Document [original]

Electron correlation in multiple ionization of

atoms and molecules by intense ultra-short laser

pulses

vorgelegt von

Diplom-Physikerin

Ekaterina Eremina

aus Kasan (Russland)

von der Fakult¨at II - Mathematik und Naturwissenschaften

der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktorin der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Mario D¨ahne

Berichter: Prof. Dr. Wolfgang Sandner

Berichter: Prof. Dr. Thomas M¨oller

Tag der wissenschaftlichen Aussprache: 25.04.2005

Berlin 2005

D 83

Abstract

The ionization dynamics of Ar and Ne atoms as well as N2and O2molecules in intense

ultra-short laser pulses has been investigated by means of highly differential correlated

electron-ion momentum spectroscopy. The main goal was to understand the detailed mech-

anism behind Non-Sequential Double Ionization (NSDI) and the influence of the atomic

and molecular structure on the final electron-electron momentum correlation after NSDI.

The experimental results for Ne can be well understood within the instantaneous

rescattering-induced electron impact ionization mechanism, while for Ar the assumption

of an additional ionization channel is necessary. For the latter the likely scenario is electron

impact excitation of the singly charged ion with subsequent electric field ionization. The

role of this mechanism in the NSDI process increases with decreasing light intensity. The

detailed analysis of the final electron momentum distributions for Ar and Ne shows that

the electric field of the light wave essentially influences the recollision process in NSDI.

Significantly different electron-electron momentum correlations are found for N2and

O2. Both molecules show new features which are not observed for atoms. The comparison

of the N2molecule and Ar atom, both having a comparable ionization potential, at similar

light intensities reveals the effect of the two nuclei.

A semiclassical analysis based on the rescattering model has been applied to the N2and

O2molecules in order to test the applicability of the electron impact ionization mechanism.

The model calculation includes the symmetry of the initial molecular state occupied by

the active valence electron. The results of this calculation qualitatively reproduce the main

features found in the experiment. Two important conclusions can be drawn. First, electron

recollision seems to be the general mechanism of NSDI for both atoms and molecules.

Second, the molecular structure decisively influences the final electron-electron momentum

correlation after NSDI.

It has been found that the bonding symmetry of the initial state of the molecule

strongly modifies the inelastic electron recollision. In contrast, the antibonding symmetry

mainly affects the first stage of NSDI during the electron transition to the ionization

continuum via tunneling. This can be interpreted as interference effects of the emitted

electronic wave packet with the bound electron at the instant of recollision as well as in

the final state of the two photoelectrons.

Zusammenfassung

Es wurde die Ionisationsdynamik der Atome Ar und Ne als auch die der Molek¨ule N2

und O2in intensiven ultrakurzen Laserpulsen mittels hochdifferentieller Elektron-Ion-

Koinzidenz-Impulsspektroskopie untersucht. Das Hauptziel der Arbeit ist, den detaillierten

Ionisationsmechanismus der nichtsequentiellen Doppelionisation (NSDI) und den Einfluss

der atomaren und molekularen Strukturen auf die Impulskorrelation der Photoelektronen

nach der NSDI zu verstehen.

Die experimentellen Ergebnisse f¨ur Ne lassen sich gut durch den Mechanismus der

instantanen r¨uckstreuinduzierten Elektronenstoßionisation erkl¨aren, w¨ahrend f¨ur Ar die

Annahme eines zus¨atzlichen Ionisationskanals notwendig ist. Das wahrscheinliche Szenario

f¨ur diesen Mechanismus ist die instantane Elektronenstoßanregung des einfachgeladenen

Ions mit folgender Ionisierung durch das elektrische Feld der Lichtwelle. Mit abnehmender

Lichtintensit¨at wird der Beitrag dieses Ionisationsmechanismus zu der Impulskorrelation

der Photoelektronen gr¨oßer. Die detaillierte Analyse der Elektronimpulsverteilungen f¨ur

Ar und Ne deutet auf einen wesentlichen Einfluss des elektrischen Feldes der Lichtwelle

zum Zeitpunkt des Elektronenstoßprozesses der NSDI hin.

F¨ur N2und O2wurden sehr unterschiedliche Elektron-Elektron-Impulskorrelationen

gefunden. Beide Molek¨ule zeigen neue Charakteristiken, die f¨ur Atome nicht gefunden

wurden. Die Rolle der zwei Kerne zeigt sich in dem Vergleich des N2-Molek¨uls und des

Ar-Atoms, beide mit ¨ahnlichen Ionisiationspotentialen, bei gleicher Lichtintensit¨at.

Um die Relevanz der Elektronenstoßionisation f¨ur die Molek¨ule N2and O2zu testen,

wurde eine semiklassische Methode eingesetzt, die auf dem R¨uckstreumodel basiert. Die

Modellrechnung ber¨ucksichtigt die Symmetrie des mit einem Valenzelektron besetzten

molekularen Anfangszustands. Die Rechnung reproduziert qualitativ die wesentlichen Ergeb-

nisse des Experiments. Daraus lassen sich zwei wichtige Schl¨usse ziehen. Erstens, in-

stantane Elektronenstoßionisation scheint der allgemeine Mechanismus der NSDI zu sein,

sowohl f¨ur Atome als auch f¨ur Molek¨ule. Zweitens, die molekulare Struktur hat entschei-

denden Einfluss auf die Impulskorrelation der Photoelektronen nach der NSDI.

Es wurde herausgefunden, dass die bindende Symmetrie des molekularen Anfangszus-

tandes den inelastischen Elektronenr¨uckstoss stark modifiziert, w¨ahrend die antibindende

Symmetrie haupts¨achlich das Elektronentunneln in der erste Stufe der NSDI beeinflusst.

In beiden F¨allen kann der Vorgang als Interferenzeffekt der beiden beteiligten Elektronen

interpretiert werden: sowohl zum Zeitpunkt des R¨uckstoßes des emittierten Elektronen-

wellenpakets mit dem gebundenen Elektron als auch im Endzustand der beiden Photoelek-

tronen.

Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Ionization in intense laser fields: fundamentals . . . . . . . . . . . . 5

2.1 Multiphoton ionization . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Above threshold ionization . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Tunneling ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Non-sequential multiple ionization . . . . . . . . . . . . . . . . . . . . 11

2.5 Free electron dynamics in the laser field . . . . . . . . . . . . . . . . . 21

3. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Setup modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 The COLTRIMS technique . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 The supersonic atomic and molecular jet . . . . . . . . . . . . . . . . 26

3.4 The Differential Pumping Stages . . . . . . . . . . . . . . . . . . . . 30

3.5 The momentum spectrometer . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Momentum resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 The laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Determination of light intensity . . . . . . . . . . . . . . . . . . . . . 47

4. Non-sequential double ionization of atoms: results and discussion 51

4.1 Electron sum-momentum distributions . . . . . . . . . . . . . . . . . 51

4.2 Electron momentum correlation . . . . . . . . . . . . . . . . . . . . . 54

4.3 Sub-threshold electron impact ionization . . . . . . . . . . . . . . . . 56

4.4 The transverse electron sum-momentum distribution . . . . . . . . . 62

4.5 Comparison with theoretical results . . . . . . . . . . . . . . . . . . . 65

5. Molecules in high-intensity ultra-short laser pulses . . . . . . . . . 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 The electron sum-momentum distributions for N2and O2. . . . . . . 75

ii Contents

5.3 e−- e−momentum correlation for N2and O2. . . . . . . . . . . . . . 77

5.4 Modeling of sequential ionization . . . . . . . . . . . . . . . . . . . . 78

5.5 NSDI model calculation for molecules . . . . . . . . . . . . . . . . . . 82

5.6 Other theoretical models and their relevance in experiments . . . . . 90

6. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A. C++ code for the data analysis . . . . . . . . . . . . . . . . . . . . . 99

B. C++ code for NSDI model calculation . . . . . . . . . . . . . . . . . 109

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

List of Figures

2.1 Schematic diagram of ionization mechanisms . . . . . . . . . . . . . . 6

2.2 ATI electron energy spectra of Xe [1] . . . . . . . . . . . . . . . . . . 7

2.3 Illustration of the Stark shift of the ionization potential . . . . . . . . 8

2.4 Schematic diagram of strong-field photoionization mechanisms . . . . 10

2.5 Ion yield dependence on light intensity for single, double and triple

ionization of Ar [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Scheme of the rescattering ionization mechanism . . . . . . . . . . . . 14

2.7 Rescattering: the electron motion and its kinetic energy . . . . . . . . 22

3.1 Schematic view of the gas-jet chamber and the differential pumping

stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Ion yield dependence on the gas source pressure for Ar+and Ne+. . 30

3.3 Schematic view of the momentum spectrometer . . . . . . . . . . . . 32

3.4 Operation principle of the delay-line anode . . . . . . . . . . . . . . . 33

3.5 The ion TOF spectrum of Ar at a light intensity of 1.5×1014 W/cm235

3.6 Two-dimensional position images for ions and electrons . . . . . . . . 38

3.7 Typical electron trajectories in the flight tube . . . . . . . . . . . . . 39

3.8 The dependence of the electron cyclotron deflection on the TOF . . . 40

3.9 The momentum conservation in single ionization of Ar . . . . . . . . 41

3.10 The scalar potential of the extracting electric field of the spectrometer 42

3.11 The dependence of the momentum on the TOF of ions and electrons 43

3.12 The dependence of the momentum resolution on the electron momentum 44

3.13 The dependence of the ion momentum resolution on the position of

the ion on the detector . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.14 Diagram of the Ti:Sa laser system used in the experiment . . . . . . . 46

3.15 Non-collinear autocorrelation traces . . . . . . . . . . . . . . . . . . . 47

3.16 Kr+ion yield as a function of light intensity . . . . . . . . . . . . . . 48

3.17 ATI photoelectron kinetic energy distribution for Ne at a light inten-

sity of 3 ×1014 W/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 The momentum distributions of Ar2+ and Ne2+ . . . . . . . . . . . . 53

