selection of an appropriate spectral range. 11.3.1 Confinement by Rapid Intensity Variation Cutoff harmonics are very sensitive to the intensity (the field amplitude) of the fundamental field. By using a very short fundamental pulse, the field amplitude can significantly change from half an optical cycle of the fundamental to the next (former) one. In this case, the maximum energy of the emitted photons changes from half an optical cycle to the next one. In particular, the cutoff photons with maximum energy can only be emitted around the times where the field amplitude is maximal. Selecting the cutoff photons can then lead to the extraction of an isolated attosecond pulse [14] if one ensures that the field has a single maximum and that the secondary maxima of the field have amplitudes that are significantly smaller than the main one. Using a 5fs fundamental pulse can fulfil this requirement provided that the field evolution in its envelop is such that the field is maximum at the time where the envelop is maximum. This requires to control the carrier envelop phase (CEP) [6] that defines the field evolution inside its envelop. When the CEP is set to zero, the field evolves as a cosine function and is maximum at the maximum of the envelop thereby ensuring that the secondary maxima (T0 /2 before and after the main one) are significantly smaller than the main one. With this CEP, a spectral selection of the cutoff harmonics has led to the emission and characterization of 250as isolated pulses [30]. In contrast, when the CEP is set to π/2, the field amplitude is zero at the maximum of the envelop and can reach twice the same maximum (in absolute value) during the pulse. In this case, even after spectral selection of the cutoff harmonics, two attosecond pulses are present. This technique has already been implemented by F. Krausz and his group by generating XUV pulses in Neon with 5fs CEP stabilized pulses and they have clearly shown that by selecting a 10eV spectral range in the cutoff


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a single 250as pulse can be selected [30]. This spectral range is limited by the difference in field amplitude between the main maximum and the secondary ones which increases with the peak intensity. This implies that very high intensities need to be used to isolate attosecond pulses with this technique. In returns, this implies that only atoms with high ionization potential (Neon, Helium which can support high intensities before ionization) can be used. Therefore the photon energy of the attosecond XUV pulse is on the order of 80eV or above. 11.3.2 Confinement by Polarization Modulation The efficiency of high order harmonic generation also depends crucially on the ellipticity of the fundamental pulse and the polarization state of a pulse can be changed during this pulse [20]. By using a pulse with a time-dependent polarization, that remains linear for a short time, it is possible to confine the XUV emission inside the temporal window where the polarization is close to linear [18]. This polarization modulation creates a “polarization gate” where the XUV emission is confined (Fig. 11.3). When the field polarization evolves from elliptic to linear and back to elliptic, the full width at half maximum of the XUV pulse (envelop of the attosecond train) is given by the delay between the two times where the ellipticity is equals to the critical ellipticity. When the width of this polarization gate, τg , is smaller than T0 /2, a single attosecond pulse is extracted [10,18,55]. As shown in Fig. 11.2, this critical ellipticity depends only weakly on the harmonic order, therefore polarization induced confinement can be effective over a broad spectral range which is a necessary condition to generate short

Fig. 11.3. Sketch of the confinement via polarization modulation. As a linearly polarized fundamental pulse is used for HHG, a train of XUV attosecond pulses is emitted. When the fundamental polarization is modulated the XUV emission can only be efficient where the polarization is close to linear and the attosecond pulse train is confined leading to a broadening of the harmonics when only few pulses are emitted

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attosecond pulses. Also, this confinement can be effective in both the plateau and cutoff region which implies that the central frequency of the attosecond pulses can be tuned by an adequate spectral filtering and a proper choice of laser intensity and gas medium. 11.3.3 Modulating Light Polarization Several techniques [3, 9, 32, 67] have been used to modulate the polarization of short pulses. To generate isolated attosecond pulses, the polarization modulation must however be performed on ultrashort pulses (typically sub 7fs) which have currently a limited energy. The polarization modulation technique must therefore be compatible with using spectrally broadband fundamental pulses and have a high throughput to limit the energy loss. The technique that we developed [67] fulfils all these requirements and is very robust. It relies on transmitting an input linearly polarized pulse (the input polarization direction is the reference direction to define the angles later) through two birefringent plates (a first thick plate and an achromatic zero order quarter waveplate) as shown on Fig. 11.4. A first thick birefringent plate is used to split the input pulse in two delayed and (linearly) cross-polarized pulses. This plate has two perpendicular

Fig. 11.4. Sketch of the two birefringent plates setup used to modulate the polarization of a short pulse. The first (thick ) plate splits the incoming pulse in two perpendicularly polarized pulses separated by a delay δ. The second plate is a zero order achromatic quarter waveplate that transforms both of these pulses in circularly (elliptically) polarized pulses with opposite helicity. The plate orientations are defined by the angle (α and β) of their neutral axis as compared to the direction of the polarization of the input pulse. The laser field evolution is also represented below, the red color representing the portion where the field is close to linearly polarized, i.e. δ ≤ εc


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neutral axes (slow and fast axes) along which the propagation is characterised by the ordinary or extraordinary index. It has therefore two optical thicknesses. When these axis are set at α = 45◦ of the input polarization direction, half of the pulse will propagate along each axis and, after the plate, the input pulse is split in two identical pulses (ordinary and extraordinary pulses), perpendicularly polarized and separated by a delay δ, proportional to the plate thickness. For a 1mm thick quartz plate and a central wavelength of the pulse at 800nm, the delay is δ = 31.77fs. After crossing the first plate, the pulse has a modulated polarization that evolves from linear along one direction (those of the fast axis) at the beginning of the pulse to a linear polarization along another direction (those of the slow axis) at the end of the pulse. For the specific case where the first plate is a (multiple order) quarter wave at the central frequency, the outgoing pulse is circularly polarized at the place where the extraordinary and ordinary pulses have the same amplitude, i.e. at the centre of the outgoing pulse. These two pulses are then transmitted through a zero order achromatic quarter waveplate having its axis at an angle β of the reference direction. For β = 0◦ , the extraordinary and ordinary pulses are both transformed in circularly polarized pulses with opposite helicity (one is left circularly polarized while the other is right circularly polarized). The sum of two circularly polarized fields with opposite helicity gives a linearly polarized field provided that these two fields have the same amplitude. This is the case only at the centre of the outgoing pulse. In the wings of this pulse, the polarisation remains circular or elliptic. The combination of these two birefringent plates allows us therefore to create a field that presents a time dependent ellipticity evolving from ε = 1 (circular polarization) in the wings of the pulse to ε = 0 (linear polarization) at the center of the outgoing pulse. For the specific case where the first plate is λ/4 at the central frequency, changing β allows us to change easily the ellipticity modulation [67] (Fig. 11.5). This angle β defines the asymptotic ellipticity without changing the ellipticity at the center of the outgoing pulse (ε = 0 remains at this time but the orientation of the polarization depends on β). For instance for β = 45◦ , the two plates have the same axes orientation and their ensemble is equivalent to a multiple order 0 or λ/2 waveplate which only change the polarization direction while keeping it linear throughout the pulse (note that when the first plate is not λ/4, the ellipticity is also kept constant throughout the pulse but is non zero which is not favourable for HHG). In between these two extreme cases, β can be continuously tuned to control the ellipticity modulation and thereby the width of the gate without changing the intensity profile (this last assumption is valid provided the maximum delay introduced by the last plate (T0 /4 = 0.66fs at 800nm) is negligible as compared to the pulse duration. For ∼5fs pulses this picture breaks and the full evolution of the field needs to be considered but it has been shown that even for such short pulses the concept of polarization gate remains valid [62, 63].

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Fig. 11.5. Temporal evolution of the intensity profile (red ) and the ellipticity (blue) of a short pulse for several angles β. The first (λ/4) plate introduces a delay of δ = 30 fs equals to the input pulse duration (τ = 30 fs) leading to a flat top intensity profile

11.3.4 Simulations of the Polarization Induced Confinement We have simulated the effect of this polarization modulation on the temporal confinement of the XUV emission [62–65] by using a numerical approach based on the semi-classical model and considering the light propagation in the gas medium. These simulations showed a clear confinement induced by the polarization modulation (Figs. 11.6 and 11.7). We also derived an analytical law [62] to estimate the duration of the polarization gate for a Gaussian shaped input pulse: τG =

εc τ 2 ln(2)δ cos(2β)

(11.1)

where εc is the critical ellipticity, τ is the Fourier limit input pulse duration (through this paper the output pulse is considered as unchirped in the interaction region). This law is valid only when the XUV emission is confined by the polarization modulation and not by other effects (such as intensity variation in the temporal profile of the fundamental pulse). For β = 45◦ , the gate width gets infinite but the XUV emission is confined by the intensity profile of the fundamental pulse. As the simulated gate width is on the order of T0 /2, a single attosecond pulse can be extracted out of the train but the XUV emission profile depends crucially on the CEP of the driving pulse [64] (Fig. 11.7). For some CEP a single attosecond pulse can be extracted while for other, two attosecond pulses are present. This arises since the XUV emission times are linked to the


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Fig. 11.6. Simulated temporal profile of the XUV (photon energy >27 eV) emission for two different gate characteristics. The red curve (full line) corresponds to the narrow gate case (considering δ = 31.4 fs with a Gaussian input pulse of width τ = 35 fs and β = 0◦ ) while the blue curve (dash) corresponds to the large gate case (β = 45◦ )

Fig. 11.7. Simulated temporal profile of the XUV emission (photon energy >27 eV) in Argon in the case of a polarization gate having a width of 1.7 fs (τ = 10 fs, δ = 13 fs). For some CEP, a single attosecond pulse is emitted while for other CEP two attosecond pulses can be emitted. The inset shows a situation where the gate is even narrower and where a single attosecond pulse is generated over a large range of CEP

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field while the gate characteristics (position and width) are imposed by the intensity profile of the fundamental pulse. Changing the CEP can therefore drive the possible emission times inside or outside the gate as we observed experimentally (see next section). These simulations also show that when an isolated attosecond pulse is extracted from the train, its temporal structure remains very close to the profile of the pulse in the train. The pulse characteristics (chirp, bandwidth, amplitude) evaluated with a linear polarization remain therefore approximately valid even after applying the polarization gate to extract one of these pulses. Similar calculations [11] have been performed by using the SFA model [35] to simulate polarization gated HHG. It was then shown that isolated attosecond pulses can indeed be extracted with this technique and that the characteristics of this isolated pulse mimic those of the attosecond pulses in a train. It was also shown that the intrinsic chirp of the attosecond pulses can be compensated by transmitting it through a thin metallic filter. In this way, isolated pulses as short as 58as [11] could be theoretically extracted for a fixed CEP. 11.3.5 Optimal Conditions for Attosecond Pulse Generation The above simulations defined the conditions under which an isolated attosecond pulse can be extracted by polarization modulation. Another important parameter is the efficiency of this emission and combining XUV confinement with high efficiency requires more specific conditions. To extract an isolated attosecond pulse, the polarization gate width needs to be smaller than (or close to) T0 /2. For any input pulse duration, this can be achieved by choosing δ properly and implies: τG =

εc τ 2 ≤ T0 /2 ln(2)δ

(11.2)

To get an efficient attosecond pulse emission, it is also necessary to have a high fundamental intensity inside the gate and this restricts the possible values of δ. Indeed, when δ is much larger than τ , the fundamental intensity profile shows a local minimum inside the gate. In return, this decreases the efficiency of the attosecond pulse emission and for δ τ , a strong intensity maximum before the gate can significantly ionize the medium and further decrease the efficiency of the process. It is therefore better to use a flat top intensity profile or an intensity profile having a maximum inside the gate. For Gaussian pulses this implies to use δ≤τ

(11.3)

Combining Equations (11.2) and (11.3) with the typical values of εc implies that pulses shorter than ∼7fs should be used to maximise the efficiency of isolated attosecond pulses generation with this technique.


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This allows optimizing the attosecond pulse emission at the single atom level and collective effects need also to be optimized. Mostly this collective response can be controlled with standard “recipes” [15, 29, 43] to optimize it. Care need however to be taken to avoid any important geometrical shift of the fundamental field (i.e. Gouy shift) in the gas medium since it is equivalent to a CEP shift and can significantly change the attosecond pulse profile.

11.4 Experimental Observation of Polarization Induced Confinement The very first experiment on HHG driven by a pulse with modulated polarization was performed in the group of Anne L’Huillier (Lund, Sweden) and they could observe clear effect of polarization induced confinement of the XUV emission [17]. However, the non-linear technique that they used to modulate the polarization of their fundamental pulse was very sensitive to intensity fluctuations and this experiment was not further continued after this successful proof of feasibility. The polarization gating scheme that we developed has been tested in several laboratory both in the spectral [59, 60, 66, 67, 71] and temporal [36, 40, 61, 71] domains. 11.4.1 Temporal Characterization of the Polarization Gate Studying the polarization gating directly in the temporal domain is not easy since HHG occurs naturally on a time scale that is smaller than the input pulse duration and the XUV envelop can therefore not directly be probed with a pulse of the same duration. To overcome this limit, the polarization gating was applied to a (40fs) long pulse that was used to generate the harmonics. The XUV temporal profile was probed by using a 9fs pulse obtained by post-compression in hollow core fiber [48]. Rather than generating isolated attosecond pulses, the goal of this experiment was to test the validity of the polarization gating approach and to check the validity of the formula (11.1) giving the duration of the polarization gate. The experimental setup was designed to perform a cross-correlation of the XUV pulse and a short (9fs) probe pulse [36, 61]. The polarization of the (40fs) input pulse was modulated by using a pair of birefringent quartz plate, the first one being a multiple order λ/4 plate with a 1.01mm thickness that induced a 32fs delay. After modulating this polarization, the pulse was focused in an argon gas cell where XUV light was generated. The fundamental beam and low order harmonics were filtered out by transmitting the beam through a 200nm thick aluminum filter. The XUV beam was then focused by a toroidal mirror in the sensitive zone of an electron time of flight spectrometer (equipped with a magnetic bottle) where it could ionize argon atoms. A second, 9fs, probe beam was focused at the

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same place and the XUV-IR delay could be varied with a delay line. These two beams were recombined by a diverging spherical mirror that transmitted the XUV beam through a centred hole and reflected the probe beam in such a way that both beams appeared to originate from the same point (i.e. the gas cell). When both pulses overlap, ionization of argon atom could occur via the simultaneous absorption of an XUV photon and absorption or stimulated emission of an IR photon. Such a process could clearly be labelled in the energy of the photo-electron and side-band appeared in the photo-electron spectrum. In the perturbative limit, the amplitude of this side-band signal is proportional to the temporal overlap between the XUV and IR pulses. As the IR-XUV delay was changed, the amplitude of the side band evolved and the evolution of this sideband signal as a function of the delay gives the intensity cross-correlation between the XUV and IR probe. Figure 11.8 shows the evolution of this signal for several gate characteristics and the cross correlation signal is clearly shorter in the narrow gate case. The deconvolution of this signal allows accessing the gate duration which was measured to be 11.5fs for the 18th side band (and also for the 16th and 20th sidebands). This measurement confirmed that polarization gating is effective over a broad spectral range and can be used with moderate order harmonics. The measured duration of the XUV envelop is also in good agreement with the narrow gate width estimated with (11.1) which is τg = 11fs. Another measurement has been performed in the temporal domain by the Saclay group [40] that used an XUV spider measurement technique [19, 40]. The basis of this technique is to create (with a dazzler) to delayed replica of

Fig. 11.8. Cross-corellation signal obtained by integrating the amplitude of the 18th sideband for several delays and for several gate configurations (large gate (open squares), no gate (full diamond ) and narrow gate (open circles))


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Fig. 11.9. Temporal profile of the 11th harmonic (and its phase) measured via the XUV spider technique with (black line) and without (grey line) polarization induced confinement

a given pulse with a slight frequency difference (a shear) between the central frequencies of the pulses. These two pulses are then used to generate harmonics and the fundamental frequency difference is apparent in the harmonic spectra since each pulse creates some harmonics at odd multiple of its central frequency. For a small shearing, the two harmonic spectra overlap and interfere together since the two emission processes are mutually coherent. The fringe pattern gives access to the dephasing between neighbouring frequencies and, once this spectral phase is known, it is possible to reconstruct the temporal profile of a given harmonic. Afterward they applied polarization gating on both replica and could also observe (Fig. 11.9) a clear confinement of the harmonic profile when the polarization gating was present even for the low harmonic (11th) that was considered. 11.4.2 Spectral Signature of Isolated Attosecond Pulses As this polarization gating approach was validated in the temporal domain and spectral domain with long pulses, further experiments have been performed in the spectral domain to observe the signature of isolated attosecond pulses. In this domain, the signature of the emission of isolated attosecond pulse should be a continuous spectrum while two attosecond pulses should lead to broad harmonics. Note that the observation of a continuous spectrum can be considered as the signature of a single attosecond pulse only when all other effects that can shift or broaden the spectrum can be ruled out [6]. The first experiments reaching the τg ∼ T0 /2 level, were performed with 9fs pulses without CEP stabilization [59, 61, 66]. Large broadenings of the spectra were observed and were attributed to the temporal confinement induced by the polarization gating. However, the acquisitions were performed by averaging the XUV spectra over several laser shots. The random CEP value can lead to strong shot-to-shot fluctuations of the harmonic spectra and even when several attosecond pulses are present, the sum of shifted har-

