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Doctoral Thesis

Coherent control of high harmonic generation using attosecondpulse trains

Author(s): Heinrich, Arne

Publication Date: 2006

Permanent Link: https://doi.org/10.3929/ethz-a-005204144

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DOCTORAL THESIS ETH NO. 16626

COHERENT CONTROL OF HIGHHARMONIC GENERATION USING

ATTOSECOND PULSE TRAINS

A dissertation submitted to the

S W I S S F E D E R A L I N S T I T U T E O F T E C H N O L O G Y Z U R I C H

for the degree of

D O C T O R O F N A T U R A L S C I E N C E S

Submitted by

A R N E H E I N R I C H

Dipl. Phys. (Universität Hannover, Germany)

born on April 22nd, 1977

in Einbeck, Germany

Accepted on the recommendation of

Prof. Dr. U. Keller, Supervisor

Prof. Dr. T. Esslinger, Co-Examiner

Dr. J. Biegert, Co-Examiner

2006

- III -

Table of contents

Publications VI

Abstract X

Kurzfassung XII

Introduction 1

High harmonic generation 5

2.1 Introduction ......................................................................................................... 5

2.2 Quantum theory................................................................................................. 10

2.2.1 Numerical solution ................................................................................. 10

2.2.2 Analytical solution ................................................................................. 13

2.3 Attosecond pulses and pulse trains..................................................................... 15

Strong field quantum path control using attosecond pulse trains .......................... 17

3.1 Introduction ....................................................................................................... 17

3.2 Attosecond pulse train assisted high harmonic generation .................................. 18

3.3 Simple man’s model .......................................................................................... 22

3.4 Summary ........................................................................................................... 25

Phase matching of attosecond pulse train assisted high harmonics ........................ 27

4.1 Introduction ....................................................................................................... 27

4.2 Numerical method ............................................................................................. 29

4.3 Macroscopic phase matching ............................................................................. 31

4.4 Summary ........................................................................................................... 40

Control of attosecond pulse train assisted high harmonic generation .................... 43

5.1 Introduction ....................................................................................................... 43

5.2 Simple man's model........................................................................................... 44

5.3 Numerical method ............................................................................................. 46

5.4 HHG dependence on:......................................................................................... 48

5.4.1 The intensity .......................................................................................... 48

5.4.2 The average photon energy of the APT................................................... 49

5.4.3 The duration of a single attosecond pulse ............................................... 53

5.4.4 Single harmonics.................................................................................... 54

TABLE OF CONTENS - IV -

5.5 Summary ........................................................................................................... 56

Enhanced vacuum-UV assisted high harmonic generation 57

6.1 Introduction ....................................................................................................... 57

6.2 Experimental setup ............................................................................................ 58

6.3 Experimental results .......................................................................................... 60

6.4 Summary ........................................................................................................... 65

Strong field effects and attosecond pulse characterization 67

7.1 Introduction ....................................................................................................... 67

7.2 Ionization of atoms in strong laser fields............................................................ 68

7.3 Temporal characterization of single attosecond pulses ....................................... 71

7.4 Temporal characterization of attosecond pulse trains ......................................... 74

7.5 Electron imaging spectrometer........................................................................... 77

7.5.1 Introduction............................................................................................ 77

7.5.2 Design of the spectrometer ..................................................................... 78

7.5.3 Momentum space image and single attosecond pulses ............................ 82

7.5.4 Experimental results ............................................................................... 88

7.5.5 Characterization of attosecond pulse trains ............................................. 92

7.6 Summary ........................................................................................................... 95

Summary and Outlook 97

Atomic units 99

Glossary 100

References 101

Curriculum Vitæ 109

Danksagung 110

- VI -

PublicationsParts of this thesis are published in the following

journal papers and conference proceedings:

Journal papers

1. K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, U. Keller, “Strong field quantum pathcontrol using attosecond pulse trains”, - Phys. Rev. Lett., vol. 92, no. 2, 023003, 2004.

2. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J.Biegert, U. Keller “Generation of intense, carrier-envelope phase-locked few-cycle laser pulsesthrough filamentation”, - Appl. Phys. B, vol. 79, pp. 673-677, 2004.

3. W. Kornelis, M. Bruck, F. W. Helbing, C. P. Hauri, A. Heinrich, J. Biegert, U. Keller, In-vited paper "Single-shot dynamics of pulses from a gas-filled hollow fiber", - Appl. Phys. B,Sonderheft der Frühjahrstagung 2004, vol. 77, no. 8, pp. 1003-1039, 2004.

4. W. Kornelis, C.P. Hauri, A. Heinrich, F.W. Helbing, M.P. Anscombe, P. Schlup, J.W.G.Tisch, J. Biegert, U. Keller, “Frequency-sheared, time-delayed XUV pulses by high harmonicgeneration in argon”, - Opt. Lett., vol. 30, no. 13, pp. 1731-1733, 2005

5. M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, U. Keller, “Large enhancement of mac-roscopic yield in attosecond pulse train-assisted harmonic generation”, - Phys. Rev. A, vol. 72,p. 013411, 2005.

6. J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. Anscombe, K. J. Schafer,M. B. Gaarde, U. Keller, Invited Paper, ”Enhancement of high order harmonic emission us-ing attosecond pulse trains”, - L. Phys. J., vol. 15, no. 6, pp. 899-902, 2005, Special issue onattosecond science and technologies, edited by Fedorov, Agostini, Paulus.

7. J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, M. B.Gaarde, K. J. Schafer, U. Keller, “Control of high order harmonic emission using attosecondpulse trains”, - J. Mod. Opt. 1-10, Preview article, May 2005.

8. A. Heinrich, W. Kornelis, M. Anscombe, C.P. Hauri, P. Schlup, J. Biegert, U. Keller, In-vited paper, “Enhanced VUV-assisted high harmonic generation”, - J. Phys. B, submittedJanuary 10, 2006, Special issue on ICOMP.

Conference papers

1. M. Bruck, W. Kornelis, C. P. Hauri, A. Heinrich, F. Helbing, J. Biegert, U. Keller, Poster,“Single-shot dynamics of hollow-fiber-compressed pulses”, - SPS Spring Meeting, Neuchatel,3-4 March, 2004

2. W. Kornelis, Mathis Bruck, C. P. Hauri, A. Heinrich, F. Helbing, J. Biegert, U. Keller, R.Lopez-Martens, J. Mauritsson, P. Johnsson, K. Varju, A. L’Huillier, M. Gaarde, K.Schafer “Time-frequency characterization of high-order harmonic emission”, - SPS SpringMeeting, Neuchatel, 3-4 March, 2004

3. J. Biegert, C. P. Hauri, A. Heinrich, F. W. Helbing, W. Kornelis, P. Schlup, U. Keller,“High field physics at ETH”, - EU XTRA Network Meeting, Ringberg , Germany, April 4-7, 2004

4. A. Heinrich, M. Bruck, C. P. Hauri, W. Kornelis, J. W. G. Tisch, J. Biegert, U. Keller, “Gastarget for efficient high harmonic generation”, - Poster ItuI42, International Quantum Elec-tronics Conference 2004 (IQEC ’04), San Francisco, USA, May 16-21, 2004

- VII – PUBLICATIONS

5. J. Biegert, M. Bruck, C. P. Hauri, A. Heinrich, F. W. Helbing, W. Kornelis, P. Schlup, U.Keller, R. Lopez-Martens, J. Mauritsson, P. Johnsson, K. Varju, A. L’Huillier, M. Gaarde,K. J. Schafer “Control of the frequency chirp rate of high harmonics”, - Poster ME62, 13thInternational Conference of Ultrafast Phenomena, Niigata, Japan, July 25-30, 2004

6. J. Biegert, C. P. Hauri, W. Kornelis, A. Heinrich, F. W. Helbing, A. Couairon, A. Mysy-rowicz, U. Keller, Postdeadline Paper, PD1, “Intense CEO-stabilized few-cycle laser pulsesfrom supercontinuum generation in filaments”, - 13th International Conference of UltrafastPhenomena, Niigata, Japan, July 25-30, 2004

7. M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, U. Keller, “Application of attosecondpulse train to quantum path control”, – Talk LTuI3, 88th OSA Annual Meeting, Frontiers inOptics, Rochester, USA, Oct. 10-14, 2004

8. C. P. Hauri, W. Kornelis, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, U. Keller,“Generation of intense few-cycle laser pulses through cascaded self-filamentation”, - TalkLMI7, 88th OSA Annual Meeting, Frontiers in Optics, Rochester, USA, Oct. 10-14, 2004

9. M. Anscombe, C. P. Hauri, A. Heinrich, W. Kornelis, P. Schlup, A. Couairon, A. Mysy-rowicz, J. Biegert, U. Keller, “Intense CEO-stabilized few-cycle laser pulses from supercon-tinuum generation in filaments”, - EU XTRA Network Meeting, Santorini, Greece, Oct. 15-17, 2004

10. K. J. Schafer, J. Mauritsson, M. B. Gaarde, A. Heinrich, J. Biegert, U. Keller, ”Theory of at-tosecond pulse train control of strong field processes”, - 340th Wilhelm und Else Heraeus-Seminar, High-Field Attosecond Physics, Obergurgl, Austria, Jan. 9-15, 2005

11. J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. Anscombe, M. Gaarde, K.J. Schafer, and U. Keller, ”Quantum path control using attosecond pulse trains”, - 340th Wil-helm und Else Heraeus-Seminar, High-Field Attosecond Physics, Obergurgl, Austria,Jan. 9-15, 2005

12. J. Biegert, P. Schlup, C.P. Hauri, W. Kornelis, A. Heinrich, M. Anscombe, U. Keller, A.Couairon, A. Mysyrowicz, “Intense phase-stable few-cycle laser pulses from self-compressionthrough filamentation”, - talk K2.2, Spring Meeting of the German Physical Society (DPG-Fruehjahrstagung), Muenchen, Germany (2005)

13. J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. Anscombe, M. Gaarde, K.J. Schafer, and U. Keller, Hauptvortrag, “Quantum path control using attosecond pulsetrains”, - 68. Jahrestagung der Deutschen Physikalischen Gesellschaft (DPG), Berlin,Germany, 4.-9. March 2005

14. C. P. Hauri, A. Heinrich, W. Kornelis, M. P. Anscombe, P. Schlup, A. Couairon, A. My-syrowicz, J. Biegert, U. Keller, “Selective high-order harmonic generation by filamentation invarious gases”, – CLEO/QELS #7, Conference on Lasers and Electro-Optics (CLEO ’05),Baltimore, Maryland USA, May 23-27, 2005

15. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, M. P. Anscombe, P. Schlup, A.Couairon, A. Mysyrowicz, J. Biegert, U. Keller, “CEO-phase-locked 5.7-fs pulses throughfilamentation”, – CLEO/QELS #7, Conference on Lasers and Electro-Optics (CLEO ’05),Baltimore, Maryland USA, May 23-27, 2005

16. W. Kornelis, A. Heinrich, C. P. Hauri, F. W. Helbing, M. P. Anscombe, P. Schlup, J. W.G. Tisch, J. Biegert, U. Keller, “Generation of frequency-sheared XUV radiation by high har-monic generation”, – CLEO/QELS #7, Conference on Lasers and Electro-Optics (CLEO’05), Baltimore, Maryland USA, May 23-27, 2005

17. A. Heinrich, C. P. Hauri, W. Kornelis, M. P. Anscombe, C. Erny, F. W. Helbing, P.Schlup, J. W. G. Tisch, J. Biegert, U. Keller, “Generation of a nearly isolated high harmonicfrom a long capillary”, – CLEO/QELS #7, Conference on Lasers and Electro-Optics(CLEO ’05), Baltimore, Maryland USA, May 23-27, 2005

18. W. Kornelis, C. P. Hauri, A. Heinrich, F. W. Helbing, M. P. Anscombe, P. Schlup, J. W.G. Tisch, J. Biegert and U. Keller, ”Frequency-sheared, time-delayed XUV pulses by highharmonic generation ”, – CLEO CG (High field laser physics and applications), CLEOEurope 2005, Munich, Germany, June 12-17, 2005

19. A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, J. Biegert, U. Keller,“Control of high-order harmonic generation using attosecond pulse trains”, – CLEO CG (High

PUBLICATIONS - VIII -

field laser physics and applications), CLEO Europe 2005, Munich, Germany, June 12-17,2005

20. C. P. Hauri, W. Kornelis, A. Heinrich, J. Biegert, U. Keller, A. Couairon, A. Mysyrowicz,“Production of selective high-order harmonics by filamentation in various gases”, – CLEO CG(High field laser physics and applications), CLEO Europe 2005, Munich, Germany, June12-17, 2005

21. U. Keller, A. Heinrich, C.P. Hauri, W. Kornelis, P. Schlup, M.P. Anscombe, J. Biegert,K.J. Schafer, M.B. Gaarde, “Control of high-order harmonic generation using attosecond pulsetrains”, - 14th International Laser Physics Workshop (LPHYS’05), Kyoto, Japan, July 4-8,2005

22. A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, J. Biegert, U. Keller,“Enhanced VUV-assisted high harmonic generation”, - ICOMP 2005 – 10th InternationalConference on Multiphoton Processes, Quebec, Canada, Oct. 9-14, 2005

23. A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, J. Biegert, U. Keller,“VUV-assisted enhancement of high harmonic generation”, - SPS Spring Meeting, Lausanne,13-14 Febuary, 2006

- X -

Abstract

The physics of atoms in strong laser fields has been a subject of intensive stud-ies in the last twenty years. When an intense laser field, with amplitude compa-rable to the Coulomb field, is applied to an atomic medium, different strongfield effects take place. Under these conditions a semiclassical model can beused to describe the phenomena. In this approach an electron is ionized by tun-neling from the ground state, and then it is accelerated in the field of the laserand driven back to the atom. The processes differ in the last step, one possibilityis that the electron rescatters elastically leading to the formation of highly ener-getic electrons, a phenomenon called above threshold ionization (ATI). On theother hand the electron can rescatter inelastically and free another electron, aphenomenon that is called nonsequential double ionization (NSDI). Alterna-tively it can recombine and generate a high energetic photon, a process knownas high harmonic generation (HHG).

In this thesis I will concentrate on high harmonic generation, the emphasis be-ing on the theoretical background and an experimental proof of how high har-monic generation can be controlled using an attosecond pulse train (APT). Inthis method an attosecond pulse train is used in combination with an intenseinfrared (IR) laser field. The timing of the attosecond pulse train with respect tothe infrared laser field can be used to microscopically select a single quantumpath contribution to high harmonic generation that would otherwise consist ofmany interfering components. The attosecond pulse train controls the release ofthe electron into the continuum and influences the yield and coherence of theharmonics generated. Depending on the chosen time delay between infraredlaser and attosecond pulse train, the harmonic spectrum is significantly en-hanced, and exhibits either a spectrally resolved plateau of harmonics, or dis-tinct harmonics in the cut-off.

- XI - ABSTRACT

To investigate, if the microscopic effect persists phase matching, nonadiabaticcalculations of the macroscopic harmonic signal created by a gas volume of he-lium atoms exposed to a strong infrared laser pulse in combination with an at-tosecond pulse train are presented. It is shown that the harmonic yield can beenhanced by two to four orders of magnitude for the optimal delays betweenthe infrared and the attosecond pulse train. Since attosecond pulse train as-sisted high harmonic generation survives and even improves phase matching asystematic study of additional degrees of freedom added by the attosecondpulse train on the single atom level is presented.

An initial experiment on attosecond pulse train assisted high harmonic genera-tion was carried out. Harmonics in the vacuum ultraviolet (VUV) spectral rangeare generated in a Xe target, and focused together with the remaining IR fieldinto a He gas jet to generate high harmonics in the extreme ultraviolet (XUV)spectral range. The harmonics generated by both fields show a clear enhance-ment of a factor of five in the cut-off region compared to the infrared-only case.In the experiment the delay between both fields could not be adjusted and theincoming vacuum ultraviolet harmonics were not temporally characterized.Therefore a direct proof that an attosecond pulse train is formed, could not beprovided.

For temporal characterization of vacuum ultraviolet or attosecond pulse trainassisted high harmonics an imaging electron spectrometer is presented. Thespectrometer maps the electron distribution of an ionization event on a phos-phor screen in such a way that the full three-dimensional electron momentumdistribution can be reconstructed. In this way new insights in strong field proc-esses not possible with conventional electron time of flight (TOF) spectrometerscan be gained. In particular the temporal structure of single isolated attosecondpulses and attosecond pulse trains can be measured. Moreover pump-probemeasurements with attosecond pulse trains, to investigate attosecond pulsetrain assisted above threshold ionization and attosecond pulse train assistednonsequential double ionization become possible with this spectrometer.

- XII -

Die Physik von Atomen in starken Laserfeldern ist ein sich seit 20 Jahren stetigweiterentwickelnder Forschungszweig. Wirkt ein intensiver Laserstrahl miteiner Feldstärke vergleichbar dem Coulombfeld auf ein atomares Mediumtreten verschiedene Effekte auf, die mit Hilfe eines semiklassischen Modellsbeschrieben werden können. In diesem Kontext tunnelt ein Elektron aus demGrundzustand ins Kontinuum. Im elektrischen Feld des Lasers wird esbeschleunigt und zum Atom zurückgetrieben. Beim letzten Schritt sindverschiedene Phänomene zu unterscheiden. Das Elektron kann elastischgestreut werden und somit hochenergetische Elektronen erzeugen. DieserProzess wird „above threshold ionization“ (ATI) genannt. Das Elektron kannaber daneben auch inelastisch gestreut werden und ein anderes Elektronionisieren. Dieser Prozess wird als „nonsequential double ionization” (NSDI)bezeichnet. Alternativ kann es in den Grundzustand rekombinieren und dabeiein hochenergetisches Photon erzeugen; ein Prozess bekannt als „highharmonic generation“ (HHG).

In meiner Dissertation habe ich mich auf HHG konzentriert, wobei derAugenmerk auf der theoretischen Beschreibung und auf dem experimentellenBeweis liegt, dass ein „attosecond pulse train“ (APT) benutzt werden kann umhigh harmonic generation zu kontrollieren. Dazu wird ein APT in Kombinationmit einem intensiven infrarot (IR)-Laserfeld benutzt. Dabei kann der Zeitpunktdes APT im Vergleich zum IR-Laserfeld benutzt werden, um mikroskopischden Beitrag eines einzelnen Quantenpfades zur HHG auszuwählen, dienormalerweise aus vielen interferierenden Komponenten besteht. Der APTkontrolliert die Ionisation des Elektrons und beeinflusst somit den Fluss unddie Kohärenz der erzeugten Harmonischen. Abhängig von dem gewähltenzeitlichen Abstand zwischen dem IR-Laser und dem APT, ist das harmonischen

- XIII - KURZFASSUNG

Spektrum signifikant verstärkt und hat entweder spektral aufgelöste Plateau-Harmonische oder ausgeprägte Harmonische im „cut-off“.