4.2 The momentum correlation of the two photoelectrons after double

ionization of Ar at 2.4×1014 W/cm2and at 1.5×1014 W/cm2. . . . 55

iv List of Figures

4.3 Projections of the e−-e−momentum correlation for Ar . . . . . . . . 56

4.4 The momentum distributions of Ar2+ at 1.1×1014 W/cm2and at

0.9×1014 W/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 The momentum correlation between the two photoelectrons after dou-

ble ionization of Ar at 1.5×1014 W/cm2and 0.9×1014 W/cm2. . . 58

4.6 The dependence of the cutoff momentum of the Ar2+ and Ne2+ mo-

mentum distributions on the light intensity . . . . . . . . . . . . . . . 59

4.7 Schematic diagram of the ionization threshold at the presence of the

electric field of the light wave . . . . . . . . . . . . . . . . . . . . . . 61

4.8 The kinetic energy of the returning electron and the instantaneous

ionization potential at the instant of recollision . . . . . . . . . . . . . 61

4.9 The dependence of the FWHM of the doubly charged ion momentum

distribution f(p⊥) on the peak electric field strength of the light wave 63

4.10 Calculated e−-e−momentum correlation for Ar at 1.5×1014 W/cm2. 67

4.11 Calculated e−-e−momentum correlation for Ar at 0.9×1014 W/cm2. 67

4.12 e−-e−momentum correlation calculated in the velocity and length

gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Electron sum-momentum distributions after double ionization for N2

and O2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 The dependence of the cutoff momentum of the N2and O2momentum

distributions on the light intensity . . . . . . . . . . . . . . . . . . . . 77

5.3 e−-e−momentum correlation for N2and O2. . . . . . . . . . . . . . 79

5.4 Calculated electron sum-momentum distribution for sequential dou-

ble ionization of O2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.5 Calculated e−-e−momentum correlation for N2at 1.5×1014 W/cm2

and for O2at 1.7×1014 W/cm2. . . . . . . . . . . . . . . . . . . . . 84

5.6 e−-e−momentum correlation calculated with g1(~

k) = 1 in Eq. (5.7) . 85

5.7 Calculated electron sum-momentum distributions of the N2and O2

model molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.8 Calculated e−-e−momentum correlation for N2at 1.8×1014 W/cm2

and at 2.5×1014 W/cm2, and for O2at 1.9×1014 W/cm2and at

2.2×1014 W/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.9 Calculated e−-e−momentum correlation for the fixed alignment of

molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.10 Relative contribution to the e−-e−momentum correlation at different

molecular orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

List of Tables

3.1 Characteristics of the supersonic atomic/molecular jet in the experiment 27

3.2 Hagena parameters for different target gases in the experiment . . . . 29

vi List of Tables

List of Abbreviations

ADK Amosov-Delone-Krainov

ATI Above Threshold Ionization

COLTRIMS COLd Target Recoil Ion Momentum Spectroscopy

CPA Chirped Pulse Amplification

e−-e−electron-electron

FWHM Full Width at Half Maximum

HHG High Harmonics Generation

KFR Keldysh-Faisal-Reiss

LCAO Linear Combination of Atomic Orbitals

LOPT Lowest-Order Perturbation Theory

MCP Microchannel Plate

MPI Multiphoton Ionization

NSDI Non-Sequential Double Ionization

OTBI Over-The-Barrier Ionization

SAE Single Active Electron

TDC Time-to-Digital Converter

TDSE Time-Dependent Schr¨odinger Equation

TOF Time Of Flight

UHV Ultra-High Vacuum

XUV eXtreme Ultra-Violet

1. Introduction

The discovery of lasers at the beginning of the sixties opened literally a new era for

researchers. Impetuous development of laser technologies has made the laser a part

of our everyday life today – from CD-ROM, laser pointer and supermarket checkout

lines (laser-based ICT ) to laser vision correction (laser medicine) and cutting of

precise patterns in glass or metal (laser-based materials processing and manufactur-

ing). Indeed lasers found very broad applications in industry, biology and medicine.

They are, however, not less important research tools. Today it is impossible to

imagine spectroscopy without lasers. In recent years more and more powerful lasers

with extremely short pulses became available. This opened a new research field of

light-matter interaction. In particular, this thesis addresses interaction of atoms and

molecules with very intense ultra-short laser pulses.

A great progress in the development of the laser techniques since the sixties of the

last century has been achieved. It started from Q-switching lasers, which delivered

few-nanosecond pulses with a laser peak power of megawatt, and has been devel-

oped through the mode-locking to the chirped pulse amplification technique

(CPA), which allows to reach a peak laser pulse intensity up to 1020W/cm2in the

femtosecond pulse regime. In the CPA scheme laser pulses are stretched, amplified,

and then compressed [3]. Due to the available high light intensity new nonlinear

nonperturbative phenomena were discovered, such as multiphoton ionization, coher-

ent emission of high frequency radiation by atoms, and laser-assisted electron-atom

collisions.

The atomic physics in intense laser fields perhaps dates back to the first experi-

mental observation of seven-photon ionization of Xenon with a ruby laser by Voronov

and Delone in 1965 [4]. Multiphoton ionization (MPI) was predicted already in

1931 by Maria G¨oppert-Mayer, a student of Max Born, who showed that ionization

of an atom can happen by absorbing many photons if the energy of one photon is not

enough to overcome the ionization potential [5]. The next remarkable phenomenon,

which has been called above threshold ionization (ATI), was discovered in Saclay

by Agostini and collaborators in 1979. They found that, at a sufficient light inten-

sity (of about 1013W/cm2), a photoelectron is able to absorb more photons than the

minimum required for MPI [6]. Another effect related to ATI is high-order har-

monics generation (HHG), where an atom responds nonlinearly to a strong laser

field emitting radiation with odd harmonics of the laser field frequency (for a recent

2 1. Introduction

review see [7, 8]). This phenomenon became very important for developing compact

(top-table) powerful high-frequency X-ray and XUV laser sources with attosecond

pulse width.

In a low-frequency strong laser field, when the electric field of a light wave com-

petes with the Coulomb field in an atom, one can consider the ionization process as

electric field ionization, i.e. tunnelling through an effective potential barrier, which is

formed by suppression of the Coulomb potential in the external electric field. Here,

the necessary condition is that the time the e−needs to tunnel through the potential

barrier is small compared to the period of the laser field. This theoretical approach

was first considered by Perelomov, Popov, and Terent’ev [9] and further developed

by Amosove, Delone and Krainov [10]. First experiments on single ionization of no-

ble gases and helium by high-intensity laser pulses showed a good agreement with

the tunnelling theory. But for multiple ionization, when more than one electron

is ionized, the experimental results showed a strong deviation from the tunnelling

approach. In particular, the experimentally observed ionization rate of doubly and

multiply charged ions was found to be many orders of magnitude larger than it was

predicted theoretically [11, 12, 13, 14]. The highly nonlinear process in a strong laser

field, which is responsible for this deviation, is now known to be non-sequential

ionization. In the last years the study of non-sequential double ionization

(NSDI) has attracted much interest of theoreticians and experimentalists. Highly

differential methods like electron and ion momentum spectroscopy and kinetic en-

ergy spectroscopy allow one to analyze the dynamics of the ionization processes (e.g.

[15, 16, 17, 18, 19, 2]). The correlation between the two photoelectrons was found to

play a decisive role in the mechanism of NSDI. Actually electron correlation effects

are important not only in atomic physics, but also for phenomena in condensed

matter physics such as high temperature superconductivity and magnetism.

For molecules, the application of intense laser fields offers a possibility to manip-

ulate inter- and intramolecular dynamics as well as orient, focus molecular beams,

control chemical reactions. In addition, intense lasers may be employed as ”soft”

ionizers for sophisticated mass spectrometry of large molecules. The development of

scientific and practical applications of the intense laser fields to molecules requires a

complete comprehension of molecular behavior in strong laser fields. One of the ba-

sic problems is the understanding of the ionization dynamics of molecules in strong

fields.

The present thesis, supervised by Prof. W. Sandner, has been performed at the

Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy in Berlin.

It is dedicated to a study of non-sequential double ionization of different atomic

and molecular systems and of the electron correlation in the ionization process. An

experimental technique based on ion and coincident ion-electron momentum spec-

troscopy, which is also known as Cold Target Recoil Ion Momentum Spec-

troscopy (COLTRIMS), was employed for this goal.

The following questions have been addressed within this work:

•detailed ionization mechanism behind NSDI;

•atom specific features of the final electron momentum correlation after NSDI;

•NSDI via instantaneous electron impact ionization under conditions where it

is energetically forbidden within the semiclassical rescattering model (below

ionization threshold);

•NSDI of molecules, comparison with atoms having similar ionization poten-

tials;

•influence of molecule specific effects on the final momentum correlation.

The present thesis is organized as follows. An overview of the main multiphoton

processes is given in Chapter 2, including the discussion of different theoretical

models developed for NSDI. In Sec. 2.5 the dynamics of a free electron in a laser

field is considered. The latter is an important part of the NSDI process.

The experimental setup used for the present investigation is described in detail

in Chapter 3. It includes the momentum spectrometer together with the electronics

for the data acquisition and the laser system. The momentum resolution of the

spectrometer and the determination of the light intensity of the focused laser beam

are also discussed in this chapter.

Chapter 4 presents the actual experimental results on NSDI of atoms. The ob-

served electron sum-momentum distributions for Ar and Ne as well as their electron

momentum correlation after NSDI will be discussed. An explanation is given for

NSDI via electron impact ionization found at light intensities where it is not ex-

pected within the semiclassical rescattering model. A qualitative comparison of the

experimental data with a semiclassical model calculation based on rescattering will

be given as well. Finally, several theoretical models and their relevance for the ex-

periment are considered.

Chapter 5 is dedicated to NSDI of molecules. N2and O2molecular systems are

considered. Electron sum-momentum distributions and the final electron momentum

correlations are presented and discussed. In this chapter a semiclassical model is de-

veloped for NSDI of molecules. The model allows one to reproduce the main features

found in the experiment as well as to understand different ionization behavior of N2

and O2. It is based on the initial-state symmetry of the molecular orbital occupied

by the active valence electrons.

4 1. Introduction

2. Ionization in intense laser fields:

fundamentals

The present chapter focuses on fundamental aspects of the ionization dynamics in

intense laser fields and gives an overview of the accumulated knowledge on this topic.

Special attention is given to the matter of non-sequential ionization, including the

rescattering ionization mechanism and various theoretical approaches. The presented

material is partly guided by several reviews devoted to multiphoton processes in

intense laser fields [20, 21, 22, 23].

2.1 Multiphoton ionization

The transition from a bound state to a continuum state by absorbing many photons

is known as multiphoton ionization (MPI). It occurs if the energy of one photon is

not enough for ionization (see Fig. 2.1 (b)). One can write down schematically the

reaction of multiphoton one-electron ionization as follows

n~ω+A→A++e−,(2.1)

where ~ωis the photon energy and nis an integer. As mentioned before, this process

was first observed by G.S. Delone and N.B. Voronov in 1965 [4], who detected

seven-photon ionization of xenon using a ruby laser. In the same year J.L. Hall

and collaborators recorded two-photon ionization of the negative Iodine ion I−[24].

Early experiments on MPI at relatively low light intensity (<1013 W/cm2) were in

good agreement with lowest-order perturbation theory (LOPT) [25, 26] where the

n-photon ionization rate is given by

ωn=σnIn.(2.2)

Here, nis the minimum number of photons needed for ionization, σnthe n-photon

ionization cross section and Ithe light intensity of the laser beam. This highly

nonlinear dependence on light intensity was proven experimentally up to n= 22 for

MPI of He [27]. However this intensity scaling breaks down at a critical intensity.

Above this critical value there is no change in the intensity dependence of the ion

yield [28]. This can be explained in terms of population depletion [29]. In general,

6 2. Ionization in intense laser fields: fundamentals

in an experiment we deal with a pulsed focused laser beam with an inhomogeneous

intensity distribution in the ionization volume and in time. For a given pulse duration

all atoms in the focus will be ionized if the light intensity is larger than the so-called

saturation intensity. The latter can be lower than the peak intensity of the laser

pulse. In order to obtain ionization up to the peak intensity of the laser pulse we

require shorter pulses.