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monic spectra can lead to an apparent continuous spectrum which is clearly not the proof for isolated attosecond pulse generation. A subsequent experiment [60] was performed by modulating the polarization of CEP stabilized 5fs pulses source in Milan (Italy). In this experiment, great care was taken to avoid any spectral broadening that was not related to temporal confinement of the XUV emission. For instance low gas pressure and moderate laser intensities were used. It was even checked that without any gate (i.e. with a peak intensity twice higher then when the polarization gating is applied) the XUV spectra showed well defined harmonics with a typical width of 1eV. The thin gas jet (400μm nozzle) was also located after the laser focus to ensure that a single quantum path (short quantum path) was dominant for the plateau harmonics and to limit the influence of the Gouy shift. In this case, a clear spectral broadening was observed as the polarization gating was applied. Figure 11.10 shows spectra obtained in the narrow gate configuration for two CEP values differing by π/2. Under these conditions, we observe either 1.6eV broad harmonics or a continuous spectrum after a π/2 CEP shift. The 1.6eV widths of the harmonics are consistent with the generation of two attosecond pulses while the continuous spectra are consistent with the emission of isolated attosecond pulses. Since changes in the CEP does

Fig. 11.10. Experimental (bottom) and calculated (top) XUV spectra generated in argon with a CEP stabilized polarization modulated pulse (τ = 5 fs, δ = 6.2 fs, β = 0◦ ) for two CEP values differing by π/2. The inset shows the corresponding temporal profile after spectral selection of all XUV photons having energies higher than 18 eV (corresponding to an Al filter transmission)


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not affect the pulse intensity profile, this spectral evolution could not be attributed to anything else but the polarization induced confinement and is considered as the signature of either an isolated attosecond XUV burst or two attosecond pulses [60]. Similar experiments were performed by generating the XUV pulses in neon and the confinement effect was observed to be even stronger with identical gate characteristics (τ = 5fs, δ = 6.1fs). This is expected since the critical ellipticity is lower for those harmonics generated in neon and therefore the gate width is decreased as compared to the argon case. In this case the XUV spectra were continuous for those CEP that maximises the efficiency and were very weak after a π/2 CEP shift and they extended from 40 to 70eV. By lowering the confinement effect (using τ = 5fs, δ = 5fs) the spectra were very similar to those presented with argon showing either broad harmonics or a continuous spectrum for different CEP. These spectra were considered as the signature of the emission of either a single attosecond pulse or two attosecond pulses and the XUV bandwidth extends from 50 to 100eV. These spectral measurements can clearly not give directly the duration of the emitted attosecond pulses but they could be estimated through simulations. For generation of XUV light in argon, the estimated temporal profile showed a single 260as pulses after spectral selection of photon energies higher than 18eV. This pulse is preceded by a weak prepulse ( 0) is used, self-focusing is reinforced and the self-focal distance is shorter than that in the unfocused case. The modified self-focal distance zf obeys the lens transformation formula [34]: zf =

zf f . zf + f

(12.2)

On the other hand, a lens with negative focal length (f < 0) will move the self-focus to longer distance along the propagation axis. In this case the selffocusing distance is still determined by (12.1) but with an altered effective critical power [2]: 2 2 0.367ka20 Pc = 0.852 + 0.0219 + Pc . (12.3) f

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Obviously, this effective critical power is higher than that for parallel Gaussian beam, since in additional to the natural diffraction, the self-focusing has to overcome the divergence imposed by the negative lens too.

12.4 Slice-by-Slice Self-Focusing of Laser Pulse Up to now, we have only discussed the self-focusing of a CW beam. However, so far, in real experiment, pulsed laser is always used. The self-focusing of a laser pulse is highly dynamic and could be described by the slice-by-slice self-focusing scenario according to the moving focus model [8, 13], or similar spatial replenishment model [15, 16]. In this scenario, the temporal shape of the laser pulse is visualized as subdivided into many thin intensity or power slices in time (or in space along the propagation axis) as seen in Fig. 12.2, and the thickness of one “slice” is at least cτ , where τ is a period of oscillation of the electromagnetic wave. This is because we are talking about intensity which is defined as the Poynting vector averaged over at least one cycle of oscillation. The propagation of each slice is approximated as the self-focusing theory of a CW beam. It is clear that the central slice has the highest power; it will self-focus at the shortest distance as illustrated in Fig. 12.2. The slice in front of the central slice would self-focus at a later position in the propagation direction according to (12.1) because its peak power is lower than that of the central slice. Consequently, a serial of self-foci is produced on the propagation axis. Note that the laser intensity at each self-focus is so high that a plasma would be produced. The importance of the plasma generation will be discussed in detail in the following sections.

Fig. 12.2. Slice by slice self-focusing. Central slice has the highest power, then it will self-focus at the shortest distance; while lower power slices self-focus at longer distances

12.5 Plasma Generation Plasma generation takes place as soon as the laser intensity at the selffocus is high enough to generate a significant amount of free electrons in

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the medium. In fact, the major fundamental physics of filamentation is the same in all optical media, be they gases, liquids or solids. The difference lies in the detail of free electrons generation. In gases, it is tunnel/multiphoton ionization of the gas molecules inside the self-focal volume resulting in the plasma [35]. In condensed matters, it is the excitation of free electrons from the valence to the conduction bands [36] followed by inverse Bremsstrahlung and electron impact ionization [37] before the short pulse is over. The well-known type of optical breakdown of the medium (generation of a spark) by longer laser pulses in the picosecond and nanosecond regimes does not occur in the femtosecond self-focusing regime because there is no enough time to sustain cascade (avalanche) ionization. For example, at one atmospheric pressure, the mean free time of electron collision is ∼1ps. This time is longer than the fs pulse duration so that only tunnel/multiphoton ionization, an ‘instantaneous’ electronic transition process, is responsible for the generation of free electrons [35] even if the full pulse is involved in the self-focusing. In the case of condensed medium, external focusing could influence the outcome significantly [14, 30, 38]. However, if we consider only those cases in which self-focusing is a dominant focusing process (with a long focal length lens or without external focusing), the filament will be formed without breakdown. The reason for this is that self-focusing induces a strongly convergent wave front only in a narrow transverse region which contains the peak power of the order of the critical power for selffocusing in the material [39]. A little amount of electrons produced due to multiphoton/tunnel transition from the valence to the conduction bands followed by at most a few cycles of collisional ionization will be enough to limit intensity increase and prevent further ionization, which is known as intensity clamping.

12.6 Intensity Clamping If we use air as an example, the free electrons are generated through tunnel ionization. Therefore, the refraction index change due to the plasma can be ω2

approximated as Δnp = − 2ωp2 (ω0 , the central frequency of the pulse). Here 0 2 the plasma frequency is given by ωp = ε0eme Ne , where e and me are the charge and mass of the electron, ε0 is the permittivity of free space and Ne is the electron density. The electron density increases very rapidly with intensity because tunnel/multiphoton ionization is a highly nonlinear process. We approximate such an increase as being governed by an effective power law according to the experimental observations [40]; i.e. Ne (t) ∝ I m where m is the effective nonlinear order of ionization. In air, m is about 8 [40]. The effective index of refraction of the central part of the slice is thus: n = n0 + Δnkr + Δnp = n0 + n2 I −

e2 βI m , 2ε0 me ω02

(12.4)

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where β is a proportionality constant. Qualitatively, it means that the free electron term would quickly catch up with the Kerr term of the self-focused slice until they are equal; i.e. until Δnkr + Δnp = 0. At this point, Kerr self-focusing balances free electron defocusing. The central part, having now an index of refraction n0 , propagates at the linear speed c. There is no more focusing and the intensity is highest at this balancing point. This is the condition of intensity clamping [22, 41, 42] because further propagation would lead to an index at the central part smaller than n0 due to the fact that the plasma contribution becomes more important. The slice will start to diverge. On the other hand, successive slices from the back part of the pulse will self-focus into the plasma of the previous self-focal region. This interaction with the plasma will naturally de-focus these slices and no self-focus is formed with these slices. That is to say, during self-focusing of a powerful femtosecond laser pulse in an optical medium, there is a maximum intensity that self-focusing can reach. In air, it is around 5 × 1013 W/cm2 [22, 43]. Intensity clamping is a profound physical phenomenon of self-focusing and filamentation. It sets an upper limit to the intensity at the self-focus not only in air but also in all optical media. We note that the plasma density in the self-focal region is always very low, roughly 3 orders of magnitude lower than that of the neutral medium [9–12, 44–46]. It is also necessary to emphasize that in the case of filamentation, the laser pulse does not degenerate or self-stretch into a thin and long line of intense light (filament). At any time, there is only one pulse propagating in space. It is not the propagation of the self-focus along the axis of propagation that gives rise to the perception of a filament. Each self-focus comes from the self-focusing of a different slice of the pulse. It is indeed the column of plasma created by the succession of self-foci that gives the perception of ‘filament’. Thus, the word ‘filament’ is now normally referred to the plasma channel left behind the laser pulse.

12.7 White Light Laser (or Supercontinuum Generation) and Conical Emission Intensity dependent refractive index does not only induce spatial deformation of the pulse, but also gives rise to changes in the temporal domain. It leads to self-phase modulation (SPM) and is connected with the spectral broadening of the pulse spectrum. The pulse frequency change due to SPM is equal to: ∂ ω0 Δn (t) ω0 ∂[Δn (t)] Δω = − z =− z , (12.5) ∂t c c ∂t where z is the propagation distance and the nonlinear refractive index is e2 Ne (t) given by: Δn (t) = n2 I(t) − 2ε 2 . We note that the electron-ion recomb0 me ω0 ination time is normally much longer than the femtosecond time scale of the

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pulse. Hence, the generated plasma could be considered as static during the interaction with the pulse. Since the front part of the pulse always sees the neutral, from (12.5) without the plasma contribution, Δω = −

ω0 z ∂I(front part) ω0 z ∂[Δn (t)] =− n2 zo > d/3 ,

(13.1)

and collisional effects dominate. The experiment performed by Hall et al. [12] gives a clearer and more direct experimental evidence of the electric field inhibition in fast electron propagation, as compared to Key and Wharton’s results. Indeed, the propagation characteristics of the medium were changed by the compression, while the electron source was kept unchanged. This is because the high intensity laser beam was focused on a layer of uncompressed cold material both in the shocked and in the unshocked target. This however is not the only important conclusion arising from such experiment, which also showed that electric fields effects may already be very important at laser intensities as low as 1016 W/cm2 . Also, it was shown that neglecting electric field effects may lead to wrong conclusions. By considering electric field effects, one could find a fast electron temperature Thot ≈ 60keV, in agreement with Beg’s law. If instead electric field effects are neglected, then the reduced penetration would be interpreted as deriving from a lower fast electron temperature, implying Thot ≈ 40keV, which is quite different from the correct result. One drawback in this experiment is that it was not possible to produce a uniformly compressed layer of large thickness, and consequently, the fast electron temperature had to be maintained low. Hence, the condition is far from the regime of direct interest for fast ignition. Let us draw a final major important conclusion. The type of results presented in this section and in the previous one (4.2) show a propagation of fast electrons which is certainly inhibited by strong electric self-generated fields, but which corresponds anyway to a macroscopic penetration over distances of

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Fig. 13.12. Experimental results at RAL after [12]. Kα yield in arb. units vs. target thickness in μm. The dashed horizontal line shows the typical noise level

several hundreds microns. Although a few different results, implying a negligible propagation and an extreme electric inhibition, were sometimes obtained (see for instance Feurer et al. [30]) we think that these results are somewhat doubtful (such discrepancy can in part be attributed to the low sensitivity of the diagnostics used in the experiments and in part to the presence of a multi-temperature distribution function).

13.6 Dependence of Propagation Inhibition Versus Material Density and Electrical Conductivity The temperature dependence of the electrical conductivity of two kinds of materials is schematically shown in Fig. 13.13, which compares the conductivity of a typical metal (Al) with the one of an insulator (plastic), vs. temperature. As temperature increases, the conductivity of metals decreases, and reaches a minimum with the saturation of the resistivity, or the Ioffe-Rigel limit, where the electron mean free path becomes equal to the inter-ionic distance, at temperatures close to the Fermi energy of metals. Then, the conductivity rises again because the material is becoming a plasma and begins to follow Spitzer’s law, that is, σ is scaled by temperature to the 3/2 power [37], σ ∝ T 3/2 . The temperature dependence of insulators is largely different. First, it increases, starting from very low values, as a result of collisional ionization of the background material, leading to Saha-like distributions of ionization states of atoms in the material. Later, the behaviour changes to the Spitzer type at a temperature range typically above a few 10eV.

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Fig. 13.13. Electrical conductivity in Al and plastic as a function of temperature. The Al conductivity is obtained from the data of cold Al, the data from Milcheberg et al [38] at intermediate temperatures, and the Spitzer model at high temperatures. The data for plastic are obtained numerically from a semiclassical model [35] using the Sesame tables, and from the Spitzer model at high temperatures

This initial phase in insulators is sometimes neglected. However it can be important in the interpretation of several recent experiments (as already explained in the previous sections). The graph in Fig. 13.13 is obtained for the constant density, equal to the initial mass density of the material. This is usually appropriate for the fast heating induced by fast electrons under isochoric conditions. The behaviour of conductivity in gases and that in foams obtained from insulating materials are similar to that in insulators, but in these two cases the initial density of the material is a parameter, which can be easily changed. In addition, due to their lower density, a given energy deposition from fast electrons results in a larger increase in temperature. Therefore, the Spitzer-regime can easily be reached when such materials are used, which in turn facilitates the analysis of experimental results. 13.6.1 Foams In order to investigate the effect of the density of the background material on the inhibition of fast electron penetration, Batani et al. [39] performed an experiments using foams. In the experiment, targets with the same areal density ρd were used, i.e. the lower foam density corresponded to the thicker targets. In this way, different targets are characterized by the same collisional penetration, and differences in experimental results can only be ascribed to those differences in collective electromagnetic effects. Clearly Fig. 13.14 shows


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Fig. 13.14. Kα yield from the Mo and Pd fluor layers vs. foam density (data from Batani et al. [39])

Fig. 13.15. Penetration range of fast electrons (in μm) vs. target density (data from Batani et al. [39])

such differences in the recorded Kα signal, with denser foam targets seeming to facilitate the propagation while less dense foams are characterized by a stronger inhibition effect. Again, as in the experiment by Pisani et al. [9], the experimental Kα yield allows us to obtain the penetration range R(ρ), e.g. by interpolating the data vs. target thickness (for a given fixed density) with the function exp(−R/R(ρ)). Here the obtained propagation range was observed to scale

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as ρ−0.5 vs. target density. A similar scaling (ρ−3/5 ) can be obtained using Bell’s law and Spitzer’s conductivity (see Fig. 13.15), with the additional assumption of a cylindrical propagation. This result was also confirmed by numerical simulations performed by Davies [40]. It should be noted that the use of Spitzer’s conductivity implies that the material is heated to a high temperature, so that the plasma state is reached. Also, since lo is scaled as ρ−1 , we have zo /lo ≈ ρ0.5 . This means that, for low density materials, zo lo , and hence, propagation conditions fall in an electric-field-limited regime. 13.6.2 Gases In order to further investigate the propagation of fast electrons and the propagation inhibition, an experiment was recently conducted using gas targets [41]. A scheme of the experimental set-up is shown in Fig. 13.16. In the case of gas target, the density of the background material could be easily varied, with the additional advantage that the use of diagnostics such as optical shadowgraphy is possible. This is because gases are transparent, unlike foams or many types of solids. The shadowgraphy allows fast electron propagation within the target to be studied in real-time. Furthermore, the study in the regime of very low densities of the background material is in itself important for the study of fast ignition. Indeed, fast electron densities nb ≈ 1023 cm−3 should be achieved by the injection of laser light of 10kJ in ≈10ps over ≈10μm. While, at present, the mechanism of the generation of such huge fast electron current is not understood, still such achieved densities are much larger than the typical coronal plasmas (ne ≈ 1021 cm−3 ). The limit within which the fast electron density is equal to

Fig. 13.16. Experimental set-up of Batani et al. [41] in which the fast electrons are produced in a conductor by ultra-intense laser, and then, are propagated across a region which has a controlled density of gas. The gas is initially neutral. Its equivalent electron density, if fully ionised, is less than the density of fast electrons created into the first metal foil


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or even larger than the background electron density in which it must propagate was also met in the gas target experiment. Indeed, the atomic gas densities used in the experiment were ne ≤ 3 × 1019 cm−3 , whereas typically nb ≈ 5 × 1020 cm−3 , as expected of the order of the laser critical density (this last number results from a fast electron energy ≈1MeV and a conversion efficiency from laser energy to fast electrons up to ≈30%, and by considering that the electron beam is produced in a region of a size comparable to the focal spot within a time of the order of the laser pulse duration). As discussed by Bell et al. [32], the limit of nfast ≥ ncold is precisely one of the conditions that forbid the transport of any substantial current of injected fast electrons. What happens in this case is that very strong inhibiting electric fields are set up within the transport medium, impeding the ballistic transport of fast electrons, and permitting a much slower, diffusive-like propagation after the fields have ionized the neutral atoms. Figure 13.17 shows a typical shadowgraphy image, showing a large cloud and straight lines probably connected to electron jets. Such jets could be due to the first fast electrons, arriving to the rear of the first foil and propagating in the gas before a large field has developed. However, the presence of jets is not too important in fast electron transport: the majority of electrons are contained in the cloud region. In the experiment, the authors find that the propagation of the majority of the fast electrons across the volume is greatly inhibited, and their velocities become sub-relativistic. The ionization rate and the collisional rate depend upon the density of the gas, and when the ionization occurs, more of the generated fast electrons make their way across the volume in a “cloud-like” structure. Figure 13.18 shows the resulting dynamics of the cloud expansion as a function of density and species. The propagation of fast electrons in the gas is limited by the need for a neutralizing return current, and by the creation of electrostatic fields due to charge separation. The condition on current neutralization of fast electron and return current gives JTOT = enb vb − ene ve ≈ 0, which implies that: 1. the maximum fast electron current density which can propagate is (ene c), and 2. background electrons are also accelerated to high velocities in such conditions. Therefore, identifying the background electrons from the material with slow electrons is no longer correct, since they rapidly become as fast as the incoming relativistic electrons. Also, the charge neutrality can be violated only at the leading edge of the propagation, over a distance in the order of Debye length λD of the fast electrons. This is also the region where the space charge electric field is large and can ionize the background gas, and thus it coincides with the width of the ionization front Δx. The electric field very rapidly ionizes the background gas, creating the free electrons needed for the neutralizing return current.