Zur Untersuchung, ob der mikroskopische Effekt „phase matching“ übersteht,werden nicht adiabatische Rechnungen des makroskopisch erzeugenharmonischen Signals von einem mit Heliumatomen ausgefüllten Volumenunter dem Einfluss eines starken IR-Laserpulses in Kombination mit einemAPT vorgestellt. Die Rechnungen zeigen, dass das harmonische Signal beioptimalem zeitlichem Abstand zwischen dem IR und dem APT um zwei bisvier Größenordnungen verstärkt wird. Da APT-assisted HHG nicht nur „phasematching“ übersteht sondern sogar verbessert, wird eine systematische Studieauf atomarer Ebene zur Abhängigkeit der Erzeugung von den zusätzlichenFreiheitsgraden des APT vorgestellt.

Ein erstes Experiment zur APT-assisted HHG wird präsentiert, wobeiHarmonische im vakuum-ultravioletten (VUV) Spektralbereich in Xe erzeugtwerden und zusammen mit dem übrig gebliebenen IR-Feld in einen He Gasjetfokussiert werden, um hohe Harmonische im extrem-ultravioletten (XUV)Spektralbereich zu erzeugen. Die von beiden Feldern gemeinsam erzeugtenHarmonischen sind gegenüber den von dem IR alleine erzeugten um dasFünffache verstärkt. In dem Experiment konnte der zeitliche Abstand nichtvariiert und die VUV Harmonischen zeitlich nicht charakterisiert werden,daher konnte kein direkter Beweis für einen APT erbracht werden.

Für die zeitliche Charakterisierung der VUV und APT-assisted HHG Strahlungwird ein abbildendes Elektronenspektrometer vorgestellt, das dieElektronenverteilung eines Ionisationsvorganges auf einen Phosphorschirmderart abbildet, dass die volle dreidimensionale Impulsverteilung rekonstruiertwerden kann. Somit können neue Einblicke in strong field Prozesse gewonnenwerden, die mit konventionellen Flugzeitspektrometern nicht möglich sind. ImBesonderen kann die zeitliche Struktur von einzelnen isolierten AttosekundenPulsen und APTs gemessen werden. Zusätzlich macht es das Spektrometer

KURZFASSUNG - XIV -

möglich Pump-Probe Messungen mit APTs durchzuführen, um APT-assistedATI und APT-asisted NSDI zu untersuchen.

- 1 -

Chapter 1

Introduction

One of the key experiments on the development of quantum mechanics was theinvestigation of the photo electric effect [1]. The inexplicable result of this ex-periment was that electrons could just be freed by light with a short wavelengthand mainly independent of its intensity. This problem was solved by Einsteinby using Plank’s quantum hypotheses [2] under which a photon has an energyof h [3]. The prerequisite for photoionization is that h I p , here Ip is the ioni-zation energy of the atom. Nevertheless in an electric field of sufficiently highintensity it is also possible that ionization occurs by absorbing two, three, andmore photons with h I p [4]. If the intensity is above 1014 W/cm2, which canbe easily generated by focusing an ultrafast amplified laser pulse and corre-sponds to one hundred times the total radiation of the sun reaching Earth, fo-cused on a spot of one square millimeter, the resulting electric field strength iscomparable to the Coulomb nuclei attraction field. An atom subjected to such astrong electric field reacts in a highly nonlinear way. Electrons initially in theground state can absorb a large number of photons (several hundreds), manymore than required for ionization. The excess energy is very large and the proc-ess is called above threshold ionization (ATI) [5]. Another process is the occur-rence of multiple ionization of atoms at intensities orders of magnitude lowerthan expected, which is called nonsequential double ionization (NSDI) [6]. Fi-nally, and probably the most interesting effect has been observed, which I willconcentrate in this thesis: The generation of high harmonics of the fundamentallaser field (HHG) [7-9], where a comb of odd high-order harmonics of the lasercarrier frequency is produced in the spectral domain, extending to photon en-ergies of 100 eV or more.

CHAPTER 1 - 2 -

A breakthrough in the theoretical understanding of HHG was the finding thatthe cut-off position in the harmonic spectra follows the universal law I p + 3U p ,[10] where I p is the ionization potential and U p = E2 4 2 , is the ponderomotivepotential, i.e. the mean kinetic energy acquired by an electron oscillating in thelaser field. Here E and are the laser electric field and its frequency. An expla-nation of this universal fact in the framework of a “simple man’s theory” wasfound shortly afterwards [11, 12]. According to this model, harmonic genera-tion occurs in the following manner: First the electron tunnels out from the nu-cleus through the Coulomb energy barrier modified by the presence of the(relatively slowly varying) electric field of the laser [13]. It then undergoes os-cillations in the field, during which the influence of the Coulomb force from thenucleus is practically negligible. If the electron comes back to the vicinity of thenucleus, it may recombine back to the ground state, thus producing a photon ofenergy Ip plus the kinetic energy acquired during the oscillation motion. Ac-cording to classical mechanics, the maximum kinetic energy that the electroncan gain is indeed 3.2U p . A fully quantum mechanical theory that is based onthe strong field approximation (SFA) was formulated soon afterwards [14]. Thesimple man’s model is included in this theory, if it is evaluated in the frame-work of Feynman path integrals [15, 16].

The theoretical description of HHG can be divided into the single-atom re-sponse, which describes the interaction of the atom with the laser field and thepropagation, where the phase matching of the radiation emitted by an ensem-ble of atoms is studied. Efficient HHG requires not only efficient radiation ofsingle atoms, but also macroscopic constructive interference of contributions ofparticipating radiation sources, i.e. efficient phase matching [17]. Neverthelessthe efficiency of HHG is generally low and typically ranges from 10-8 to 10-5.

The properties of the harmonic emission make it a unique source of XUV radia-tion, used in a growing number of applications ranging from atomic [18, 19]and molecular [20, 21] spectroscopy to solid-state [22, 23] and plasma [24, 25]physics. The harmonic radiation produced is intense enough to induce nonlin-ear optical processes in the XUV range [26, 27]. Furthermore and probably mostimportant is that HHG provides a means to produce attosecond pulses. A peri-odic train of attosecond pulses [28, 29] is generated, if the relative phases of the

- 3 - INTRODUCTION

harmonic peaks can be locked to each other. Clearly the carrier frequency ofthese attosecond bursts is significantly higher (in the XUV) than that of thedriving pulses and therefore the femtosecond barrier can be breached. Trains ofpulses with a duration as short as 130 attoseconds have been demonstrated [30-33]. Also the generation of single attosecond pulses by HHG is feasible usingfew-cycle driving pulses [34-36] and single pulses with a duration of 250 atto-seconds have been measured [36]. Apart from the generation of attosecondpulses, their temporal characterization is indispensable to provide proof of atto-second pulses and trains. Several measurement schemes have been proposed,where the majority is based on the generation of electron wavepacket replicasof the attosecond pulses by photo-ionization of a gas in the presence of an in-tense IR dressing field. Among the many good review articles about HHG Iwould like to point out these two references [37, 38].

In the framework of this thesis a new scheme for the control of strong fieldprocesses like ATI, NSDI and HHG is introduced and studied in detail. An ex-perimental setup to control HHG is built and first results are obtained. Anelectron imaging spectrometer (EIS) is designed and built, which is capable toimage the full three-dimensional electron momentum distribution of a strongfield event, to study strong field processes and to temporally characterize singleattosecond pulses as well as APTs.

The thesis is organized as follows: In chapter 2, I give an introduction to HHGthrough a theoretical description and I investigate the possibility of the genera-tion of attosecond pulses. A new scheme to control HHG on the single atomlevel by APTs is introduced in chapter 3. The influence of phase matching in apossible experimental realization is investigated in chapter 4. Since APT-assisted HHG survives and is even improved by phase matching, a systematicstudy of the additional degrees of freedom added by the APT on the singleatom level is presented in chapter 5. The experimental setup is described andfirst experimental results are presented in chapter 6. An EIS is presented inchapter 7, which can be used to study strong field processes and also to tempo-rally characterize APTs as well as single attosecond pulses. Throughout the the-sis atomic units are used and the relationship to the SI system is given in theappendix.

- 5 -

Chapter 2

High harmonic generation

HHG in gases in the multi-photon regime has been studied since the end of the60’s. In this regime the harmonic intensity diminishes rapidly with the har-monic order and only a few orders can be observed. In contrast, at high intensi-ties, a typical harmonic spectrum exhibits a fast decrease for the first few har-monics, followed by a long plateau, which ends up with rapid cut-off [7-9].HHG is a source of coherent radiation in the extreme ultra-violet (XUV) spectralrange, but the striking fact is the possibility to generate single attosecond pulses[35, 36] and APTs [16, 39, 40].

2.1 IntroductionHigh harmonics are generated through the interaction between an intense laserand a gas. The interaction region can be a pulsed gas jet, a windowless cellcontaining a gas at static pressure or a gas-filled hollow fiber [41]. Belowthreshold harmonics can be generated through the interaction between laserpulses and gases in the multi-photon regime, where the generation efficiencydecreases exponentially with increasing order. The exponential decrease in effi-ciency in the multi-photon regime can be well-explained using perturbationtheory. For intensities typically ranging from 1013 - 1014 W/cm2, the exponentialdecrease is observable followed by the harmonic plateau, where the harmonicefficiency stays roughly constant with increasing order, before reaching the cut-off, where a rapid decrease of the harmonic efficiency is observed [42, 43]. A

CHAPTER 2 - 6 -

typical high harmonic spectrum is shown in figure 2.1, which displays thesethree different behaviors.

Figure 2.1: Typical harmonic spectrum. At low orders an exponential decrease isobserved, followed by a region of roughly constant efficiency, the plateau, beforethe cutoff is reached, where the efficiency drops rapidly.

The perturbative multi-photon picture breaks down for intensities larger than1013 W/cm2. The beginning of the plateau is a sign for the onset of the tunnelionization regime, where the laser electric field is strong enough to deform theatomic potential well, such that a barrier is formed through which the electronscan tunnel (Figure 2.2(b)).

The prevailing and widely accepted simple and intuitive model for high har-monic generation is the simple man’s model [11, 12, 14], which is applicable inthe tunnel ionization regime. The model is a semi-classical model, meaning thatthe quantum aspects, like tunneling and ionization are incorporated into themodel, but that the motion of the electron in the driving laser field is treatedclassically. The three steps are as follows (Figure 2.2):

1. The electron is born into the continuum through tunnel ionization withzero kinetic energy.

- 7 - HIGH HARMONIC GENERATION

2. The electron is treated like a free classical particle, which is acceleratedby the driving laser field thereby accumulating kinetic energy.

3. Depending on the laser field when the electron was born, there is a prob-ability for it to return to the vicinity of the atomic potential. There it caneither scatter or recombine producing high-energy free electrons orphotons, respectively.

Figure 2.2: Simple man’s model. The atomic potential (a) is deformed by the elec-tric field of the driving pulse (b) and the electron can tunnel through the barrier(b). It is then accelerated in the laser field and accumulates kinetic energy (c). De-pending on the phase of the laser field, when the electron was born, it is possiblefor it to return to the vicinity of the atomic potential, where it recombines, emittinghigh-energy photons.

HHG is a periodic process with a period TL/2, where TL is one cycle of thedriving laser field. The system has inversion symmetry and thus the ionizationand recombination can occur every half-cycle of the laser field. This periodicityin time results in a spectral frequency comb with peaks separated by twice thefrequency of the driving laser field. Because the fundamental laser field is in-cluded in the frequency comb with order q = 1, the harmonics will only appearas odd multiples of the driving frequency.

The energy of the electron and thus also the frequency of the emitted XUV ra-diation upon recombination, depends on the phase of the driving field whenthe electron is born into the continuum (see different electron trajectories in fig-ure 2.3). If the ionization takes place before the optical field reaches its maxi-mum at = / 2 , it will be accelerated such that it never returns to the atomicpotential and no harmonics are generated. If, instead, the electron is born whenthe optical field is at its peak, it will return to the ion after half an optical cycle,

CHAPTER 2 - 8 -

but with zero kinetic energy. Free electrons, which are created later, however,return with high kinetic energy, and harmonic radiation is produced upon re-combination.

Figure 2.3: Different electron trajectories depending on when during the laser fieldthe electron is born in the continuum (dashed lines). The driving field is displayedas the solid line.

Atmax

0.6 the electron reaches the maximum energy before returning to theion core and E

max= 3.2 U

p [44]. The existence of the cut-off [10] in the harmonic

spectrum is thus explained by this maximum possible photon energy [45],which can only be reached from a single trajectory. For lower order harmonicsin the plateau, however, two, or more, different trajectories are possible thatreach the same final energy. The plateau harmonics are therefore generated byelectrons that have spent different times in the continuum, which affects therelative phase of the emitted light [32]. The two dominant trajectories are usu-ally called the “short” and the “long” trajectory, depending on the time theelectron spent in the continuum.

The simple man’s model is semi-classical and can therefore only be used toqualitatively explain the process of HHG. For a quantitative treatment, a fullquantum mechanical approach has been developed [14] based on the time-


dependent Schrödinger equation (TDSE). The close analogy between the classi-cal and the quantum mechanical pictures is further illustrated by the use ofFeynman’s path integrals to describe the electron propagation [16].

So far, only the microscopic aspects of HHG, where the interaction of the driv-ing laser field with a single atom was described. Macroscopically, HHG consistsof the interaction of the laser beam, which in general does not present a spa-tially homogenous intensity distribution, with an extended gas medium, whichcontains a large number of single atomic emitters and whose density might notbe constant over the interaction region. When the XUV emission of the singleemitters adds coherently, phase matching is said to occur. To achieve properphase matching, the optimization of different parameters is necessary: the posi-tion, profile, size and range of the laser focus plays an important role as well asthe density distribution, pressure and extent of the gas medium, to mentiononly the most important. Phase matching occurs as a complicated interplaybetween all these variables and can be visualized in phase-matching maps [46].

High harmonic radiation can extend to very high orders, depending on thedriving laser intensity, gas species and target conditions. The highly energeticphotons, which can be produced, extend from the XUV into the soft X-ray re-gion. Harmonic radiation in the wavelength region between 4.37 and 2.33 nm,which is called the water window, has been produced [47, 48]. This region is ofparticular interest for imaging biological material as light of these wavelengthsis more strongly absorbed by carbon than by water, thus providing high con-trast for biological samples in aqueous solutions. Recently, even harmonics withenergies up to 1.3 keV ( 1nm ) have been generated [49]. The efficiency is high-est in Xe and decreases going to He, in contrast the highest photon energies areobtained in He.

HHG is a source of coherent radiation in the XUV range and preserves all de-sirable properties of the driving laser pulse. It has good spatial and temporalcoherence [50-52], it is directional [53-55], it is emitted in short pulses of femto-second or even attosecond duration and has thus enormous potential as a toolfor time-resolved XUV spectroscopy [18], plasma interferometry [56] andpump-probe experiments [57].

CHAPTER 2 - 10 -

2.2 Quantum theoryHigh harmonics are generated by the interaction of an atom with a strong laserfield. The theoretical description is a challenging task and is in general simpli-fied to a single active electron (SAE) atom in a strong electric field. In this ap-proximation the role of the electronic structure inside the atom is encoded in aneffective (for instance Hartree-Fock) static potential felt by the (active) electroninteracting with the laser field.

There are two main approaches to this problem. First the exact full numericalsolution of the TDSE, which not only makes the interpretation of the results dif-ficult, but is also very intensive in computer time. Second an analytical solutionprovides a more straightforward interpretation and disentangles the differentcontributions, but is always based on more or less strong approximations. Thenext two subsections are a short tutorial how to simulate HHG in both ways.

2.2.1 Numerical solution

The TDSE can be numerically solved in different ways. I will concentrate on thesplit-operator fast Fourier transform (FFT) method, which is very efficient, be-cause highly optimized FFT-routines are available. The Hamilton operator H foran atom in a linear polarized laser field in x-direction Ex t( ) in the dipole ap-proximation is given by:

H =2

2

Z*

r ++ E

xt( ) x, (1)

where the electric field strength E in atomic units is connected to the intensityI [W/cm2] by E = 5.342 10

9I and the effective atomic charge Z* can be cal-

culated for a Coulomb potential by Z*= 2I

p. The soft-core parameter ,

which smoothes the singularity of the Coulomb potential has to be chosen suchthat the ground state energy matches the energy of the regular Coulomb poten-tial. The TDSE

id

dtt( ) = H t( ) (2)


is integrated over time t in a formal way to calculate the wave function t( ) :

t( ) = exp i dt ' H t '( )0

t

( ) 0( ) . (3)

The integral can be approximated by a sum with small time steps t witht = N t . This approximation is justified, if by varying t the result is un-changed. The equation is be further simplified by changing the summation to aproduct:

N t( ) = exp i H2k 1

2t t

k=1

N

0( )

= exp iH2k 1

2t t 0( )

k=1

N. (4)

The operators in the Hamilton operator do not commute therefore their tempo-ral order should to be conserved. Nevertheless for the further calculation theHamilton operator is split into the momentum part and the rest. The error ofthis approximation is of second order in the time steps t( )

2 and one product ofthe upper equation takes the following form:

exp iH2k 1

2t t = exp i

2

2+

Z*

r +E

2k 1

2t x t

exp i

2

4t exp i

Z*

r +E

2k 1

2t x t exp i

2

4t

.

(5)

The physical interpretation of the equation is that the electron propagates fort 2 field free, then interacts for t with the potential and propagates for t 2

field free, again. With this approximation the complete products reads:

CHAPTER 2 - 12 -

N t( ) exp i

2

4t exp i

Z*

r +E

2k 1

2t x t exp i

2

2t

k=1

N

exp i

2

4t 0( )

.(6)

Since the contribution of the first and last term is negligible compared to themain part, a further simplification can be done, leading to the final result:

N t( ) exp iZ

*

r +E

2k 1

2t x t exp i

2

2t

k=1

N

0( ) . (7)

This formula can be computed very efficiently, because the operators are multi-plications in real space and in momentum space, which are connected by FFT.To compute the upper formula the wave function for the atom is transformed inmomentum space by FFT, multiplied by the propagation operator, transformedback to real-space by FFT, multiplied by the operator with the Coulomb and la-ser field and so on. To get the high harmonic spectrum the time-dependent di-pole moment must be calculated:

x t( ) = d3r t( ) x t( ) . (8)

The integration can be simplified due to rotational symmetry around the laserpolarization x and the integration boundaries can be small in y and z-direction,because HHG happens close to the atom. The classical oscillatory motion of theelectron E

2 gives a hint for the integration boundaries in x-direction.

x t( ) = 4 dx dy0

dz0

t( ) x t( ) (9)

A more accurate way, since HHG takes place near the atom, is to calculate thespectrum from the acceleration rather than from the dipole moment, which isvery sensitive to errors in the wave function for large x.


d

dtt( ) x t( ) = i t( ) H ,x t( )

d2

dt2

t( ) x t( ) = t( ) H , H ,x t( ). (10)

For a spherically symmetric potential like He

t( ) H ,x t( ) = t( )2

2,x t( ) = t( )

xt( ) . (11)

Then the acceleration is

a t( ) =d

2

dt2

t( ) x t( ) = t( ) H ,x

t( ) . (12)

Substituting in for H

a t( ) = t( )x

Z*

r +t( ) E t( ). (13)

Finally the time-dependent dipole spectrum is calculated by the Fourier trans-form a ( ) = FT a t( )( ) via

x t( ) = FT1

a ( )2

. (14)

2.2.2 Analytical solution

The analytical solution is based on the strong field approximation (SFA) andwas first presented by Lewenstein et al. in reference [14]. The Hamilton operatorfor an atom in a linearly polarized laser field in x-direction in the dipole ap-proximation reads:

H =2

2

Z*

r+ E t( )sin t( ) x, (15)

CHAPTER 2 - 14 -

where E(t) is the pulse envelope and Z* is the effective atomic charge. To sim-plify the calculation the slowly varying envelope approximation is used. There-fore the vector potential is:

A t( ) =E t( )

cos t( ). (16)

To calculate the time-dependent dipole moment, transitions from the groundstate 0 to the continuum states in momentum space p have to be approxi-mated. For a Coulomb potential the transition matrix element is given by:

r 0 =Z *( )

3 2

exp Z * r( )

d p( ) = p r 0 = i27 2 Z *( )

5 2

p

p + Z *( )3 2

. (17)

The time-dependent dipole moment in the tunnel ionization regime, where theKeldysh parameter = I p 2U p <1 , is:

x t( ) = ia*

t( ) dt '0

t

g t '( )E t '( ) dp dx

*p A t( )( )dx

p A t '( )( )( )

sin t '( )expi

2dt ''

t '

t

p A t ''( )( )2

+ 2Ip

+ c.c.