2.2 Above threshold ionization

At light intensities >1013 W/cm2the low-order perturbation theory is no longer

valid because of the strong coupling of the atomic states with the pulsed laser field,

which leads to the AC-Stark shift. In this regime of light intensity an electron can

absorb more photons than the minimum needed to overcome the ionization bar-

rier. This process is known as above threshold ionization (ATI) and is depicted

schematically in Fig. 2.1 (c). Experimentally ATI was first observed by P. Agostini

Fig. 2.1: Schematic diagram of ionization mechanisms: a) one-photon ionization; b) mul-

tiphoton ionization by nphotons; c) above-threshold ionization by (n+s) pho-

tons.

et al. (1979) [6]. In the photoelectron energy spectrum of six-photon ionization of

Xe at 1013 W/cm2they found a second energy peak separated from the expected

first one by the photon energy. An example of ATI spectra of Xe is shown in Fig.

2.2. After this discovery energy-resolved photoelectron spectra have been studied

in detail for different atoms and molecules and with lasers at different wavelengths

[25, 30, 1, 31, 32]. Despite its obstructions perturbation theory has been applied to

ATI [33] and has been shown to be sufficient to account for experimental results in

a certain intensity range [25]. The ionization rate then has a more general form than

2.2. Above threshold ionization 7

Eq. (2.2)

ωn+s∝In+s,(2.3)

where sis the number of excess photons absorbed. The photoelectron energy can

be calculated from the extended Einstein photoeffect formula

E= (n+s)~ω−Ip(2.4)

with Ipthe ionization potential.

Fig. 2.2: ATI electron energy spectra of Xe at λ= 1064 nm with 130 ps laser pulses [1].

(a) I= 2.2×1012 W/cm2; (b) I= 1.1×1013 W/cm2.

Another remarkable feature of ATI is the suppression of the low-energy peaks

in the photoelectron spectra. This effect occurs at increasing laser intensity (see the

first two peaks marked with dashed red arrows in Fig. 2.2). The reason for this

suppression is the AC-Stark shift of the energies of atomic states in the presence of

8 2. Ionization in intense laser fields: fundamentals

the external field. For low laser frequencies the AC-Stark shifts of the lowest bound

states are not significant and can be neglected (e.g. a Nd-YAG laser with ~ω= 1.17

eV). In contrast, the shift of Rydberg and continuum states is characterized by

the electron ponderomotive energy Up. The latter is an important parameter

for many processes in intense laser fields. Upis defined as the kinetic energy of the

electron quiver motion in a laser field averaged over an optical cycle and is given by

Up=e2E2

4mω2,(2.5)

where mis the mass and ethe charge of an electron, E0is the electric field strength

and ωthe frequency of the laser. The ionization barrier is boosted by Upin the laser

with field

I1I2

Energy

-Ip

withoutfield

Intensity

with field

I1I2

Energy

-Ip

withoutfield

Intensity

Fig. 2.3: Illustration of the Stark shift of the ionization potential depending on the

laser intensity. At the intensity I1five-photon ionization occurs, whereas at

the higher light intensity I2one photon more is needed for ionization. The

ionization threshold is increased by Up.

field (Fig. 2.3) and the final photoelectron energy is given by

E= (n+s)~ω−(Ip+Up).(2.6)

Fig. 2.3 illustrates the AC-Stark shift of the ionization threshold. One can see that

ionization by absorption of n= 5 photons, which is possible at the laser intensity

2.3. Tunneling ionization 9

I1, is energetically forbidden at I2. Here, one photon more is needed to ionize the

atom. However, in experiment we deal with a smoothly varying intensity in the

focus during the laser pulse, so that the corresponding energy peak at I1will not

completely disappear. Fig. 2.2 demonstrates well this evolution of the main energy

peak (marked with the small red arrow) with increasing laser intensity.

Despite the suppression at low energy, the positions of the ATI peaks in Fig. 2.2

do not change with increasing light intensity. The reason is the following. Due to

the intensity gradient of a focused laser beam, a freed electron experiences a force

−∇Up. Accelerated by this force in the laser field, the electron regains exactly its

ponderomotive energy deficit, which appears due to the ionization potential shift by

Up[34]. This is possible for long laser pulses (1 ps) when the laser intensity changes

slowly compared with the time, the photoelectron needs to leave the focal spot. As

a consequence, the photoelectron peaks appear at the same energy as predicted in

Eq. (2.4). For sub-picosecond laser pulses (<1 ps), however, there is no time for

the photoelectron to acquire the full amount of energy during the ponderomotive

acceleration in the laser field before the end of the pulse. Therefore, the ATI peaks

become shifted towards lower energies. Such a shift as well as a broadening of the ATI

energy peaks with decreasing laser pulse width has been observed experimentally

[31]. Additionally, fine structures appear in the photoelectron energy spectrum when

sub-picosecond laser pulses are used [35, 36, 37]. The origin of these structures has

been attributed to resonant MPI processes. The latter occur for excited states when

ac-Stark shifted into resonance during the laser pulse[37]. In the long-pulse regime

the substructures in the ATI electron energy spectrum cannot be resolved due to

the ponderomotive acceleration of the photoelectron in the intensity gradient of the

focused laser beam.

2.3 Tunneling ionization

Using laser techniques, based on the Chirped Pulse Amplification (CPA) scheme

[3], it has become possible to produce very intense ultrashort light pulses with a

light intensity of the focused laser beam in excess of 1014 −1015 W/cm2. Such a

light intensity corresponds to an electrical field strength of about 108−109V/cm.

This is already comparable with the strength of the Coulomb field in an atom and

is considered as a strong field. In this case perturbation theory is no more valid.

A more useful approach is obtained by considering the strong-field photoionization

process as tunneling through an effective potential barrier. The latter is formed

by the suppression of the atomic potential in the laser electric field as shown in

Fig. 2.4. Such an approach is valid if the oscillation period of the external field

is long enough in comparison with the time the electron needs to tunnel through

the potential barrier. This quasi-stationary theory was first worked out by Keldysh

10 2. Ionization in intense laser fields: fundamentals

a) b)

-Ip-Ip

e-e-

V(z) V(z)

z z

laser pulse

a) b)

-Ip-Ip

e-e-

V(z) V(z)

z z

laser pulse

Fig. 2.4: Schematic diagram of strong-field photoionization mechanisms: a) tunneling;

b) over-the-barrier ionization.

(1965) [38] and then developed further by many theoreticians [10, 9, 39, 40, 41, 42].

Using a quasi-stationary model in the low-frequency limit first Perelomov, Popov,

and Terent’ev [9] and then Amosov, Delone and Krainov [10] have found a tunneling

ionization rate (ADK rate) which is given by

ω=ω0exp[−2(2Ip)3/2

3E].(2.7)

Here, Ipis the ionization potential, Eis the laser field strength and ω0is a slowly

varying function of E,Ipand Z, the atomic core charge. The atomic unit system

(}=me=e= 1) is used.

As the laser intensity is further increased a critical value of the intensity is even-

tually reached. Beyond this value the Coulomb potential is so strongly suppressed

by the external field that the ground state is no longer bound. An electron then just

”flows over the top” of the barrier as shown in Fig. 2.4 (b). This process is known as

over-the-barrier ionization (OTBI) and occurs at 1.4×1014 W/cm2for atomic

hydrogen. The critical intensity for OTBI is defined by [21]

IcOT BI =c0I4

32Z2,(2.8)

with cthe speed of light and ε0the dielectric constant of vacuum.

As a quantity indicating the transition from multiphoton ionization to the low-

frequency tunneling limit Keldysh has introduced the adiabaticity parameter γ.

2.4. Non-sequential multiple ionization 11

The latter can be defined as the ratio of the mean time the electron needs to tunnel

through the potential barrier [38] to the period of the laser field oscillation, which

can be expressed as

γ=ωp2mIp

eE =sIp

2Up

,(2.9)

where Ipis the field free atomic ionization potential and Upis the ponderomotive

energy. For γ > 1, multiphoton absorption is adequate to describe the ionization

process. In contrast, for γ < 1 the quasistatic tunneling ionization is most appropri-

ate. The transition between the two approaches, multiphoton and tunneling, takes

place at γ= 1. It occurs over a rather narrow range of light intensity or frequency. In

the intermediate regime with γ∼1 the ionization process has a complicated char-

acter and is difficult to model. In fact multiphoton ionization and tunneling are two

limiting cases of one ionization process. Moreover, the formula (2.7) was obtained

with a short-range atomic potential, while for real atoms it is rather long-range. Ac-

tually, for real atoms the laser field effect on a freed photoelectron is accompanied

by the Coulomb force of the ionic core. An analytical solution of such a complex

problem is highly complicated [43]. Experiments in optical, infrared and ultraviolet

wavelength ranges have confirmed qualitatively the exponential dependency of the

ionization rate found in (2.7) at γ < 1 and also at γ∼1 [44, 45, 12, 46].

From Eqs. (2.9) and (2.5) follows that the Keldysh parameter depends on the

frequency and is inversely proportional to the square root of intensity of the field. In

contrast to the quite high laser intensities needed to obtain tunneling ionization in

the optical region (e.g. >1014 W/cm2for an 800 nm Ti:Sa laser), microwave tunnel-

ing ionization of highly excited Rydberg atoms has been obtained in considerably

weaker fields [47, 48].

2.4 Non-sequential multiple ionization

The formation of doubly charged ions in multiphoton ionization was first observed

for the alkaline-earth atoms Ba and Sr with two valence electrons on the outer shell

[49]. Later it was also found for the rare-gas atom Kr ionized by Nd:YAG laser pulses

[50]. The light intensity necessary to obtain multielectron ionization via absorption

of many photons is about 1013 W/cm2and higher.

At the time of these experiments, theoretical methods allowed one to calculate

the total ionization rate for multi-electron ionization in intense laser fields based on

the so called single-active-electron approximation (SAE) [10, 41, 42, 51, 52].

In this approximation, correlations between electrons in a many-electron atom are

included only via the field-free initial state wave function. The ionization dynamics is

only governed by the outermost electron while the other electrons are ”frozen”. The

latter only contribute to the effective atomic potential (Hartree-Fock potential). In

12 2. Ionization in intense laser fields: fundamentals

this way the SAE approximation assumes that multi-electron ionization is a stepwise

process (sequential ionization). For instance double ionization can be written as a

two-step process

A+n1~ω−→ A++e−,(2.10)

A++n2~ω−→ A++ +e−,(2.11)

where n1and n2are the numbers of absorbed photons. With the new generation

Fig. 2.5: Ion yield dependence on light intensity for single (), double (•) and triple

(N) ionization of Ar using Ti:Sapphire laser radiation (800 nm, 80 fs) from [2] 1.

The two black curves are the ion yields calculated using ADK tunneling rates

for single and double (sequential) ionization. The red curve is for non-sequential

double ionization.

of high-power, stable laser systems with ultra-short pulses (of the order of fs) and

high repetition rates it became possible to measure total ion yields for higher charge

states for all rare-gas atoms and also for He with high precision. These SAE-based

1see also the web site http://gomez.physics.lsa.umich.edu/nsdidetail.html.

2.4. Non-sequential multiple ionization 13

approaches to double and multiple ionization were found to be no more applicable

for the description of such experiments. Although the experimentally found depen-

dence of the total ionization rate on light intensity for singly charged ions was in

perfect agreement with sequential ADK model, the results for double ionization in

a certain intensity range showed an ion yield which is many orders of magnitude

higher than theoretically predicted [12]. Only at high laser intensity, where single

ionization is saturated, double ionization becomes sequential (see Fig. 2.5 for Ar).