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Finally, free electrons are set in motion and establish a return current which cancels the fast electron current. The establishment of the return current and the cancellation of the positive charge left behind by the fast electrons takes a time of the order of ≈λD /ve , where ve is the drift velocity of the background electrons. This process is slow because the free background electrons are (at least initially) slow and strongly collisional, and collisions inhibit the return current. Since, however, no further propagation of the fast electrons is possible before the charge separation is cancelled, the fast electron current is finally forced to move with a velocity close to the return velocity of background electrons, i.e. vcloud ≈ ve . This gives a slow velocity and a strongly inhibited propagation. Images obtained using the Optical Transition Radiation (OTR) from the second foil confirm this strong inhibition. (See Part II of our work for the use of OTR emission as a diagnostic of fast electron propagation) The process is qualitatively explained in Fig. 13.19. This experiment may indeed put some light also on the problem of inhibition of electron transport through the front surface of the target, a problem of great current interest to the modelling community. As it is well known, the prepulse creates a preplasma on the target front side, which is characterized

Fig. 13.17. Typical shadowgraphic image obtained in an experiment on fast electron propagation in gases [41]. On the right, the target scheme. The white arrow shows the position where the CPA laser beam is focused. In this particular shot, the probe beam is sent 20 ps after the CPA beam is sent on the Ti foil front side


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by low electron density and fast electrons may not be able to penetrate into the solid because they are stopped by the electrostatic fields produced by charge separation, and there is need for ionization, and a return current, the creation of an ionization wave. It also shows the potential difficulties in transmitting the fast electron beam through a density gap (here the gap at the Ti foil/gas interface). This situation may indeed also be met in cone interaction experiments when the electron beam must propagate from the Au foil on the back of the cone tip to the non-yet fully compressed Deuterium-Tritium (DT) plasma behind [7]. The character of this ionization wave (and its velocity) may sensitively depend on the number of available electrons. This may explain why the cloud velocity is sub-relativistic in the gas while it was of the order of c in the solid, as observed by Gremillet et al. [42] and Borghesi et al. [43]. However, conceptually the mechanism is the same. Indeed this also could explain why

Fig. 13.18. Cloud dimension in μm vs. time delay between main and probe pulses. Atomic densities corresponding to pressures are 1019 cm−3 (He 30 bar, empty squares, solid line), 3.2 × 1019 cm−3 (He 80 bar, full squares, dashed line), 1019 cm−3 (Ar 30 bar, empty circles, solid line), 2.8 × 1019 cm−3 (Ar 70 bar, full circles, dotted line). The data are interpolated with curves of the type r(t) = rmax (1−exp(−t/to )) where the rise time is 5 ps for all pressures and gases. Average velocities over 60 ps are much less than the speed of light (c/30 to c/10)

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in Gremillet’s paper the shadowgraphy images showed that the electron cloud transversally (i.e. along the surface of fused silica) was much larger than the laser focal spot, i.e. of the order of a couple hundreds microns against a focal spot diameter of a few tens microns. This could indeed originate from the motion of the fast electrons “along” the surface of the targets before they penetrate into the target. Such surface motion is due to large fields created on the front surface, as a consequence of the inability of the plasma to create a sufficient neutralizing return current. The effect was also observed in some PIC simulations and sometimes called the fountain effect.

13.7 Conclusions In this chapter, by referring to experiments and analyses which we have been involved in, we have reviewed some important points related to the generation of very large relativistic currents of “fast” electrons in laser-matter interactions at ultra high intensities, discussing their propagation in matter. In particular, we have described the problems of fast electron generation, which are conversion efficiency of laser energy into energy of the fast electron beam, scaling of fast electron “temperature” vs. laser intensity, and the shape of the electron distribution function, and we have discussed collisional effects vs. collective effects in propagation due to electric and magnetic fields, including the dependence of collective effects on material characteristics. We have seen ample evidence that ultra high intensity lasers can produce considerable quantities of energetic (≥1MeV) electrons with an efficiency that can exceed 30% for intensities on the order of 1019 W/cm2 . We have emphasized that the electrons so produced have an initial spatial and energy distribution that can be difficult to be determined experimentally, although there have been recent experiments addressing this issue. We have also seen that the fast electron penetration into dense materials is not simply a function of their collisions with ions, but rather is more

Fig. 13.19. A model for fast electron penetration in gases. The electrostatic field is large in the Debye shield where it produces field ionization of the background material and accelerates the generated free electrons


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often constrained by electrostatic forces arising from the necessity of localized charge neutrality. Indeed, one of the surprising aspects of the recent experimental results is that the fast electron penetration is a strong function not only of the density of the material, but of its conductivity and, more surprisingly, on the relative density of the generated fast electrons to the density of unbound electrons in the target material. The new findings of very high current transport in neutral materials arising from laser ionization have been introduced, which prove to be much more complicated than previously predicted. In this new field, researchers are now examining how to make such measurements in materials with densities much larger than normal density, and at temperatures that are more relevant to fast ignition. In the next chapter (Part II of our report), we will describe in more detail some aspects related to the heating of background material induced by fast electrons, the propagation geometry of the fast electron beam (including the issue of possible instabilities), the formation of electron bunches, the refluxing of electrons in the target, and, finally we will discuss the direct evidence of strong electric fields leading to the inhibition of fast electron propagation. Acknowledgement. The authors wish to acknowledge the precious collaboration with F. Pisani, A. Bernardinello, V. Masella, M. Manclossi, F. Canova, A. Morace, J. Davies, M. Koenig, F. Amiranoff, L. Gremillet, C. Rousseaux, T. Hall, P. Norreys, D. Neely, V. Malka, J. Santos, H. Popescu, V. Tikhonchuk, M. Lontano, M. Passoni, T. Cowan, M. Key, S. Wilks, and R. Stephens. Dedication. The authors wish to dedicate this work to the memory of Richard Snavely who pioneered and contributed greatly to the experimental developments in this field. He was not only a brilliant scientist but also an extraordinary human being. His premature death has been an incredible loss for our scientific community.

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D. Batani, R.R. Freeman, S. Baton F. Pisani et al., Phys. Rev. E, 62, R5927 (2000) F. Beg et al., Phys. Plasmas, 4, 447 (1997) M. Key et al., Phys. Plasmas, 5, 1966 (1998) T. Hall et al., Phys. Rev. Lett., 81, 1003 (1998) J.J. Santos et al., Phys. Rev. Lett., 89, 025001 (2002) (originally OTR emission has been described by I. Frank and V. Ginzburg, J. Phys. USSR 9, 1945) R.R. Freeman, D. Batani, S. Baton, M. Key and R. Stephens, Fusion Sci. Technol., 49, 297 (2006) K. Wharton et al., Phys. Rev. Lett., 81, 822 (1998) F. Brunel, Phys. Rev. Lett., 59, 52 (1987) C.E. Max “Physics of the coronal plasma in laser fusion targets”, in LaserPlasma Interactions, edited by R. Balian, J.C. Adam, North Holland Pub., Amsterdam (1982) William L. Kruer “The Physics of Laser Plasma Interactions” Wokingham: Addison-Wesley (1988). S.C. Wilks, W.L. Kruer, M. Tabak, and A.B. Langdon, Phys. Rev. Lett, 69, 1383 (1992) A.P. Fews et al., Phys. Rev. Lett., 73, 1801 (1994) T. Tan et al., Phys. Fluids, 27, 296 (1984) E L. Clark, K. Krushelnick, M. Zepf, et al., Phys. Rev. Lett., 85, 1654 (2000) J R. Davies, Phys. Rev. E, 65, 026407 (2002) D. Batani, Laser Part. Beams, 20, 321 (2002) S.R. de Groot et al., Relativistic Kinetic Theory: Principles and Applications, North Holland Publishing Company (1980); D.B. Melrose et al., Journal of Plasma Physics, 62, 233 (1999); T. Phillips et al., Rev. Sci. Instrum., 70, 1213 (1999); P.A. Norreys et al., Phys. Plasmas, 6, 2150 (1999); K.W.D. Ledingham et al., Phys. Rev. Lett., 84, 899 (2000). M. Manclossi, J.J. Santos, J. Faure, et al., J. Phys. IV, 133, 499 (2006); D. Batani, M. Manclossi, J.J. Santos, et al., Plasma Phys. Contr. Fusion, 48, B211 (2006) P.K. Patel et al., CLF annual report 2003/2004, p. 36 (2004) G. Malka and J.L. Miquel, Phys. Rev. Lett., 77, 75 (1996) D.W. Forslund et al., Phys Rev A, 11, 679 (1975) T. Feurer, W. Theobald, R. Sauerbrey, et al., Phys. Rev. E, 56, 4608 (1997) I.H. Hutchinson Principles of Plasma Diagnostics, Cambridge University Press, Cambridge (1987) A. Bell, et al., Plasma Phys. Control. Fusion, 39, 653 (1997) R. Decoste et al., Phys. Fluids, 25, 1699 (1982). D. Bond et al., Plasma Physics, 24, 91 (1982) D. Batani et al., Phys. Rev. E, 51, 5725 (2000) J.R. Davies et al., Phys. Rev. E, 56, 7193 (1997) L. Spitzer, The Physics of Fully Ionised Gases, Wiley Interscience, New York (1962) H.M. Milchberg, R.R. Freeman, S.C. Davey, and R.M. More, Phys. Rev. Lett., 61, 2364 (1988) D. Batani, A. Antonicci, F. Pisani, et al., Phys. Rev. E, 65, 066409 (2002) J. Davies, Phys. Rev. E, 68, 056404 (2003) D. Batani, S.D. Baton, et al., Phys. Rev. Lett., 94, 055004 (2005) L. Gremillet et al., Phys. Rev. Lett, 83, 5015 (1999) M. Borghesi et al., Phys. Rev. Lett, 83, 4309 (1999)

14 The Transport of Relativistic, Laser-Produced Electrons in Matter – Part 2 Dimitri Batani1 , Richard R. Freeman2 , and Sophie Baton3 1

2

3

Dipartimento di Fisica “G. Occhialini”, Universit` a di Milano Bicocca, Piazza della Scienza 3, 20126 Milano, Italy Mathematical and Physical Sciences, 425 Stillman Hall, The Ohio State University, Columbus, OH 43210-1123, USA Laboratoire LULI, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Summary. This second part of our review is devoted, in particular, to the description of the heating of a background material induced by fast electron propagation, the propagation geometry of fast electron beams (including the issues of possible collimation and filamentation instability), the formation of electron bunches, the refluxing of electrons in the target and finally, the direct detection of electric fields connected to fast electron propagation. General conclusions on the topic of fast electron propagation are presented.

14.1 Introduction As we have observed in the first part of our review, the propagation of relativistic electrons in dense matter is not as simple as one may think at first, being governed by collisional effects on one side, and by the effects of selfinduced electric and magnetic fields on the other side. In this second part of our review, we address other important physical phenomena related to fast electron propagation. First, we describe the results of experiments aimed at measuring the heating induced in a background material by fast electrons travelling in the target. The importance of this subject is clear. After all, in the context of fast ignition, the real important question is not how the fast electrons propagate, but where they deposit their energy and how much energy is deposited. An additional problem here is that describing the heating also implies dealing with the response time of the material, i.e. how fast the energy deposited by the fast electrons is converted into thermal energy and how fast the electrons and ions in the background material attain thermal equilibrium conditions. After this, we describe the propagation geometry of the fast electron beam in the material, its direction and pointing stability, its angular divergence (opening angle), and some finer but very important aspects like the possible collimation of the electron beam, the possible filamentation of the beam as it travels in the target and whether this phenomenon arises as a result of a Weibel-like instability or as a consequence of ionization issues. Another important aspect, which has been observed in several recent experiments, is that fast electrons are not injected continuously into the target;

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rather, they are injected (and travel) in the form of electron bunches. This behaviour arises because of the effect of ponderomotive forces and other absorption mechanisms, which act periodically on the electron population in the region where the laser is absorbed and the fast electrons are generated. Experiments have also shown how the electrons reflux in the target. These important refluxing effects arise as a consequence of huge electrostatic fields produced by space charge separation that effectively confine the bulk of the fast electron population inside the target, and as a consequence of the fact that the generated fast electrons have such a large average kinetic energy that the penetration range becomes much larger than the typical target thickness used in most experiments performed thus far. In other words, when fast electrons arrive at the rear side of the target, they continue to maintain most of their kinetic energy; however, they cannot escape into vacuum because of the space charge. Therefore, they are forced to move inside the target, thereby resulting in refluxing. This has very important effects on the physics as well as the performance of diagnostics used in experiments. We then describe the results of some recent experiments that address the problem of the direct detection of electric fields. After all, since the existence of huge electrostatic and inductive fields that cause the refluxing of electrons, the electric inhibition of propagation and the setting up of the return current is predicted theoretically, it is important to directly show the presence of these fields and measure their strengths. This has been performed by using the recently developed proton-deflectometry diagnostics. We finally present some general conclusions on the topic of fast electron propagation in matter. Further, let us recall that due to the large amount of experimental work, which is already available on this subject, we have not attempted to present an exhaustive review here. Rather, we mainly deal with our own experiments and describe their results.

14.2 Heating of Background Material Induced by Fast Electrons It is clear that the feasibility of fast ignition depends not only on the propagation of fast electrons in general, but also more specifically on their capability to deposit their energy and heat the background material. Several experiments have been performed in this regard using optical or X-ray diagnostics. Martinolli et al. [1] have measured the time and space variations in the reflectivity of the rear side of a target irradiated on its front side by an ultrahigh-intensity laser beam. Figure 14.1 shows a time series obtained for two Al targets of different thicknesses. A probe beam (τ = 300fs) was used and gave a temporal resolution better than 1ps. The diagnostic was based on the fact that the heated material has a larger resistivity and hence a lower reflectivity. In Fig. 14.1, it is observed that the heated region expands with time.