, (18)

where g(t) is the probability amplitude for the ground state. The integrationover momentum space is performed with the stationary phase method, whichhas a close relationship with the Feynman path integral approach. Finally thetime-dependent dipole moment can be calculated from

x t( ) = 4 2a*

t( ) d+

3 2

0

t

a t( )E t( )dx

*p

st,( ) A t( )( )

dx

ps

t,( ) A t( )( )sin t( )( )sin4

Ss

t,( )

, (19)

where = t t ' represents the time the electrons spend in the continuum and0.001 is a normalization constant needed for the numerical calculation of the


integral. The only non-vanishing stationary momentum ps t,( ) is given in x-direction:

ps

t,( ) =2E t

2

2sin

2cos

22t( ) (20)

and the corresponding stationary action Ss t,( ) is:

Ss

t,( ) = Ip

2

4E

2t

2sin

2

2cos

2

22t( )

+1

43

E2

t2

+ sin ( )cos 2t( )( )( ). (21)

This method gives a nice physical interpretation and reproduces the simpleman’s model, when analysed using the stationary phase principle.

2.3 Attosecond pulses and pulse trainsThe minimum pulse duration Tmin achievable with a specific laser source withwavelength is determined by the oscillation period of the light, which is de-termined by its wavelength. For near-infrared pulses around 800 nm, the opti-cal cycle period is 2.7 fs. To enter the attosecond domain it is therefore essentialto generate radiation of higher frequency i.e. shorter wavelengths. High har-monic radiation provides significantly shorter wavelengths in the extreme ul-traviolet (XUV) domain and sufficient bandwidth to synthesize attosecondpulses [28].

Tmin ,( ) (22)

The understanding of HHG through the simple man’s model indicates thatharmonics are generated in short bursts every half-cycle of the laser. Also thespectral structure of the harmonic plateau (Figure 2.1) allows for the formationof an APT, if the spectral phase relation between the harmonics is appropriate.Harmonics in the plateau, if added coherently with suitable relative phases,

CHAPTER 2 - 16 -

provide the broad spectral signature of a very short pulse, the attosecondbursts. However, because the spectrum consists of several peaks – the harmon-ics – an overall broad temporal envelope is present, which limits the duration ofthe APT [28, 29, 58-62].

An APT is the coherent superposition of several plateau harmonics and as-suming constant harmonic amplitude, it can be described by:

= ei q t

ei

q,

q odd

(23)

whereq= q

L is the phase of the qth harmonic and

L is the phase of the

driving laser pulse. The generation of APTs illustrates the connection betweentime domain, spectral domain, simple man’s model, individual harmonics andfinally attosecond pulses. The simple man’s model works in the time domainand predicts the emission of ultrashort bursts of radiation every half-cycle ofthe driving laser pulse. The generation of single attosecond pulses relies ontemporally confining one of the pulses from the APT. This has been achieved bydriving HHG with few-cycle driving pulses [34-37, 59]. The generation of thehighest cutoff harmonics with a few-cycle driving pulse is naturally confined tothe central half-cycle of the pulse, where the electric field is maximal. In thiscase the cutoff emission will only be generated during one half-cycle (see Figure2.3) and careful filtering of this spectral region results in a single attosecondpulse. As the effect depends on the electric field rather than the intensity, thismethod requires control of the absolute phase, or carrier-envelope offset phase[63]. If the absolute phase is shifted by / 2 the highest intensity in the pulsewill occur twice, resulting in not one but two attosecond pulses.

- 17 -

Chapter 3

Strong field quantum path control

using attosecond pulse trains

APTs have a natural application in the control of strong field processes. In com-bination with an intense infrared laser field, the pulse train can be used to mi-croscopically select a single quantum path contribution to a process that wouldotherwise consist of several interfering components. The calculations demon-strate this by manipulating the time-frequency properties of high harmonics atthe single atom level. [64]

3.1 IntroductionIt has recently been experimentally demonstrated that a train of pulses as shortas a few hundred attoseconds is produced when several odd harmonics of anintense IR laser field are phase-locked [30]. The periodicity of the resulting APTis half the fundamental IR cycle. This periodicity makes the APT a natural toolfor controlling strong field processes driven by the IR laser. The efficacy of thiscontrol mechanism can best be appreciated in the framework of the successfulsemiclassical description of intense laser-matter interactions. In this picture, theamplitude for any strong field process can be expressed as a coherent sum overonly a few quantum orbits. These space-time trajectories follow a sequence ofrelease into the continuum (ionization), acceleration in the IR field, and returnto the ion core, where the electron can either rescatter or recombine. When anAPT is used in combination with an IR laser, the short duration of the attosec-

CHAPTER 3 - 18 -

ond pulses fixes the ionization to a particular time in each IR half cycle and al-lows us to select which quantum paths are available for the electron to follow.

The quantum orbits contributing to each harmonic are characterized by theirtime of release into the continuum and their kinetic energy upon return to theion core. The two most important orbits are those that have travel times

1and

2 less than one optical cycle. Each orbit contributes to the dipole moment with

a phase j=

j ( ) Up( ) , where

j ( ) is the phase coefficient for the orbit

j, is the return energy in units of the ponderomotive energy Up, and is

the IR laser frequency. For harmonics in the plateau, 1 and

2 differ by more

than an order of magnitude, giving rise to very different time-frequency be-haviors [65, 66]. Macroscopic phase matching often favors one phase behaviorover the other [29, 67, 68], but at the single atom level both components are al-ways present and interfere. By using an APT to steer the electron to one trajec-tory or another, the interference can be eliminated and the spectral properties ofthe harmonics can be chosen.

3.2 Attosecond pulse train assisted high harmonicgeneration

We calculate the time-dependent dipole moment for a helium atom interactingwith an IR fundamental field centered at 810 nm and an APT by solving thetime-dependent Schrödinger equation in the single active electron approxima-tion [69, 70]. The total electric field is given by

E t( ) = EL

t( )sin t( ) + Eh

t td( ) sin q t t

d( ) + q( ) ,q

(24)

where the IR field envelope, EL

t( ) , is a cosine function with a full width at halfmaximum (FWHM) in intensity of 27 fs, and the APT envelope, E

ht( ) , is a cos4

function with a FWHM of 14 fs. We use an idealized train with odd harmonics11–19 and relative phases

q= 0 , resulting in a train of 270 as pulses [13]. For

most of the calculations, the IR peak intensity is 4x1014 W/cm2 and the APTpeak intensity is 1013 W/cm2 [71]. The intensities have been chosen such that theionization is enhanced by a factor of 20–25 for all values of the delay td, which

- 19 - QUANTUM PATH CONTROL USING APTS

means the APT dominates the ionization step. The ionization enhancement islinear in the APT intensity.

Our main result is summarized in figure 3.1. The dipole spectrum plotted is cal-culated from a ( ) 4 where a ( ) is the Fourier transform of the electron’sacceleration [70]. For reference, the harmonic spectrum generated by the IRpulse alone is shown in blue in every panel. The plateau harmonics in thisspectrum are not resolved. This is because the intensity dependence of the di-pole phase gives rise to a characteristic time-dependent frequency (a chirp) foreach quantum path. The chirp is negative, with a rate bj proportional to

jI

LT

L

2 , where TL and IL are the IR pulse duration and the peak intensity, re-spectively. These chirps induce a spectral broadening. Since

j is large for the

long quantum paths, the bandwidth of each harmonic is larger than 2 for a 27fs IR pulse.

Figure 3.1 also shows harmonic spectra generated by the IR pulse in combina-tion with an APT for various values of the delay time td in units of the IR periodTL. The temporal overlap between the IR field and the APT is shown in the in-sets. The presence and the timing of the APT alter both the strength and spec-tral resolution of the harmonics. At some delays the plateau harmonics are en-hanced by 1 to 2 orders of magnitude, while at others the enhancement is com-pletely absent. This variation in yield comes in spite of the fact that, as men-tioned above, the ionization is enhanced at all delays. The yield and the resolu-tion of the harmonics follow a progression as a function of td. At td=-0.25 (Figure3.1(a)), the harmonics in the plateau region are enhanced and spectrally re-solved. The resolution then decreases as td increases, so that at td=-0.16 (Figure3.1(b)), the harmonics are enhanced but not spectrally resolved. After this, theresolution increases again, and at td=-0.094 (Figure 3.1(c)) the spectrum is againwell resolved. The enhancement and spectral resolution then gradually disap-pear, beginning with the high-energy harmonics and moving to lower orders.At td=-0.063 (Figure 3.1(d)), the spectrum is enhanced only through the 49thharmonic, and the enhancement is completely absent by the time td is 0.063(Figure 3.1(e)), except for the strong response at the train harmonics which isevident in all the spectra.

CHAPTER 3 - 20 -

Figure 3.1: Harmonic spectra generated by a 4x1014 W/cm2 IR pulse alone (bluelines), or in combination with a 1013 W/cm2 attosecond pulse train (red lines), forfive different values of delay between the two pulses. The delays are in units of theIR cycle, (a) td=-0.25, (b) td=-0.16, (c) td=-0.094, (d) td=-0.063, (e) td=+0.063.

The most striking result in figure 3.1 is the spectral resolution of the plateauharmonics at the two ‘‘good’’ delays, td=-0.25 and td=-0.094. The presence of theAPT seems to have eliminated the longer quantum paths, selecting only theshort quantum orbit to contribute to the harmonic generation process. This


short path selection can be confirmed by analyzing the time-frequency behaviorof the harmonics generated at these two delays. The time-dependent frequencyof each harmonic is evaluated by windowing its spectrum and transforming tothe time domain [70]. Each harmonic exhibits a single, linear frequency chirp.This would not be the case if there is more than one quantum path contributionto the dipole moment.

Figure 3.2: (a) Chirp rates versus harmonic order for three different peak intensitiesof the IR pulse; IL=3x1014 W/cm2 (open squares), IL=4x1014 W/cm2 (solid squares),and IL=5x1014 W/cm2 (open circles). (b) Comparison of chirp rates of the full calcu-lation (symbols) at the two good delays (see text) of those of the simple model(lines) for an IR intensity of IL=4x1014 W/cm2. We show both the downhill (solidsquares/ solid line) and the uphill (open circles/ dashed line) results.

In figure 3.2(a) the chirp rate as a function of the harmonic order for a delay oftd=-0.094 is plotted for three different IR intensities. The magnitude of the chirprate increases with harmonic order for all three intensities. This is characteristicof the short quantum path, since

1 increases as a function of the return energy,

CHAPTER 3 - 22 -

while2 decreases. Furthermore the magnitudes of the harmonic chirps match

very well those calculated for the short quantum path in the semiclassicalmodel. Finally the chirp rate is slowly dependent on the intensity. As the inten-sity changes between 3x1014 W/cm2, 4x1014 W/cm2 and 5x1014 W/cm2, the chirprates for the plateau harmonics change by only 20% . This is also typical of theshort quantum path; bj is proportional to the product

jI

peak, and since

1 de-

creases with increasing intensity for a given harmonic [65, 72], the chirp ratevaries only slowly with intensity. Again, this is opposite to the behavior ex-pected for the long quantum path. This slow intensity dependence means thatthe spectral narrowing and small chirp values that are observed in these singleatom calculations will be relatively insensitive to the intensity variations in areal laser focus. Therefore phase-matching can be fulfilled for all regions in thefocus simultaneously, which should make APT-assisted HHG experimentallyobservable.

3.3 Simple man’s modelTo understand why varying the APT delay selects a particular quantum path,the classical electron dynamics in the combined IR + APT electric field is ex-amined. The standard semiclassical model of harmonic generation [16] is modi-fied as follows. First, the ionization step is assumed to be entirely due to theAPT. At higher IR intensities, which approach saturation, this assumption willbreak down. Second, the tunnel-ionization condition is modified such that theelectron trajectories begin with zero initial velocity. Since ionization proceedsvia one-photon absorption, the electron is released with a nonzero kinetic en-ergy. This is estimated as E

kt0( ) = E

0+ q V t

0( ) where E0 is the ground stateenergy, q is the average energy of the APT ( q =15 here) and V t

0( ) is theheight of the combined Coulomb + IR field barrier at the release time t0 [73].The subsequent electron dynamics are the same as in the standard model; theacceleration is imparted by the IR field, and only trajectories that return to theion core give rise to harmonic generation.

The new feature of the APT + IR trajectory calculations is the inclusion of theelectron’s initial velocity v t

0( ) as an additional degree of freedom. There arenow two families of classical trajectories the electron can follow, corresponding


to v t0( ) in the same direction as the IR field (uphill with respect to the com-

bined potential because of the electron’s negative charge), or in the opposite di-rection of the field (downhill with respect to the potential). The two families oftrajectories are shown in figure 3.3 where the return energy of the electron isplotted as a function t0. Each family is characterized by a range of release timesleading to either a short trajectory (solid line) or a longer trajectory (dotted line)that can include several returns. There are also release times that lead to no re-turning trajectories. Since the individual attosecond pulses have a duration ofabout 0.1TL, they can select a small range of release times and thereby deter-mine which trajectories contribute to the spectrum.

Figure 3.3: Return energy as a function of the release time. Left hand curve: uphilltrajectories; right hand curve: downhill trajectories. The IR intensity isIL=4x1014 W/cm2. On each curve the long quantum path

2 is shown as a dotted

line and the short quantum path 1 is shown as a solid line. The five labels (a)-(e)

correspond to the panels in figure 3.1.

This simple model can be used to qualitatively explain the trends in harmonicyield and spectral resolution that are discussed in connection with figure 3.1.When the electron is released at -0.25 (as in figure 3.1(a)), only the short uphilltrajectories contribute to the harmonic spectrum. These

1 contributions lead to

a well-resolved spectrum, which is enhanced compared to the IR laser alone. As

CHAPTER 3 - 24 -

the release time increases, there is a region around t0

0.16 (Figure 3.1(b))when only long trajectories contribute to harmonic generation, leading to en-hancement of the spectrum with unresolved individual harmonics. For t0 be-tween approximately -0.125 and 0 (as in figures 3.1(c) and (d)), only shortdownhill trajectories will contribute to the spectrum, leading to well resolvedharmonics. As the release time increases toward zero, the energy of the return-ing electrons decreases, meaning that only the low energy harmonics will beenhanced and resolved. Finally, when electrons are released after t0=0, there areno returning trajectories initiated by the APT, and the spectrum is not en-hanced.

The precise positions of the two families of trajectories are sensitive to the initialvelocity. The simple model predicts the optimal delays for 1 selection to beslightly too large for the uphill trajectories and slightly too small for the down-hill trajectories. This means that our estimate of the initial velocity is somewhattoo large. Reducing the initial energy by 10%–20% would give better agreementbetween figures 3.1 and 3.3.

Our analysis can quantitatively reproduce the harmonic chirp rates found in thefull calculations. In figure 3.2(b) the chirp rates for the uphill and downhillshort and long trajectories are calculated by integrating the classical action forthe electron in the IR field [74]. The curves have an upper and a lower branch,corresponding to the

1 (upper) and

2 (lower) contributions. The comparison

of the 1 path chirp rates to those found in the full calculations at the good de-

lays is excellent.

Quantum path selection can also be applied to other strong field processes. Forexample, the high-energy portion of the above threshold ionization (ATI) spec-trum results from the elastic rescattering of electrons that return to the ion core[20]. As with harmonic generation, varying the delay of the APT would allowfor the study of individual path contributions to the spectrum [75]. In particu-lar, when the IR laser is elliptically polarized, the ATI spectrum shows severalplateaus which can be attributed to different quantum paths [16, 76]. In addi-tion, the extra degree of freedom introduced by the APT, the initial electronvelocity, will cause new effects. The calculations indicate that the photoelec-trons resulting from uphill and downhill trajectories have different angular


distributions. The uphill trajectory also yields a large number of orbits with lowreturn energies (less than Up). These make little contribution to the harmonicspectrum, and they are omitted from figure 3.3, but they should rescatter veryefficiently.

3.4 SummaryAPTs are natural tools for controlling strong field processes. This is becausetheir periodicity automatically matches that of the driving IR laser, allowing forthe selection of different quantum path contributions. As a first application ofthis method, calculations are presented that demonstrate the manipulation ofthe time-frequency characteristics of high harmonics at the single atom level.

- 27 -

Chapter 4

Phase matching of attosecond pulse

train assisted high harmonics

Nonadiabatic calculations of the macroscopic harmonic signal created by a gasof helium atoms exposed to a strong infrared IR pulse in combination with anAPT are presented. The harmonic yield can be enhanced by two to four ordersof magnitude for optimal delays between the IR and the APT pulses. The largeenhancement is related to the change in the IR-intensity dependence of both theharmonic strength and phase induced by the presence and timing of the APT.This leads to enhancement of the harmonic yield and improved phase matchingconditions over a large volume. [77]

4.1 IntroductionIn the previous chapter HHG by a single He atom was studied. The timing ofthe APT relative to the IR pulse is used to select the time at which an electron isreleased into the continuum determining also the space-time trajectory theelectron will follow in the continuum. The yield and the coherence properties ofthe harmonics were improved when the APT launched the electron along theshortest quantum path.

The harmonics generated by a macroscopic number of helium atoms exposed tothe combination of a strong IR laser pulse and an APT both represented by fo-cused laser beams is studied in this chapter. The aim is to investigate whether

CHAPTER 4 - 28 -

the single atom effects of the APT can still be discerned after propagation andphase matching in the macroscopic nonlinear medium. Phase matching is ingeneral imposed by a balance between the IR intensity dependence of the har-monic strength and phase, and the geometrical variations of the Guoy phase ofthe combined driving fields. Of particular interest are the following issues:

1) What is the consequence of the change in the relative phase over the spatialextent of the medium, imposed by the difference in carrier frequency andthereby focusing characteristics of the two pulses; and

2) What is the interplay between the single atom quantum path selection im-posed by the APT and the macroscopic quantum path selection imposed byphase matching?