There is a characteristic ”knee” structure in the ion yield curves for double and

triple ionization between the two intensity ranges. At this point sequential ioniza-

tion starts to dominate over non-sequential ionization as the intensity is further

increased (Fig. 2.5). This failure of the theory to predict the experiment correctly,

suggests that two electrons are removed from a neutral atom simultaneously rather

than in a sequential way

A+n~ω−→ A++ +e−+e−(2.12)

This ionization process is called Non-Sequential Double Ionization (NSDI). It

was observed first for Xe [11], then for He [12, 14, 53] and other rare-gas atoms

[54, 55, 56], and meanwhile also for some molecules [57, 58, 59]. The observation of

NSDI is a strong evidence of electron correlation in strong field physics.

In order to explain NSDI several mechanisms were proposed.

•”Shake-off”

Fittinghoff et al. [12] proposed a ”shake-off” mechanism for non-sequential

ionization. This mechanism is known to be responsible for double ionization

by absorption of a single high-energy photon (~ωIp) [60]. Here, an electron

is removed during ionization in the laser field so fast that the remaining elec-

trons cannot adjust adiabatically to the new eigenstates. Thus some of them

may become excited to a higher-energy state or directly shaken off into the

ionization continuum.

•Rescattering

Rescattering (also referred to as the ”simple-man model”) was proposed orig-

inally by Kuchiev [61] under the name ”antenna model”. He assumed that an

ionized electron is driven in the laser field acting as an antenna. The latter

absorbs the energy which is then shared with a bound electron via correlation.

This idea was extended by Schafer et al. [62] and Corkum [63] to a three-step

rescattering model as presented in Fig. 2.6.

In the first (bound-free) step, an electron is liberated from its parent atom by

tunneling or via over-the-barrier ionization. The probability for these processes

is maximum near the extrema of the oscillating electric field in the laser pulse.

14 2. Ionization in intense laser fields: fundamentals

Fig. 2.6: Rescattering ionization mechanism: after tunneling through the effective barrier

(1.), electron e1moves in the electric field of the light wave during one optical

cycle (2.), until it is driven back to the parent ion core where it can ionize a

bound electron e2by an inelastic collision (3.).

In the second (free-free) step, the free electron is accelerated by the electric

field of the laser away from the remaining ionic core. As the phase of the field

reverses the electron can be driven back to its parent ion core with some ad-

ditional energy acquired from the field.

When the electron returns to the core (most effectively in linearly polarized

light) a third step takes place: an inelastic scattering of the energetic electron

by the core leads to ionization of a second electron.

•Collective Tunneling

A further mechanism for NSDI was suggested by Eichmann et al. [64]. Here,

two electrons tunnel together through the Coulomb barrier which is suppressed

in the laser field. From an analytical model calculation and numerical solution

2.4. Non-sequential multiple ionization 15

of a 1D Schr¨odinger equation it has been concluded that collective two-electron

tunneling ionization does exist in a strong electric field if both electrons stay

at equal distance from the nucleus. However, the total ionization rate for two-

electron collective tunneling was found to be too small to account for the

observed NSDI in a strong laser field [64]. Rather, collective tunneling can

become a dominant ionization mechanism in a static (or quasi-static) field

with a very high field strength or for half-cycle pulses, when there is no time

for rescattering.

First guess-work about the mechanism behind NSDI came from experiments where

the dependence of the total yield of doubly charged ions on the polarization of

the laser beam was studied [54, 65]. It was found [54] that NSDI is suppressed in

elliptically polarized light and the ”knee” structure completely vanishes at circular

polarization. This effect can be understood within the rescattering model while the

”shake-off” and collective tunneling fail to explain it. For Ar, the experimentally

observed dependence of NSDI on wavelength [13] also suggests rescattering as the

main ionization mechanism. Furthermore the semiclassical rescattering model was

successfully used to explain some effects related to NSDI, such as high harmonics

generation (HHG) [62, 63] and the existence of a ”plateau” in ATI photoelectron

energy spectra [32, 66]. Elastic scattering of the active electron on its parent ion core

contributes to the emission of high-energy ATI photoelectrons while recombination

of the electron with the ion results in the release of its kinetic energy plus the

ionization energy in the form of short-wavelength photon emission (HHG).

At first sight the semiclassical rescattering model predicts the appearance of an

NSDI threshold at a certain critical laser intensity. For intensities lower than this

value the returning electron can no longer get enough energy from the laser field to

ionize the second bound electron directly. Thus one would expect an abrupt change

in the intensity dependence of the ratio of double to single ionization rates. However,

no such threshold was found experimentally [14, 67].

The validity of any of the ionization mechanisms cannot be revealed based on

the integral measurements of the total ion yields only. These measurements are in-

tegrated over all final kinetic energies, yielding just the number of ions or electrons,

and thus do not give any idea of the dynamics of the ionization process and the cor-

relation between photoelectrons. In order to obtain more insight into the underlying

physics, differential methods are necessary, for example, photoelectron spectroscopy

or correlated measurements of the energy and angular distribution of the photo-

electrons. In differential measurements the energy or momentum of every electron

(ion) can be detected. Unfortunately, the first experimental results on electron spec-

troscopy based on electron-electron and electron-ion coincidence measurements for

Xe [53] have not been sufficient to reveal the mechanism of non-sequential ionization.

Considerable progress in the study of multiple ionization in high-intensity laser

16 2. Ionization in intense laser fields: fundamentals

fields by differential methods was made, when the momentum imaging technique was

applied to the problem. The momentum distributions of doubly charged He and Ne

ions were measured with so-called Cold Target Recoil Ion Momentum Spectroscopy

(COLTRIMS) [16, 17]. These experiments were soon extended to a kinematically

complete momentum analysis of the final state of the photoelectrons after ionization

[68, 69, 70, 71, 72]. The pioneering experimental works have been done by the group

of R. D¨orner (Frankfurt University) in collaboration with Marburg University and

the group of J. Ullrich (MPI, Heidelberg) in collaboration with our group (MBI, W.

Sandner). Based on these measurements and also on highly resolved electron kinetic

energy distributions measured in coincidence with doubly charged ions [18, 19, 2, 73],

the mechanism of non-sequential ionization of atoms in high-intensity laser fields

was identified as rescattering. However, the question of an ionization threshold for

the rescattering mechanism is still open. Moreover, the ionization rates found in

experiment cannot be reproduced quantitatively. The theoretical treatment of the

problem of atomic ionization in high-intensity laser fields is quite complicated. The

laser field is too strong to allow perturbative methods to be applied. Furthermore,

one has to deal with a quantum many-body atomic system and a nonseparable

Coulomb correlation between electrons. There are several theoretical approaches

which have been used to interpret the experimental results on NSDI of atoms in

high-intensity laser field. Most of them are applied to the He atom as the simplest

correlated two-electron system.

•Numerical Integration of the TDSE

One ab initio approach is the attempt to solve the time-dependent Schr¨odinger

equation (TDSE) for a few-electron system directly. For Nelectrons this in-

volves a partial differential equation with 3Nspatial variables to be solved

over a realistically large space-time domain. For He, the assumption of an infi-

nite nuclear mass and of a linearly polarized light wave efficiently reduces the

problem to that of a five-dimensional time-dependent partial differential equa-

tion. A remarkable progress on numerical integration of the five-dimensional

Schr¨odinger equation for He has been achieved in recent years using massively-

parallel computing (Cray T3D) [74, 75, 76, 77, 78]. However, in these calcu-

lations the laser wavelength was limited to 390 nm. That is shorter than the

typical wavelength used in experiments (800 nm). Just recently a numerical

TDSE calculation for He at 780 nm has been performed [79]. As a result, a

time delay between single and double ionization has been found which sup-

ports the idea of rescattering for NSDI of He. Unfortunately, till now there was

no success, using this theoretical method, to obtain the final state electron mo-

mentum correlation or energy distribution. This would be highly desirable for

comparison with experimental data.

•Intense-field Many-body S-matrix Theory (IMST)

2.4. Non-sequential multiple ionization 17

In this systematic approach the quantum mechanical S-matrix theory is ap-

plied to the NSDI problem. The main Feynmann diagram of this approach can

be interpreted in terms of the rescattering picture. The full transition matrix

amplitude (S-matrix) is defined by three essential contributions:

–transition of an electron from a bound state to a so-called Volkov state in

the continuum (wave function of a free e−interacting with a plane-wave

laser field);

–propagation of the ionized electron in the laser field with a vector poten-

tial ~

A(t), which is described by the so-called Volkov propagator (time-

evolution operator); here, the strong field approximation (SFA) is used,

where the Coulomb interaction of the propagating electron with the re-

maining ion core and with the bound electrons is neglected as well as the

influence of the laser field on the bound electrons;

–electron-electron interaction, which leads to the ionization of a second

electron

Two different forms for the latter and crucial contribution to the S-matrix

element have been considered. The first form includes the e−-e−Coulomb in-

teraction and ignores the interaction with the ion [80, 81, 82, 83, 84]. In the

other form e−-e−contact interaction localized at the position of the ion core

is assumed [85, 86, 87]. In the earlier calculations any interaction between the

two photoelectrons has been neglected in the final state after they reached the

continuum.

The S-matrix calculations by A. Becker and Faisal have so far provided the

closest agreement with experimentally observed double ionization rates [80,

88, 89].

A good agreement was found between S-matrix calculations involving the

rescattering (correlated energy-sharing) picture [82] and experimental mea-

surements of the photoelectron energy distribution for NSDI of He [19]. There

was also agreement that ”shake-off” can be ruled out as the main NS ioniza-

tion mechanism for atoms in low-frequency laser fields.

The quantum-mechanical S-matrix analysis qualitatively reproduces the ex-

perimental results for the recoil-ion momentum distribution and for the cor-

related electron momentum distribution found for NSDI of Ne [85, 86, 87]. In

contrast, for other rare-gas targets like He and Ar the theoretical calculations

[81, 82, 84, 85] are not sufficient to explain the experimental data. Thus, a

consideration of additional ionization mechanisms is required. One such mech-

anism will be discussed later in this section.

The comparison between the model implying an e−-e−Coulomb interaction at

18 2. Ionization in intense laser fields: fundamentals

the instant of electron rescattering and the other model using a contact inter-

action reveals that the latter matches the experimental data better [86, 90, 91].

Despite the generally successful application of the IMST to interpret particular

experimental results, there are some controversies concerning several points.

In particular, the importance of the Coulomb repulsion between the two outgo-

ing photoelectrons in the final state is still unclear. Some calculations indicate

that this interaction is essential to explain the recently observed ”back-to-

back” emission of two photoelectrons in the plane perpendicular to the electric

field of the linearly polarized light wave [92]. In other NSDI calculations the

effect of the e−-e−Coulomb repulsion in the final state becomes visible in the

e−-e−momentum correlation spectrum if the transverse momentum (perpen-

dicular to the light polarization) of one of the electrons is restricted to a small

value [87, 91]. In contrast, for a nonrestricted transverse momentum p1e,⊥, tak-

ing into account the final state e−-e−repulsion results in an e−-e−momentum

correlation which does not agree with experiment.