14 Relativistic Electron Transport 2

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Fig. 14.1. Reflectrometry of the rear side of the target using a 300-fs probe beam. The heated region corresponds to a drop in reflectivity and expands with time [1]

Unfortunately, it was observed that reflectivity is quite insensitive to temperatures in the range of 1–100 eV, i.e. in the range attained in the experiment. Further, the analysis was somewhat complicated by hydro effects, i.e. by the expansion of the material on the rear side during the recording of the experimental data. A better constraint on the experimental temperature could be obtained by simultaneously analysing the thermal emission from the rear side of the target. The main problem with this approach is that target emission at significantly early times is completely dominated by optical transition radiation (OTR) emission (see Sect. 14.3). Consequently, the signal at later times (a few tens of picoseconds after CPA laser irradiation) must be considered and back extrapolation must be performed again using hydro models. Further, using optical/near-UV diagnostics, Kodama et al. [2] have tried to measure the thermal emission from the rear side of the target. They have developed a very sophisticated diagnostic (HISAC [2]), which is based on a bundle of optical fibres connected to a streak camera. A 2D image of the sample is formed on the bundle top. The fibres are arranged such that the 2D image is divided into 1D sections on the slit of the streak camera. Hence, several 1D (x,t) images corresponding to different y-sections are obtained by using the streak camera. After the image has been recorded, software reconstruction allows us to obtain the continuous time evolution of the 2D image. The signal is then analysed with the usual greybody approach. Using such diagnostics, Kodama et al. have determined temperatures higher than the expected values. However, it is not entirely clear whether the recorded signal corresponds only to thermal emission and how the problem of OTR emission at early times is solved. Indeed if some OTR is

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present in the assumed thermal emission, it would imply a bigger signal and hence a larger-than-expected temperature (see Sect. 14.3 for a discussion on OTR). Stephens et al. [3] have performed the same type of measurements in the extreme UV (XUV) range by using a multilayer spherical mirror operating at hν ≈ 70eV (λ ≈ 18nm). Although the principle used is the same as that of optical diagnostics, these measurements are much more sensitive in the range of a few eV and approximately 100eV (simply because the peak of blackbody emission at these temperatures is closer to the observation wavelength). Hence, the brightness temperature of the rear side of the target could be obtained. The drawback of the experiment was the absence of time resolution. Therefore, the XUV images were time integrated and the brightness temperature was obtained in a time-averaged manner. Hence, a hydro calculation was necessary in order to recover the initial temperature, corresponding to fast electron heating, from the time-integrated signal. Figure 14.2 shows a typical image obtained in the experiment. Figure 14.3 shows the plot of the experimental measurements vs. the Al target thickness. In general, the results are compatible with those obtained using optical diagnostics. Finally, Martinolli et al. [4] have tried to measure the temperature inside the target by using hot Kα diagnostics. The principle of the diagnostics is based on the detection of shifted Kα lines. Such a shift of resonance lines is due to ionization (and the difference in screening because of missing electrons, i.e. shift in energy levels). Hence, shifted emission can be used as temperature diagnostics; furthermore, it possesses the advantage of having high transparency through deep plasma. Figure 14.4 shows a typical X-ray spectrum obtained in the experiment (performed at the Rutherford Appleton Laboratory). Taking into account

Fig. 14.2. XUV image of the rear side of the target obtained using a multilayer spherical mirror operating at hν ≈ 70 eV [3]


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Fig. 14.3. Rear-side temperature by XUV imaging vs. target thickness in μm [3]

Fig. 14.4. Emission spectrum from a layered Al/Cu target showing the cold and hot Kα lines of Al and the cold Kα line of Cu [4]

the ratio between the shifted line and the cold Kα line, Martinolli et al. have obtained a typical temperature of ∼20eV for a target thickness of less than 50μm. Such an experimental set up can potentially measure the heating inside the material with high accuracy. However, here, the fast electron beam simultaneously acts as the pump (i.e. it heats the material and induces its thermal ionization) and the probe (i.e. fast electrons induce hot Kα emission from ionized atoms). Therefore, any delay between energy deposition and ionization and between energy deposition and changes in electrical conductivity results in a strong reduction in hot Kα emission. This is expected because of the finite electron-ion relaxation time. Another reason for the reduced hot/cold Kα emission may be due to the fact that while hot Kα emission only originates from the central heated region, which is crossed by the fast electron beam, cold Kα emission also originates from a much larger spatial region where the number of fast electrons is sufficient to produce Kα emis-

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sion but not sufficient to induce target heating. Therefore, the interpretation of these experimental results significantly depends on the spatial (2D) and temporal details of the fast electron beam and those of the target material. More recently, at ILE, H. Nishimura et al. [5] have used the same type of diagnostics, but with a much thinner target thickness. The experiment was performed with a Ti:Sa laser (800nm, 100mJ, 135fs) at an intensity of 2 × 1017 W/cm2 (Thot = 40keV). Two types of targets were considered: 1) Al with Mg coating (thickness in the range of 0–2.2μm) and 2) chlorinated plastic with CH coating (0–2μm). A curved crystal X-ray spectrometer was used to detect Kα emission from Al or Cl. A typical spectrum is shown in Fig. 14.5. Figure 14.6 shows the variation in the recorded X-ray spectrum of an Al target vs. the thickness of the CH overcoating. With an increase in the overcoat thickness on Cl, the Kα, Kβ and Heα lines monotonically decrease at approximately the same slope, while the bulk electron temperatures at around 120eV are almost constant. The fact that the K and He lines decrease with approximately the same slope is evidence that the measured heating is in fact directly related to fast electron propagation. In conclusion, it is interesting to observe Fig. 14.7, which shows the experimental data obtained at LULI and at Rutherford. These data show a temperature increase of the order of 1–2eV per joule of incident laser energy, at least for sufficiently thin targets. Such results are qualitatively in very good agreement with those obtained at LLNL (300eV for 500J) and with the results of the integrated fast heating-implosion experiment performed at ILE (400eV for 300J [6]) where the temperature increase

Fig. 14.5. Typical X-ray spectrum obtained by Nishimura et al. [5]


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Fig. 14.6. Behavior of intensity of clorine lines as a function of target thickness after Nishimura et al. [5]

Fig. 14.7. Experimentally measured temperature induced by fast electron propagation in the background target material (eV) vs. propagation layer thickness (μm). The data are obtained from experiments conducted at LULI and Rutherford

was measured with respect to the increase in the neutron yield from fusion reactions. In all these experiments, the values appear to converge around a number of the order of 1eV/J. Of course it would be highly speculative to extrapolate these data to a real (and very different) fast ignition regime; however, it would indeed yield a temperature of 5–10keV for a 10-kJ laser beam.

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14.3 Propagation Geometry of the Fast Electron Beam 14.3.1 Pointing and Divergence Another key point is to evaluate the typical divergence of the produced electron beam and its beam pointing with respect to the laser beam (and/or target normal). Several diagnostics have been used for this evaluation. Figure 14.8 outlines the general nature of the experiments recently undertaken by Stephens et al. [3]. In these experiments, a thin Cu fluor layer was buried in a planar Al target at increasing distances from the front side of the target. Figure 14.9 shows a typical image recorded on the bent crystal imaging system, and the plot shows how the size of the image varied with the position of the fluor layer. In this experiment, the number of recorded Kαα photons is assumed to be proportional to the number of fast electrons passing through the fluor layer. Figure 14.10 shows the results of various shots of differing total energies (each shot normalized by the laser energy) obtained from two different lasers with nominally similar intensities but presumably differing values of prepulse. There are several essential elements presented in this figure: (1) The data are largely independent of the type of laser used (compare the data from LULI and RAL); (2) the fast electrons that traverse the target do so at a large angle, that is, there is no apparent collimation; and (3) the minimum size of the Kα recording obtained with the fluor layer at or near the front side of the target is at least 5 times the size of the laser spot size on the target. The same experiment also shows the effects of the pointing stability of the fast electron beam. The small spot on the bottom-left side of Fig. 14.9(a) represents the bright Bremsstrahlung emission due to the Al plasma on the front side. Therefore, it gives a rather precise indication of the laser focus

Fig. 14.8. Schematic of the measurements of the propagation of fast electrons through Al at room temperature. The bent Bragg crystal imaging system was capable of a resolution of the order of 10 μm at the target; the data were reproducible for a variety of laser energies and even laser types (see text). The initial spread of the electron beam at the front side of the target is larger than the size of the laser spot


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Fig. 14.9. (a) X-ray image from a 10-μm Al/100-μm CH/20-μm Ti/10-μm Al target at LULI showing the Bremsstrahlung emission from the Al layer (small spot, lower left) and the Kα emission from the Ti layer; the camera view is at an angle of 30◦ to the surface normal and the camera vertical is rotated by 23◦ . (b) Propagation angle relative to the surface normal calculated from the relative position of the two spots for a series of films with CH thicknesses varying from 50 – 200 μm

Fig. 14.10. Kα spot diameter at half-max intensity as a function of Al thickness. The black line is a linear fit to the data, showing a spreading angle of ∼40◦ . The X’s indicate targets with thicker backlayers to limit refluxing. The distribution of Ti Kα spot diameters is similar (circular spots in inset) but covers a shorter range; both sets of data extrapolate to a large initial diameter. The open squares show the diameter calculated with a Monte Carlo model [7]

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position. The larger spot indicates the Cu Kα emission. From such images, it is possible to reconstruct the fluctuation in beam pointing (Fig. 14.9(b)). Another experimental technique that demonstrates the rapid transverse spreading of the fast electrons as they propagate through the target employs an analysis of the optical emission as the fast electrons cross the target/vacuum interface on the back side (see Fig. 14.11). Such radiation is mainly due to Optical Transition Radiation (OTR) since a sudden variation in the refreactive index is observed when the fast electrons cross the interface [8]. Figure 14.12 shows typical time-integrated and time-resolved images of the emission at the rear side of the target. OTR emission produces a bright emission that is strongly localized in space (bright point surrounded by less intense signal) and time (the first intense signal in time); however, thermal emission and the emission due to the arrival of the shock/heat wave on the rear side of the target are much more spread in space and time. Figure 14.13 shows gated images of the rear side emission as a function of target thickness. Such images allow the typical divergence of the fast electron beam to be reconstructed, as shown in Fig. 14.14.

Fig. 14.11. Schematic layout of rear-side self-radiation diagnostics for fast electron transport. This technique uses a streak camera to differentiate between the emission from the heated/shocked target and the prompt OTR/synchrotron radiation due to fast electron interactions on the rear surface of the target

Fig. 14.12. Typical time-integrated (left) and time-resolved (right) images of the emission at the rear side of the target


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Fig. 14.13. Typical temporally gated images of the emission at the rear side of the target as a function of target thickness

Fig. 14.14. FWHM of the optical signal as a function of thickness, showing the full propagation angle of ∼35◦ . Note the large angle of transport (figures from Santos et al. [8])

The important element in these results is that by using a technique quite different from the Kα imaging technique, the spreading of the fast electron beam was reproduced, except for very thin targets. [In this case, the size of the OTR source was of the order of 2 times the laser focal spot, while the minimum size of the the Kα source was as large as 100μm. This is probably due to the fact that Kα emission can also occur due to refluxing electrons] However, it must be noted that there is an important difference. Although, in general, it is true that Kα imaging and OTR can be used to determine the opening angle of the fast electron beam, in principle, these two diagnostics consider different components of the fast electron distribution. OTR is dominated by the contribution of very fast electrons, while Kα is mainly produced by lower-energy electrons (indeed the cross section for inner-shell ionization has a peak just above E = EK , where EK is the ionization energy of the inner shell). New experimental techniques have been recently employed to gain a more detailed understanding of this rapid transverse spreading. One of these is shown in Fig. 14.15, where the bent Bragg crystal imaging system is configured to view the target from the side, specifically near the very front side of the target. This technique requires special materials for the target [7] (in this

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Fig. 14.15. Schematic of a technique for recording the penetration of fast electrons into a target from the side. The technique requires a special material – a solid solution amalgam of 50% Cu and 50% Al. The opening angle of the bent Bragg crystal imagining system is shown greatly exaggerated. In practice, the opening angle is of the order of 1◦

Fig. 14.16. Typical result obtained using the side view technique, schematically shown in Fig. 14.15. The fast electrons are observed to b.e rapidly attenuated along the horizontal direction; however, they are spread along the transverse direction near or along the surface of the target. The Cu/Al alloy target is viewed side-on from the near edge

case, a solid solution amalgam without grains and comprising 50% Cu and 50% Al). The targets have dimensions of 2mm×2mm×200μm with the laser impinging on the thin dimension approximately 50μm from the observation side. Figure 14.16 shows a representative result of the measurements described in Fig. 14.15. These results are, again, independent of the type of laser used or the energy of the laser. In this particular example, the RAL laser (0.5ps,


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Fig. 14.17. Focusing difference between an ideal diffraction-limited beam (lowestorder Gaussian) and a more realistic focus spot with non-negligible energy spread out over an area many times the ideal spot size. All the focal spot measurements at various large laser facilities around the world tend to be more like the realistic focal spot, which is schematically represented in this figure

∼50J nominally 100TW) was focused to approximately 10μm. The laser was focused at approximately 50μm from the nearest edge. In this figure, two orthogonal line-outs have been drawn that suggest strong initial spreading at the surface of the target with a strongly attenuated penetration along the horizontal axis. It is observed that the width of the Kα image is substantially larger with respect to the lateral dimension than the measured laser spot size; this is in agreement with the side view. One entirely plausible explanation for the very large entering spot size of the laser is that the laser is not ideal by any means. Figure 14.17 elucidates the difference between a lowest-order Gaussian focused beam and the more probable realistic beam from an existing highly amplified, short-pulse laser system. 14.3.2 Possible Collimation of the Electron Beam In the previous section, we have observed how the fast electron beam appears to expand with an opening angle of the order of 35−40◦ . Although such behaviour can be reproduced by Monte Carlo simulations (see Fig. 14.10), it certainly results from the competition between collisional and electromagnetic effects in fast electron propagation. The universal opening angle, which

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has been observed in experiments, is probably simply the result of the fact that the intensity range (and the laser and fast electron beam characteristics) used in most experiments is quite similar. Therefore, approaching a regime of direct interest for fast ignition (laser pulses of 10kJ in 10ps) could result in dramatic changes. In particular, efficient heating of the core of an inertial confinement fusion (ICF) target requires the electron beam to remain collimated up to its final absorption zone, i.e. up to a distance of several hundreds of microns. This can only be achieved through the pinching effect of the beam-driven magnetic field that completes with multiple scattering. The first indirect indication of magnetic field effects in fast electron propagation inside solid materials is probably mentioned in the study of Tatarakis et al. [9]. They focused the laser at ≈1018 W/cm2 on solid targets having thicknesses of approximately 50μm and observed the target with shadowgraphy techniques. The images showed the formation of some localised plasma on the rear side of the target (see Fig. 14.18); this process was related to the arrival of the fast electrons (because it appears very early and with a small dimension). The region involved in this process had a dimension comparable to that of the laser focal spot, which suggested that the electron propagation was in the form of a beam and not isotropic. Moreover, the location of the plasma expansion at the rear side did not always correspond to the exact focal spot position, but could be shifted, thereby indicating that the electron beam could sometimes travel in a direction other than that along the normal to the target. This effect could be due to the strong magnetic fields localized

Fig. 14.18. Front- and rear side-expansion of a laser-illuminated 50-μm Al target (from Tatarakis et al. [9])


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on the front side of the target that probably changed the propagation of the electron beam. A much more direct evidence for fast electron beaming inside the target material was provided in the study of Gremillet et al. [10] and also in a similar study of Borghesi et al. [11]. Here, we describe the first study only. In this experiment, the laser, operated at 2ω in order to eliminate the prepulse, was focused onto a 500-μm-wide fused silica platee. Each target was coated with a 10-μm Al foil to prevent any transmission of the laser light. A probe beam at λ = 1.057μm allowed 2D transverse imaging (shadowgraphy) with spatial and time resolutions of the order of 5μm and 600fs, respectively. The region was ionised up to a fraction of the critical density and it becomes opaque to the probe light at 1.057μm. By varying the time delay between the probe and the main beam, it was possible to analyse the ionisation dynamics occurring within the glass slab. The shadowgraphic images of the target at three successive times are presented in Fig. 14.19, corresponding to an intensity of ≥1019 W/cm2 on the target. We can clearly see narrow (≈ 20 μm) well-collimated long jets originating from the focal spot. Their lengths correspond to velocities very close to the velocity of light. The shape of a roughly isotropic cloud centred on the focal spot and expanding at a velocity of roughly c/2 is also visible. At lower laser intensities, (Fig. 14.20, ≈ 1018 W/cm2 ) the jets and the cloud are also clearly visible, although their velocities are lower. The 10-μm Al coating and the high-contrast ratio of the laser ruled out the presence of any laser light within the target. Moreover, hard X-rays effects have been ruled out by firing onto the targets with a vacuum gap. This process could completely stop the fast electrons by electron inhibition, but it would not stop hard X-rays. In the experiments, the cloud and the jets completely disappeared, thus showing that these jets consist of relativistic electrons. The observation of filamented hot electron structures clearly indicates the role of self-generated magnetic fields. It has long been known that the interplay of magnetic focusing and pressure effects may result in a self-guiding regime. In the case of solid-density targets, the first numerical predictions of

Fig. 14.19. Shadowgraphy images for laser intensity ≈1019 W/cm2

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Fig. 14.20. Shadowgraphy images for laser intensity ≈1018 W/cm2 . The arrow indicates the direction of the laser pulse

such long-scale (up to 250μm) collimated transport were obtained by Davies et al. [12]. However, in reality, the origin of the jets and the isotropic cloud remains to be fully understood. Due to the high target resistivity, electrons are not able to penetrate the target unless they ionise the background material to produce free electrons that can be made available for the return current. This process probably takes place on a small-localised region, possibly that connected to the nonhomogeneous shape of the focal spot (laser hot spots). Once a low-resistivity channel has been created, electrons tend to flow inside it, thereby further increasing the local heating and reducing the resistivity. This phenomenon, predicted by Haines several years ago, is known as electrothermal instability [13], and it may play an important role in the formation of the jets which are then sustained by the strong magnetic fields. Further, laser hot spots could contribute to electron jet formation. Indeed, localised very high intensity regions inside the laser focal spot could produce local populations of very energetic electrons with dynamics different from that of the less-fast electrons originating in the surrounding lower-intensity regions of the focal spot. These hot spots could be the regions where the insulator begins to breakdown (giving rise to the electro-thermal instability) or where the magnetic fields begin to develop. However, there is also another problem, which is not fully discussed in these papers. When detecting an ionization region by means of shadowgraphy diagnostics, it must be considered that a very small number of electrons is capable of producing a shadowgraphy signal. The critical density of an infrared probe beam corresponds to 1021 cm−3 ; however, at much lower densities (≈ nc /100), the probe rays may be refracted outside the imaging optics from which they are collected and hence may produce a shadow in the image. In other words, even if the jets observed by shadowgraphy are definitely real, they are likely to contain a negligible number of fast electrons, and hence, to play a negligible role in the transport of fast electrons in matter. This as-


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sumption has been confirmed by several 2D-PIC simulations. For this reason, other experimental studies have tried to use other types of diagnostics (such as Kα imaging, OTR, etc), which are more directly related to the passage of fast electrons. 14.3.3 Possible Filamentation of the Electron Beam, Weibel-Like Instability and Ionization Issues The role of pinching and the possibility of stable electron propagation over macroscopic distances finds a counterpart in the possible role played by Weibel-like instabilities and the resulting filamentation of the electron beam. Although there have been several claims of observations of filamentation instability in the propagation of fast electron beams in matter (particularly, Weibel-like instability), we think that most of these observations are either rather inconclusive or have been obtained only in insulating targets. A satisfactory experimental result has been obtained by Stephens et al. [3] and is shown in Fig. 14.21. In their study, the Kα images obtained from Cu fluor layers buried in Al or CH (the latter covered with a thin Al layer for consistent photon-electron coupling) are compared. It is observed that the total target thicknesses obtained in the two cases are nearly equal. It is clearly evident that in the case of a metallic propagation layer, the electron beam remains quite small and spatially homogeneous. However, in the case of plastic targets, it appears to be larger and affected by beam breaking.