To answer these questions nonadiabatic calculations of the macroscopic har-monic response of a gas of helium atoms to the combined IR and APT focusedlaser pulses are performed, by solving the Maxwell wave equation (MWE) cou-pled with the TDSE, which is solved in the SFA. There are indeed phasematching conditions where the single atom quantum path selection has a verylarge impact on the generated harmonics. First, the choice of quantum pathstrongly influences the coherence properties of the macroscopic harmonics forinstance their spectral bandwidths, in agreement with the single atom results.Second, and more importantly, the harmonic yield can be enhanced by two tofour orders of magnitude over most of the harmonic spectrum, for the optimaldelays between the two driving pulses. This is a much larger enhancement thanthat of the single atom response. The large enhancement is due to the change inthe IR intensity dependence of the harmonic strength and phase, caused by thepresence and timing of the APT. At low IR intensity, the single atom enhance-ment of the harmonic signal is much larger than at high IR intensity. Thismeans that atoms at the edge of the IR spatial profile contribute to the harmonicsignal almost as much as atoms in the center of the IR beam, and a large volumeeffect is achieved. In addition, phase matching is improved, since the harmonicphase varies much less with IR intensity when the APT is present. Finally, thevalidity of using the SFA as the single atom basis for our macroscopic calcula-tion is discussed.

- 29 - PHASE MATCHING OF APT-ASSISTED HHS

4.2 Numerical methodThe MWE is solved in three dimensions in the slowly evolving wave approxi-mation (SEWA), which is valid for pulses with durations as short as a single cy-cle see reference [37] for details . This approach is similar to that outlined in ref-erence [78], in which the wave equation has the following form, in a coordinatesystem that moves with the driving pulse:

2E

L ( ) +2i

c

EL ( )z

=G ( )i

c( )E

L ( ) , (25)

2E

h ( ) +2i

c

Eh ( )z

=2

0P

nl ( ) +G ( )i

c( )E

h ( ) , (26)

where all are functions of the cylindrical coordinates r, z.

These equations are solved in the frequency domain, by space-marching all thefrequency components of the driving field, E

L ( ) , and the generated fieldE

h ( ) , through the ionizing nonlinear medium. At each point z along thepropagation direction, the temporal pulse shape of the driving field, E

Lt( ) , is

found by Fourier transforming EL ( ) . The atomic response to the driving field

EL

t( ) at each point in the radial direction is calculated in the time domain bysolving the TDSE, as described below. This yields the time-dependent dipolemoment x t( ) and the free electron density n

et( ) . The source terms G ( ) (ow-

ing to free electron dispersion) and Pnl ( ) (the nonlinear polarization) are then

found by transforming back into the frequency domain.

G ( ) = Fe

2n

et( )

0m

ec

2E

Lt( ) , (27)

Pnl ( ) = F n

0n

et( )( )dnl

t( ) (28)

are used to propagate all the electric fields forward in z and n0 is the initial gasdensity. Absorption is included for all frequencies above the ionization thresh-old of helium, using the absorption coefficients ( ) of reference [79]. The ab-sorption coefficient is proportional to the density of neutral atoms.

CHAPTER 4 - 30 -

For this study, the free electron density from ionization rates is calculated as de-scribed by Ammosov, Delone, and Krainov ADK [80] , and the SFA is used tocalculate the time-dependent dipole moment, in a nonadiabatic form as inequation 1 of reference [78]. In this description, the time-dependent dipole mo-ment is an integral over quantum trajectories that each corresponds to an elec-tron which is released into the continuum at time t’, accelerates in the laserfield, and returns to the core at a later time t. During the continuum propaga-tion the electron acquires a phase given by the quasi classical action, S t ',t( ) .Only trajectories that correspond to stationary points of the action are included.When calculating these stationary points, S

stt ',t( ) , and a vector potential A t( )

which does not include contributions from the APT is used:

Sst

t ',t( ) = t t '( ) Ip

1

2p

st

2t ',t( ) t t '( ) +

1

2A

2t ''( )dt ''

t '

t

. (29)

In this equation Ip is the atomic ionization potential and the stationary value ofthe momentum p

stt ',t( ) is given by

pst

t ',t( ) =1

t t 'A t ''( )dt ''

t '

t

. (30)

Neglecting the influence of the APT during the continuum propagation is con-sistent with the interpretation presented in reference [64] for the strong fielddynamics in the combined IR and APT fields: the ionization step is dominatedby one photon ionization driven by the APT, and the continuum dynamics arecontrolled by the IR field, whose ponderomotive energy is several orders ofmagnitude larger than that of the APT. In the propagation code, A

IRt( ) is cal-

culated as the Fourier transform of the product of the combined propagatingdriving field E

D ( ) and a spectral window function which suppresses the highfrequency components. The Fourier transform of the full E

L ( ) is used in theSFA for the ionization step, and to calculate the ADK ionization rates.

To perform this calculation the following values are assumed. The driving fieldat the entrance of the medium is the same as in chapter 3. The focusing has beenchosen to model the experiment described in chapter 6. At the entrance to thenon-linear medium, the IR and the APT are described by Gaussian beams with


confocal parameters bIR=0.5 cm and bXUV=7.5 cm, respectively. The position ofthe focus of each beam relative to the center of the gas jet can be chosen inde-pendently. In all the calculations shown below, the focus of the APT has beenkept at the center of the medium. The factor of 15 difference in the confocal pa-rameters is chosen to ensure that the waists of the two beams are similar (about25 microns) over the 0.1 cm length of the medium. This maximizes the volumeof atoms that experience the combined fields. The relative phases of the har-monics in the train are chosen such that, in the absence of dispersion effects, theharmonics are exactly phase locked in the focus of the XUV beam, and slightlyout of phase at the beginning and at the end of the medium.

In a typical calculation, 8000 points in time and frequency, 301 points in the ra-dial direction (spanning 100 microns), and 251 points in the propagation direc-tion spanning the length of the medium (0.1 cm) are used. For the choice of pa-rameters in this study, it is only necessary to evaluate the source terms ap-proximately every 10 steps in the z direction as was also noted in reference [78] .To further speed up the calculation, perturbation theory is used to calculate theharmonic response at the lowest intensities.

4.3 Macroscopic phase matchingFigure 4.1 presents macroscopic harmonic spectra generated in a 0.1 cm long jetof helium atoms with a density of 2x1018 atoms/cm3 which corresponds to 80mbar at room temperature, when the IR laser beam is focused 0.2 cm before thecenter of the jet (zIR=0.2 cm). In this focus position, the phase matching condi-tions favor the contributions from the short quantum trajectory see for instancereferences [29, 52]. This can be seen from the spectral widths of the harmonicsgenerated by the IR pulse alone (thin solid curve), which are small in the pla-teau region and increase with the harmonic order.

The thick solid line shows the spectrum generated by the combined IR and APTpulses, when the relative delay between the two pulses, in the center of the me-dium, is _0.095. The delay is measured in units of the IR optical cycle, and td=0means that the attosecond pulses in the train coincide with the zero crossing ofthe electric field in each half-cycle. The delay td=_0.095 was found to be optimal

CHAPTER 4 - 32 -

for quantum path selection of the short trajectory contribution in single atomcalculations based on numerical integration of the TDSE within the SAE ap-proximation [64]. The SFA single atom calculations reproduce this result. Theeffect of the APT pulse on the macroscopic signal is a very large enhancementover most of the spectral range. The harmonics between 39 and 51 with photonenergies from 60–80 eV, for instance, are enhanced by three to four orders ofmagnitude. The shape of the spectrum is also significantly different with itsconstant decrease as the photon energy increases. The bandwidths of the indi-vidual harmonics are still characteristic of the short trajectory contribution.

Figure 4.1: Harmonic spectra generated by a 5x1014 W/cm2 IR pulse alone (thinsolid lins), or in combination with a 1013 W/cm2 APT, for two different values ofthe delay between the two pulses. The delays are, in units of the IR cycle, td=-0.095(thick solid line) and td=+0.185 (dashed line). The confocal parameters of the IR andthe XUV beams are 0.5 cm and 7.5 cm, respectively. The IR focus is located 0.2 cmbefore the center of the gas jet, the XUV focus is in the center of the jet.

The origin of the large enhancement seen in figure 4.1 is illustrated in figure 4.2,showing the single atom strength (a) and phase (b) for harmonics 39 (circles), 49(squares), and 59 (diamonds) as the peak intensity of the IR pulse varies be-tween 1 and 5x1014 W/cm2. The strength and phase generated by the IR pulse


alone (open symbols) are compared with that generated in combination withthe APT (closed symbols). Each point is the result of a single atom calculationwith a 27 fs IR pulse with a different IR peak intensity, and the same APT peakintensity of 1013 W/cm2. This is justified because the harmonic strength variesonly linearly with the APT intensity, as opposed to its nonlinear dependence onthe IR intensity. The harmonic phase also varies slowly with APT intensity [64].

Figure 4.2: Comparison of the IR intensity dependence of the single atom harmonicstrength and phase, when the harmonics are calculated with the IR pulse alone(open symbols), or in combination with a 1013 W/cm2 APT (filled symbols). Weshow three harmonics, the 39th (circles), the 49th (squares), and the 59th (diamonds).The calculations were performed within the SFA.

The APT has several effects on both the strength and the phase of the harmon-ics. First, the harmonics up to approximately the 59th are all enhanced by severalorders of magnitude, for all intensities below the peak intensity used in the

CHAPTER 4 - 34 -

macroscopic calculation. Second, the enhancement is in general much larger atlow intensity than at high intensity—the strength is essentially flat as a functionof intensity once a harmonic has reached the plateau. This means that atoms atthe edge of the IR radial profile, which in the absence of the APT contributevery little to the macroscopic signal, now contribute to the harmonic signal asmuch as atoms in the center of the IR beam. Therefore a large volume contrib-utes to the harmonic yield. To achieve this volume effect it is very importantthat the IR and the APT beams overlap in the radial dimension through most ofthe medium, which necessitates the much longer confocal parameter of the APTbeam. Third, the enhancement is in general larger for the lower harmonics thanfor the higher harmonics. This gives rise to the altered shape of the two-colorspectrum, with the strength decreasing as a function of the harmonic order.And finally, the characteristic quantum path interference, which in the IR alonecalculations gives rise to the rapid variation of the harmonic strength with theIR intensity, vanishes when the APT is present. This is consistent with the in-terpretation that the APT selects one quantum path out of several. This absenceof interference is also clear in the intensity dependent phase, which becomessmoother and flatter when the APT is present. Phase matching is thus also im-proved by the presence of the APT.

The single atom enhancement of the harmonic strength is a result of two effects,both of which can be discussed within the framework of the semiclassicalmodel of harmonic generation. First, the initial ground state to continuum tran-sition is a one-photon process for the APT but a multi photon process for the IR.And second, the timing of the electron’s release into the continuum with thenatural release time for the short trajectory means that the electron is muchmore likely to return to the core, compared to when it is released throughoutthe cycle as in tunnel ionization. The first of these effects is delay independentand could be achieved by a pulse consisting of just one harmonic of the IR field,whereas the second effect depends on the delay between the two pulses and re-lies on the short duration of the individual attosecond pulses in the train.

That the relative delay indeed matters for the macroscopic harmonic signal canbe seen from the dashed curve in figure 4.1. This has been calculated with adelay of td=+0.185, at which is no quantum path selection in the SFA. The en-


hancement is now much smaller than at td=_0.095, and at the end of the plateauthere is no enhancement compared to the IR alone result. There is, however,still a substantial enhancement of the harmonics between 30th and 50th. This ismostly driven by the single atom enhancement and the resulting volume effectdiscussed in the previous paragraph.

Figure 4.3: The delay dependence of the 43rd and the 53rd harmonics, shown withcircles and squares, respectively, driven by the combined IR and APT pulses. In (a)and (c) the yield after propagation through a 1 mm gas jet is shown, for two differ-ent positions of the IR focus, zIR=-0.2 cm (a) and zIR=0 cm (c). The atomic density is4x1017 per cm3. All other parameters are as in figure 4.1. In (b) and (d) the singleatomic yields calculated with the SFA (b), and with the SAE (d), using peak inten-sities of 4x1014 W/cm2 for the IR and 1013 W/cm2 for the APT.

Figure 4.3(a) explores in more detail the delay dependence of the macroscopicyield of harmonics 43 and 53 generated in the phase matching conditions usedin figure 4.1, but at a lower atomic density 4x1017 per cm3. The yields have beennormalized for comparison. The harmonics from approximately 35 to 63 all ex-

CHAPTER 4 - 36 -

hibit similar behavior with respect to the delay. The yield for each harmonicvaries by more than an order of magnitude as the delay is changed, and isstrongly peaked for delays between 0.35 and 0.4 (or _0.15 and _0.1). The higherharmonics are peaked at earlier delays than the lower harmonics, and exhibitmore structure as the delay varies.

The variation with delay originates in the single atom delay dependence. Seefigure 4.3(b), where the same two harmonics, calculated within the SFA, areshown. The position of the peak at t

d0.1 (or 0.4 in the figure) can be ex-

plained using the analysis of reference [64]. The electrons released by the APTin general have a small initial velocity. If the electron released close to _0.1 ini-tially moves downhill with respect to the laser potential, it can return to thecore only by following the short quantum path. It is characteristic of the shorttrajectory that the return energy increases when the electron is released earlierand returns later. This leads to the shift of the optimum delay for the higherharmonics compared to the lower ones. The width of the enhancement peak iscomparable to the 270 as duration of the individual pulses in the APT.

Next the influence of the phase matching conditions on the two-color harmonicgeneration is explored. Figure 4.4 shows a macroscopic harmonic spectrum cal-culated with the same parameters as in figure 4.1 except the position of the IRlaser focus, which is now in the center of the gas jet. At this position of the fo-cus, the phase matching conditions favor contributions from the longer quan-tum paths. This causes the spectral bandwidths of neighboring harmonics tospread and overlap, as can be seen in the IR alone curve (thin solid line).

The spectrum generated by the IR in combination with the APT, with a relativedelay of td=_0.095, is shown with thick solid line. The presence of the APT inthis case both enhances the harmonic yield and changes the spectral structure ofthe individual harmonics due to the quantum path selection of the short trajec-tory contribution. The APT is thus able to impose a different kind of phasematching on the generated harmonics. The dashed line shows the result of arelative delay of td= +0.125. The enhancement at this delay is much lower, be-tween one and two orders of magnitude. The highest harmonics between 50and 65 are barely enhanced at td= +0.125, and their spectra are broad and noisy.The dependence of the yield on the delay when zIR=0 is very similar to the de-


pendence found for zIR=_0.2 cm, as is shown in figure 4.3(c). The macroscopicvariation of the yield again closely reflects the single atom behavior, both withrespect to the position of the peak enhancement, and the difference between“good” delays and “bad” delays. It is interesting how slightly the position ofthe IR focus relative to the gas jet matters when the APT is present, which is instark contrast to macroscopic harmonic generation driven by an IR pulse alone.The changes in the harmonic strength and phase driven by the quantum pathselection of the APT apparently dominate the build-up of the harmonic radia-tion in the macroscopic medium. However, it is clear from the difference in theenhancement between figures 4.1 and 4.4 that the influence of the quantumpath selection is highest when the phase matching conditions work togetherwith it (as in figure 4.1) rather than against it (figure 4.4).

Figure 4.4: Harmonic spectra generated by a 5x1014 W/cm2 IR pulse alone (thinsolid line), or in combination with a 1013 W/cm2 APT, for two different values ofthe delay between the two pulses. The delays are, in units of the IR cycle, td=-0.094(thick solid line) and td=+0.125 (dashed line). The confocal parameters are 0.5 cmand 7.5 cm, respectively. Both beams are focused in the center of the gas jet.

In the phase matching configurations discussed above, the number of photonsgenerated in each harmonic in the IR alone case is low. This is because of theshort IR confocal parameter 0.5 cm, the short medium length and the intensity,

CHAPTER 4 - 38 -

of course, which is far below the saturation intensity of 1015 W/cm2. The IRspectrum in figure 4.1 for instance only corresponds to a few tens of photonsper pulse in each of the harmonics between 35 and 55. With the large enhance-ment of the APT top spectrum in figure 4.1, we get more than 105 photons in the35th harmonic, 104 photons in the 45th harmonic, and 103 photons in the 55thharmonic. Since the enhancement comes from the spatial overlap between thetwo beams, or in particular from the region where the APT is strong, it scaleswell with an increase of the interaction region.

Increasing the confocal parameters of both beams, as well as the length of thegas jet, by a factor of 2 gives almost an order of magnitude increase in the num-ber of photons generated both in the IR alone and in the two-beam case. In thecalculation, the only limitation to keep increasing the interaction region is theabsorption of the 17th and 19th harmonics, which are just above the ionizationthreshold in the field-free helium atom. Using the field-free values for the ab-sorption cross section in helium, i.e., the calculation does not take into accountdynamical effects such as ac Stark shifts arising from the interaction of the atomwith the strong IR field. Increasing the IR intensity would somewhat increasethe number of photons in the IR alone case, but only marginally increase thenumber of photons in the two color case. This is because the enhancementdriven by the APT strongly depends on the initial ionization step being drivenby the APT. If the IR intensity is increased to the point where it dominates theionization step, the enhancement and the quantum path selection is no longerdominant.

Finally, the validity of using ADK rates and the SFA as the single atom basis forthe macroscopic calculation is discussed. First, the ionization rates are onlyused to calculate the number of free electrons in the medium and thereby therefractive index and the harmonic strength is calculated within the SFA. Forcomparison with the SFA results shown in figure 4.2(a), figure 4.5 shows thesingle atom intensity dependence of harmonics 43 (a) and 59 (b) calculated bydirect numerical integration of the TDSE within the SAE approximation. TheAPT pulse has a duration of 14 fs and a peak intensity of 1013 W/cm2, and the IRpulse has a duration of 27 fs and a peak intensity which varies up to 5x1014

W/cm2. Two different delays, td=_0.094, where the short trajectory is selected


(filled circles), and td= +0.063 where no trajectories are selected by the APT(squares) are shown and also the dependence on the IR intensity in the absenceof the APT (open circles).

Figure 4.5: IR intensity dependence of the single atom strength of harmonics 43 (a)and 59 (b), calculated by numerical integration of the TDSE within the SAE ap-proximation, for two different delays between the IR and APT pulses. The delaysare, in units of IR cycles, td=-0.094 where the short trajectory is selected (filled cir-cles) and td=+0.063 where no trajectories with returns within one cycle are selected(filled squares). The strength of the harmonics driven by the IR pulse alone isshown with open circles.

In figure 4.5, the magnitude of the enhancement at td= _0.094 is similar to,though marginally smaller than, that found with the SFA (figure 4.2). Also thelarger enhancement at low intensity, the larger enhancement of the lower har-monics than the higher harmonics, and the absence of interference in the inten-sity dependence, are correctly reproduced by the SFA (figure 4.2) compared to

CHAPTER 4 - 40 -

the full calculation (figure 4.5). The SAE also predicts, like the SFA, that there issome enhancement of the lower order harmonics even at delays where no tra-jectories are selected at least no trajectories that return within one cycle of thefield, as seen from the td= +0.063 curves. The macroscopic calculations based onthe SFA to approximately predict the outcome of an experiment, in terms of theenhancement of the yield and the change in the spectral characteristics of theharmonics, when the delay between the IR and the APT is chosen to select theshort quantum path. However, in one aspect an experiment is expected to differfrom the result presented in figures 4.3(a) and 4.3(c), namely in the dependenceon the relative delay. Figure 4.3(d) shows the single atom yield calculatedwithin the SAE as a function of the relative delay. In addition to the peak in theyield around td=0.4 (or td=_0.1) there is clearly a second, equally strong, peakcentered around td=0.25. This peak corresponds to electrons that are releasedand initially move uphill with respect to the potential, before scattering on thecore and then being accelerated in the characteristic short trajectory manner[64]. This second peak is absent in the SFA prediction.