Another point is the use of different gauges in the S-matrix treatment of the

NSDI problem: i. the length gauge [86, 90, 91], and ii. the velocity gauge

[81, 82, 92]. Some approximations in the S-matrix calculation obviously break

gauge invariance since the results are quite different for the two gauges.

The crucial point, however, is that the above mentioned S-matrix approaches

still miss the inclusion of the interaction of the returning electron in the inter-

mediate state as well as of the two final state electrons with the ion core.

•Semiclassical and Classical Approaches

Different semiclassical approaches have been developed on the basis of the

simple rescattering model (e.g. [67, 93, 94, 95, 96]).

In these approaches the ionization probability of the first electron by tunneling

is determined by the ADK formula (2.7). The further evolution of the freed

electron in the electric field of the laser is described using classical mechanics.

The only free parameter in the rescattering model by Corkum is the impact

parameter. This is determined by the spread of the electron wave packet trans-

verse to the electric field of the light wave.

The original rescattering picture [62, 63] neglects several important physical

effects and thus, cannot predict quantitatively the double ionization yield. In

particular, it does not take into account the Coulomb potential of the parent

ion. Later it has been shown that the the ion’s attracting Coulomb potential

together with the laser field plays an important role for the NSDI yield (so-

called Coulomb focusing) [93, 94, 95].

Because of spreading a large part of the electron wave packet misses the ion

at the first return. However, due to the Coulomb attraction, the electron tra-

jectories can be focused (especially after multiple returns) onto the parent ion

2.4. Non-sequential multiple ionization 19

and thus, the probability of electron collision-assisted ionization increases sig-

nificantly.

Alternatively the returning electron may be trapped temporarily by the parent

ion into Rydberg orbits. In this state the electron can gain additional energy

during ”soft” collisions with the ion until it finally ionizes a bound electron [95].

For certain electron trajectories in the continuum, the returning electron does

not directly ionize a second bound electron. The collision then only leads to

an excitation of the ion followed by laser-induced ionization. The probability

for such a scenario depends on the energy of the recolliding electron and on

the impact excitation cross section. The semiclassical theory for He includ-

ing both, impact ionization and excitation plus the Coulomb focusing effect,

predicts well the experimentally observed double to single ionization yield ra-

tio [95].

For He and Ar impact excitation dominates the total inelastic scattering cross

section over the whole range of electron impact energies. This fact is a possible

reason for the discrepancy between experimentally found momentum distrib-

utions of He2+ and Ar2+ on one hand and Ne2+ on the other hand [97].

Despite its simplicity, the three-dimensional classical trajectory analysis for

NSDI of atoms in linearly polarized light is able to reproduce qualitatively the

characteristic double-hump structure of the electron sum-momentum distrib-

ution as well as the electron momentum correlation in the final state found in

COLTRIMS experiments [98, 99, 100].

Apart from this semiclassical approach, a pure classical analysis of the final

stage of NSDI has been developed which also reveals main features of the elec-

tron sum-momentum distributions [101]. The idea here is that the rescattering

of one electron on the ionic core produces a highly excited two-electron com-

plex, which is next doubly ionized in the laser field. The main requirement for

two electron escape is the formation of a saddle in the Coulomb potential in the

presence of the electric field of the laser; the relevant parameters for the model

are the total energy of the excited complex and the time of its formation.

•One-dimensional quantum mechanical approaches

As mentioned above, the full time-dependent 3D two-electron problem of the

He atom in a laser field is still not solved for relevant experimental conditions,

i.e. for near-infrared light [79]. For linearly polarized light the field is acting

only in one direction. This gives an opportunity for testing various 1D quan-

tum mechanical models, where the motion of both electrons is restricted to

one dimension along the polarization direction [102, 103, 104, 105, 106].

In such a 1D He model atom both, the electron-electron and the electron-

nucleus interaction are described by the soft Coulomb potential V(x) = −1/√x2+ 1.

The time-dependent two-electron wave function is represented numerically on

20 2. Ionization in intense laser fields: fundamentals

a flat grid in the presence of the oscillating laser field [104, 107, 108, 109, 106,

110].

1D numerical simulations have the advantage of providing quite easily infor-

mation on the time evolution of the correlated two-electron wave function in

momentum and coordinate space.

1D numerical calculations are able to reproduce qualitatively the ”knee” struc-

ture in the experimentally observed yield of the doubly charged ions as a func-

tion of light intensity [105, 111]. Within this model for NSDI it also has been

shown that the two photoelectrons are emitted preferentially into the same

direction. This feature is consistent with experimental results. Furthermore

the experimentally observed ion momentum distribution can be reproduced in

such a calculation [104]. In general, all 1D calculations support the validity of

the rescattering model.

The main shortcoming of a 1D model is that it cannot describe angular effects

of electron emission and that it is valid only for linearly polarized light. More-

over, the repulsion between electrons is overestimated due to the restriction of

the electron motion to only one dimension. As a consequence, the final state

electron momentum distributions differ from the experimental results (for de-

tails see [23]).

A comparison between 1D quantum mechanical and fully classical treatments

of the electron dynamics in a strong laser field has shown, that the main dy-

namical features of double ionization of a quantum mechanical two-electron

system are reproduced well, provided a classical ensemble of two-particle tra-

jectories is used [106]. This indicates that the classical description of multiple

ionization is a good approximation for more complex atomic systems in the

classically allowed phase space.

Most theoretical approaches strongly support the recollision scenario for NSDI of

atoms. However, till now only a few calculations have been done for the differential

yield of the correlated final state electron momenta, which have been measured in

COLTRIMS experiments [82, 90, 91, 100, 104]. Most of these calculations [82, 100,

104] have been done for the He atom, while experimentally correlated two-electron

momentum distributions have been obtained only for Ar and Ne so far. Available

theoretical results for Ar and Ne [90, 91] presently cannot explain the difference in the

electron momentum correlations of these two atomic species found experimentally.

Thus, on one hand experiments on He and, on the other hand, more sophisticated

calculations for other atomic systems are desirable.

2.5. Free electron dynamics in the laser field 21

2.5 Free electron dynamics in the laser field

Since electrons driven by the oscillating electric field of a light wave play a key role

in double ionization of atoms in strong laser fields, let us now consider the dynamics

of a free electron in the laser field. We restrict ourselves to a classical description

of an electron in linearly polarized light. This simplification is justified because

NSDI is most efficient at linear polarization and most differential measurements

have been performed under these conditions [16, 17, 18, 19, 68, 69, 73]. Moreover,

for low-frequency ultra-short intense laser pulses both the classical analysis and the

quantum mechanical treatment of double ionization lead to similar results [106].

We start with an electron which is ionized in the electric field of the light wave via

tunneling in the first step of NSDI. For simplicity, we consider the electron motion in

the strong field approximation, i.e. we neglect the electron interaction with the ionic

core. A free electron in the electromagnetic field of a laser pulse ( ~

E,~

B) is exposed

to the Lorenz force ~

F=e(~

E+~v ×~

B),(2.13)

with ethe electron charge and ~v its velocity. Within the considered regime of NSDI

with a light intensity of the laser field of about 1013−1016 W/cm2and the wavelength

of about 800 nm, the electron velocity vis small compared to c, the speed of light

(nonrelativistic regime). This fact allows one to neglect the second term in (2.13)

and take into account only the electric field of the light wave, which is then described

as ~

E=~eE0(t) cos ωt, (2.14)

where ωis the frequency of the field, ~e is a unit vector. In a linearly polarized light

wave the electron experiences a force only along one axes parallel to the field

m˙vx= 0, m ˙vy= 0, m ˙vz=eE0(t) cos ωt. (2.15)

Let us assume that the electron starts from the position z= 0 at time t0with

zero velocity. Tunneling occurs most probably near the top of the laser pulse which

consists of few optical cycles (see Fig. 2.6). Here, the amplitude change E0(t) is very

small after one optical cycle that is relevant for the electron return to the ion core.

Thus after the integration of Eq. (2.15) we obtain the electron velocity along the

field and the position of the electron z

vz(t) = e

mωE0(sin ωt −sin ωt0),(2.16)

z(t) = e

mω2E0(cos ωt0−cos ωt −ω(t−t0) sin ωt0).(2.17)

Whether the electron comes back to the ion or not depends on the initial time t0

when the motion starts. In Fig. 2.7 (a) the position z(t) of the electron is plotted

22 2. Ionization in intense laser fields: fundamentals

Fig. 2.7: a) Electron motion along the electric field of the light wave obtained from Eq.

(2.17) for different t0: (1) for an electron which does not return; (2) for a single

and (3) for multiple electron returns. b) electron kinetic energy at the instant

of rescattering trin terms of Upas a function of t02.

for three different values of t0. The latter have been chosen such that the electron

never returns (1), returns once (3), or returns several times (2) back to the ion.

Analyzing Eq. (2.17) for a certain t0we can obtain the return time trand then the

corresponding electron kinetic energy Ekin using (2.16). The latter is an important

parameter for collisional ionization. As is well known the mean quiver energy of a

free electron in an electric field is given by the ponderomotive energy Up(see (2.5)).

For instance, the ponderomotive energy of a free electron in a laser field with a

wavelength of 800 nm and with a light intensity of 1014 W/cm2is about 6 eV. The

classical analysis considered here shows that the maximum kinetic energy an electron

can gain upon recollision is 3.17Up(see Fig. 2.7 (b)). If the maximum kinetic energy

of the electron Ekin,max is larger than the ionization potential of the parent ion I+

direct impact ionization of a second bound electron can occur. Since Ekin,max

depends on the ponderomotive energy Upand thus on the intensity of the light

wave, an ionization threshold is expected to occur in the classical rescatterig picture

at a certain intensity, where the electron kinetic energy reaches I+

Double ionization in a strong laser field upon inelastic recollision may happen in

a second way: via collisional excitation of a bound electron, which is subsequently

field ionized from the excited state, or alternatively, via capture of the recolliding

electron into an excited state. The released energy leads to excitation of the bound

electron. Finally, the two bound electrons get ionizied from their ecited states by

the electric field of a light wave.

2Rottke in [112]

2.5. Free electron dynamics in the laser field 23

As mentioned above experiments on Ar indicate that impact excitation may

contribute significantly to double ionization [69, 70]. Taking into account this mech-

anism also allows one to reproduce the double to single ionization yield ratio for He

[96].

For the scenario with capture – field ionization the lifetime of the excited state

has to be long enough to survive until both electrons can be freed simultaneously

by the increasing electric field of the light wave.

After the two electrons are in the continuum, they oscillate and start to drift in

the electric field of the light wave until the laser pulse has passed by. The remaining

doubly charged ion moves in a similar way in the opposite direction.

Thus the final electron drift momentum for an e−which starts with zero velocity

is given by

pz=−2eqUp(tr) sin ωtr.(2.18)

Namely this final electron momentum after the acceleration in the laser field can be

measured in COLTRIMS experiments in coincidence with the drift momentum of

the corresponding doubly charged ion.

In this chapter we have considered the fundamental aspects of ionization dynamics

of atoms in intense laser fields based on the present knowledge of the subject from

the experimental and theoretical point of view. We have also addressed the open

questions and the perspectives for future theoretical as well as experimental investi-

gations of the ionization dynamics of atoms and molecules in strong laser fields. In

the next chapter we will concentrate on a detailed description of our experimental

setup.