Fig. 14.21. X-ray photons from Cu fluor layers embedded in Al or CH (the latter covered with a thin Al layer for consistent photon-electron coupling), ≈30◦ off-axis view, 1250 μm × 1250 μm images, front layer: (left) 130-μm Al or (right) 130-μm CH + 11-μm Al, medium layer: 20-μm Cu, final layer: 16-μm Al

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Clearer evidence, along the same line, is provided in the experimental results shown in Fig. 14.22. Such results have been obtained at the LOA Laboratory by using a Ti:Sa laser focused on plastic or metallic targets [14]. The laser operates in the chirped-pulse amplification mode at 820nm delivering full width at half maximum (FWHM) linearly polarized pulses at 40fs with on-target energies of up to 1.5J. The laser beam is focused using an f/5 off-axis parabola at normal incidence onto thin foils of Al and CH with thicknesses ranging from 10 to 100μm. The waist of the focal spot is 6μm, resulting in focused intensities of the order of 1020 W/cm2 with a contrast ratio better than 10−6 . The diagnostics used in the experiment is the recording of visible emission at the rear side of the target (due to OTR and CTR, see Sect. 14.3.1); therefore, it is primarily related to significantly fast electrons [However, in [14], it was noticed that in the case of insulating transparent targets such as CH, a very important contribution comes from Cherenkov emission]. It is evident that in the Al target, the fast electron beam is homogenous and even when the target thickness is increased. In the plastic target, the beam is initially homogeneous; however, as the thickness is increased, the beam begins to become larger and breaks into large filaments. For even larger thicknesses, the beam spreads considerably and the typical scale length of the filaments is reduced. The fact that such filamentation is observed in insulators alone strongly suggests that this is not the result of a Weibel-like instability. Such instability does indeed develop as a result of the magnetic forces between two counterstreaming currents and it is therefore a volume phenomenon. This should

Fig. 14.22. Optical emission images obtained using Al and plastic targets of different thicknesses [14, 15]


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not be very different in insulators and conductors. Indeed, the fast electrons crossing a plastic target transform it very rapidly into an ionized (conducting) material due to collisions, field ionization and rapid induced heating. Therefore, the Weibel instability is expected to grow in a similar way in conducting and insulating targets. Another phenomenon called ionization instability is more likely to occur. This instability occurs only in insulators and only at the beam edge (ionization front). Hence, the filamentation could be due to an ionization front instability. This has been theoretically described in the paper by Krasheninnikov [16] in the context of ultra-high-intensity laser-plasma interactions. It originates from the interplay between the local fast electron density, the strength of the (ionizing) electrostatic fields and the local speed of the ionization front.

14.4 Formation of Electron Bunches The spectral analysis of light emitted from the rear side of the target(described in Sect. 3) showed that it was largely dominated by the second harmonic of the laser light (i.e. green around 532nm), as shown in Fig. 14.23 (from [17]). The emission of light, as well as the dominance of the above mentioned component, continues up to a target thickness of 1mm. Initially, the presence of the second harmonic at the rear side of very thick targets, which can never be penetrated by laser light, was surprising. Accurate

Fig. 14.23. Top: experimentally recorded spectrum of the light emitted from the rear side of a 914-μm Al target irradiated by a laser beam at 1019 W/cm2 , showing the strong 2ω peak, bottom: spectral plots for two different target thickness

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shielding of the path of collected light (as well as timing measurements) assured that such light was really coming from the rear side and was not scattered inside the experimental chamber. The interpretation of harmonics emission at the rear side of a target depends on coherent optical transition radiation (CTR) from the back side of the target. This is based on the fact that electrons are produced (and travel in the target) in bunches. Each bunch produces OTR when crossing the metal/vacuum interface on the rear side of the target. The interference between light emitted from successive bunches results in a very strong peaked harmonics emission. Hence, CTR is very important because it clearly shows that (at least some part of the) fast electrons are produced and travel in the target in bunches. Such an observation is quite important because it can provide some insight into the physics of fast electron generation due to the interaction between a laser and solid targets. The dual experimental problems of hydrodynamic mass flow during and before the arrival of the ultra-intense beam on one side, and, on the other side, the degree of penetration of the laser beyond the linear critical density (a result of laser boring into plasma as well as relativistic effects) and the subsequent distorsion of the laser/plasma interface, give rise to a fundamental uncertainty regarding the nature of the acceleration process that generates the fast electrons. Possible mechanisms are the so-called (a) vacuum heating (or the Brunel effect), where p polarized light impinges directly upon the steep plasma interface; (b) resonant absorption, where the laser excites a plasma wave at the critical density, and the breaking of this wave ejects fast electrons both into the vacuum and into the target; (c) excitation of plasma waves in sub critical regions by the laser; and (d) ponderomotive effects. If we restrict ourselves to densities well above the linear critical density, the excitation of plasma waves by Raman scattering can be ignored, which eliminates the mechanism (c) mentioned in the above paragraph. In an attempt to elucidate the relative role of the remaing processes, Baton et al. [17] have made a systematic recording of ultra-short electron bunches in laserplasma interactions at relativistic intensities (of the order of 1019 W/cm2 ). Since mechanisms (a) and (b) of fast electron generation produce bunches at ω (the laser frequency), while mechanism (d) produces bunching at 2ω, it should be, in principle, possible to differentiate between them by examining the relative strengths of the harmonics of the CTR signal. Figure 14.24 shows an example of such a harmonic signal [18]. From the analysis of CTR signal, it could be deduced that only the highest end tail of the distribution was responsible for the CTR signal (mean energy ∼2MeV) with only 10−4 of the total fast electrons contributing to the bunched, tightly collimated signal. From the data of the form shown in Fig. 14.24, using a heuristic model based upon several physical assumptions, Baton et al. have determined that for 20J, delivered in 0.4ps with a peak intensity of 3 × 1019 W/cm2 , the generation of approximately 60% of the 2MeV electrons was a result of ponderomotive acceleration; the rest was a result of vacuum


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Fig. 14.24. Harmonics of the CTR radiation [18]. This measurement technique holds substantial promise for determining the relative contributions of vacuum heating to ponderomotive heating in the generation of fast electrons at the surface of the target

acceleration and/or resonance absorption. Although the modelling and assumptions included in this analysis need further refinement, the technique shows great promise for determining the fundamental nature of fast electron generation.

14.5 Refluxing Refluxing has often been discussed as an important mechanism in fast electron transport in matter and it has been used as a computational tool to explain several experimental results. The principle of refluxing is based on the fact that after crossing a relatively thin target (lo d, where lo is the penetration range), fast electrons still have a considerable amount of energy. However, the vast majority of fast electrons are not free to escape the target because of the strong electrostatic fields arising due to charge separation, which effectively confine them in the target. Therefore, they must be recycled in the target and move back and forth (refluxing takes place at both the rear and front sides). Several early indirect indications of electron refluxing are observed in many experimental results obtained with various diagnostics. One of these is given in Fig. 14.25 [8], which shows the signal recorded on the rear side of the target (OTR) as a function of target thickness (see Sect. 14.3). The rapid drop observed after 35μm could be due to causes other than refluxing. However, this was one of the first experimental indications of the importance of refluxing (apart from those arising as a result of modelling of experiments with computer codes) Direct evidence of electron refluxing was first obtained by Martinolli et al. [4]. They recorded Cu Kα yields from Al/Cu/Al layers and changed the thickness of the Al rear layer (see Fig. 14.26). If refluxing were not to occur, increasing the rear-layer thickness would only slightly reduce the Cu Kα emission by means of the absorption of emitted X-rays. However, if refluxing

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Fig. 14.25. OTR signal recorded by Santos et al. [8] as a function of target thickness (aluminium)

Fig. 14.26. Cu Kα intensity vs. Al rear-layer thickness. The black circles indicate experimental data (in 1011 ph/sr); the lines, Monte Carlo simulations with (solid line) and without (dashed line) electron refluxing

were to occur, the reduction would be stronger because a lesser number of the recirculating electrons would reach the Cu layer after being stopped in the thicker rear layer. The simulations of fast electron propagation and Kα emission were performed by adding reflective boundary conditions as limiting cases of more complicated situations at the rear side (simple refluxing) and at both the rear and front sides (multiple refluxing). The real situation (see Fig. 14.26) probably lies somewhere between the limiting cases of single-pass refluxing (no refluxing) and multiple refluxing (giving an energy efficiency of approximately 15% and a temperature of approximately 400keV). The last value is quite different from those obtained without refluxing (approximately 25%), which shows the importance of accurately considering such a phenomenon. Another demonstration of refluxing was given in the experiment by Nishimura [5], which has been described before. The Cl Heα line was observed only from the rear-surface region (the target is sufficiently thick to


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Fig. 14.27. Refluxing demonstrated in the experiment by Nishimura et al. The Cl Heα line observed from the rear surface region of a 110-μm chlorinated plastic target, with (right) or without (left) a 1-μm CH overcoating [5]

block X-rays from the front side) (see Fig. 14.27). The results were obtained for 55-μm- and 110-μm-thick chlorinated plastic with or without a 1-μm CH overcoating. These results indicate a strong suppression of Cl Heα emission with the introduction of the CH overcoating; the suppression is much greater than that expected due to the (negligible) absorption of the emitted X-rays in 1-μm-thick plastics. Most of the energy appears to be deposited in plasmas on the front and rear sides, rather than in the main body of the target. Such a result has been explained on the basis of the refluxing of hot electrons at the rear side of the target, thereby implying an increase in the heating of the rear layer (and front layer), but lesser bulk heating.

14.6 Direct Evidence of Electric Fields Although fast electron propagation dynamics is considered to be largely dominated by the effects of electric and magnetic fields, these fields have never been detected until recently. The first direct observation of electric fields associated with fast electron propagation was obtained by Batani et al. [19] by using proton radiography. This is a recently developed diagnostics, which employs laser-generated protons as a point-like source for backlighting [20], as shown in Fig. 14.28; it is used to obtain radiographic images (point projection imaging) of the gas onto a stack of radiochromic films. Notice that in principle, proton radiography can be obtained either because of a mass difference in the crossed path (different proton absorption) or because of the

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deflection of protons due to fields. This is indeed the case in the experimental setup used in [21], in which the propagation of fast electrons (see Sect. 13.6.2 in the first chapter) and the fields in a gas was studied. Due to its low mass density, the gas is completely transparent to energetic protons so that proton trajectories can be altered by fields alone. Further, note that proton radiography is sensitive to only quasi-static fields, the rapidly oscillating fields (at laser or plasma frequency) being averaged out over time. This is indeed the case of the electrostatic fields produced by charge separation, which are considered important in fast electron propagation in gases. In the proton radiography experiment, one critical parameter was the distance between the gas jet and the proton target because the presence of a residual atomic density on the rear side of the proton target was an essential reason for the deterioration of the proton beam (maximum obtained energy). The maximum energy measured with the gas was of the order of 5–6MeV as compared to 12MeV or more without the gas. Figure 14.29 shows typical proton radiography images [22]. We clearly see a hemispherical shape, more pronounced at 100 Bars than at 15 or 30 Bars. This appears to indicate the presence of a very strong electrostatic field located at the ionization front (for comparison, see the classical shadowgraphy images in Fig. 13.17 in our first chapter). In this case, the background gas pressure (N2 ) has been changed, thereby showing an increased penetration at higher pressure; this is in agreement with the shadowgraphy results. Moreover, the size of the region is in qualitative agreement with the shadowgraphy results. The formation of proton images due to proton deflection gives a clear and direct evidence of the presence of very strong fields (quasi-static magnetic and electric fields) in the gas. Indeed, the stopping power of the gas is clearly negligible for protons with these energies. However, in this exper-

Fig. 14.28. Experimental scheme for proton radiography


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Fig. 14.29. Proton radiography images obtained with N2 gas at pressures of 15, 30 and 100 Bars. The dimension of the images considering the magnification is approximately 2 mm. All images correspond to a time of approximately 20 ps after the arrival of the main laser beam on the electron target (i.e. they are formed by protons with energies of approximately 3 MeV)

iment, the time resolution was poor, and in particular, did not allow the initial fast evolving phase to be temporally resolved. The proton images have been reconstructed by using a ray-tracing code, which simulates the experimental setup and considers detailed arrangements (distances). Based on the shadowgraphy results, we assumed a hemisphere expanding according to the law r(t) = ro (1 − exp(−t/to )). Further, in accordance with the theoretical model used in [21], we assumed an electric field, which is concentrated at the hemisphere edges and radially decays exponentially over a typical distance of the order of a few fast-electron Debye lengths (i.e. a few microns). Employing these assumptions, we calculated the values of the maximum electrostatic field as ≈1011 V/m and the fall-off distance as ≈10μm, which are in fair agreement with the theoretical expectations. In conclusion, this novel diagnostics applied to the study of fast electron propagation in gases confirms that the propagation velocity and the propagation distance increase with the gas pressure. This appears to confirm the role played by the density of the background medium in establishing a return current and the essential roles of charge separation and electrostatic fields.

14.7 Conclusions In the present and previous chapters, we have performed experiments and analyses in order to understand how very large currents comprising relativistic electrons are transported through materials. As discussed in the introduction, a complete understanding of the physics of electron transport is essential for fast ignition. We have attempted to demonstrate several key concepts: 1. There is ample evidence that ultra-high-intensity lasers produce considerable quantities of energetic (≥1MeV) electrons with efficiencies that can

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exceed 30% for intensities of the order of 1019 W/cm2 . This is definitely good news for the fast ignition concept, showing that the required fast electron source will probably be achievable with the next generation laser systems. In this discussion, we have also emphasized that the fast electrons have an initial spatial and energetic distribution that cannot be easily determined by experiments, although there have been recent experiments addressing this issue using either Kα diagnostics or OTR, and coherent transition radiation in particular. It was found that the fast electrons are generated by a combination of ponderomotive effects and vacuum heating. Currently, this problem of understanding exactly how the electrons are produced and what determines their spatial and momentum distributions (including their angular divergence) remains a question of maximum interest. 2. Fast penetration of electrons into dense materials is usually constrained by electrostatic forces, developing as a result of charge space, and it is not simply a function of their collisions in the material. Indeed, one of the striking aspects of the recent experimental results is that the fast electron penetration is a strong function not only of the density of the material but also of its conductivity, and more surprisingly, it depends on the relation between the generated fast electron density and the density of electrons in the target material. One particularly interesting and important electrostatic phenomenon consists in the refluxing of electrons. Many, if not most, of the experiments performed using planar targets are complicated due to the refluxing of the electrons. Experimental techniques to minimize refluxing effects were presented. It is also possible that one of the least expected, but perhaps most useful, experimental aspects of the studies of the generation and transport of high-energy electrons by lasers is a result of refluxing: the generation of intense, bright beams of protons at the back surface of the targets. This phenomenon is useful not only for understanding energy transfer within the target but also for developing some amazing applications. We have discussed some of these applications and outlined an extremely promising avenue of research in using these protons for the detection of electric and magnetic fields within a laser plasma. 3. The evidence obtained is that fast electrons fail to self pinch in normal materials at room temperature; however, they may simultaneously undergo surprisingly complicated behaviours when they enter the target material, including ionization instabilities or Weibel-like instabilities at the front surface, which may yield anomalously large heating in the front-most layers of the targets or in the lower-density plasma generated by the laser prepulse. Several mechanisms could be responsible for this phenomenon, which is currently the subject of considerable experimental


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and theoretical study. There is some evidence of filamentation in certain experiments, but a broad analysis considering all the parameters of the experiment suggests that these filaments are composed of electrons with the highest energy that are generated by the laser and a negligible portion of the overall current. The study of very high current transport in neutral materials arising from laser ionization is new, and has proven to be much more complicated than originally anticipated. Studies are being conducted in this to determine how to perform such measurements in materials with densities much larger than their normal densities and at temperatures that are more relevant to fast ignition. Acknowledgement. The author wishes to acknowledge the precious collaboration with F. Pisani, A. Bernardinello, V. Masella, M. Manclossi, F. Canova, A. Morace, J. Davies, M. Koenig, F. Amiranoff, L. Gremillet, C. Rousseaux, T. Hall, P. Norreys, D. Neely, V. Malka, J. Santos, H. Popescu, V. Tikhonchuk, M. Lontano, M. Passoni, T. Cowan, M. Key, S. Wilks, and R. Stephens.