The origin of this discrepancy is the simplified description of the ground stateto continuum transition in the SFA compared to the SAE, and the absence of theatomic potential once the electron is in the continuum. In the SAE calculations,the electron that is ionized by the APT is released with a small, delay-dependent initial velocity, and its motion close to the core depends on the de-tails of the combined atomic and laser potential. In contrast, in the SFA the sad-dle point trajectories correspond to electrons released with zero initial velocity.These electrons can only move downhill in the laser potential and therefore noquantum path selection corresponding to the uphill electron is present. In anexperiment an enhancement versus delay curve is expected to look like that infigure 4.3(d), rather than figure 4.3(a).

4.4 SummaryHHG by a macroscopic number of helium atoms interacting with a combinationof a strong IR laser pulse and an attosecond pulse train was studied. The yieldof many harmonics can be simultaneously enhanced by several orders of mag-nitude compared to the IR-only case, when the timing of the APT is such that


the shortest quantum trajectory is selected. In this case the spectra of the indi-vidual harmonics are well resolved, independent of the phase matching condi-tions. The large enhancement is due to the change in the IR intensity depend-ence of the harmonic strength and phase caused by the presence of the APT. Inparticular, the enhancement is much larger at low IR intensity than at high IRintensity. This means that atoms at the edge of the IR radial profile, which inthe absence of the APT contribute slightly to the harmonic signal, contributesignificantly to the macroscopic harmonic signal and a large volume effect isachieved. Phase matching in the macroscopic medium thus not only permits theobservation of the single atom quantum path selection driven by the timing ofthe APT, but actually enhances its effects.

The SFA single atom results are compared to the more accurate results calcu-lated within the SAE, and identified a range of conditions where our SFA basedpredictions are valid. The missing uphill electron trajectory in the SFA, and theoverestimation of the enhancement of harmonics below approximately the 30th,indicates the need for large scale nonadiabatic MWE-TDSE solvers that incor-porate more accurate solutions of the TDSE into the propagation [81].

- 43 -

Chapter 5

Control of attosecond pulse train

assisted high harmonic generation

APT can be used to select the quantum path contribution to HHG at the atomiclevel. The previous chapter has shown that the single atom effects are still pre-sent after propagation and therefore I restrict myself in this chapter to singleatom effects, study additional degrees of freedom added by the APT to thesystem and construct an optimal APT for HHG.

5.1 IntroductionThe control of HHG requires that the APT has the same periodicity as the proc-ess of HHG and that its timing is locked to the driving IR laser. Both facts arenaturally fulfilled by synthesizing the APT from harmonics of the IR laser.Through the APT, tunnel ionization is replaced by one-photon ionization and isfixed to a particular point in each IR half cycle, which allows to decouple theionization step from the IR field and to choose the quantum path in the contin-uum. By changing the delay of the APT relative to the IR field and therebysteering the electron wavepacket to one orbit or another, both the intensity andspectral width of the generated XUV burst can be altered. At some delays theplateau harmonics are enhanced by one to two orders of magnitude, while atothers there is no enhancement at all. This variation in yield comes in spite ofthe fact that the ionization is enhanced at all delays.

CHAPTER 5 - 44 -

The APT adds additional degrees of freedom to HHG, because the parametersof the APT can be varied in a controlled manner.

• The delay between IR and APT alters the intensity and the spectral widthof the generated harmonics.

• The average photon energy of the APT determines the initial velocityand the quantum path of the electron.

• The intensity of the APT determines the number of electrons releasedinto the continuum.

• The duration of an attosecond pulse in the train selects the number ofquantum paths in the continuum.

For example if an APT ionizes an atom by one-photon absorption, the electronhas an initial kinetic energy (velocity). Since only electrons, which come back tothe atom can contribute to HHG, the initial velocity can either be in the direc-tion of the IR polarization uphill with respect to the potential or downhill,which gives rise to two sets of return energy curves.

These new degrees of freedom and their influence on APT-assisted HHG areexplored. As shown in chapter 4 APT-assisted HHG survives and is even en-hanced through phase matching and therefore I restrict myself to single atomeffects in this chapter. The calculations are based on the numerical integrationof the TDSE in SAE approximation. The aim is to study the dependence on theIR and the APT intensity, on different average photon energies of the APT andon the duration of a single attosecond pulse in the train. Finally, results onwhich harmonics in the APT are important for APT-assisted HHG are shown.

5.2 Simple man's modelTo understand the influence of the APT on HHG, the classical electron dynam-ics in the combined APT + IR electric field is examined (Chapter 3.3). It has tobe highlighted that only trajectories that return to the atom give rise to HHG.This is important since the electrons are born with an initial velocity and can

- 45 - CONTROL OF APT ASSISTED-HHG

have velocities perpendicular to the IR laser polarization, which prevents a re-turn to the atom.

The new feature of the APT + IR trajectory calculations is the inclusion of theelectron’s initial velocity v t0( ) as an additional degree of freedom. The effect ofthe initial velocity is demonstrated in figure 5.1, which shows the return energyof the electron ionized from a He atom as a function of release time. There arenow two families of classical trajectories per IR-cycle the electron can follow,corresponding to v t0( ) in the direction of the IR laser polarization (uphill withrespect to the combined potential because of the electron’s negative charge), orin the opposite direction (downhill with respect to the potential). Each family ischaracterized by a range of release times leading to either a short trajectory or alonger trajectory that can include several returns. There are also release timesthat lead to no returning trajectories. The yield and the width of the harmonicsfollow a progression as a function of delay td , which is given in units of the IRcycle. Figure 5.1 shows for an APT with an average photon energy corre-sponding to the 15th harmonic, that, at td = 0.25 and td = 0.094 , just the shortquantum orbits contribute to HHG, which leads to an enhanced spectrum withcleanly resolved harmonics in the following called “good” delays. There is a re-gion around td = 0.16 where only long trajectories contribute to HHG, leadingto an enhanced spectrum with unresolved individual harmonics. For td be-tween approximately -0.125 and 0, only short downhill trajectories contribute tothe spectrum leading to an enhanced spectrum with cleanly resolved harmon-ics, again. Finally, when the electron wavepacket is released after td = 0 , thereare less returning trajectories initiated by the APT. These predictions are in verygood agreement with the solution of the TDSE.

Larger initial velocities from higher photon energies increase the splitting of thecurves as illustrated in figure 5.1. One interesting feature is, that the harmonics19 to 27 have return curves after a zero crossing of the electric field, whereas forthe tunnel case, no returns are possible. The photon energy of these harmonicsis greater than the ionization threshold of He, therefore the electron always hasan initial velocity. For these ionization times the electron starts to travel againstthe electric field of the laser and after a short time is driven back to the atom.No indication of these quantum orbits giving a significant contribution to HHG,

CHAPTER 5 - 46 -

was found. Since the individual pulses in the APT have durations of about0.1TL, they can select a small range of release times and thereby determinewhich trajectories contribute to the spectrum. Also visible in figure 5.1 is thelowering of the cutoff energy compared to the tunnel-ionization case, which be-comes stronger for higher initial velocities.

Figure 5.1: Return energy as a function of the release time for different averageharmonic orders q of the APT. The solid line shows the tunnel-ionization case;earlier curves: downhill trajectories; later curves: uphill trajectories. For q =19 to27 the average photon energy is above the ionization threshold of He.

5.3 Numerical methodAn atom interacts with a combination of the fundamental IR field and an APT,where both are linearly polarized in x-direction. The three dimensional TDSE inthe length gauge is numerically solved, where V r( ) is a one-electron pseudopotential

id

dtr ,t( ) =

2

2+V r( ) xE t( ) r ,t( ) , (31)

E t( ) = El t( )sin t + Eh t td( ) sin q t td( )( )q

. (32)

The harmonic intensity is calculated via I ( ) = A( )2

/ 4 , where A( ) is theFourier Transform of the acceleration a t( ) , which is given by:


a t( ) = r ,t( )r

V r( )cos( ) r ,t( ) + E t( ). (33)

In these equations, is the central frequency of the IR, corresponding to 810nm. The IR field envelope, El t( ) , is taken to be a cos-function with a FWHM inintensity of T = 27 fs. The APT envelope, Eh t( ) , is taken as a cos4-function with aFWHM of 14 fs. The parameters to study are the delay td between APT and IRfield, the intensity (El and Eh), the APT average photon energy q and the du-ration of a single attosecond pulse by varying the order and number of har-monics included in the APT.

If not stated otherwise, the IR intensity is 4 x 1014 W/cm2 in the calculations. TheAPT is transform-limited consisting of five or ten consecutive odd harmonics ofequal intensity with an average photon energy around the ionization thresholdof He. Calculations performed for APTs with experimentally determinedphases have given similar results at slightly shifted delays and therefore I re-strict myselves in the following to the idealized APT consisting of pulses withduration of T/2N, where N is the number of harmonics involved. If not statedotherwise the APT consists of harmonics 11-19, therefore its average harmonicorder is q =15 with pulse duration of 270 as. For APT-assisted HHG the ioni-zation of the generation gas must be dominated by the APT over the IR field.Therefore He is chosen as generation gas, because it can survive high IR inten-sities. The APT intensity is estimated for a typical 1 kHz Ti:Sapphire laser sys-tem. An IR pulse energy of 1 mJ and a conversion efficiency of 4 x 10-5 for the15th harmonic and the same for neighboring harmonics are assumed. Since theharmonics are added coherently to form the APT, the intensity of each individ-ual harmonic is a factor of N2 lower than that of the APT. For all calculationspresented in this paper we have chosen a peak intensity of 1 x 1013 W/cm2 atwhich the ionization by the APT is at least 5-times, typically 15-times, strongerthan by the IR field alone.

CHAPTER 5 - 48 -

5.4 HHG dependence on:

5.4.1 The intensity

The IR intensity is the most important parameter for tunnel-ionization HHG,since it determines the ionization probability and the cutoff. In the case of APT-assisted HHG the ionization probability is not determined by the IR intensityanymore and therefore the high harmonic yield doesn’t change with IR inten-sity. After ionization the dynamics are dominated by the IR laser and thereforethe cutoff follows max = I p + 3.2U p . This is demonstrated in figure 5.2, whichshows calculations performed at four different IR intensities. If the IR intensityreaches 5 x 1014 W/cm2 the ionization by the laser becomes visible. The plateauharmonics are a sum of APT-assisted and tunnel-ionization harmonics, whichalso have contributions of the long quantum paths and therefore are not nicelyresolved.


Figure 5.2: APT-assisted harmonic spectra for a delay of td

= -0.094 and four peakintensities of the IR pulse from 2 to 5 x 1014 W/cm2.

The dependence on the APT intensity is exploited in figure 5.3, which showscalculations for APT intensities of 0.5 x 1013 W/cm2 and 1.0 x 1013 W/cm2. Thegenerated high harmonic yield differs by a factor of two as expected from aone-photon process.

5.4.2 The average photon energy of the APT

Next the effect of the average photon energy q of the APT is explored, whichis important because of two facts: First the one-photon ionization cross-sectionis energy dependent and second the electron gets an initial velocity and there-fore its dynamics in the continuum are changed. Both effects are also dependenton the delay td since the IR field is also involved.

CHAPTER 5 - 50 -

Figure 5.3: APT-assisted harmonic spectra for a delay of td

= -0.25 for two APT in-tensities 0.5 x 1013 W/cm2 and 1.0 x 1013 W/cm2.

Figure 5.4 shows the ionization probability as a function of delay for two differ-ent APTs with q =15 and 21. The uncertainty in the ionization probabilities re-sults from a dependence on the delay between APT and IR laser. A maximumof the ionization probability for the APT with q =15 is visible at the peak of theIR electric field since there the below-threshold harmonics are able to ionize.For the APT with q = 21 the opposite behavior is visible since the energeticoverlap of the harmonics with the states of He (Stark shift) is worse than for theIR free case.

Figure 5.4: Ionization probabilities for APTs consisting of five harmonics with av-erage harmonic orders of q = 15 and 21 as a function of delay t

d between APT and

IR laser.


Figure 5.1 shows that the good delays change with q . Therefore the spectra fora series of delays for HHG assisted by an APT consisting of five harmonics withq =15 and by an APT of five harmonics with q = 21 are calculated. In figure5.5, where the good delays for an APT with q = 21 are shown. These delays areshifted relative to the q =15 due to the larger initial velocity of the electronwavepacket from td = 0.25 to td = 0.188 and from td = 0.094 to td = 0.063 ,but the intensity of the generated harmonics is comparable.

Figure 5.5: Harmonic spectra generated by an APT consisting of the harmonicsq = 21 for two delays of t

d= -0.188 and t

d= -0.063 .

The influence of q on the harmonic spectrum can be seen in figure 5.6, wherethe average harmonic order of an APT consisting of five harmonics for theirgood delays from q =11 to 27 was scanned, keeping the APT temporal enve-lope. The spectra show that the APT-assisted HHG works best for an APT with

CHAPTER 5 - 52 -

average photon energy around or below the ionization threshold of He (15 ). Ifthe average photon energy of the APT is far above, e.g. 27 , the generatedelectron wavepacket has a large initial velocity, which leads to a fast spread andtherefore a reduced overlap with the atom for HHG.


Figure 5.6: APT-assisted harmonic spectra for different average photon energiesq of the APT and the corresponding good delay oft

d= (-0.156, - 0.063, - 0.031, 0.0) for q = (11,19, 23, 27) . The harmonics of the APT can

be identified, particularly in the q = 23 and 27 cases.

The harmonics in the APT with q = 27 are all well above the ionization thresh-old of He. The ionized electron wavepacket has an initial velocity, which ispartially perpendicular to the polarization direction of the IR laser, which re-duces the overlap of the electron wavepacket with the ground state wavefunc-tion. This effect is partially compensated in the case of He, because the ioniza-tion occurs via one photon from an s-state, which results in a p-like electronmomentum distribution, which is peaked in the direction of the laser polariza-tion. Therefore most of the electron wavepacket can come back to the atom andgenerate harmonics.

5.4.3 The duration of a single attosecond pulse

The short duration of a pulse in the APT is the key to selecting a single quan-tum path at the atomic level. Therefore the possibility of using more harmonicsfor the APT was exploited, leading to shorter attosecond pulses, which shouldallow for selective enhancement of isolated parts of the harmonic spectrum. Acalculation with an APT consisting of ten odd neighboring harmonics with av-erage harmonic order of q=21 , which results in pulses of 135 as duration wasperformed. The calculation shows that it is not possible to selectively enhanceparts of the spectrum in this way and the results for the two good delays areshown in figure 5.7. The reason is the sensitivity of the quantum path on the

CHAPTER 5 - 54 -

initial velocity, see figure 5.1. Although a shorter pulse fixes the ionization to ashorter period, and therefore enhances just a few harmonics, the return curvesfor each harmonic in the APT are shifted depending on the initial velocity. At afixed ionization time several harmonics are generated. A possible solution toget a selective enhancement is to chirp the APT [82] such that the paths mergetogether.

Figure 5.7: Harmonic spectra generated by an APT consisting of the ten harmonicsq = 21 for the two delays of t

d= -0.25 and t

d= -0.094 .

5.4.4 Single harmonics

The photon energy of harmonics below 15th isn’t sufficient to ionize He withoutthe IR laser, but the below threshold harmonics are nonetheless important forAPT-assisted HHG. The electron wavepacket, which is generated by these har-monics, has a small initial velocity and is therefore well confined. The above


threshold harmonics generate an electron wavepacket with a larger initial ve-locity that tends to spread faster, reducing the overlap with the atom when it isdriven back by the IR field.

Figure 5.8: (a) Ionization probability of He for an IR field and a single harmonicfrom 7th to 31st and (b) corresponding generated harmonic intensity integrated fromthe 32nd-80th harmonic.

To investigate this influence, the spectra for HHG by He interacting with a sin-gle harmonic from the 7th to the 31st with a FWHM of 14 fs and an IR field arecalculated. Although the generated harmonics are not well resolved, an en-hancement of harmonic yield is present. The ionization probability changeswith input harmonic order (Figure 5.8(a)). To get a measure of the enhancementin harmonic yield, the harmonic intensity from the 32nd-80th harmonic is inte-grated, therefore all harmonics sent in are not included (Figure 5.8(b)). The re-sult shows, that an APT consisting of the 9th to 19th harmonic is optimal for

CHAPTER 5 - 56 -

APT-assisted HHG in He. The average photon energy of the APT is 14 , whichis slightly below the ionization threshold of He. Therefore it is preferable forAPT-assisted HHG to start with low energetic electrons.

5.5 SummaryAPT assisted HHG has been studied and it has been investigated how HHG atthe single atom level is affected by the APT parameters. The IR field dominatesthe dynamics of the electron in the continuum and ionization by the APT is aone-photon process. For best HHG efficiency, the average photon energy of theAPT should be around ionization threshold of the generation gas and to en-hance selective parts of the harmonic spectrum, it is not sufficient to just use ashorter attosecond pulse, but probably chirp it in an appropriate way. All thecalculations in this chapter, carried out for He, can be adapted to other gases byappropriate scaling of the APT average photon energy and the IR and APT in-tensity.

- 57 -

Chapter 6

Enhanced vacuum-UV assisted

high harmonic generation

APT-assisted HHG has been studied in detail in the previous three chapters.This chapter presents the experimental realization with the previously deter-mined optimal parameters for the APT [83]. The setup does not allow to changethe delay between APT and IR field, nevertheless, as demonstrated in chapter 4,it should anyway show an enhancement. This could be proven by generatingXUV harmonics around 90 eV in He using a combination of vacuum ultravioletVUV harmonics, generated in a Xe capillary, and the strong infrared (IR) laserpulse. With no changes in the IR input energy or the configuration of the Hetarget, the collinearly focused combination of the two fields changed the spec-tral properties and increased the yield of the XUV harmonics compared to thosegenerated with the IR field alone. [83, 84]

6.1 IntroductionAlmost since its discovery, there has been an extensive effort to improve the ef-ficiency of HHG. Most of these studies have focused on improving the macro-scopic phase matching of the generated radiation, and it was demonstrated thatbetter phase matching in a noble gas jet can be achieved with different focusinggeometries and by varying the position of the gas jet relative to the laser focus[67, 85-87]. In particular, conversion efficiencies of about 10–5 in the 30 nm [88]and 10-7 in the 13 nm [89] region were obtained using high power IR laser

CHAPTER 6 - 58 -

pulses and a loose focusing geometry. Improved phase matching in hollow-corefibers [90] and filaments [91] has been demonstrated, but is ultimately limitedby reabsorption [71, 92]. The techniques of adaptive laser pulse shaping in thefrequency [41, 93, 94] and spatial [95, 96] domain have also been used to spec-trally control and increase the yield of HHG. In addition quasi-phase matchinghas been extended from the visible and near infrared into the soft-X-ray regime[97]. In contrast to these methods, which rely on improved phase matching,Kim et al [98] have recently enhanced the single-atom HHG response by usingan orthogonally polarized two-color (fundamental and second harmonic) driv-ing laser.