24 2. Ionization in intense laser fields: fundamentals

3. Experiment

In this chapter we consider in detail our experimental setup used for the present

investigation. Our aim has been to study the ionization dynamics of atoms and

molecules in an intense laser field.

The setup consists mainly of three parts: the atomic (molecular) gas-jet source

chamber, differential pumping stages and the interaction chamber (Fig. 3.3). A su-

personic atomic or molecular beam is formed in the gas-jet source chamber, it is

collimated by apertures separating the differential pumping stages, and finally in-

teracts in the main chamber with a focused laser beam. We will also discuss the

momentum resolution of the spectrometer and the issue of light intensity determi-

nation in a focused laser beam.

3.1 Setup modifications

In the course of the present work the previously existing experimental setup has

been significantly modified. In particular, a supersonic gas jet source together with

a beam collimation system has been incorporated. Additionally, two turbomolecular

pumps have been installed. Moreover, a new detection system has been integrated.

The latter includes two position sensitive MCP detectors for ions and electrons as

well as the data acquisition electronics which allows coincident detection of ion-

electron pairs. A new software (CoboldPC) has been applyed for data acquisition

and analysis.

3.2 The COLTRIMS technique

The experimental technique we used is based on COLTRIMS (Cold Target Recoil

Ion Momentum Spectroscopy) (for a review see Ref. [15, 113]). It was developed from

recoil-ion momentum spectroscopy (RIMS) [114, 115], a powerful tool for the inves-

tigation of the dynamics of atomic collision reactions, e.g. atoms interacting with

electrons, ions or photons. It provides the possibility of high-resolution measurement

of the recoil-ion momentum in combination with a large detection solid angle close

to 4π. However, due to thermal motion a room-temperature static target limits the

momentum resolution to a few atomic units. This corresponds to a kinetic energy

26 3. Experiment

of ∼40 meV. The realization of a cold supersonic gas-jet target in the COLTRIMS

technique allowed one to achieve a momentum resolution of only a few per cent of

1 a.u. 1corresponding to kinetic energies at the 1µeV level.

The combination of COLTRIMS with an electron imaging technique (based on

the same principle as the recoil-ion detection), which is also known as ”Reaction

Microscope” [116], enabled the coincident detection of the momenta of recoil-ions

and electrons, and thus, made possible a complete kinematical analysis of atomic

or molecular reactions. In particular, the coincident ion-electron momentum spec-

troscopy was successfully applied to study the ionization dynamics of atoms in high-

intensity laser fields [16, 17]. In the present work its application has been extended

to molecular systems.

A reaction microscope includes a well collimated target beam of cold atoms or

molecules, which interacts with a projectile beam (in our case a laser beam) at

some point. After the interaction, the charged target fragments (ions and electrons)

are extracted from the interaction region by a weak homogeneous electric field.

After acceleration in opposite directions the ions and the electrons pass a field-free

drift region until they reach position-sensitive microchannel plate (MCP) detectors

[117]. From the detected time-of-flight (TOF) and the position of a fragment on

the detector its momentum vector after the interaction can be reconstructed. The

detailed description of our momentum spectrometer will be given below. Here we

just note that the spectrometer geometry plays a decisive role for the momentum

resolution.

3.3 The supersonic atomic and molecular jet

A spatially well localized atomic/molecular target is realized by supersonic expansion

of gas from a high-pressure gas source into vacuum through a small nozzle. The

important condition for a supersonic expansion is that the pressure ratio between

the gas source and vacuum P0/Pv>2.1 [118].

In the free-jet isentropic expansion the free enthalpy hof the gas is converted

into the kinetic energy of directed motion in the gas-jet2

h0−h=v2

2,(3.1)

where h0is the stagnation enthalpy per unit mass. During the expansion the gas

cools down from the temperature of the gas source T0to some temperature TT0.

11 a.u. corresponds to the momentum of an electron bound in the ground state of a hydrogen

atom.

2his enthalpy per unit mass.

3.3. The supersonic atomic and molecular jet 27

Tab. 3.1: Typical values of speed ratio S, temperature in the atomic/molecular beam T

and the spread in momenta ∆punder different experimental conditions.

Gas T0, K P0, bar S T, K ∆p, a.u.

Ar 300 1 12.9 4.5 1.4

300 2 18.8 2.1 1.0

Ne 300 1 8.4 10.6 1.6

77 1 22.7 0.4 0.3

N2300 1 6.0 29.0 3.0

300 4 9.8 10.8 1.8

O2300 1 6.2 27.4 3.1

300 4 10.2 10.1 1.9

For an ideal gas dh =cpdT and thus the squared velocity after the expansion is

v2= 2 ZT0

cpdT. (3.2)

Taking into account TT0and cp= (γ/(γ−1)(R/W)) for an ideal gas we obtain

for the final jet velocity [118]

v=s2R

Wγ

γ−1T0,(3.3)

where Ris the molar gas constant, γis the specific heat ratio cp/cvand Wis the

molar molecular weight. For example, the mean velocity of Ar-atoms in the jet after

expansion is 560 m/s at a source temperate T0= 300 K. This corresponds to a

momentum of 18.6 atomic units.

The quality of the jet is characterized by the so-called speed ratio, S=v/p2kT/m,

which is defined as the mean velocity divided by the thermal spread in velocities.

Using Eq. (3.3) we get a simple expression for the speed ratio

S=sγ

γ−1T0

T.(3.4)

The typical values of the speed ratio, the corresponding temperature in the jet after

expansion from a nozzle of 20µm diameter, and the thermal spread in momenta in

the atomic/molecular beam are presented in table 3.1 for target gases used in the

experiment under different conditions (pressure P0and temperature T0) of the gas

28 3. Experiment

Fig. 3.1: Top view of the gas-jet chamber and the differential pumping stages.

jet source. Minimum spread in momenta of 0.3 a.u. is reached for neon as a target

gas using a pre-cooled gas jet source (with liquid nitrogen at T0= 77 K) at P0= 1

bar.

The other characteristic parameter of the supersonic free-jet expansion is the

Mach number. This is the mean flow velocity divided by the speed of sound

M=v

pγRT/W .(3.5)

For a supersonic expansion M1. The latter condition is also known as the ”zone of

silence”, where the expansion properties become independent of the background gas

pressure Pvin the vacuum chamber. The atomic/molecular beam is extracted from

this region by a skimmer. The diameter and shape of the nozzle and the skimmer

design determine the free-jet properties.

In our experiment a nozzle with a diameter of 20µm can be operated with the

pressure of 1 −20 bar (usually, in experiment 1 −2 bar). The nozzle is mounted

on a translation stage, which allows for the adjustment of the nozzle position in the

plane perpendicular to the atomic beam axis (Fig. 3.1). The nozzle design enables

cooling of the gas before expansion down to the temperature of liquid nitrogen.

A cone-shaped skimmer with a 0.5 mm diameter and a very sharp edge of the

opening at the tip is located at a distance of 10 mm from the nozzle. The skimmer

wall serves as a boundary between two differential pumping stages. The gas-jet

chamber is pumped by two turbomolecular pumps with pumping speeds of 300 l/s

and 500 l/s respectively. The background pressure in the jet chamber is smaller than

1×10−3mbar when the beam source backing pressure is 1 bar.

Under certain conditions dimers, trimers or clusters may be formed in the gas-

jet during expansion. The clustering depends on the gas used, the pressure P0,

3.3. The supersonic atomic and molecular jet 29

Tab. 3.2: Hagena parameters Γ and condensation parameters cfor target gases in our

experiment at the gas-jet source conditions P0= 2 bar and T0= 300 K.

Gas Ar Ne N2O2

c1650 185 528 1400

Γ 90 10 28 76

the temperature T0in the beam source, and the diameter of the nozzle d. Cluster

formation in a gas-jet can be estimated by using the semi-empirical scaling law of

Hagena [119, 120, 121]:

Γ = cd[µm]0.85P0[mbar]

T0[K]2.29 (3.6)

where Γ is a dimensionless scaling parameter and cis the condensation parameter,

which depends on the gas. The table below includes cvalues for the gases used

in our experiment and the corresponding estimated maximum values of the Γ pa-

rameter. Based on available experimental data clusters start to be observed when

Γ = 100 −300. Therefore we do not expect any cluster formation under our experi-

mental conditions (T0= 300 K, P0= 1 −2 bar).

The formation of dimers and trimers for monoatomic gases can be avoided if

the corresponding semi-empirical dimensionless parameters Cand Ddo not exceed

specific critical values [122, 123]

C≡P0

ε/σ3d

σ0.88 T0

ε/k−2.3

<15,(3.7)

D≡P0

ε/σ3d

σ0.4T0

ε/k−2.4

<0.1.(3.8)

Here, εand σare gas-specific Lennard-Jones parameters. In our case the upper limits

of the parameters for Ar are: C= 11; D= 0.05, and for Ne: C= 1.5; D= 0.006. At

lower nozzle temperature (T0<160 K) clusters form with higher probability. For

example, at T0= 162 K and P0= 2 bar we detected singly charged Ar-dimers after

multiphoton ionization in the reaction microscope.

Fig. 3.2 shows the ion yield dependence on the nozzle pressure P0for Ar+and

Ne+. The ion yield on the y-axis is measured using a ratemeter which counts the

number of the singly charged ions per second detected by the MCP. It increases

steeply at low pressure (1−2 bar). With the pressure increasing further the ion yield

first becomes saturated (P0: 2−4 bar) and then decreases slightly (P0>4 bar). The

main reason for this behavior is scattering of the target jet on the background gas

30 3. Experiment

Fig. 3.2: Ion yield dependence on the gas source pressure for Ar+ions with pre-cooled

nozzle (T0= 152 K) and for Ne+ions at room temperature.

in the gas-jet chamber at high pressure. This leads to a decrease of the gas density

at the focal spot of the laser beam.

3.4 The Differential Pumping Stages

The first differential pumping stage is separated from the gas jet chamber by the

supporting wall of the skimmer (Fig. 3.1). It is pumped by a turbomolecular pump

with a pumping speed of 300 l/s. The background pressure in this stage is about

5×10−8mbar. A small slit with adjustable width (20 −500µm) and fixed hight

of 3 mm is located at a distance of 480 mm from the skimmer. It collimates the

atomic beam and thus determines the final momentum spread in the beam along

the xdirection (Fig. 3.3). The slit serves to separate the two differential pumping

stages. The second stage (370 mm long) is pumped with an identical pump (300 l/s)

as the first one. At the end of the second pumping stage there is an aperture with

the diameter of 5 mm, which separates this stage from the main chamber. Finally,

a well collimated atomic beam with low density enters the interaction chamber. At

the point of interaction with the laser beam the gas density in the atomic beam is

∼109atoms/cm3.

3.5. The momentum spectrometer 31

3.5 The momentum spectrometer

The momentum spectrometer employed in the experiment is schematically shown

in Fig. 3.3. After passing the beam collimation chambers the cold atomic/molecular

target beam (y-axis) intersects the focused laser beam (x-axis) at right angle in the

center of the interaction chamber.3

The rest gas pressure in this chamber has to be as low as possible, to reduce

the contribution from background gas ionization in the focused, high intensity laser

beam. The base pressure in our ultra-high vacuum (UHV) chamber was kept at

about 3 ×10−10 mbar using several pumps: a cryopump with the pumping speed

of 1500 l/s, a turbopump with 500 l/s pumping speed and a couple of titanium

sublimation pumps, which were operated from time to time.