References 1. E. Martinolli, M. Koenig, F. Amiranoff, et al., Phys. Rev. E, 70, 055402(R) (2004) 2. R. Kodama, K. Okada, and Y. Kato, Rev. Sci. Instrum., 70, 625 (1999); Ryosuke Kodama, K. Okada, and H. Setoguchi, Proc. SPIE, 4183, 917 (2001) 3. R.B. Stephens, et al., Phys. Review E, 69 066414 (2004) 4. E. Martinolli, M. Koenig, S.D. Baton, et al. Phys. Rev. E, 73, 046402 (2006) 5. H Nishimura, Y Inubushi, M Ochiai, et al., Plasma Phys. Control. Fusion, 47 B823 (2005) 6. R. Kodama, et al., Nature 412 798 (2001) 7. R.R. Freeman, D. Batani, S. Baton, M. Key and R. Stephens, Fusion Sci. Technol. 49, 297 (2006) 8. J.J. Santos. et al., Phys. Rev. Lett., 89, 025001 (2002) (originally OTR emission has been described by I. Frank and V. Ginzburg, J. Phys. USSR 9, 1945) 9. M. Tatarakis et al., Phys. Rev. Lett., 81, 999 (1998) 10. L. Gremillet et al., Phys. Rev. Lett., 83, 5015 (1999) 11. M. Borghesi et al., Phys. Rev. Lett., 83, 4309 (1999) 12. J.R. Davies et al., Phys. Rev. E, 56, 7193 (1997) 13. M.G. Haines, Phys. Rev. Lett., 47, 917 (1981) 14. M. Manclossi, J.J. Santos, D. Batani, et al., Phys. Rev. Lett., 96, 125002 (2006) 15. D. Batani, M. Manclossi, J.J. Santos, et al., Plasma Phys. Contr. Fusion, 48, B211 (2006) 16. S.I. Krasheninnikov, et al., Phys. Plasmas 12, 073105 (2005) 17. S.D. Baton, et al., Phys. Rev. Lett., 91, 105001 (2003) 18. H. Popescu, S.D. Baton, et al., Phys. Plasmas, 12, 063106 (2005) 19. M. Manclossi, D. Batani, D. Piazza, et al., Rad. Effects Defects Solids, 160, 575 (2005)

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20. M Borghesi, et al., Plasma Phys. Control. Fusion, 43, A267 (2001); M. Borghesi, et al., Rev. Sci. Instrum., 74, 1688 (2003); M. Borghesi, et al., Las. Part. Beams, 20, 269 (2002) 21. D. Batani, S.D. Baton, et al., Phys. Rev. Lett., 94, 055004 (2005) 22. S.D. Baton, D. Batani, M. Manclossi, et al., Plasma Phys. Contr. Fusion, 47 B777 (2005)

15 Ultrafast, Ultrahigh Intensity Lasers: Challenges and Perspectives Koichi Yamakawa Japan Atomic Energy Agency (JAEA), 8-1 Umemidai, Kizu, Kyoto 615-0215, Japan, e-mail: [email protected]

Summary. Recent progress in the generation of multiterawatt optical pulses into the 10-fs range is reviewed. A design, performance and characterization of a compact titanium-doped sapphire laser system based on chirped-pulse amplification, which has produced a peak power of 0.85-PW (850-TW) with 33-fs pulse duration is described. Further extension of the systems to the exawatt power level is aso outlined.

15.1 Introduction Modern high-power lasers can now access to extraordinary high intensities of over 1020 W/cm2 in extremely short durations of 10-fs at unprecedented high repetition rates of 36 000 shots per hour (10-Hz) [1, 2]. At such intensities the electron velocity in the laser field becomes relativistic and exhibits highly nonlinear motion, thus making it possible to investigate entirely new classes of physical effects. Potential applications of these lasers include the generation of ultrafast x-ray radiation [3–7], ultrahigh-order harmonic generation [8– 18], photo-ionization pumped x-ray lasers [19, 20], optical field ionization xray lasers [21–24], laser wakefield particle acceleration [25–29], laser induced nuclear photophysics [30, 31], laboratory-based astrophysics [32, 33] and fast ignitor fusion [34, 35]. The technique of chirped pulse amplification (CPA) has opened new avenues for the production of high-energy ultrashort duration pulses without optical damage to amplifiers and optical components [36, 37]. The combination of CPA and ultrabroad-band solid-state laser materials has made it possible to produce terawatt and even one hundred terawatt femtosecond pulses with ever increasing average powers [38–41]. The CPA technique consists of four basic components: (1) a short pulse oscillator, (2) a pulse expander, (3) an amplifier, and (4) a pulse compressor. In ultrafast CPA, a short pulse is generated by a mode-locked laser oscillator and is temporally stretched by an antiparallel grating pair pulse expander [42]. The low energy and long duration chirped pulse is then amplified to a high energy commensurate with the saturation fluence of the solid state laser amplifiers. The amplified pulse is then compressed to a transform-limited short pulse of high peak power with a parallel grating pair compressor [43].

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Over the past 20 years, peak powers from terawatt CPA systems have steadily increased from less than a terawatt initially to a present record of 1500-TW (1.5-PW) [44] as shown in Fig. 15.1. At the same time, pulse durations have steadily decreased from greater than a picosecond to sub-20 femtoseconds [45]. Extremely modest amounts of energy can now be used to achieve multiterawatt peak powers. At the 20-fs pulse duration, for example, only ∼20-mJ of energy is required to achieve a peak power of 1-TW. Because much less energy is required to reach the same peak power, the size of the laser system can be significantly reduced and the repetition rate of the system can be very high. For instance, the size of the inertial-confinementfusion Nd:glass laser system such as the NOVA laser at the Lawrence Livermore National Laboratory (LLNL) [46] requires a 200 meter long building to produce 100-TW peak powers in 1-ns duration and the system cannot be operated at more than 1 shot per hour. On the other hand, 100-TWclass Ti:sapphire laser system occupies an area of only ∼15-m2 and produces the same peak power at 10 shots per second. This capability is especially significant because it allows reliable, high repetition rate, ultrahigh peak power lasers to become realistic laboratory tools for investigations requiring ultrahigh intensities. In addition the high average power is also desired to produce high fluxes of energetic particles or x-rays and to allow signal averaging techniques to be applied to relativistic laser/matter investigations. In this chapter I review the evolution of CPA into the 10-fs range. As an example, the design, performance and characterization of a compact fourstage Ti:sapphire CPA laser system at the Japan Atomic Energy Agency is

Fig. 15.1. Representative history of the evolution of CPA peak power over the past 10 years. Notations indicate location of laboratories which produced the results

15 Ultrafast, Ultrahigh Intensity Lasers: Challenges and Perspectives

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described. This system is designed to produce 30-fs pulses with peak power of one petawatt. I also discuss further extension of the systems to the exawatt power level.

15.2 Ultrafast CPA Techniques The progression of the CPA systems with respect to pulse duration, is illustrated in Fig. 15.2. The CPA technique has been demonstrated with a variety of laser materials such as Nd:glass [44,47–54], alexandrite [55,56], Ti:sapphire (Ti:Al2 O3 ) [38–41,45,57–62], Cr:LiSAlF [63,64], Yb:glass [65], Yb:KGW [66] and Yb:YLF [67]. These materials all have relatively large saturation fluences of the order of joules per square centimeter or even more, relatively long upper state lifetimes and broad bandwidths. While the first generation of CPA systems were based on Nd:glass amplifiers and generated high energy picosecond pulses, the relatively narrow bandwidth of Nd:glass has limited amplified pulse duration to a few 100’s of femtoseconds (refer to the group of Nd:Glass in Fig. 15.2). To date, pulses as short as 450-fs with a peak power of 1.5-PW have been generated by using a large scale, single-shot-perhour, inertial-confinement-fusion, Nd:glass laser [44]. While Nd:glass amplifiers have good energy storage and can easily be scaled to large volumes, they are in general limited to low repetition rates and low average power operation because of the poor thermal characteristics of laser glasses. Nevertheless, a terawatt laser with a repetition rate of 1-Hz has been built using a flashlamp-pumped Nd:glass slab power amplifier [68]. Using larger gain bandwidth materials such as Ti:sapphire [69] and Cr:LiSAF [70], however, permits the amplification of sub-100 femtosecond

Fig. 15.2. Representative history of the evolution of terawatt CPA pulse duration

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pulses from the Kerr-lens mode-locked oscillators (refer to the group of Broadband Materials in Fig. 15.2) [71–77]. In particular, Ti:sapphire has several desirable characteristics including a high saturation fluences (∼0.9J/cm2 ), a high thermal conductivity (46-W/mK at 300K) and a high damage threshold (>5-J/cm2 ) for producing high-peak and high-average power pulses [69]. Its gain bandwidth of ∼230-nm at Full width at half maximum (FWHM) could in principle support transform limited pulses of ∼3-fs. Recently pulses shorter than two optical cycles have been generated directly from Kerr-lens mode-locked Ti:sapphire oscillators by using prism pairs and double chirped mirrors in combination with and without a semiconductor saturable absorber mirror [78, 79]. As for the amplification system, Sartania et al. have demonstrated the generation of 5-fs, 0.5-mJ pulses at a 1-kHz repetition rate using the technique of hollow fiber based pulse compression and ultrabroadband chirped mirrors [80]. Although Ti:sapphire amplifier systems for the generation of pulses with duration of around 20-fs have been demonstrated [81, 82], the amplification of 20-fs pulses to energies greater than one joule has only recently been accomplished [41, 62]. The difficulty lies in the control of two major effects: high-order phase distortion in the amplification chain and gain narrowing in the amplifying media. To obtain near transform limited pulses through the CPA chain, the phase (group delay) of the pulse must be nearly constant over the broad bandwidth. We can expand the spectral phase φ(ω) in a Taylor series about the carrier frequency ω0 : ∂φ 1 ∂ 2 φ (ω − ω0 ) + (ω − ω0 )2 φ(ω) = φ(ω0 ) + ∂ω ω0 2! ∂ω 2 ω0 1 ∂ 3 φ 1 ∂ 4 φ 3 + (ω − ω ) + (ω − ω0 )4 + . . . (15.1) 0 3! ∂ω 3 4! ∂ω 4 ω0

ω0

The coefficient of the third term is the second-order-dispersion. The coefficients of the fourth and fifth terms are third- and fourth-order-dispersions, respectively. Terawatt level pulses with durations of 100-fs – 1-ps have been produced with the elimination second- and third-order-dispersions by a number of laser systems. For ultrashort pulse systems (≤20-fs), however, the fourthorder-dispersion must be eliminated. This concern has been addressed by a number of groups, which have proposed and demonstrated dispersive optical systems that are capable of controlling dispersion up to fourth order [83–86]. For example, tests of the system described by Lemoff and Barty [84] indicate that broadening during the amplification and recompression of a 10-fs Gaussian pulse would be limited to less than 1-fs. With such a system, amplification of 10-fs optical pulses is therefore limited primarily by gain narrowing during amplification and the bandwidth of the optical components in the amplification chain.


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The stretch factor in the expander–compressor combination system plays a key role in determining how close to the theoretical maximum quantum efficiency (TMQE), the amplification can be operated. In principle, if one considers the quantum defect and non-radiative losses, Ti:sapphire lasers which are pumped by frequency-doubled Nd:YAG lasers can achieve energy extraction efficiencies that approach ∼60% of the pump pulse energy. Most Ti:sapphire CPA systems, however, achieve only 10–30% efficiency. To safely achieve TMQE, an amplifier must operate above the saturation fluence of the laser material [87] but below the intensity dependent damage threshold of the optical components in the amplification chain. For instance, to achieve efficiencies near the theoretical limit of Ti:sapphire, the fluence needs to reach ∼2-J/cm2 (twice the saturation fluence of Ti:sapphire). Saturating the gain in the amplifiers also stabilizes the pulse-to-pulse amplitude fluctuations. When operating at these fluences, the intensity in the amplifier must remain low to avoid intensity dependent breakdown of dielectric materials and self-focusing and self-phase modulation that degrade spatial and temporal pulse qualities upon compression. These effects typically occur at around an ∼5-GW/cm2 for stretched pulses in the ns-range. Dividing saturation fluence by maximum intensity without nonlinear effects, we find that the pulse duration in the amplifier must be at least 200-ps and 1-ns for Ti:sapphire and Nd:glass, respectively in order to safely reach one times saturation fluence. Most Ti:sapphire CPA lasers use stretched pulse duration that are equal to or below 200-ps and are, therefore, unable to operate at the high fluences necessary for efficient amplification. In this paper we will present a Ti:sapphire system which stretches the pulses to order 1-ns and achieves final energy extraction efficiencies of >90% of TMQE. Recent progress in ultrafast CPA systems has also utilized regenerative pulse shaping to counter gain narrowing (refer to the gourp of Regenerative Pulse Shaping in Fig. 15.2) [88,89]. By including a frequency dependent filter to eliminate gain narrowing, regenerative pulse shaping allows the production of very short duration and high energy amplified pulses. With this technique the pulse duration of the amplified compressed pulses has been reduced by a factor of 2, reaching now 16-fs at a 10-TW level [45].

15.3 Design and performance of a 0.85-PW, 33-fs Ti: Sapphire Laser System As an example, the design and performance of a compact four-stage Ti:sapphire CPA laser system at the Japan Atomic Energy Agency is described in this section. The laser system has produced peak powers of 0.85-PW in 33-fs duration [90]. A schematic of the laser system is shown in Fig. 15.3. The system consists of a 10-fs Ti:sapphire oscillator, a cylindrical-mirror-based pulse expander, a regenerative amplifier incorporating regenerative pulse shaping,

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Fig. 15.3. Schematic of a petawatt Ti:sapphire laser system

a 4-pass preamplifier, a 4-pass power amplifier, 3-pass booster amplifier and a vacuum pulse compressor. 15.3.1 Laser System Front End Seed pulses were derived from an all-solid-state mirror-dispersion-controlled (MDC) Ti:sapphire oscillator which is capable of producing ∼10-fs pulses [91]. The repetition frequency of the oscillator is 82.7-MHz. The pump laser is a 5-W frequency-doubled cw diode-pumped Nd:YVO4 laser (Spectra-Physics, Millennia). Stable mode locking can be achieved with 4-W of pump power. The FWHM bandwidth of the pulses is ∼120-nm. The interferometric autocorrelation of the pulses has been measured using a dispersion-balanced interferometric autocorrelator which is capable of measuring pulses down to 5-fs in duration. The FWHM pulse duration of the mode-locked pulses is typically 10-fs assuming a sech2 envelope. In addition the phase and amplitude noise characteristics of the Ti:sapphire laser were greatly improved by using the diode-pumped solid state laser as a pump source [92]. Before amplification, the pulses from the oscillator were stretched by a factor of 100 000 in an all-reflective, cylindrical-mirror-based pulse expander [84]. The expander consists two gold-coated 1200-groove/mm ruled gratings, two cylindrical mirrors with a 1-m radius of curvature, a roof mirror and a horizontal image inverter. This design allows the compensation of dispersive phase errors up to fifth-order and eliminates spatial inhomogeneities. In order to calculate the dispersive characteristics of the expander, materials in the amplifier chain and the compressor, we use a dispersive ray-tracing analysis [84]. By calculating the phase distortions of the bulk material such as Ti:sapphire, BK7 glass, KD*P, dielectric coatings and other dispersive elements in the laser system, we were able to determine the optimum settings of the grating separations and grating incidence angles for the expander and compressor


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that compensate the phase distortions and allow the pulse to recompress close to the transform limit. The bandpass of this expander is roughly 100nm. The FWHM duration of the output of the expander is over 1-ns. Note that in this arrangement the pulse double passes the expander, i.e. is incident upon grating surfaces 8 times, before exiting. After passing through the expander and Faraday isolator, the stretched pulses are amplified in the regenerative amplifier. 15.3.2 Regenerative Pulse Shaping If the saturation is negligible, the amplified spectrum of the pulse, Iout (ω), is determined by [93], Iout(ω) = Iin(ω) [T(ω) G(ω) ]N ,

(15.2)

where Iin(ω) is the input spectrum of the pulse, G(ω) is the frequencydependent small signal gain, N is the number of passes, and T(ω) is the frequency-dependent single-pass transmission function of the amplifier. In most terawatt CPA systems, a net gain of >108 is necessary to reach the desirable output pulse energy. Even though Ti:sapphire is the broadest bandwidth material as described in Section 15.2, because of the frequency dependent gain profile G(ω) , amplification leads to gain narrowing of the pulse spectrum. A reduction in gain bandwidth not only reduces the stretched pulse duration but also results in a longer pulse after compression. The spectrum which results from seeding the amplifier with infinite bandwidth, zero energy pulses has been termed the “gain narrowing limit” of the system. Systems which include frequency dependent, lossy elements, such as polarizers and Pockels cells, will require more total small signal gain to achieve the same output energy and will thus experience more gain narrowing. For this reason, regenerative amplifiers tend to limit amplified bandwidths more than multipass amplifiers. Therefore, conventional wisdom is that multipass amplifiers are better than the regenerative amplifier in terms of spectral narrowing. Equation (15.2), however, suggests three ways to alleviate gain narrowing. First, the gain bandwidth of the amplification medium could be increased. A number of groups have attempted to increase the effective gain bandwidth of the amplification medium through mixed gain media with different peak gain wavelengths in Nd:glass [93, 94]. In this way pulses as short as 275-fs have been generated in a Ti:sapphire/Nd:phosphate and Nd:silicate glass hybrid laser [94]. Second, while narrower spectra will experience less percentage narrowing, for fixed input energy, a larger input bandwidth will always result in a larger output spectrum. This fact is exploited by reshaping the pulse spectrum before amplification by increasing Iin(ω) . Multiterawatt pulses with duration of 30-fs have been produced [57] in this manner by placing a spatial mask in the focal plane of the expander. A third possibility is to introduce a frequency dependent loss during amplification with greater attenuation at the peak of the gain profile than in