A new method for enhancing HHG is experimentally demonstrated. It modifiesboth the process on the single atom level [64] and also the macroscopic phasematching, but does not rely on shaping the IR laser pulses, increasing their en-ergy or modifying the gas target. The initial tunnel ionization step in the HHGprocess is assisted with single VUV photon ionization, which results in a largevolume effect: Since the dependence of tunnel ionization on the IR intensity ishighly nonlinear, IR-driven HHG is confined to a small volume around thepeak of the IR beam. In contrast, VUV photon ionization is a linear process andis therefore also effective in the low-intensity wings of the IR beam, which leadsto HHG in a much larger volume. This results in enhanced HHG at moderate IRintensities [77].

6.2 Experimental setupA first experimental demonstration of enhanced HHG in the XUV through acombination of strong low order VUV harmonics and an IR laser field is dem-onstrated. A schematic of the experimental setup is shown in figure 6.1. Thecollinear configuration makes it easy to align and is particularly suitable forfurther applications. VUV harmonics are generated by loosely focusing a 29 fs(FWHM) IR pulse with a central wavelength of 785 nm and an energy of 0.7 mJinto the Xe-filled capillary with a spherical silver mirror (ROC = –1.5 m). Thecapillary has an inner diameter of 800 m and a length of 50 mm and the focusis placed at the end of the capillary for maximum signal. Since the focus of theIR laser is five times smaller than the diameter of the capillary, we observe no

- 59 - ENHANCED VUV ASSISTED-HHG

guiding of the IR beam [99]. Xe is filled into the capillary from one end at thelaser entrance, at pressures between 0 and 25 mbar, and is differentiallypumped through the other end to pressures below 10-3 mbar. The advantage ofthis configuration is that the capillary acts as a differential pumping stage, re-ducing the gas flow into the chamber compared to a pinhole of the same size.The drawback is that the resulting pressure gradient makes it difficult to de-termine the precise gas distribution in the interaction region.

Figure 6.1: Experimental configuration. The IR laser generates VUV harmonics upto the 19th order in a Xe-filled capillary, which co-propagate and are used to ma-nipulate XUV HHG in a pulsed He jet. The resulting VUV and XUV harmonicsspectra are recorded using an XUV spectrometer. An aperture (Ap) is used to con-trol the IR intensity in the interaction region.

With this setup, harmonics up to the 19th order were generated, and the IR peakintensity was estimated to be 1 1014 W/cm2 in the interaction region. The gen-erated VUV and the IR co-propagate 85 cm in vacuum through a variable ap-erture (which is 62 cm away from the Xe capillary exit) and a differentialpumping section, used to minimize the VUV absorption, to a spherical silvermirror (ROC = –300mm) with an estimated reflectivity of ~5% around the 15th

harmonic. Both beams are tightly focused into the He target and the variableaperture is used to adjust the IR energy such that the ionization is greatly en-hanced when the VUV harmonics are present. The focusing geometry wassimulated and a condition such that the focus sizes of the VUV and IR are ap-proximately the same was chosen. The focus is placed before the gas jet for

CHAPTER 6 - 60 -

maximum signal. Harmonics were generated with the IR alone with a cutoffaround order 69 in He, from which the IR peak intensity was estimated to be~5 1014 W/cm2 in the interaction region, which corresponds to a focus size ofabout 50 m diameter. This cut-off does not change significantly with the Xegas present (Figure 6.2), i. e. with the presence of the VUV harmonics. The Hegas target consists of a pulsed, T-shaped tubular jet with an inner diameter of260 m and a length of 2 mm, where the laser passes along the tube length. Apiezoelectric plunger synchronized with the laser fills the target from a gas res-ervoir with a backing pressure of up to 3 bar. The interaction region is about1 mm long with a pressure of several tens of mbar. The additional gas loadraises the backing pressure in the vacuum chamber to almost 10-2 mbar. Thegenerated VUV-assisted high harmonics propagate 1.7 m through the vacuumchamber with the Xe target and a differential pumping section to an XUV spec-trometer (McPherson) fiber-coupled to a CCD camera.

6.3 Experimental resultsResults of our measurements are shown in figure 6.2, where the spectral inten-sities measured for the input VUV (left) and generated XUV harmonics (right)are shown as a function of Xe pressure within the capillary. The VUV harmon-ics were completely absorbed by the He and were measured using the sameconfiguration but with the He jet switched off. The strengths of the VUV har-monics are observed to increase up to a Xe pressure of 15 mbar, followed by adecrease at higher pressures due to reabsorption in the Xe. The high harmonicsgenerated within the He jet are observed to follow the VUV harmonic strength,illustrating that the VUV harmonics do increase the XUV yield. By adjusting theIR intensity, an enhancement of up to a factor of five in VUV-assisted HHG ascompared to the yield with only the IR field present was observed.

To obtain a quantitative assessment of the influence that the Xe-generated VUVharmonics have on the XUV yield in He, the areas under each correspondingspectra for the VUV harmonics (harmonic orders 11–19) and the generated XUVharmonics (orders 43–79), for a range of different Xe pressures were integrated.The backing pressure of the He jet was maintained at 3 bar. Figure 6.3 showsthe distribution of spectral areas, where the relative errors in both spectrum in-


tegrals were estimated to be 30% accommodating background corrections. If theHHG process were initiated solely by single-photon ionization we would ex-pect a linear increase in HHG photon yield with increasing VUV strength. De-spite the large distribution of data points in figure 6.3, a monotonic increasingtrend is clearly visible, supporting the VUV-assisted single-photon ionizationmodel.

Figure 6.2: Enhancement of VUV-assisted HHG. The harmonic spectra of the VUVharmonics (left) and the XUV harmonics (right) are shown as a function of Xe pres-sure, for a He jet backing pressure of 3 bar. The VUV harmonics strength dimin-ishes for high pressures due to reabsorption. The harmonic plateau was experi-mentally not accessible due to the configuration of the measurement setup.

The generated XUV harmonics in figure 6.4 show a variation of the spectralwidth and a blue shift with increasing Xe pressure keeping all other parametersconstant [100]. The blue-shift is caused by the blue shifted IR pulse on propaga-tion through the ionized Xe in the capillary. We have measured an IR pulse

CHAPTER 6 - 62 -

with SPIDER after propagation through a gas cell filled with Xe. The pulse en-velope, the energy and the divergence of the IR pulse were not significantly af-fected, but the center of the spectrum was shifted by 7 nm after propagationthrough 25 mbar of Xe compared to the Xe-free case in close agreement with theobserved blue shift of the XUV harmonics. The spectral widths of individualharmonics within each HHG spectrum show a markedly different trend withand without VUV-assisted ionization, as shown in figure 6.4(b). Without theVUV harmonics, the higher orders exhibit a larger spectral bandwidth, whereasthe VUV-assisted harmonics show a decreasing bandwidth with increasing or-der. We also expect the coherence properties of the XUV harmonics to be differ-ent compared to tunnel-ionization-based HHG.

Figure 6.3: Integrated areas under the spectra of XUV harmonics 43–79, as a func-tion of the VUV harmonics 11–19, for different capillary Xe pressures, as labeled byeach point.

The experimental demonstration that VUV harmonics can be used to enhanceHHG leaves open the exact mechanism by which this comes about. Theprevious theoretical studies have shown that phase locked VUV harmonics (the11th to 19th) which form an APT can be used to enhance HHG [64]. The largeenhancement results from the change in the IR-intensity dependence of the


harmonics caused by the APT’s presence and timing relative to the IR field. AnAPT would allow us to select the timing of the ionization of the electron intothe continuum such that its ionization coincides with the release time tocontribute to HHG. The enhancement is expected to be greatest underconditions that would otherwise yield low photon numbers, and enhancementsof up to 2–4 orders of magnitude are predicted. At favorable delays, an APT-assisted HHG should result in a comb of clearly resolved harmonics ofincreasing width for increasing order. However, significant enhancement ispredicted in all cases, even at unfavorable delays where the ionization timefixed by the APT does not allow the returning electron wave-packet toefficiently recombine with the parent ion.

The use of all the VUV light from the Xe capillary, and not just orders above the11th, raises the question of how efficient the various harmonics are at ionizingHe. At low IR intensity harmonics below the 13th cannot ionize the He via a(possibly resonantly enhanced) single photon process. However, at higher IRintensities this picture is expected to be strongly modified due to the distortionof the Coulomb potential by the strong field. To investigate this effect, we havecalculated the relative efficiency of ionization with each of the harmonics fromthe 5th to the 13th separately in the presence of a strong IR field of different in-tensities. To separate the VUV-driven ionization from tunnel ionization causedby the IR field an approximate TDSE for two fields (IR + VUV) was solved thatrestricts the ionization to be driven just by the VUV field, but propagates theelectron thereafter in the full atomic potential modified by the strong IR field.The harmonic fields are assumed to have a 5 fs FWHM Gaussian pulse shape.The ionization yield from each harmonic is normalized to the ionization yield ofthe 9th harmonic. Figure 6.5 shows that at a relatively low intensity the 13th har-monic ionizes about 1010 more than the 5th. However, by the time the peak in-tensity reaches 4 1014 W/cm2, near the intensity used in this experiment, thedistribution flattens out considerably, and the 13th harmonic ionizes only about100 times as much as the 5th.

CHAPTER 6 - 64 -

Figure 6.4 (a) XUV harmonic spectrum for a He jet backing pressure of 3 bar, withand without VUV-assistance in the XUV HHG. (b) The spectral widths as a func-tion of harmonic order.

Since there are typically many more 5th harmonic photons than 13th harmonicphotons in a HHG spectrum, all of the Xe harmonics play a role in initiatingHHG in He, especially where the IR intensity is highest. Since harmonics below


the 9th are not expected to be phase locked to the higher ones, the harmonicsemitted by the Xe capillary will not form an APT. Therefore, the current ex-periment lacks control over the precise ionization time and we benefit onlyfrom an increase in the single atom response and possibly an increase in thevolume over which the harmonics are efficiently phase matched.

Figure 6.5: Relative He ionization probability as a function of harmonic order fordifferent IR intensities, where the ionization yield of the 9th harmonic is normal-ized in each curve.

6.4 SummaryFirst results of enhancement in harmonic yield due to VUV-assisted HHG in Heare presented. This novel scheme modifies both the process on the single atomlevel and also the macroscopic phase matching, but does not rely on shapingthe IR laser pulses, increasing their energy or modifying the gas target. The ini-tial tunnel ionization step in the HHG process is replaced with single VUV-photon ionization, which results in a large volume effect. It is predicted thationization with APTs in the VUV regime could result in even higher enhance-ments by several orders of magnitude.

- 67 -

Chapter 7

Strong field effects and attosecond

pulse characterization

Strong field effects like HHG, ATI and NSDI can be described in the frameworkof the simple man’s model and are comparable to scattering processes in highenergy physics since the electron is driven back to the atom with a high kineticenergy. To get insight into the processes the knowledge of the electron trajecto-ries involved, respectively, their momentum distribution and not just their en-ergy is important. Therefore, an electron imaging spectrometer (EIS) was de-signed and the transformation off the recorded image to momentum space isdiscussed. For the temporal characterization of single attosecond pulses andAPTs the pulse envelope is transferred into a continuum electron wavepacketreplica through single-photon ionization of an atom [30, 36, 101-104]. Com-monly a laser field is used as an ultrafast phase modulator on the wavepacketto shear or streak it in momentum space or energy, respectively.

7.1 IntroductionThe research of strong field processes is a challenging task since the effects havetimescales of femtoseconds or even attoseconds. The commonly used pump-probe scheme is hard to adapt in the case of harmonics, because suitable opticsin the XUV spectral range are missing and the number of photons generated issmall. Also the detection system is usually limited to the spectrum of photons,the energy of the electrons or the number of ions, which only gives limited in-

CHAPTER 7 - 68 -

sight into the process. The temporal characterization of single attosecond pulsesand APTs relies on the generation of an electron wavepacket replica of thepulses. Both a semi-classical [105, 106] and a quantum model [14] have beendeveloped for the description of the production of the electron wave packet. Forthe characterization of APTs which is based on interferences in electron spectra,a quantum mechanical description, the strong field approximation (SFA) [14], isneeded. To get a more intuitive picture of the single-photon ionization in thepresence of a dressing laser field, the interaction can be divided into two steps:first the absorption of the XUV photon and subsequently the acceleration of theelectron in the dressing laser field, which can be treated using classical me-chanics [105, 106].

In this chapter I will discuss the theoretical basis for the characterization of sin-gle attosecond pulses and APTs. The EIS spectrometer is introduced and a fastdirect transformation of images to momentum space is developed. First ex-perimental results on ionization are presented, the characterization of singleattosecond pulses and APTs using the EIS is discussed and a possible parame-ter range is determined.

7.2 Ionization of atoms in strong laser fieldsAssuming the harmonic field Eh t( ) and the dressing field EL t( ) are delayed by

, the transition amplitude a ( ) from the ground state to the continuum statep for times large enough for both fields to have vanished, becomes:

a ( ) = i dt d*

p t( ) A t( )( ) Eh

t( )( )exp i Ipt dt ' p

2t '( ) / 2

t( ) , (34)

where p t( ) = p + A t( ) is the instantaneous momentum of the free electron in thedressing field with A t( ) being its vector potential such that the dressing fieldE

Lt( ) = A / t .

The interpretation of equation (34) is straightforward: the transition amplitude ais the sum of all possible electron trajectories which lead to a final momentump when the fields have vanished. The harmonic field injects the electron into

the continuum at a time ti with momentum p , with a probability amplitude

- 69 - STRONG FIELD EFFECTS & ATTOSECOND PULSES

proportional to the harmonic field amplitude at time ti and the dipole transi-

tion matrix element d* , which depends on the electron momentum p t

i( ) justafter ionization. Subsequently the electron is accelerated in the dressing fieldand because its final momentum is known to be p after the dressing field hasvanished, due to momentum conservation p t

i( ) = p + A ti( ) .

The exp-function in equation (34) is the phase for each time t and is composedof the phase I

pt accumulated in the fundamental state up to time t and of the

phase dt ' p2

t '( ) / 2t

+ accumulated in the continuum with the presence of thedressing field. The latter term corresponds to the integral of the instantaneousenergy p

2t '( ) / 2 of a free electron in the dressing field from an ionization time

t to the observation time. Using

p2

t( )2

=p

2

2+ p A t( ) +

A2

t( )2

= Ekin+ p A t( ) +

A2

t( )2

(35)

and rearranging the exponential term yields:

a ( ) = i dt d*

p t( ) A t( )( ) Eh

t( )( )exp i (t)+ Ekin+ I

p( ) t( )( )+

, (36)

t( ) = dt ' p A t '( ) + A2

t '( ) / 2( )t

. (37)

The dressing laser field induces a temporal phase modulation t( ) on the elec-tron wavepacket in the continuum. In addition, due to the scalar productp A t( ) , a spatial dependence of the phase modulation is introduced. In the

majority of experiments the dressing field is linearly polarized and temporallylong enough to apply the slowly varying envelope approximation. ThenE

Lt( ) = E t( )cos t( ) and consequently A t( ) = E t( ) / sin t( ) and the phase can

be written as :

t( ) = 1t( ) + 2

t( ) + 3t( ) , (38)

with

CHAPTER 7 - 70 -

1 t( ) = dt 'U p t '( )t

,

2 t( ) =8EkinU p t( )

cos( )cos t( ),

3 t( ) =U p t( )

2sin 2 t( ).

(39)

Here the ponderomotive potential of the electron in the dressing laser field attime t is U

pt( ) = E

2t( ) / 4

2( ) and the observation angle is the angle betweenthe final electron momentum p and the laser polarization direction E t( ) . While

1t( ) varies slowly i.e. on the timescale of the dressing laser field envelope

EL

t( ) ,2

t( ) and 3

t( ) oscillate at the laser field frequency and its secondharmonic 2 . Because Ekin is much larger than U p in most cases,

2t( ) domi-

nates except for observation angles close to = / 2 , where the term cos( ) issmall. The total phase modulation reaches large values for small observationangles, since the energy of the final state E

kin is large.

Figure 7.1: Phase modulation (t) induced on the electron wavepacket by a line-arly polarized 800 nm dressing field with a moderate intensity of 9x1012 W/ cm2

(Up

0.54eV ) and for a final electron energy Ekin

= 100eV . t( ) is shown for dif-ferent observation angles and its modulation is large for as large as 30° . (Fig-ure taken from [107])


In figure 7.1 t( ) is shown for different observation angles and a large phasemodulation can be observed for as large as 30° .

7.3 Temporal characterization of single attosecondpulses

The first proposal for attosecond pulse characterization was the attosecondstreak camera [102], where the dressing laser field is used to map the temporalproperties of the harmonic pulse to the photoelectron spectrum by using its de-pendence on the ionization instant t

i. Assuming that the harmonic pulse is

shorter than the dressing field optical cycle, it is then possible to evaluate theduration of the pulse just from the energy, i.e. with a electron TOF spectrome-ter. Provided the measurement of the electron spectrum can be angularly re-solved like in an EIS, also its chirp can be reconstructed.

The electron is injected into the continuum at a time ti with an initial energy

Ekin= p

2/ 2 , which corresponds to the harmonic photon energy minus the ioni-

zation potential: Ekin= q I

p, where the influence of the dressing field on the

ionization has been neglected. Neglecting the influence of the ionic potential onthe motion of the electron in the continuum, the electron momentum after ioni-zation is only affected by the dressing field and reads:

p t( ) = A t( ) + p A ti( ). (40)

On the one hand the electron oscillates in the dressing field, which is describedby the first term in equation (40), which vanishes as the dressing laser pulseends. The final electron momentum, which can be experimentally accessed, isthus determined by the ionization time t

i and yields:

p = p ti( ) = p A t

i( ). (41)

The evolution of the electron momentum is shown in figure 7.2, where theelectron comes into the continuum at time t

i, where the dressing electric field is

zero. While the laser field lasts, the electron carries out a quiver motion in thefield, whose amplitude decreases with the laser field amplitude.

CHAPTER 7 - 72 -

Figure 7.2: Momentum of the electron after ionization at time ti in the presence of

an 800 nm dressing field (dashed line). The electron momentum immediately afterionization p is assumed to be parallel to the laser polarization. (Figure taken from[107])

After ionization, the initial electron momentum p can have any angle withthe polarization direction of the dressing field. Plotting the final electron mo-mentum p as a function of this angle in polar coordinates [34, 35, 102] gives inthe absence of a dressing field a circle with radius p centered on p = 0 (seedashed circle in figure 7.3). The dressing field displaces the circle from p = 0

by A ti( )according to equation (41) for a certain ionization time t

i (see full line

circle in figure 7.4). This shift occurs along the direction of the laser polarizationand for a linearly polarized dressing field the circle thus oscillates back andforth around p = 0 with changing t

i. The dressing field does not only change

the final electron energy but in general also deflects the electron from its initialpropagation direction just after ionization.