The laser beam is focused onto the target atomic/molecular beam by a spherical

mirror with a focal length of 100 mm in back reflection. The focal spot diameter in

the focal plane is estimated to be 10µm4. The interaction volume is determined by

the focal spot diameter and by the width of the atomic beam along the x-axis (laser

beam propagation direction). It is approximately 2.5×10−9cm3. The atom/molecule

density in the interaction volume is about 1.5×109−3.5×109cm−3. Therefore the

mean number of atoms/molecules in the interaction region is 3 −8. A low target

density is important for coincident ion and electron momentum spectroscopy for

reasons that will be discussed later.

After photoionization in linearly polarized 30 fs laser pulses, electrons and ions

are accelerated in opposite directions (along the z-axis) in an applied static electric

field with a field strength of 1 −7 V/cm. This electric field is generated by a system

of twenty metal rings equally separated from each other. The applied voltage is

equally divided between the rings so that the electric field is nearly homogeneous.

After extraction by this field the charged particles are flying to their corresponding

detectors in field-free tubes. They are detected with position-sensitive microchannel-

plate (MCP) detectors.

The geometry of a whole flight tube is chosen such that we can reconstruct with

high precision the initial momentum vector of the particle after ionization from its

detected time-of-flight and its position on the detector. For ions the ratio between

the length of the field-free drift path and the acceleration length was chosen as 2 : 1

(0.2 m and 0.1 m respectively). The lengths ratio for electrons in our experiment

was 1 : 1 (0.1 m for each).

Electrons emitted from the focal spot with a large kinetic energy into a large solid

angle usually miss the detector. The application of a weak homogeneous magnetic

field parallel to the extraction electric field allows one to detect high-energy electrons

3This coordinate system will be used throughout the work.

4FWHM of a Gaussian function

32 3. Experiment

Fig. 3.3: Schematic view of the momentum spectrometer including the supersonic atomic

(molecular) jet.

3.5. The momentum spectrometer 33

Fig. 3.4: Operation principle of the delay-line anode: signals from the electron avalanche

arrive at two ends of each delay-line (xand y); the time difference between

them is measured (symbolized by the clocks).

emitted within a solid angle of up to 4π. This magnetic field is generated by a couple

of Helmholz-coils. By adjusting the distance between the coils equal to their radius

a sufficiently homogeneous magnetic field can be achieved in a quite large spatial

region. In our experiment the radius of the magnetic field coils was 0.42 m and the

magnitude of the magnetic field reached 4 −20 G. The deviation from homogeneity

of the magnetic field ∆B

Bover the flight path of the electron from the laser focal

spot to the electron detector is less than 1%. The influence of the magnetic field on

electrons and on ions is defined by the Lorenz-force ~

F=q[~v ×~

B]. However, for an

electron and an ion with the same momentum, for instance 1 a.u., the corresponding

velocity of the ion (e.g. Ar+), and therefore the Lorenz-force, is 1.4×10−5times

smaller than that of an electron. In fact we can neglect the influence of the magnetic

field on ions.

Both, electrons and ions, are detected with identical commercial position-sensitive

MCP detectors [124]. Their active detection area is 80 mm. Each detector consists of

a pair of micro-channel plates (MCP) where the time-of-flight of the charged particle

is picked off, and a delay-line anode for position decoding. An optimal MCP detec-

tion efficiency is achieved for ions at a kinetic energy of 2.2 keV and for electrons

at 200 eV. An avalanche of secondary electrons emerging from the MCP creates a

signal on the delay-line anode. The time resolution is limited only by the electronics

used usually to 0.5 ns. A delay-line anode consists of two couples of parallel wires

wound spirally in many turns along two directions perpendicular with respect to

each other. Fig. 3.4 shows the detection principle of a delay-line anode. The position

is decoded from the arrival time difference of the signal an electron cloud induces in

the wire at both ends of the corresponding line. Both coordinates xand ycan be

34 3. Experiment

calculated as [112]

x= (tx1−tx2)vsignal,(3.9)

y= (ty1−ty2)vsignal,(3.10)

where (tx1−tx2) and (ty1−ty2) are the respective time differences and vsignal is

the signal velocity. The latter can be determined from the time a signal needs to

travel from one end of the line to the other one (for our detector about 1 mm/ns).

A position resolution of 0.25 mm is achieved with a time resolution of 0.5 ns of the

time-to-digital converter we used.

The timing signals from each MCP and the corresponding delay-line anode are

amplified by means of a differential amplifier (RoentDek DLATR6), converted to

standard ECL pulses by constant fraction discriminators, and then measured by a

fast time-to-digital convertor (TDC) with a time resolution of 0.5 ns (LeCroy TDC

3377). Additionally the ion time-of-flight and position were measured by a second

TDC (Philips Scientific 7186) with a higher resolution of 100 ps. The signal from

the laser pulse served as the reference time for each event. Since the electron time-

of-flight is typically less than 500 ns and the ion TOF amounts to several tens of

microseconds, the ion TDC was started by a suitably chosen high precision constant

time delay after an electron arrived at the electron MCP. Data from the TDCs were

only accepted by the data acquisition system if at least one electron and one ion

reached the respective MCP within a laser shot. The TDCs used were integrated

in a CAMAC crate. The data were read out by a CAMAC controller, which was

connected to a PC via an interface card. By means of a special commercial software

”CoboldPC” (ReontDek Handles GmbH) we controlled the data acquisition and

stored the raw data in a so-called List-Mode-File (LMF) event by event. This allows

one to replay an experiment off-line. CoboldPC also enables the full data analysis

and graphic representation of the analyzed or raw data. A typical ion TOF spectrum

for argon as a target gas is presented in Fig. 3.5. Beside the main peaks of the three

Ar+isotopes and Ar2+, arising from ionization of the Ar gas jet in the laser focus,

other peaks are found. The quite broad peaks of O+, N+

2, and H2O+originate from

photoionization of the background gas. The sharp lines of O+

2and H2O+come from

ionization of the gas jet, which was not entirely consisting of Ar atoms. Moreover,

due to the high repetition rate of the laser pulses (100 kHz), atoms/molecules are

ionized every 10µs. This gives rise to the peaks (Ar+)∗, (Ar2+)∗, and (H2O+)∗coming

from the previous/next laser pulse, and thus repeating their respective actual peaks

with a time difference of 10µs. The ion and electron position images are shown in Fig.

3.6. Ar+and Ar2+ ions (Fig. 3.6 (a)) can be identified as the two maxima extended

along the x-axis, i.e. the direction of the atomic beam. In the projection onto the

detector plane, ions start their motion from the point C(laser beam focal spot) and

move along the arrow until they hit the detector. Electrons move helically in the

applied magnetic and electric fields. In the detector plane an electron trajectories is

3.5. The momentum spectrometer 35

a circle, Cis the starting point (Fig. 3.6 (b)). As mentioned above we detect only

one electron and one ion per laser shot. The data acquisition is initiated only if both

signals are detected.

Ultimately, for double ionization, in our experiment the momentum of a doubly

charged ion and that of one of the two photoelectrons are measured. The momentum

Fig. 3.5: Typical ion TOF spectrum obtained from strong field ionization of Ar as target

gas using a Ti:Sa laser system at a light intensity of 1.5×1014 W/cm2. An

explanation for every peak is given in text.

of the second electron can be calculated from the momentum conservation law

~pA+n}~

k=~pi++ +~pe1+~pe2.(3.11)

A neutral atom Aabsorbs effectively nphotons each with a momentum }~

k. The

momentum of one photon at 800 nm wavelength is 4.15 ×10−4atomic units. This

is small compared to the typical experimental momentum resolution. Depending

on the ionization potential and the final charge state of the ion from several tens

up to several hundreds of photons are absorbed by a target atom. But even for 200

effectively absorbed photons the whole photon momentum transfer (0.08 a.u.) is still

smaller than our accuracy of ion momentum determination (see below). Therefore

we can exactly calculate, within the spectrometer resolution, the momentum of the

36 3. Experiment

second electron from the measured momenta of the doubly charged ion and one of

the two photoelectrons

~pe2,z =−(~pi++,z +~pe1,z),(3.12)

~pe2,x =−(~pi++,x +~pe1,x),(3.13)

~pe2,y =−(~pi++,y +~pe1,y) + ~pA,y.(3.14)

Here, it has been taken into account that the initial momentum components of the

atom in the well collimated atomic beam are negligibly small along the xand zaxes.

For instance, for Ar as a target gas, with the temperature of the gas source 300 K and

the pressure 1 bar, the maximum momentum in the jet along x-axis is estimated to

be 2.8×10−4a.u. and 9 ×10−5a.u. along the z-axis. For the atomic beam direction

(y-axis) we cannot neglect the momentum of the neutral atom and the residual

thermal momentum spread in the beam. Even in a pre-cooled (77 K) target jet the

momentum of an atom is about 9 a.u. Fig. 3.6 (a) shows the two-dimensional ion

image on the detector. Ar+and Ar2+ ions created in the jet are resulting in two

maxima extended in the xdirection on the right side of the detector. The spectrum

is observed at an atomic beam source temperature of 300 K and a pressure of 1 bar.

At these conditions the thermal velocity distribution of atoms in the beam results

in a quite large momentum spread of ions along the beam propagation direction

(x-axis) of 2 a.u. for Ar+ions and 3 a.u. for Ar2+. The spread in momenta along

the y-axis, which is actually determined in the ionization event, is much narrower:

0.6 a.u. for Ar+and 1.1 a.u. for Ar2+. This can be obtained from Fig.3.6 (a) if one

divides the widths of the xand ycoordinate distributions by the corresponding ion

TOF.

The calculation of the momentum of one electron from the measured momenta

of the other electron and of the doubly charged ion is problematic because we can-

not decide whether the detected ion-electron pair arose from one double ionization

event or not. The best solution of this problem would be to ensure that only one

atom/molecule is ionized per laser shot. As was mentioned before, there are 3 −8

target atoms/molecules in the interaction volume in our experiment. The experi-

mental conditions were chosen such that the repetition rate of the laser pulse was

100 kHz with 8000 photoelectrons detected per second ( i.e. 8 kHz measured by

a ratemeter during the experiment). It follows that the probability to ionize one

atom per laser shot and to detect the corresponding electron is not higher than

0.08. Under these conditions the contribution from false ion-electron coincidences

can be estimated to be about 8%. This value can be derived from the momentum

conservation for single ionization, where the narrow peak around zero due to true

coincidences (pi+,z +pe,z = 0) is sitting on the broad background resulting from false

coincidences. Integrating over the main peak and over the background we obtain the

number of true and false events.

3.5. The momentum spectrometer 37

The momenta of the detected ions and electrons are calculated from their mea-

sured time-of-flight and from the positions where the particles hit the detectors

using a classical analysis of their motion in the extraction electric field ~

Eand in the

homogeneous magnetic field ~

Bfor electrons

mi~

˙v=q~

E, (3.15)

me~

˙v=e~

E+e[~v ×~

B].(3.16)

Since both the electric and the magnetic fields are directed along the z-axis (Fig.