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the wing. The net amplification on each round trip is small, typically 2–3. Therefore, gain narrowing can be compensated on each round trip by inserting a linear filter that selectively attenuates the pulse spectrum after each transit of the amplifier. This method is practical for systems that employ multiple passes of single gain medium since the gain narrowing on any one pass is also relatively small. This is conveniently accomplished in regenerative amplification schemes by placing a frequency dependent attenuator inside the regenerative amplifier cavity. We have tested birefringent filters, spatially masking in the dispersive end of a cavity incorporating a prism pair, angletuned thin etalons, angle-tuned air-spaced etalons, and thin film polarilzer etalon as spectral filters. In the sample amplification system, angle-tuned thin etalons are used in the Ti:sapphire regenerative amplifier to broaden the amplified spectrum beyond the gain narrowing limit. A layout of the above mentioned ultrabroadband Ti:sapphire regenerative amplifier is shown in Fig. 15.4. The regenerative amplifier is a stable TEM00 cavity and the resonator is l.8-m long and uses two cavity mirrors [61]. The cavity consists of a 10-m radius of curvature concave dielectric mirror, and a 20-m radius of curvature convex dielectric mirror. The MgF2 anti-reflection (AR) coated Ti:sapphire crystal is 7-mm long with 0.15wt.% doping. The crystal is end pumped with a frequency-doubled Q-switched Nd:YAG laser that produces 7-ns pulses at a 10-Hz repetition rate. A 50-cm focal-length lens focuses the pump beam onto the Ti:sapphire crystal. Pulse injection in the regenerative amplifier is achieved by an intracavity Pockels cell placed between thin film dielectric polarizers. The thin film dielectric polarizers have a single-pass transmission (Tp > 98%) from 700- to 950-nm. The Pockels cell is coated with a sol-gel material. A high voltage pulse generator capable of producing up to ∼6-kV pulses with a FWHM of 8-ns is used to drive the

Fig. 15.4. Schematic of a regenerative amplifier with a spectral filters. Pol 1, 2, thin film polarizers; M1, −20 m dielectric mirror; M2, +10 m dielectric mirror


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Pockels cell. A pulse is injected after reflecting off of one of the polarizers and passing through the Pockels cell which is pulsed with a half wave voltage coincident with the optical pulse arrival. After 12 round trips, the pulse is ejected from the cavity, having fully depleted the gain, by once again pulsing the Pockels cell to have a half wave voltage and reflecting off of the other polarizer. The amplified pulse energy is typically ∼13-mJ. Two 3-μm thick etalons are used in transmission and are angle tuned so as to be off resonance (highest attenuation) around the peak of the gain profile centered at 790-nm. It should be noted that for a given thickness etalon which has been tuned to anti-resonance, there corresponds one value of single pass gain which gives a maximally flat spectrum. Higher values of gain produce narrower spectra and lower values produce double peaked spectra. The single pass gain is conveniently adjusted by changing the pump energy to the regenerative amplifier. The total output can be held constant by adjusting the number of cavity round trips. The amplifier output energy was adjusted to be approximately 8-mJ. When the stretched pulse was amplified in the regenerative amplifier without a spectral filter, the spectrum narrowed to 28-nm as shown in Fig. 15.5. However, by using the etalons to produce a frequency dependent attenuation and selectively amplifying the wings of the spectrum, the spectrum of the amplified pulse was broadened to 82-nm FWHM (Fig. 15.5). These etalons can however produce significant cubic phase error [45]. The delay is largely quadratic over most of the bandwidth of the pulse indicating that the predominant phase distortion is cubic. This predominantly cubic phase of the etalons as well as the cubic phase distortion of the high damage threshold mirror coatings has been compensated by slight alteration of the grating angle of incidence in the compressor.

Fig. 15.5. Spectra for amplified pulses from the regenerative amplifier (a) with and (b) without the thin solid etalons

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Fig. 15.6. Spectrum of two color pulses centered at 765 nm and 855 nm amplified simultaneously

In addition, it should be noted that modified filters also permitted the simultaneous amplification of two colors. Pulses centered at 765-nm and 855nm have been amplified simultaneously without unwanted amplification at the peak of the gain profile as shown in Fig. 15.6. Such pulses lead to compact sources of high power, tunable mid-infrared light via difference frequency mixing [95]. It should also be possible to produce energetic, multi-wavelength, femtosecond pulses with this technique. 15.3.3 Multipass Amplifiers Performance The 8-mJ output beam from the regenerative amplifier is enlarged by a Galilean telescope to an approximately 6-mm diameter. Further amplification is accomplished in a 4-pass preamplifier. This amplifier uses a 20-mm diameter, 15-mm long, Ti:sapphire crystal (0.15wt.% doping) with MgF2 AR coatings on both faces. The amplifier is pumped with 532-nm pulses from a frequency-doubled, Q-switched Nd:YAG laser that produces, 690-mJ, 7-ns pulses at a 10-Hz repetition rate. The beam from the pump laser is split into two outputs, which are then relay imaged to opposite faces of the amplifier crystal. Relay imaging optics with a demagnification (M = 0.67) provide spatially uniform pump beams with diameters of ∼6-mm at both faces of the crystal. The signal and pump beams propagate in a near collinear manner to maximize the gain and absorption, respectively. Since thermal lensing occurs in the Ti:sapphire crystal at ∼7W average pump power, the beam diameter on the last pass is decreased to ∼4.5-mm. For high efficiency, the pulse fluence on the last pass in the amplifier was designed to be ∼1.6-J/cm2 .


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The resulted output pulse energy was ∼320-mJ. This amplifier provides total saturated gain of 40. The small signal gain in the amplifier has also been measured to be 3.7. The output of the preamplifier is up collimated to an ∼18-mm diameter with a Galilean telescope and then introduced into the 4-pass power amplifier. This amplifier uses a water cooled 40-mm diameter 25-mm long Ti:sapphire crystal with anti-reflection coatings on both faces and is pumped with a custom built Nd:YAG laser which is capable of producing ∼7J of 532-nm radiation at 10-Hz. Output of conventional Ti:sapphire CPA systems, is typically limited in energy and average power to few hundred millijoules and several watts. This is primarily due to the limited energy and average output power of commercially-available, flash-lamp-pumped Nd:YAG lasers which produce maximum outputs of order ∼1-J and ten watts. Nevertheless, an ultrashort pulse Ti:sapphire laser which generates pulses of ∼25-TW peak power and ∼30-fs duration at a 10-Hz repetition rate pumped by three Nd:YAG lasers [39] has been demonstrated. In addition, a high energy amplifier system based on single-shot Nd:glass pumped Ti:sapphire has also delivered ∼1.1-J pulses with 120-fs duration [60]. Because the first laser system requires large number of pump lasers, however, cost and complexity are high. The second system is limited in repetition rate, because the Nd:glass pump laser can only be fired every 4 minutes. To obtain greater than 3-J of energy from the Ti:sapphire disk at 10-Hz requires a pump laser with >6-J of energy and >60-W of average power at 532-nm. For this purpose we have developed a flash-lamp-pumped high energy and high average power Nd:YAG laser. The pump laser uses Nd:YAG rod amplifiers up to a diameter of 12-mm in the master oscillator - power amplifier (MOPA) configuration. The goal of this type of pump laser design is to extract as much energy as possible and to ensure the high frequency conversion efficiency while maintaining a uniform spatial beam profile. This had been previously accomplished with relay-imaged amplifier chains in large scale Nd:glass laser systems [96]. In this architecture a master oscillator generates tens of nanosecond pulse of several hundred millijoules that is then spatially shaped and split into parallel chains of single-pass rod amplifiers of increasing size. Faraday isolators and Pockels cells are used to isolate pulses from propagating backward down the laser chain. The amplifier chains are separated by relay telescopes or spatial filters that reimage a beam forming aperture at several places through the amplifier chains. Our custom pump laser uses a similar architecture and consists of an oscillator, a preamplifier chain, two power amplifier chains and two frequency doubling crystals. 18-ns pulses from a long cavity singlelongitudinal-mode Nd:YAG oscillator, are passed through a Faraday isolator and the serrated aperture [93]. The pulses are then relayed with a spatial filter (magnification; M = 1.0) through Nd:YAG amplifier rods of 9-mm diameter. The filter uses a 1.3-mm pinhole in the far field of the spatial filter. A rel-

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atively long duration pulse compared with most Q-switched Nd:YAG lasers enables the Ti:sapphire amplifiers to be pumped at high fluence level without intensity-dependent, 532-nm damage to the Ti:sapphire crystal. Next, the 470-mJ output pulses pass through two Pockels cells and are split into two beams. Each beam is further amplified in four 12-mm-diameter Nd:YAG rod amplifiers. Energies up to 7-J per beam in the IR have been achieved. Four relay imaging optics per arm (M = 2.0 and M = 1.0) are used to relay the images into those 12-mm rod amplifiers and the frequency doubling crystals. Faraday isolators and Pockels cells are used to prevent parasitic oscillation (PO) and amplified spontaneous emission (ASE) between the amplifiers. Ninety degree rotators placed between first and second 12-mm rod amplifiers, third and fourth 12-mm rod amplifiers are also used to cancel the birefringent depolarization introduced by the amplifiers. The IR outputs from the power amplifier chains are frequency doubled to 532-nm using type II KD*P crystals. The conversion efficiency defined as the green energy output from the crystal divided by the IR energy input to the crystal was ∼50% which yields output pulse energy of ∼3.5-J per pulse at 532-nm.

Fig. 15.7. (a) Nd:YAG pump beam profile imaged on the Ti:sapphire power amplifier crystal. (b) Vertical and (c) horizontal cross-sections of the profile, respectively


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Figures 15.7 (a) and (b) show vertical and horizontal cross sections of a typical pump beam profile image on the Ti:sapphire amplifier crystal, respectively. The saturation characteristics for the frequency-doubled Nd:YAG laser pumped power amplifier are shown in Fig. 15.8. The three curves are the calculated efficiencies of the power amplifier as a function of pump pulse fluence at different input pulse fluences based on a Frantz-Nodvic simulation [87]. Squares, circles and a triangle represent the measured efficiencies in the power amplifier. With 6.4-J of pump light incident upon the crystal the amplifier has produced 3.3-J of 800-nm radiation. This amplifier provided a total saturated gain of 10 and a small signal gain of ∼5. Under these conditions, this amplifier has reached 90% of the theoretical maximum conversion efficiency of 532-nm pump light to 800-nm radiation. Nearly identical extraction efficiencies have also been obtained from a similar laser at the University of California, San Diego [62]. These results agree well with our model calculation [97]. One important note in operating a Ti:sapphire amplifier with such high fluences is that the amplified pulse spectrum is reshaped and red-shifted due to saturation. In this case, it is not desirable to center the wavelength of the pulses at 800-nm in the regenerative and pre-amplifier stages, since the effect of gain saturation causes the pulse spectrum to show an appreciable red-shift with respect to the peak of the gain spectrum of the power amplifier. Instead, a broad amplified bandwidth at high-energy output was obtained by shifting the spectrum of the input toward the short wavelength side of the desired

Fig. 15.8. Measured amplified spectrum after the regenerative amplifier (solid line), the 4-pass pre-amplifier (dashed line) and the 4-pass power-amplifier (dotted line), respectively

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Fig. 15.9. Measured amplified spectra after the regenerative amplifier (solid line), the 4-pass pre-amplifier (dashed line) and the 4-pass power-amplifier (dotted line), respectively

output. This was accomplished by slight tuning the incidence angle of one of the etalons in the regenerative amplifier (Fig. 15.9). 15.3.4 Pulse Compression and Temporal Characteristic The output of the power amplifier passes through relay imaging optics with a magnification (M = 3.3). This provides a spatially uniform beam for the pulse compression gratings and collimates the beam diameter to an approximately 50-mm. The vacuum pulse compressor consists of two parallel, goldcoated, 1200-grooves/mm, ruled gratings (Richardson Grating Labs.), which had a measured diffraction efficiency of ∼91%. A fraction of the compressor output was sent to a single-shot autocorrelator, which utilized a 50-μm BBO doubling crystal. The FWHM of the measured pulse duration is 18.7-fs. A typical autocorrelation trace and an amplified spectrum after the compressor are shown in Fig. 15.10. The duration of the transform limit, as calculated from the measured, amplified spectrum after the compressor is 17-fs. The high degree of agreement suggests that the compressed pulses are nearly transform limited. The transmission of the compressor, including the multilayer dielectric- and gold-coated turning optics, was ∼57%, yielding a compressed output pulse energy of 1.9-J, which implies a peak power for the laser pulse in excess of 100-TW. Because gain saturation in the amplifier stabilizes the pulse-to-pulse amplitude fluctuation, the energy stability is typically less than ±2%. The final output energy that is achievable from the laser system is limited by the damage threshold of the pulse compression gratings. The first grating has been operated at a fluence


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Fig. 15.10. Measured and calculated autocorrelation. The solid line is the calculated, transform-limited autocorrelation based on the measured, amplified spectrum after the pulse compressor. The circles represent the measured autocorrelation of the 18.7 fs pulse. Inset: measured amplified spectrum after compression

of 80-mJ/cm2 , an average power density of 0.8-W/cm2 and a peak power density of 2.5-TW/cm2 , respectively, with no indication of damage on the grating. In high-field physics experiments, a low-intensity pedestal (pre- and postpulses) and/or ASE would create a low density plasma in advance of the main laser pulse and thus significantly alter the physics of the laser/matter interaction [98–100]. Therefore detailed characterization and control of the temporal shape and phase of the laser pulse is crucial to the study of highintensity laser-matter experiments. Techniques for the contrast measurement of high-intensity ultrashort laser pulses are now widely used. A slow scanning high dynamic range cross-correlation between the fundamental and its second harmonic pulse, has enough dynamic range (∼108 ) suitable for the detection of very low intensity ASE. The cross-correlation can also distinguish prepulses from postpulses [101]. The arrangement of the cross-correlator is that of a Type I phase-matched, noncollinear geometry which incorporates SHG and THG nonlinear crystals. The SHG and THG signals were obtained with 1-mm and 500-μm thick KDP crystals, respectively. The THG signal was then recorded by a standard photomultiplier tube. A computer controlled stepping-motor was used to vary the delay between the two cavity arms (up to ±160-ps). Calibrated neutral density filters were also used to obtain the THG signals at a different attenuation level. The signal from the PMT was time gated (50-ns) to avoid any other long-time-scale noise and averaged over ten laser shots in a Boxcar integrator. The cross-correlation signal of the compressed pulses is shown in

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Fig. 15.11. (a) High dynamic range cross-correlation trace of a compressed pulse. Each point of the cross-correlation trace corresponds to an average of ten laser shots. (b) Cross correlation trace at a time scale different from (a). SHG-FROG trace and transform-limited pulse calculated from the measured spectrum are also shown

Fig. 15.11(a). Each point of cross-correlation trace with a time resolution of 670-fs corresponds to an average of ten laser shots. The detection limit of this apparatus is approximately 10−8 . The measured contrast is of the order of 10−6 limited by ASE mainly coming from the regenerative amplifiers. ASE can be easily suppressed by two orders of magnitude by using a solid-state saturable absorber or a nonlinear induced birefringence with a preamplifier before the pulse expander [102, 103]. Figure 15.11 (b) shows the cross correlation trace at a time scale (±1ps) different from Fig. 15.11 (a). The time resolution is about 80-fs due to group velocity mismatch in the doubling and tripling crystals. The retrieved pulse obtained from the SHG FROG measurement [104] and the compressed pulse calculated from a Fourier transform of the measured spectrum are also shown


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in this figure. The cross-correlation and FROG results are in good agreement at 10−4 which is limited by the dynamic range of the CCD camera used for the FROG measurement. 15.3.5 Wave-Front Characteristic and Correction From the point of view of high intensity laser applications, focusability of the laser beam is one of the most important features of the high-peak power CPA system. On the general assumption that temporal and spatial shape of the laser focus are sech2 and Gaussian, respectively, the peak intensity is determined as follows, IPeak =

EL 2 · , τ πw2

(15.3)

where EL , τ and w are energy of the laser pulse, FWHM pulse duration and 1/e2 radius of the focus, respectively. When the 100-TW-class laser pulse is focused by an f /3 parabola ideally, the maximum focused peak intensity, in the absence of any distorsions, would be expected to reach the 1020 W/cm2 level. This is difficult to realize in typical high peak power CPA laser systems for the following reasons: thermal lensing due to the high average power operation and relatively low optical quality of the large diameter Ti:sapphire crystals used in the amplifiers, and low optical fidelity of the compressor gratings that can distort the wavefront of compressed laser pulses. Wavefront distortion results in focal plane aberrations when the laser pulse is focused, which appears in both the enlargement of the focal spot size and the lowering of the energy content in the spot. The focused peak intensity achieved relative to the ideal case is the Strehl ratio and can be decreased to values below 0.2. Consequently, such intensities are limited to the 1019 W/cm2 level in typical repetition-rated CPA laser systems. Recently the adaptive optics have been applied to several high-peak power CPA laser systems for wavefront correction [105–107]. With these devices in their multipass amplifiers and after compressors, near diffractionlimited focusing has been obtained with Strehl ratios as high as 0.8. More recently, a generation of 1020 W/cm2 intensity was reported and used for laser nuclear reaction within a tightly-focused 5-μm2 area operating at 10Hz [108], and a highest peak intensity of 0.7 × 1022 W/cm2 has also been generated by focusing 0.1-Hz intermittent Ti:sapphire laser pulses with an f /0.6 parabola [109]. In this setup ∼f /1 tight-focusing geometry, e.g. a focal length of ∼3-cm, is employed to focus the laser beam and it is practically limited for a number of laser applications. Furthermore the focusing and evaluation conditions of the laser beam in the former laser system was not reported in detail, and the latter system would benefit from a higher repetition rate that is beneficial for many applications. Based upon these considerations, we have implemented the adaptive optics system and multi-purpose