Although the semiclassical model gives a good intuitive picture of the dynam-ics of single-photon ionization in the presence of a dressing laser field, it doesnot contain interferences between parts of the electron wavepacket emitted atdifferent times t

i but having the same final energy, which are well described in

the quantum mechanical picture. Assuming a linearly polarized dressing field


and that the photoelectron spectrum is observed in a single direction , themaximum streaking effect is reached by adjusting the delay between the atto-second field and the dressing field such that 2

/ t2= E

kin/ t presents a

maximum.

Figure 7.3: Final electron momentum in the absence of a dressing field (dashed cir-cle) and in the presence of an 800 nm dressing field (solid circle). The circle is dis-placed by the dressing field according to the value of the vector potential at thetime of ionization t

i. (figure taken from [107])

In figure 7.4 the phase modulation t( ) and the final electron energy Ekin

t( ) areshown for an optimal synchronization with a dressing field with high intensity.On the right axis the broadening of the photoelectron spectrum is shown, fromwhich the temporal duration of the attosecond pulse can be deduced. Thehigher the intensity of the dressing field, the higher the streaking effect be-comes, because for higher intensities the ponderomotive potential U

p is larger.

If the intensity becomes too high however, the dressing field also starts to ionizethe atom by multiple-photon absorption. The streaking effect is most importantin the direction of the laser polarization = 0 .

The attosecond streak camera method was used in [36] to measure single XUVharmonic pulses with a duration of 250 as . To produce the single attosecondpulses using HHG, it was necessary to stabilize the absolute phase of the driv-ing IR pulse, which was also used as dressing laser field.

CHAPTER 7 - 74 -

Figure 7.4: The phase modulation t( ) (top graph) and the final electron energyE

kint( ) (bottom graph) for a dressing laser field with I=5x1013 W/cm2 and ob-

served along the polarization direction = 0 . The attosecond pulse is shown at aposition chosen for maximum streaking in energy as illustrated by the broadeningof the energy spectrum on the right axis. (figure taken from [107])

An extension of the attosecond streak camera method is chronocyclic tomogra-phy [108-110], which allows for a full reconstruction of the attosecond pulseusing a limited number of streaked spectra. This is possible because, in additionto the width, the detailed shapes of the streaked spectra contain informationabout the temporal structure of the pulse, including higher order terms in thespectral phase.

7.4 Temporal characterization of attosecond pulsetrains

Reconstruction of attosecond harmonic beating by interference of two-photontransitions (RABITT) [30-33] is used to characterize trains of attosecond VUVpulses. The presence of a dressing field induces the formation of sidebands inthe photoelectron spectra by the additional absorption/emission of an opticalphoton during the ionization process by the VUV photon. Therefore each side-band contains a contribution from the two adjacent harmonics. Second-orderperturbation theory predicts that the interference term between the contribu-


tions to the qth-order sideband (here q is even) from adjacent harmonics is pro-portional to:

cos(2 +q 1 q+1

+atomic

f), (42)

whereatomic

f is the atomic phase, which is the intrinsic phase of the matrixelements for above-threshold, two-photon ionization.

atomic

f can be extractedfrom simulations or calculated for some atomic species [111]. When scanningthe delay between the APT and the dressing field, the sideband signal pre-sents oscillations with frequency 2 , which corresponds to half the dressinglaser period. The phase contribution

q 1 q+1 describes the spectral phase dif-

ference between the harmonic q-1 and q+1 and with a measurement of the side-band signals for different orders it is therefore possible to determine the relativephases

q of the different harmonic orders by concatenation of the measured

phase differences.

The knowledge of the relative phases q of the harmonics can be used together

with their spectral intensities Iq to reconstruct the average pulse in the attosec-

ond pulse train, which then reads:

E(t) = Iqe

iq t+q

q odd

. (43)

The APT pulse characteristics such as spectral phase and spectral features of theVUV pulses formed by the single harmonics are not taken into account andtherefore the reconstruction does not include the variations of the pulse char-acteristics over the pulse train. A full pulse train characterization as predicted tobe possible with FROG CRAB [112] or a recently proposed method using adia-batic phase expansion [113].

The first measurement of an APT was carried out using RABITT [31] and 250-asbursts spaced by half the laser period were observed [30]. Later RABITT hasbeen used to measure the phase differences between a large number of con-secutive harmonics demonstrating that the different harmonic orders are notemitted at the same time and don’t exhibit a flat phase relation [32]. Thereforean attosecond pulse train composed of a number of harmonics is not automati-

CHAPTER 7 - 76 -

cally transform-limited and the spectral phase needs to be shaped to producethe shortest possible attosecond bursts. Shaping of the spectral phase to pro-duce nearly transform-limited attosecond bursts has recently been demon-strated [33]. Compression from 480 as to 170 as, close to the transform limit of150 as, has been achieved.

A typical RABITT acquisition is shown in figure 7.5 for harmonics 13 to 23. Thesidebands appear in-between the harmonics and their intensity is modulatedwith half the dressing laser period.

Figure 7.5: Measured RABITT spectrum for harmonics 13 to 23 generated in argonusing a 35 fs 800 nm pulses and with an 800 nm 35 fs dressing laser field.

Fitting a cos-function to the oscillations or performing a Fourier analysis of thesideband gives access to the pair-wise phase difference of the harmonics. Thisinformation can be used to reconstruct the average attosecond pulse in the trainaccording to equation (43). A sample reconstruction of an attosecond burst isshown in figure 7.6 for harmonics 11 to 19, the sideband modulation is shownat the left and the reconstructed 250-as bursts are shown on the right.


Figure 7.6: Sideband modulation for harmonics 11 to 19 (left, top to bottom) andreconstructed 250-as burst (right). (Figure taken from [31])

7.5 Electron imaging spectrometer

7.5.1 Introduction

Strong field processes happen on timescales of femto- or even attoseconds,which makes their investigation challenging. The detection system is usuallylimited to the spectrum of photons, the energy of the electrons or the number ofions. Therefore an electron imaging spectrometer (EIS) was designed and builtto study HHG, ATI and NSDI. All effects can be described in the framework ofthe simple man’s model and are comparable to scattering processes, because theelectron is driven back to the atom with a high energy. The knowledge of theelectron trajectories involved or their momentum distribution makes new in-sights about these processes possible.

The EIS maps the electron distribution of an ionization event onto a two-dimensional phosphor screen and gives the possibility to retrieve the full three-dimensional electron momentum distribution. For fast reconstruction of themomentum images, a direct transformation has been developed and first ex-

CHAPTER 7 - 78 -

perimental results on ionization of Xe are presented. The EIS can also be used asan ion TOF spectrometer, for the temporal characterization of single attosecondpulses as well as APTs and for pump-probe measurements with APTs to inves-tigate APT-assisted ATI and APT-assisted NSDI.

In this thesis an EIS is designed to image high energetic electrons of up to200 eV in order to temporally characterize single isolated attosecond pulsesaround 90 eV. This energy corresponds to a wavelength of 13 nm and is chosen,because good Mo/Si multilayer mirrors with reflectivities of up to 70% arecommercially available. A new effusive gas target (glass needle) with no cool-ing requirement is developed to allow for a highly localized and dense gas me-dium in the interaction region, which allow image acquisitions with short inte-gration times. The use of the EIS as a replacement for an electron TOF spec-trometer for RABITT measurements in the VUV is investigated.

7.5.2 Design of the spectrometer

An EIS is comparable to a large capacitor with the top plate replaced by a MCP.If a voltage is applied to one plate and a charged particle is created, it gets ac-celerated to the MCP that enhances its signal and visualize it on the attachedphosphor screen attached. To make the spectrometer suitable for attosecondpulse characterization, a laser must be focused between the plates and gas mustbe injected in the focus. A schematic drawing of the EIS with the laser and theneedle is shown in figure 7.7. The laser is focused with a mirror (ROC = -75mm) between the plates directly in front of the tip of the needle, where theelectrons are generated and extracted to the MCP. By a high speed CCD cameraa picture of the electron distribution on the phosphor screen is taken.

The electric field and therefore the arrangement of the plates in the EIS must becarefully designed and simulated. The electric field can be homogenous like inthis case or be designed with electrical lenses, setups known as velocity mapimaging (VMI) spectrometers [114]. The disadvantage of the VMI spectrometersis the detection of high energetic electrons, because high voltages have to beapplied to form the suitable lenses. The EIS consists of an array of five roundmetal plates placed in front of a MCP with a voltage on each plate equal to thepotential at its position. The diameter of the last three plates is 90 mm and the


distance a to the front of the MCP is a = 56 mm. The ratio of diameter to dis-tance between two plates is chosen to produce a nearly homogeneous field,which is the main requirement. The additional plates enhance the homogeneityof the field, leading to a fivefold increase to a total mean homogeneity of morethan 98,0 %. The main distortion of the electric field comes from the mirrormounted on a piezo stage, which is connected to ground potential (In the fol-lowing just piezo).

Figure 7.7: A schematic drawing of the EIS with the laser and the effusive gas tar-get. The laser is focused with a Mo/Si mirror (ROC=-75mm) between the plates di-rectly in front of the tip of the needle, where the electrons are generated and ex-tracted to the MCP. A picture of the electron distribution on the phosphor screen istaken by a high speed CCD camera.

Figure 7.8 shows a schematic setup with the plates, one electrode at groundpotential to simulate the piezo and the calculated electron trajectories. Theplates are made of stainless steel and have a thickness of 0.5 mm [115]. The totalvoltage between the MCP and the outermost plate is maximum 10 kV. The dis-tance of the plate to the piezo is chosen such that the field strength is not higher

CHAPTER 7 - 80 -

than 1 kV/mm to prevent arcing. The gas used to characterize the attosecondpulses is injected between the plates with a glass needle. Since this is an isolatorit gets charged due to the electrons produced in the interaction region. In orderto avoid this the needle is covered with a layer of silver and a voltage is ap-plied, which corresponds to the potential of its position in the EIS minimizesthe distortion of the electric field. The needle has the advantage, compared to askimmed gas jet, that no cooling is required. The high gas density in the inter-action region provides a high counting rate, making short integration timespossible. Furthermore it is cheaper compared to a skimmed gas jet, whichwould require additional turbo pumps.

The main distortion of the electric field is caused by the piezo, which introducesan error between the first and the second plate of the order of 15 %. To mini-mize this error to less than 5 % the interaction region is moved to the height ofthe second plate.

Figure 7.8: Electron trajectories in the EIS with an starting kinetic energy of 80 eVand the generation region is 6.5 mm away from the first plate (Cut through thedetector plates).

It is important that no harmonics hit a plate, because then electrons are pro-duced, which destroy the signal on the MCP. To quantify the order of magni-tude of the involved parameters, 5000 V on the bottom plate are sufficient to re-cord a spectrum generated by harmonics with an energy of 90 eV with a reso-lution of about 1 eV. The intensity has to be chosen such that no ionization or

Piezo stage

MCP

Metal plates


Stark shifts take place, which translates into the inequality 8Up< q I

p, where

Ip is the ionization energy. For He the maximum intensity is 1.2 x 1014 W/cm2 forheavier atoms even higher [116]. Figure 7.8 shows that moving the interactionregion to the second plate the piezo stage doesn’t strongly disturb the electricfield anymore such to be considered as homogeneous.

Figure 7.9: Potential surface with the electron trajectories in the EIS (Cut throughthe detector plates). The potential in the EIS shows a linear increase through theMCP, which corresponds to constant electrical field strength.

A last but not less important step concerns the treatment of the image on thephosphor screen of the MCP that needs to be transformed to momentum space.A standard way is the Abel transformation. This technique has at least twomain drawbacks, first it is computer time intensive since it is an indirect trans-formation and second the different electron trajectories are not treated correctly.To overcome these limitations in the following I present a direct transformationbased on the real electron trajectories.

CHAPTER 7 - 82 -

7.5.3 Momentum space image and single attosecond pulses

A different and fast direct transformation has been developed especially for thetemporal reconstruction of attosecond pulses and therefore the focus is alwayson the reconstruction of the momentum added by the laser to the electron. Thetransformation is derived from the classical equation of motion [117, 118] and inthe following the electric field in the EIS is considered to be constant. Three dif-ferent momenta are considered in the calculation, the initial momentum p , themomentum of the driving laser p

L and the total momentum P = p + p

L. After

the laser pulse has vanished, the motion of an electron between the plates isgiven by the following equations:

x px,t( ) = p

xt +

E

2t

2,

y Py,t( ) = P

yt,

z Pz,t( ) = P

zt.

(44)

The position on the MCP can be obtained by calculating the arrival time on theMCP and using E

kin= p

x

2+ P

y

2+ P

z

2( ) 2 (For ionization by qth harmonic withoutIR laser E

kin= q I

p). The coordinates on the phosphor screen are given by the

following equations:

yD

Py, P

z( ) =P

y

E2aE + 2E

kinP

y

2P

z

22E

kinP

y

2P

z

2( ) ,

zD

Py, P

z( ) =P

z

E2aE + 2E

kinP

y

2P

z

22E

kinP

y

2P

z

2( ) ,(45)

where a is the flight distance to the MCP plate. These equations give one elec-tron trajectory for a definite ionization event. In order to extract from the posi-tion of the screen the momentum the equations must be inverted. In this casethere are two possible electron trajectories with different initial conditionsleading to the same spot on the screen. These trajectories belong to an electronwith an initial momentum in the direction of the MCP and one with momentumaway from it. Therefore the inversion is not unique, which results in the fol-lowing set of momenta:


px

1

y, z( ) =

4Ekin

aE + 2Ekin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )( ) E2

y2+ z

2( )

2 aE + 2Ekin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )( )

px

2

y, z( ) =

4a2E

kin+ y

2+ z

2( ) 2Ekin

4Ekin

aE + Ekin( ) E

2y

2+ z

2( ) aE( )2 a

2+ y

2+ z

2( )

, (46)

Py

1

y, z( ) =yE

2 aE + 2Ekin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )( )

Py

2

y, z( ) = yaE + 2E

kin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )2 a

2+ y

2+ z

2( )

, (47)

Pz1

y, z( ) =zE

2 aE + 2Ekin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )( )

Pz

2

y, z( ) = zaE + 2E

kin+ 4E

kinaE + E

kin( ) E2

y2+ z

2( )2 a

2+ y

2+ z

2( )

. (48)

In order to test the inversion algorithm, the kinetic energy of electrons ionizedjust by harmonics can be calculated and compared to the expected photon en-ergy minus the ionization energy. As discussed before, there are nearly alwaystwo trajectories leading to one position on the MCP, which makes the uniquereconstruction impossible. Nevertheless the edges of the electron distributionbelong to just one trajectory, which reflects mathematically in P

y1

= Py

2

. Usingthis approximation a unique transformation is possible and given by the fol-lowing set of equations:

px

y, z( ) = E2a

2+ y

2+ z

2

2 a2+ y

2+ z

2

a , (49)

CHAPTER 7 - 84 -

Py

y, z( ) = yE

2 a2+ y

2+ z

2

, (50)

Pz

y, z( ) = zE

2 a2+ y

2+ z

2

, (51)

Ekin

y, z( ) =E

2a

2+ y

2+ z

2a( ). (52)

These simple formulas, which are explicit and easy to calculate, only depend onthe flight distance a to the MCP and the effective field strength E, which is notexactly voltage/distance, but it can be calibrated with an electron spectrumgenerated by harmonics with a knows photon energy.

E =2E

kin

a2+ y

2+ z

2a

. (53)

The EIS can also be used as an ion TOF spectrometer. When opposite voltagesare applied to the plates, ions hit the MCP causing a voltage drop, that can berecorded. The temporal resolution is about 5 ns, perfectly suitable ion detectionsince the difference in the TOF of a single charged and a doubly charged He ionis in the range of 500 ns. Furthermore the use of the EIS as an ion TOF spec-trometer gives an easier and probably more accurate way to calculate the effec-tive electric field strength. In fact:

E =2m

iona

t2n

, (54)

where mion is the mass of the ion, n is the charge state and t is the flight time tothe detector. The reconstruction of the electron momentum distribution fromthe image on the screen is possible with the previously given formulas. For at-tosecond pulse characterization the momentum added by the laser must be ex-tracted from the total momentum consisting of the initial momentum and themomentum added by the laser. The ionization due to a high harmonic pulsewithout a dressing laser field is schematically shown in figure 7.10 for the easi-


est case of a circular initial momentum distribution. The distribution exhibitsvery pronounced and sharp edges.

Figure 7.10: Schematic picture on the phosphor screen of the ionization of He by a90 eV, single 250 as pulse without the driving laser.

Figure 7.11: Schematic picture on the phosphor screen of the ionization of He by a90 eV, 250 as single attosecond pulse in the presence of the driving laser.

Distance / cm

Distance / cm

Intensity / arb. units


Distance / cm

Distance / cm

CHAPTER 7 - 86 -

If the atoms are also subjected to the driving laser and the high harmonic pulseis centered in the maximum of the electric field of the driving pulse, the electrondistribution gets shifted and smeared out along the polarization axes of thedriving laser (Figure 7.11). This smearing can be understood in the followingway: The homogeneous distribution (figure 7.10) is shifted and multiplied withthe intensity of the attosecond pulse, which can be schematically seen in fig-ure 7.12.

Figure 7.12: Cut through figure 7.11 on the polarization axes of the laser.

The momentum added by a linearly polarized driving laser to the initial mo-mentum can be calculated from a cut in the laser polarization through the im-age on the screen:

pLy

y,z( ) = yE

2 a2+ y

2+ z

2

1

y2+ z

2

2 q Ip( ) E

2a2+ y

2+ z

2

2 a2+ y

2+ z

2

a , (55)

pLz

y,z( ) = zE

2 a2+ y

2+ z

2

1

y2+ z

2

2 q Ip( ) E

2a2+ y

2+ z

2

2 a2+ y

2+ z

2

a . (56)

The duration of the attosecond pulse can be calculated with linearly and circu-larly polarized light. The calculation with circularly polarized light is muchsimpler, but the generation of a harmonic pulse locked in time to the electricfield of a circularly polarized laser pulse is a real challenge. In the case of line-


Distance / cm


arly polarized light this is naturally fulfilled, if the generating pulse is also usedfor characterization. For circularly polarized light the following formula givesthe relation between the rotation of the circle to the pulse duration for a har-monic pulse shorter than one driving laser cycle:

THH=

1+

2( )T

L

2, (57)

where1 is the angle between maximum and half maximum on one side,

2on

the other side and the pulse duration of the driving laser is TL. The calculationof the pulse duration with linearly polarized light is more complicated, because,if ti is the ionization time, the momentum is given by the following formula:

pL

ti( ) = dt' E t'( )

ti

. (58)

This integration has to be performed numerically, because the driving pulse isrecorded by SPIDER and has no analytical expression. Nevertheless for an ex-perimental reconstruction a full scan of the delay between the attosecond pulseand the driving pulse has to be performed.

Figure 7.13: Comparison of the attosecond pulse form with the picture on thephosphor screen.

To investigate the accuracy of the pulse reconstruction, the electron distributiongenerated by a given attosecond pulse and streaked with the IR is calculated


Distance / cm

CHAPTER 7 - 88 -

and compared with the attosecond pulse. Figure 7.13 shows the result of thiscalculation. The electron distribution is found to be slightly broader than thepulse, which is due to the bandwidth of some eV of the harmonic leading to anelectron distribution with smooth edges. Therefore the pulse duration calcu-lated from the outer wings of the distribution is always a bit longer than theoriginal one.

The spectrometer works with electrons with a maximum kinetic energy of200 eV. The resolution in energy around 90 eV is around 1 eV, which corre-sponds to an error of 150 microns on the screen. For attosecond pulse charac-terization with a driving laser intensity of I=5x1013 W/cm2 the smearing of theelectron distribution from maximum to half maximum is around 3.0 mm.Therefore the relative error should be less than 10 % and can be decreased withhigher intensities.

7.5.4 Experimental results

The EIS has been built according to the design parameters given in the previoussections. One of the key features is the effusive gas needle, which replaces thetypically used expensive and complicated skimmed gas jet. The benefits of theneedle are the high pressure and the localized interaction region. This is espe-cially useful for the characterization of harmonic pulses, because the ionizationhas no threshold and therefore the background pressure must be as low as pos-sible.

The gas profile of the needle has been recorded by flowing gas through theneedle and measuring the pressure with a skimmer connected to a pressuregauge. The needle is mounted on a xyz-translation stage and moved in front ofthe skimmer to record the spatial pressure distribution. Figure 7.14 shows theresult of a scan in xy-direction and it can be nicely seen that the gas distributionis highly localized, due to the dimensions of the needle. In fact its length is 5 cmand the diameter is 65 microns, therefore the gas is guided in the needle for along distance and forms at the tip of the needle a collimated gas beam.


Figure 7.14: Gas distribution of the needle after a distance of 0.5 mm in z-direction.

Figure 7.15 shows the pressure distribution in the z-direction, which is the dis-tance from the skimmer. It is highly localized and drops rapidly with increasingdistance from the needle. With this data the interaction volume is estimated tobe about 22 1( )mm3 12.6mm3 .

Figure 7.15: The pressure profile in z-direction in the maximum of the radial pres-sure distribution (Figure 7.14).

For the reconstruction of ionization events in general, NSDI or even APT-assisted NSDI, it is important to measure the charge state of the produced ions.

CHAPTER 7 - 90 -

To perform this measurement the EIS has to be used as ion TOF spectrometer.By reversing the voltage applied to the plates and so getting ions accelerated tothe MCP. The phosphor screen is covered with a conducting layer and the volt-age is monitored. If an ion hits the MCP the voltage drops and from the flighttime to the MCP the charge state of the ion under investigation can be deter-mined. Figure 7.16 shows a screen shot of the oscilloscope, while recording thedrops resulting from the ionization of Xe. Two clearly distinguishable drops inthe voltage belonging to Xe+ and Xe++ formation can be observed.

This feature is very useful since due to the possibility of correlating the inset ofdouble ionization to the electron momentum distribution, the process of doubleionization can be easier found. In the case of APT-assisted NSDI the peak ofXe++ should alternatively vanish or being enhanced depending on the delaybetween APT and IR laser field. Together with the electron image on the screenthe responsible quantum paths can be found, which could finally end the dis-cussion if the slow or fast electrons are responsible for NSDI [119].

Figure 7.16: Screen shot of an oscilloscope of the Ion-TOF signal taken with the EISfor the ionization of Xe.

However the main test for the EIS is to acquire a real electron image. For thispurpose IR laser pulses of an amplified Ti:Sapphire laser system deliveringpulses of 1 mJ, 30 fs at a repetition rate of 1 kHz are focused between the plates.


The needle is filled with Xe at a pressure of 10 mbar keeping the backing pres-sure below 10-6 mbar during the measurement. To control the intensity in thefocus, an aperture is used to cut part of the incoming beam.

The results are presented in figure 7.17, where the laser intensity increases fromleft to right. The top left picture shows a center ring and additional outer ringsseparated by one IR photon energy each. The innermost circle belongs to ioni-zation with a number of photons right above the ionization threshold, in thiscase eight IR photons. The outer rings are ATI rings, where an electron absorbsmore photons than required for ionization. The signal vanishes several timesalong a ring and from this number the initial angular momentum of the groundstate can be determined. The top middle picture shows an ionization imagetaken with a higher intensity and more ATI rings become visible. The ionizationis still caused by multi photon ionization. The right upper picture shows twospecial features. First the picture is getting asymmetric in the direction of laserpolarization, which is a signature of the onset of tunnel ionization. Second, inthe middle of the picture two coils show up, which coincide with the first dou-bly ionized Xe atoms [120]. The left lower picture is more asymmetric and thecoils are clearer. At this intensity the ionization starts to be dominated by tun-neling. The middle lower picture shows additional structures on both sides ofthe laser polarization. These structures belong to electrons, which are ionized,driven back to the ion core and scattered in this direction. Also, these structurescarry information about the ion and can be used for the investigation of ions instrong laser fields. The last picture shows the ionization in the tunnel regime,where still structures of multi photon ionization are visible, which are causedby the leading edge of the pulse with lower intensity. A last point to mention isthe line visible perpendicular to the laser polarization, which is a result of spacecharge effects in the interaction region. The ions act like a positive lens spread-ing the charges to a line [121]. This effect can be avoided by reducing the pres-sure in the needle and therefore reducing the ions in the interaction region.

CHAPTER 7 - 92 -

Figure 7.17: Pictures of the phosphor screen for the ionization of Xe for differentintensities. The pictures show the transition from the multi photon to the tunnelregime.

One concern during the EIS design has been the influence of the needle on theelectric field in the spectrometer. Since the gas density drops rapidly with in-creasing distance from the tip, the needle must be placed close to the interactionregion. Therefore the needle can distort the electrons trajectories leading tosmeared and deformed images on the screen. During operation the inset ofdistortion on the electrons due to the needle can be very accurately observed inthe electron distributions on the screen, however choosing appropriately dis-tance from the interaction volume this problem can be eliminated.

7.5.5 Characterization of attosecond pulse trains

As pointed out in the previous sections the EIS has originally been designed forcharacterization of single attosecond pulses around 13nm, which is done bystreaking of the electron distribution generated by the harmonics with a stronginfrared (IR) field. In this case the electron distribution is moved by more thanseveral millimeters and therefore the requirement on the energy resolution isnot demanding. However characterization of APTs of low order harmonics 9th

till 19th requires the use of the RABITT technique, which relies on the interfer-


ence in the electron spectra (Figure 7.16). The energy resolution needs to bebetter for RABITT and the following calculation addresses this in the energyrange around 20 eV. In the calculation the following assumptions are made:

1) The electrons are generated in a point and therefore the dimensions of thefocus is neglected.

2) The electric field inside the spectrometer is homogenous and no imaging er-rors of the camera are taken into account.

3) Only the outermost electron trajectories are counted.

The first assumption is valid, because the focus of the VUV harmonics is tinyand the gas well localized. The second assumption holds and has been verifiedin the previous section and imaging errors are minimized by using a high qual-ity camera objective. The last assumption is valid and is experimentally provenby figure 7.16, where clearly can be distinguished between the different ringsand therefore each can be treated independently. With these simplifications thefollowing formula, which connects the coordinates y and z on the screen withthe electron energy can be used:

Ekin y, z( ) =E

2a2+ y2

+ z2 a( ), (59)

where E is the electric field strength in the spectrometer, a is the flight distancefrom the ionization to the MCP.

The electric field strength, respectively the voltage on the plates of the EIS, hasto be chosen such that the most energetic electron to be characterized hits theMCP. The active area is 40 mm in diameter, because of symmetry the furtherdiscussion is limited to one half of the screen and assumes z=0. The 19th har-monic should be near the edge of the screen, therefore at 15 mm. The maximumkinetic energy of electron ionized from a Kr atom by the 19th harmonic of an IRpulse with photon energy around 1.6 eV is given by:

19 1.6-14.0( )eV =16.4eV. (60)

CHAPTER 7 - 94 -

The required electric field strength for a flight distance a = 56 mm is around 220V/cm and therefore in total 1540 V.

Fig. 7.18: Energy distribution on the phosphor screen plotted against pixel of theCCD camera.

The minimum spatial resolution to distinguish the different rings from eachother, which can be separated with the camera attached to the MCP, has to bedetermined. The camera has an array of 1280 pixels, where around 1000 can beused for imaging the active area of the MCP. Therefore one pixel covers 40 mi-crons. In order to determine the energy resolution of the spectrometer equation(60) is rewritten in the following way:

Ekin P( ) =E

2a2+

4 106

5.292P

2

a , (61)

where P is the pixel on the camera and Figure 7.18 shows the kinetic energydistribution on the MCP against pixel for the previously determined parame-ters. To determine the energy spread per pixel, the derivative in P is taken:

d

dPP( ) =

E

2

4 106

5.292

2

P a2+

4 106

5.292P

21

. (62)


Figure 7.19 shows the energetic part of one IR photon per pixel. The IR photonenergy spread per pixel must be below 0.3, which is the minimum to distin-guish between two maxima. The calculation shows, that there should be about3-times as much pixels than minimally required. In summary with the assump-tions made, the energy resolution of the EIS is sufficient to characterize APTsaround 20 eV.

Fig. 7.19: Energetic part of an IR photon per pixel of the CCD camera.

7.6 SummaryAn electron imaging spectrometer (EIS) for electron energies of up to 200 eV hasbeen designed and built to study HHG, ATI and NSDI. The spectrometer mapsthe electron distribution of an ionization event onto a two-dimensional phos-phor screen and gives the possibility to retrieve the full three-dimensional elec-tron momentum distribution. The knowledge of the electron trajectories in-volved, respectively the momentum distribution, makes the interpretation ofthe processes much easier. For fast reconstruction of the momentum images adirect transformation has been developed and first experimental results onionization of Xe are presented. A new effusive gas target with no cooling re-quirement has been developed to produce a highly localized and dense gasmedium in the interaction region, which makes high counting rates possible.Also the energy resolution of the EIS in the VUV is sufficient for RABITT meas-urements [122]. Furthermore the EIS can be used as an ion TOF spectrometer to

CHAPTER 7 - 96 -

distinguish between different charge states of ions. Numerous experiments arepossible with the EIS but the main purpose is the temporal characterization ofsingle attosecond pulses and pump-probe experiments with APTs to investigateAPT-assisted ATI and APT-assisted NSDI.

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Chapter 8

Summary and Outlook

In this thesis an introduction to HHG, the theoretical description and the possi-bility of attosecond pulse generation is presented. A new scheme for the controlof strong field processes like ATI, NSDI and HHG is introduced and studied indetail. By combination of an APT with an intense IR laser field, the timing of theAPT with respect to the infrared laser field selects microscopically a singlequantum path contribution to HHG that would otherwise consist of many in-terfering components. The APT controls the release of the electron into the con-tinuum and improves the yield and coherence of the harmonics generated. De-pending on the chosen time delay between IR laser and APT, the harmonicspectrum is significantly enhanced, and exhibits either a spectrally resolvedplateau of harmonics, or distinct harmonics exclusively in the cutoff. The influ-ence of phase matching in a possible experimental realization is investigatedand nonadiabatic calculations of the macroscopic harmonic signal created by agas of helium atoms exposed to a strong infrared IR pulse in combination withan APT are presented. The harmonic yield can be enhanced by two to four or-ders of magnitude for the optimal delays between the IR and the APT pulses.The large enhancement is due to the change in the IR-intensity dependence ofboth the harmonic strength and phase caused by the presence and timing of theAPT. This leads to enhancement of the harmonic yield and improved phasematching conditions over a large volume. Since APT-assisted HHG effect sur-vives and even improves phase matching a systematic study of APT-assistedHHG on the single atom level is presented and an optimal set of parameters forAPT-assisted HHG is determined. Based on this theoretical work an experi-mental setup to control HHG has been built and first results on APT-assisted

CHAPTER 8 - 98 -

HHG have been obtained. VUV harmonics are generated in a Xe target, and fo-cused together with the remaining IR field into a He gas jet to generate highharmonics in the XUV. The harmonics generated by both fields show a clearenhancement of a factor of five in the cutoff region compared to the IR-onlycase. The experimental setup did not allow the delay between both fields to bechanged. Furthermore the input VUV harmonics were not temporally charac-terized; therefore a definite proof that an APT was present has not been estab-lished, yet.

An electron imaging spectrometer (EIS) was designed and built to study strongfield processes like HHG, ATI and NSDI, which can be described in the frame-work of the simple man’s model and are comparable to scattering processes inhigh-energy physics since the electron is driven back to the atom with high ki-netic energy. To get insight into the processes, the knowledge of the electrontrajectories involved, respectively the momentum distribution, is importantrather than just the energy. The EIS maps the electron distribution of an ioniza-tion event on a two-dimensional phosphor screen and gives the possibility toretrieve the full three-dimensional electron momentum distribution. For fastreconstruction of the momentum images, a direct transformation is developedand first experimental results on ionization of Xe are presented. The main pur-pose of the EIS is the temporal characterization of single attosecond pulses aswell as APTs and pump probe measurements with APTs to investigate APT-assisted ATI and APT-assisted NSDI.

As a future experiment the quantum path selection by an APT can be used todefine a high-resolution attosecond clock. By using an APT to launch electrontrajectories with only a single return to the ion core could advance attosecondmetrology beyond its present limits. When the short trajectory is selected, thereturn energy, as reflected in the emitted harmonic frequency, serves as a veryhigh resolution attosecond ‘‘clock.’’ The harmonics 31st through 63rd, corre-sponding to return energies between 1 and 3Up, receive their primary contribu-tions from trajectories having return times spread over a 400 as interval, withthe higher energies returning later. The 16 harmonic intervals thus define a 25as clock division. By using a higher IR intensity the number of harmonics be-tween 1 and 3Up can be further increased, yielding even finer divisions.

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Atomic units

Physical quantity Symbol Atomic units Système international

Electron mass me

1 9.1093897 x 10-31 kg

Elementary charge E 1 1.6021773 x 10-19 C

Speed of light c 137.036 2.99792458 x 108 m/s

Angular momentum 1 1.05457266 x 10-34 Js

Bohr radius a0=

40

2

mee

2

1 0.529177 Å

A.u. of time0=

a0

c1 2.4188843 x 10-17 s

A.u. of energy mee

4

40( )

2

1 27.2116 eV

A.u. of electric fieldstrength

e

40a

0

21 5.14221 x 1011 V/m

A.u. of intensity c

20

e

4 a0

2

2

1 3.51x1016 W/cm2

- 100 -

Glossary

ADK rate (Ammosov, Delone, Krainov) Ionization rate

APT Attosecond pulse train

ATI Above threshold ionization

EIS Electron imaging spectrometer

FFT Fast Fourier transform

FWHM Full width at half maximum

HHG High harmonic generation

IR Infrared

MWE Maxwell wave equation

NSDI Nonsequential double ionization

RABITT Reconstruction of attosecond beating by interference of two-photon transition

SEWA Slowly evolving wave approximation

SFA Strong field approximation

TDSE Time-dependent Schrödinger equation

VUV Vacuum ultraviolet

XUV (EUV) Extreme ultraviolet

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Curriculum Vitæ

Personal Data

Name: Arne HeinrichDate of birth: April 22nd, 1977Place of birth: Einbeck, Germany

Nationality: German

School Education

1983 – 1987 Elementary school Kreiensen1987 – 1989 Orientation school Greene1989 – 1996 Comprehensive secondary school Bad Gandersheim

06/1996 General qualification for university entrance

University Education

10/1996 – 11/2001 Study of physics at the University of Hannover, Germany11/2000 – 11/2001 Diploma thesis at the Institute for Theoretical Quantum Optics

Supervisor: Prof. Dr. Maciej Lewenstein

Subject: Nonsequential double ionization of helium inlow-frequency laser fields

2/2002 – 5/2006 PhD thesis at Institute of Quantum Electronics, ETH Zurich,Switzerland

Supervisor: Prof. Dr. Ursula Keller

Subject: Coherent Control of High Harmonic Genera-tion using Attosecond Pulse Trains

Military Service

3/1998 – 11/1998 Munster, Germany

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Danksagung

Vielen Dank, dass Ihr es bis zur letzten Seite meiner Dissertation geschafft habt.Ich gehe mal davon aus, dass Ihr die vorherigen 109 Seiten auch gelesen habt.Im Besonderen gilt mein Dank:

Ursi, „Ich habe viel von Dir im Allgemeinen und über mich selbst gelernt.“

Jens, „Wir haben die meiste Zeit zusammengearbeitet und Du hast mir gezeigt,wie man eine erfolgreiche wissenschaftliche Karriere macht.“

Prof. Dr. Tilman Esslinger, dass Sie das Korreferat meiner Dissertationübernommen haben und für das Interesse an dem Experiment.

John, „Your supervision helped me till the end of my thesis.”

Christoph, Wouter und Flo, „Ja, Ihr seid die alte Sub-10 und Ihr habt es einfachgemacht!“ Petrissa und Christian, „Ihr werdet mich nicht so schnellloswerden.” Mirko und Florian, „Keine Sorge!” Annalisa, „I had a great timeduring and after work.” Amelle, „You showed me the French way, merci.”

Rosmarie für die ganze Hilfe und die netten Gespräche.

Rachel, „Du hattest immer gute Laune.“ Mathis, „Wir sind viele Kilometergelaufen und haben zu viel Zeit am Verstärker verbracht.“ Lukas, „Wir habenschon ein paar coole Touren gemacht… und ich denke, dass Kolja auch baldanfangen sollte.“ Markus, „Ich habe Dir viel Arbeit gemacht und Du hast mirdafür gezeigt, wie man richtig sichert.“ Thomas, “Wir hatten viel Spaß.“ Gertund Toni, „Die Sauna war so schnell gebaut.“

Ken, Mette and Johan, “I had a great time in Louisiana.” Mitsuko, “You alwayscheered me up.”

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Deran (Ist Oranje diesmal besser?), Anastassia (Nicht zu verwechseln mit IhrerNamensvetterin), Andreas (Der Weg aus Graubünden ist zu weit.), Gabi(Immer gut drauf.), Simon (Schwer zu verstehen.), Vinci Sergio (Immerwillkommen.), Valeria and Chris (You belong together.), Peter (DasVersuchskaninchen.), Andreas (Immer mit der Letzte), Mad Max (Der Name istProgramm.), Adriano (Coole Sau und der schnellste Snowblader ever.), Aude-Reine (Die halbe French Connection.), Dirk und Beni (Beavis and Butthead,yeah dudes!), Matthias (Schnellster Ruderer der Gruppe), Paolo (Der dritteItaliener), Silke (Doch in der Schweiz geblieben.), Rüdiger (Alles Gute!),Konstant… Kostas (Your name is too complicated.), Marcel und Harry (Das„Dynamische Duo“ der Werkstatt.), Birgit (Welcome back!), Felix (Man nehmeeinen Standard Achromaten.), Alex (Der gelassenste Mensch.), Arne(Nervensäge), Steve (Der beste Laufer der Gruppe), Edith (Die High-PowerFrau).

Ich danke meiner lieben Familie, die mich nicht nur während meinerDoktorarbeit, sondern mein ganzes Leben begleitet und unterstützt hat.

Vielen Dank, Arne.

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