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f /3 off-axis parabola into our 100-TW laser system in order to generate an ultra-relativistic intensity of over 1020 W/cm2 at a 10-Hz repetition rate [110]. Our adaptive optics system consists of a deformable mirror, a wavefront sensor and a personal computer for closed-loop operation [111]. The deformable mirror is based on a bimorph structure, whose substrate consists of two-piezo disks, 31 electrodes, where the mirror is glued on the top of the substrate. The dimension of electrodes area is 40-mm × 40-mm square, and arrangement of the electrodes is like many sectors in three concentric circles of 9-, 25.6-, and 42.8-mm diameters, respectively. A surface shape of each part of the mirror is deformed by applying a high-voltage to each corresponding electrode independently. By controlling the high voltages applied on 31 electrodes, the mirror surface can be modified arbitrarily. The clear aperture and stroke of the deformation of the mirror surface was 55-mm and ±4-μm, respectively. The wavefront sensor is the Shack-Hartmann type, which consists of a 40 × 40 microlenslet array coupled with a charge-coupled device (CCD) camera. A distortion of the wavefront on each small part of the laser beam is converted into a positional variation in the focal plane. The accuracy and maximum dynamic phase range of the sensor are 0.1-μm and 10-μm, respectively. The personal computer calculates the wavefront from the sensor CCD image, determines the shape of the mirror surface, and applies a high voltage to each electrode of the deformable mirror which establishes feedback control in the adaptive optics system. After pulse compression, the ultrashort laser pulse was reflected by the deformable mirror first, and propagated to the 98%-reflectance beamsplitter (Fig. 15.12). The leakage of the beamsplitter was split into two beams by a wedge that was placed out of vacuum, and delivered to the ShackHartmann sensor for the wavefront measurement and far-field monitor with long (f = +3.2m) focal length lens associated with a CCD camera, respectively. Our closed-loop adaptive optics system was capable to operate with higher repetition rates than that of the laser system 10-Hz, and the wavefront was corrected within a few seconds after several tens of feed-backs once the system was started to operate. Fig. 15.13 shows the measured wavefronts and far-field images of the laser pulse before and after wavefront correction. The Strehl ratio of the laser pulse was significantly improved from 0.20 to 0.80 by correction, which appears clearly in the reduction of the large pedestal around the focal spot between the far-field images [Fig. 15.13 (a) and (b)]. The RMS of wavefront distortion was reduced from 0.485λ to 0.102λ, that can be seen in Fig. 15.13 (c) and (d), and astigmatism measured from wavefront data was significantly reduced to one-tenth by correction. From these results, it is concluded that the adaptive optics successfully corrected the wavefront of the laser pulse, and 4 times improvement of the focusability was expected at focus in the vacuum chamber. Then we measured the focal spot with the off-axis parabola (OAP) in the chamber (also indicated in Fig. 15.12). The spot image of the attenu-


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Fig. 15.12. (a) Experimental setup. G, second compressor grating mounted on a stepping-motor-driven linear stage; DM, deformable mirror; BS, 98%beamsplitter; P, 5 μm-thick pellicle; OAP, off-axis parabola; M, removable mirror; MO, microscope objective (N A = 0.4); F, neutral density and interference filters; SM CCD, spot monitor CCD camera; W, wedge; L, f = +3.2 m plano-convex lens; FF CCD, far-field monitor CCD; WFS, wavefront sensor; V, variable leak valve; PC, personal computer; HV, stack of high-voltage power suppliers for deformable mirror. (b) Electrodes arrangement of the Bimorph deformable mirror. 40 × 40 mm square area is divided by 30 electrodes. There is another one electrode that is placed on the lower piezo disk and covers all of the electrode area, which defines the fundamental curvature of the DM surface

ated laser pulse was analyzed by a microscope objective lens associated with a 12-bits CCD camera. On the focal spot measurement a sliding mirror was inserted just before the focal point to form a vertical image point and to image the spot to the monitor. To eliminate the chromatic aberration on the spot images, an interference filter (λcenter = 800−nm with 1.5-nm bandwidth) was inserted into the imaging path just before the CCD camera. To prevent air breakdown and optical damage of the lens and CCD camera by the high intensity focused beam, the beam attenuation line was used. Since the laser pulse energy was attenuated after amplifier stages under full power operation, beam qualities, i.e. beam profile and wavefront distortion were

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Fig. 15.13. Results of the wavefront correction. (a) and (b) the far-field images focused by an f = +3.2 m lens, (c) and (d) the measured wavefronts before and after correction, respectively. The Stehl ratio of (a) 0.2 and (b) 0.8, and RMS wavefront distortion of (c) 0.485λ and (d) 0.102λ, respectively. Astigmatism in distortion is calculated to be (c) 2.25λ and (d) 0.211λ, respectively

conserved. Therefore the attenuation itself had no effect on the qualities of the focal plane, such as temporal profile, focal spot profile, spot size and its energy content. The measured focal spot image after wavefront correction was shown in Fig. 15.14. The 1/e2 size of the focal spot was 6.33 ± 0.11μm, which is very close to the diffraction-limited one of 6.29-μm. The Strehl ratio was calculated to be 0.72 in this case. Based on the separate measurements of the laser energy and pulse duration, the laser-focused peak intensity was calculated to be 1.19 ± 0.19 × 1020 W/cm2 with the laser pulse energy of 600 ± 30mJ. This Strehl ratio is slightly smaller than 0.80 of the previous far-field measurement. This degradation is considered to originate from the quality of the optics placed after the 98%-beamsplitter, such as the pellicle, guiding mirrors, and OAP. The optical field ionization (OFI) yield of helium as a function of laser intensity was also measured in order to calibrate and confirm the focused


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Fig. 15.14. Measured focal spot image after wavefront correction. The diameter and energy content of 1/e2 focal spot were 6.33 ± 0.11 μm and 60.3%, respectively

peak intensity in situ. OFI of atoms is one of the most fundamental ones, and its ionization ratio is utilized as a reference of the laser peak intensity. Fittinghoff and co-workers have investigated the OFI of helium atom in detail, including the polarization state of the laser pulse, and have been first to utilize the OFI yield as “direct intensity metering” tool [112–114]. Especially the tunnel-ionization yield of He+ ion was precisely measured and compared with results calculated by the ADK (Ammosov-Delone-Krainov) model [11], based on a quasi-classical tunneling theory within ±15% accuracy [116]. Figure 15.15 shows the He+ yield as functions of both laser energy and least-squares-fitted peak intensity. The factor of intensity calibration obtained from Fig. 15.14 was 2.02 ± 0.30 × 1020 W/cm2 per joule of energy. From the energy scaling of the laser pulse, the maximum peak intensity was calibrated to be 1.21 ± 0.18 × 1020 W/cm2 at the laser pulse energy of 600-mJ, which well agrees with the calculated intensity from optical parameters with an accuracy of 16%. This is the first result, to our knowledge, that a calculated intensity based on independent optical measurements has been confirmed by a direct OFI intensity calibration of helium.

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Fig. 15.15. Intensity calibration by OFI of helium as a function of laser intensity. The ion source volume was limited by the 300μm mobile slit to improve the signalto-noise ratio. The filled and open circles are the experimental ion yields of He+ and He2+ , respectively. The solid and dotted lines are the ion yields calculated from the ADK theory. The factor of intensity calibration is 2.02 ± 0.3 × 1020 W/cm2 per joule of energy

15.3.6 High-Field Atomic Photoionization On the basis of the peak-intensity calibration confirmed above, an OFI experiment in the fully relativistic intensity of ∼1020 W/cm2 was performed. In this study, krypton and xenon gases were selected, which have many ionic charge states around the intensity regime. The setup of the experiment is identical to that used for the helium OFI. Figure 15.16 (a)–(d) shows the typical TOF mass spectra of krypton and xenon ions at the gas pressure of ∼−107 Torr and at the laser peak intensity of 0.8 × 1020 W/cm2 . These spectra were the sum-averages of 2000 laser shots. Our measurements were concentrated on the highly-charged, inner-shell ions over Kr9+ or Xe9+ , respectively, since outer-shell ions (Kr+ –Kr8+ or Xe+ –Xe8+ , respectively) are completely saturated at such a high intensity. In the mass spectrum shown in Fig. 15.16 (b), Kr21+ peaks were not visible due to the peak overlap of the ionizing con-


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Fig. 15.16. TOF spectra of multiple-charged ions of (a) and (b) 40 krypton, (c) and (d) xenon at the laser peak intensity of 0.8 × 1020 W/cm2 . An appearance of Ar8+ in (d) is due to the result of mixed gases in order to determine the relative laser intensities among the gases

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taminations of O4+ (C3+ ). The maximum observed krypton ion charge state was +24 [Fig. 15.16 (b)], which is significantly higher than +19 observed in the previous OFI study with the laser intensity of 2.6 × 1019 W/cm2 [117]. The maximum detected xenon charge state was +26 [Fig. 15.16 (d)], which is the same as that observed previously but the signal is clearly enhanced in this study. Since there is a large gap of ionization energy between Xe26+ and Xe27+ (857-eV to 1495-eV), it is understood that Xe26+ would be the maximum observed charge state. These results are therefore solid evidence of the increase of laser peak intensity by wavefront correction. We also investigated ionization dynamics of a Xe atom in the extensive intensity range from 1013 to 1018 W/cm2 . Ion yields of Xe+ to Xe20+ were compared with the results from the ADK model. Unexpected ionization probabilities for lower charge states and also no interplay between the inner- and outer-shells by screening are inferred. Suppression of nonsequential ionization towards higher intensity and few optical cycle regimes is also proved [118]. These new findings have given us a hint of how many-electron systems behave in such fields. 15.3.7 Toward the Petawatt To scale the system to peak powers up to the petawatt level requires larger size gain media, higher energy pump lasers and larger diameter gratings. In this section, we show how to access to the petawatt level in the femtoscond range. To do so, a 3-pass booster amplifier with a 80-mm diameter, 33-mm long Ti:sapphire crystal (Crystal Systems Inc.) is added to the 100-TW Ti:sapphire CPA laser chain [90]. One major problem with a large aperture Ti:sapphire disk amplifier is ASE and parastic oscillation across the amplifier disk at high-energy pump fluence. Parasitic oscillation is due to the Fresnel reflection at the transverse material interfaces of the gain medium. Above the parasitic oscillation threshold, transverse spontaneous emission lases and the gain is clamped, no additional energy can be stored in the amplifier, and the amplifier efficiency is thus significantly reduced. For Ti:sapphire, techniques for parasitic mode suppression include using absorbative polymer thermoplastic claddings to lower the disk edge reflectivity [119]. The thermoplastic (Cargille Laboratories, Inc.) has a refractive index of 1.6849 for 800-nm light and the thus Fresnel reflection at the Ti:sapphire interface is estimated to be ∼0.048%. According to our model calculation, no parasitic oscillation occurs across the input face of the crystal until the transverse gain is reached to ∼2100, corresponding to a pump fluence of ∼5.6 J/cm2 . Furthermore, pulse stretching and compression for the petawatt laser system are also considered to produce ultrashort laser pulses. In order to compensate for the phase distortion of the materials up to fourth order in the laser chain, we have chosen a 1200-grooves/mm ruled grating (Richardson


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Grating Labs.) in the Offner triplet stretcher and 1,480-grooves/mm holographic gratings (HORIBA, Inc.) in the compressor based on the mixed grating scheme [85, 86]. Diffraction efficiencies of the compressor gratings are critical in terms of the available output pulse energy after compression in CPA systems. The diffraction efficiency of the compressor gratings has been reported to be higher for 1480-groove/mm holographics gratings than for 1200-groove/mm ruled gratings, thus the overall efficiency of the compressor should be greater. The compressor will consist of four gold-coated, holographic gratings in vacuum. The sizes of the gratings are 220-mm × 165mm for the first and last gratings and 420-mm × 210-mm for the second and third gratings, respectively. The diffraction efficiency of these gratings was measured to be greater than 92% over the 100-nm bandwidth (centered at 800-nm), and thus the overall efficiency should be greater than 75%. In the stretcher, the optimized incident angle and perpendicular separation between gratings are 8.5◦ and 627-mm, respectively. In the compressor, the incident angle and separation are also calculated to be 24.4◦ and 544-mm, respectively. Under these conditions, the residual fifth-order dispersion would broaden the initial 20-fs pulse to only 30-fs duration. A Nd:glass pump laser used in the petawatt laser system has a single-pass master oscillator power amplifier (MOPA) architecture, operated at a few shots per hour. In the system, 25-ns laser pulses from a single-longitudinalmode 1064-nm Nd:YAG master oscillator were amplified up to 800mJ by a Nd:YAG rod pre-amplifier of 9mm diameter. The laser energy of the pulses was further increased in a chain of 16-, 25-, 45- and 64-mm diameter Nd: silicate glass rod amplifiers to approximately 150-J. The IR outputs from the amplifier chains are frequency doubled to 532-nm using a couple of type II KD*P crystals. The output characteristic for the frequency-doubled Nd:glass laser pumped booster amplifier are shown in Fig. 15.17. The solid curve is a calculated efficiency of the booster amplifier as a function of pump pulse fluence based on a Frantz-Nodvic simulation [87]. Circles represent the measured efficiency in the booster amplifier. With 65-J of ∼50-mm-diameter pump light incident upon the Ti:sapphire crystal the amplifier has produced 37.4-J of 800-nm radiation. Under this condition, this amplifier has reached to near 60% of conversion efficiency of 532-nm pump light to 800-nm radiation. The spatial profile of the amplified beam is nearly flat-toped. Since the low repetition rate of the laser system (one shot per every 20 minutes), no degradation of the spatial profile is observable without any cooling system for the Ti:sapphire crystal. The thin etalons in the regenerative amplifier can be tuned to produce pre-weighted spectrum from the regenerative amplifier to compensate for spectral shifting that occurs during 3-pass amplification. As a result, the amplified spectrum is increased to 61-nm FWHM after amplification. The duration of the transform limited pulse is calculated to be ∼20-fs. Before being sent to the compressor, the beam was up-collimated to ∼120mm in diameter with relay imaging optics. A fraction of the compressor

348

K. Yamakawa

Fig. 15.17. Experimental and numerical energy conversion efficiencies in the booster amlifier. The solid line is the calculated result and the points indicate experimental results

Fig. 15.18. Measured autocorrelation trace of the compressed pulse. The pulse duration is 32.9 fs (FWHM)

output was sent to a single-shot autocorrelator, which utilized a 100-μm KDP doubling crystal. A typical autocorrelation trace of the compressed pulse is shown in Figure 15.18. The measured pulse duration, as calculated from the measured, amplified spectrum is 32.9-fs (FWHM). The transmission of the compressor was ∼76%, yielding a compressed output pulse energy of 28.4-J, which implies a peak power for the laser pulse of 0.85-PW. The


349

pedestal appeared in the autocorrelation trace is primarily due to the fifth order phase distortion. This situation is currently being corrected by simply placing the acousto-optic programmable dispersive filter (AOPDF) without any modification of the optical layout in the laser chain [120]. Spatial quality of amplified laser pulses is 1.2 and 3.2 times diffraction limited in the horizontal and vertical planes, respectively. With a diamondturned f /1 parabolic mirror (optical quality of λ/6rms) and a deformable mirror, the laser pulses can be expected to realize a focused intensity of 1.2 × 1023 W/cm2 with this beam quality [121]. This intensity will access entirely new classes of physics and applications such as direct laser acceleration of relativistic protons and ions [122].

15.4 Route to the Exawatt The potential scalability of ultrafast CPA architectures described in this section to higher energies and shorter pulse duration is an important issue considering peak powers to reach the exawatt level. We have completed a design study of a four-segmented Ti:sapphire amplifier with a full appeture of ∼200-mm-diameter for peak powers of a multi-tens of petawatts. Each crystal segment is separated by the thermoplastic claddings in order to suppress the transverse ASE. With ∼1-kJ of ∼170-mm-diameter, frequency-doubled Nd:glass pump laser light incident upon the crystals the amplifier should be able to produce ∼37.4600-J of 800-nm radiation. A pulse compressor with ∼1-m size gold coated diffraction gratings which are now routinely manufactured at LLNL can handle this output energy [44]. Thus the peak power of the compressed pulse would reach to ∼25-PW in 20-fs duration with current CPA technologies. Alternatively, optical parametric chirped-pulse amplification (OPCPA) is another candidate for the generation of ultrahigh peak power ultrashort duration laser pulses [118, 119]. Its major advantages include high gain, high contrast and high beam quality while maintaining ultra-broad spectral bandwidth. A multi-terawatt OPCPA system pumped by a high energy Nd:glass laser has been developed to produce pulses longer than 100-fs [125]. A prospect of generating peak powers up to me, terawatt-class, few-cycle (

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