3.3) the electron motion along z-axis is determined by the electric field only while

in the xy-plane the magnetic field alone acts on the electron

me˙vz=−eE, me˙vx=evyB, me˙vy=−evxB. (3.17)

From a classical calculation for an electron and an ion which first move in a homo-

geneous electric field and then in a field-free space we obtain the relation between

the time-of-flight to the detector and the initial momentum component parallel to

the extraction electric field pk(along the z-axis). Thus for electrons

Te=pek

eE +√2m

eE seEl +p2

2m+L√2m

2qeEl +p2

,(3.18)

where lis the length of the acceleration stage, Lthe length of the field-free tube

and Ethe electric field strength. The corresponding time-of-flight for ions is

Ti=−pik

qE +√2m

qE sqEl +p2

2m+L√2m

2qqEl +p2

,(3.19)

where qis the ion charge. In the experiment with the above-mentioned parameters

(E: 1 −7 V/cm, l: 10 cm) we usually have for ions the condition

2mqEl. (3.20)

Expanding Eq. (3.19) into a Taylor-series with respect to p2

i,k/(2mqEl) we obtain

Ti=−pik

qE +√2m(2l+L)

2√qEl +p2

2mqEl√2m(2l−L)

4√qEl

+p2

2mqEl2√2m(3L−2l)

16√qEl +p2

2mqEl3√2m(6l−15L)

96√qEl +... (3.21)

38 3. Experiment

Fig. 3.6: Two-dimensional position images showing the position distributions for ions

(a) and for electrons (b) where they hit their detectors.

Furthermore, as we mentioned before, for ions the field-free path Lis twice as long

as the path lin the acceleration electric field. Therefore the third term in Eq. (3.21)

vanishes. Finally, we have

Ti= 2s2ml

qE −pik

qE +∅

"p2

i,k

2mqEl#2

,(3.22)

where only the first two terms are significant. Thus, under our experimental con-

ditions, the ion time-of-flight depends almost linearly on the momentum of the ion

after photoionization.

An electron in a magnetic field cycles with a cyclotron frequency ω= (eB)/me.

In the xy-plane it moves along a circle starting at point C(laser beam focal spot)

and hitting the detector at H(Fig. 3.6 (b)). The initial momentum of the electron

in the xy-plane ( ~pe⊥) determines the length of the helical trajectory. As an example,

Fig. 3.7 shows twelve typical electron trajectories at an extraction electric field

E= 250V/m and magnetic field B= 5Gobtained using the simulation program

SIMION. Electrons are emitted from the focal spot with a kinetic energy of 12eV

and an emission angle which is varying from 0 to 360 degree (in xz-plane). The

momentum component of an electron ~pe⊥perpendicular to magnetic field ~

B(which

coincides with the light polarization direction) can be calculated if we know the

time-of-flight of the electron Te, the cyclotron frequency ωand the distance Rfrom

the electron starting point Cto its end point at the detector H

|~pe⊥|×2|sin ωTe

2|=meωR, (3.23)

where R=p(x−xC)2+ (y−yC)2. Here, (x, y) are actual coordinates of an elec-

tron at the detector and (xC, yC) can be obtained from the maximum Cof the

3.5. The momentum spectrometer 39

Fig. 3.7: Electron trajectories at an extraction electric field E= 250V/m and a magnetic

field B= 5Gin xy-plane (left) and in xz-plane (right). Red points are the end

points of the trajectories, where the respective electron hits the detector surface.

electron position image. Many electrons in Fig. 3.6 (b) start and arrive at the de-

tector at the same point flying an integer number of cycles ωTe= 2nπ, n = 0,1,2, ...

This can be seen in Fig. 3.8, where Ris plotted versus the electron time-of-flight.

The two maxima at R= 0 correspond to electrons with closed trajectories. At

these points we cannot resolve ~pe⊥since the sin term, and thus Rin Eq. (3.23) be-

come zero. The cyclotron frequency ωis easily determined from the time difference

between two points with R= 0 where the number of events is maximum (Fig. 3.8).

The ion momentum component transverse to the extraction electric field, which

coincides with the light polarization direction, can be obtained from the position of

the ion on the detector (Fig. 3.6 (a)). Ions with a transverse momentum pi⊥= 0

arrive at the detector in the center of the corresponding ion distributions (Ar+or

Ar2+). Therefore the ion momentum components along the xand yaxes can be

calculated using the coordinate distribution and the corresponding TOF of the ion

pi,x =x−xc

mi,(3.24)

pi,y =y−yc

mi,(3.25)

where xand yare actual coordinates of the ion on the detector, Tiis its time-of-

flight and miis the ion mass. xcand yccorrespond to the center of the respective

coordinate distributions (Ar+or Ar2+). In general, the transverse ion momentum is

a sum of the momentum of the respective atom in the atomic gas jet pAand the

momentum transfer δp through photoionization

pi,x =pA,x +δpx,(3.26)

pi,y =pA,y +δpy.(3.27)

40 3. Experiment

Fig. 3.8: Cyclotron deflection Rversus the electron TOF. The period of the a cycle is

Tc= 2π/ω.

The momentum component pi,y is mainly determined in the ionization event, since

pA,y in a well collimated beam propagating along the x-axis is negligibly small. Thus,

pi,y =δpywith uncertainty 5.7×10−4a.u.5. This is much smaller than the typical

measured values of pi,y. As can be seen in Fig. 3.6, the ion coordinate distribution

along the x-axis is much broader than that along y. This indicates that the pi,x

momentum distribution, in contrast to pi,y, is determined by the remaining thermal

momentum distribution of atoms in the jet pA,x. The latter cannot be neglected.

3.6 Momentum resolution

In general, the ion momentum resolution is determined by the time-of-flight (0.5ns)

and the position resolution (0.26 mm for each delay-line). It also depends on the

target jet quality. The ion and electron momentum resolution in the longitudinal

direction with respect to the light polarization direction also depends on the extrac-

tion voltage. Similarly, the transverse momentum resolution for electrons depends

on the magnitude of the applied magnetic field. The longitudinal ion momentum

resolution can be derived from the momentum conservation condition for single

ionization. A sharp peak at zero momentum is expected for the sum-momentum

(pik+pek). The density plot of the latter versus the electron longitudinal momentum

pek(Fig. 3.9 (a)) yields some useful information. A narrow distribution around zero

of the sum-momentum is observed. The width of the distribution depends mainly

5this value is estimated using Eq. (3.3) and taking into account the jet geometry in the experi-

ment

3.6. Momentum resolution 41

Fig. 3.9: (a) The momentum conservation of single ionization of Ar parallel to the light

polarization direction represented as a density plot of (pik+pek) versus pek; (b)

cut AB through the two-dimensional distribution in (a) along the line pek= 0.

on the ion momentum resolution when the electron momentum is equal zero. Thus

we can estimate ∆pikfrom a cut through the two-dimensional distribution along

the line pek= 0 (Fig. 3.9 (b)). The width of this distribution (FWHM) is about

0.12 a.u. This value can be taken as the upper limit for ∆pik. The deviations from

(pik+pek) = 0 for pek⩾1 a.u. (Fig. 3.9 (a)) indicates that the extraction electric

field is not entirely homogeneous and thus the TOF - momentum transformations

(3.18) and (3.19) are not exact. The deviations for pek⩽−1 a.u. occur due to high-

energy electrons starting into the direction opposite to the electron detector. These

electrons are flying up to the edge of the applied electric field, where the field is

inhomogeneous.

Fig. 3.10 shows the scalar potential of the extraction field throughout the yz-

plane of the spectrometer and a comparison with the extraction potential of an

ideally homogeneous field along the z-axis (y= 0), which was used to derive the

ion and electron TOF in (3.19) and (3.18), respectively. It is evident that the main

difference between the ideal and real scalar potentials appears at the transition

between the extraction region and the field-free drift part (z= 100 and 300 mm).

The real field distribution in the spectrometer was determined by means of the

”SIMION 3D” simulation program [125]. In SIMION electrostatic or static magnetic

fields are computed by solving the corresponding Laplace equations. The program

also allows one to simulate the motion of any charged particle in an electric or/and

magnetic field of a given geometry. In Fig. 3.11 we compare the TOF dependence on

the initial momentum for ions and electrons derived from the SIMION simulation

42 3. Experiment

Fig. 3.10: The scalar potential of the extracting electric field in the yz-plane and along

z-axis (small graph) of the spectrometer derived from the SIMION simulation

for the real spectrometer geometry. It is compared with the ideal extraction

potential (black curve in the small graph) used in Eqs. (3.18) and (3.19).

3.6. Momentum resolution 43

Fig. 3.11: Comparison of the TOF dependence on the initial momentum derived from a

SIMION simulation (red curve) and calculated from (3.18) and (3.19) (black

curve): (a) for ions; (b) for electrons.

on one hand and from calculations with the formulae (3.18), (3.19) on the other

hand. For ions the results coincide. But for electrons emitted with large negative

momenta (backward from the detector) the discrepancy is quite strong for pek<−1

a.u., which is marked as a dashed line in Fig. 3.11 (b). The consequence of this

discrepancy can also be seen in Fig. 3.9 (a) as a deviations from zero momentum for

pek⩽−1 a.u. as discussed above.

Since the essential discrepancy for the TOF-momentum dependency appears for

pek<−1 a.u. we can use the formula (3.18) in a limited range of momenta for an

estimation of the resolution in the longitudinal electron momentum. The momentum

resolution is mainly determined by the resolution in time-of-flight (0.5 ns). It there-

fore depends on the electron momentum pek(Fig. 3.12). ∆pekis almost constant

for electrons emitted with negative momenta (better than 0.01 a.u.) and degrades

for positive momenta up to 0.06 atomic units.6The same momentum resolution de-

pendence has been obtained by the SIMION simulations (red curve in Fig. 3.12).

For ions, the effect of the extraction field inhomogeneity is that the ion TOF does

not depend on the longitudinal momentum alone (3.19) but becomes a function of

both pikand pi⊥. Fig. 3.13 clearly demonstrates the dependence of pikon the position

on the detector, and thus on pi⊥for Ar+ions. The left pannel of Fig. 3.13 shows

the distribution of Ar+ions on the detector where the equal narrow areas at three

different coordinates xi,(i= 1,2,3) are marked. On the right side of Fig. 3.13 the

momentum distribution (pek+pik) is plotted at pek= 0 (similar to Fig. 3.9 (b)) only

6In Fig. 3.11 (b) and Fig. 3.12 electrons with positive momenta are emitted towards the electron

detector and those with negative momenta start in the opposite direction.

44 3. Experiment

Fig. 3.12: Electron momentum resolution estimated using Eq. (3.18) for two acceleration

electric field strengths: 1.2 V/cm and 2.4V/cm; comparison with the SIMION

simulations (red curve).

Fig. 3.13: Left: two-dimensional ion image for Ar+ions; right: sum-momentum pek+

pAr+kat pek= 0 (identical to Fig. 3.9 (b) for three different positions xion

the ion detector image.

3.7. The laser system 45

for those Ar+ions which hit the detector within one of these three narrow areas. As

can be seen, the (pek+pik) distributions resulting from the different xpositions on

the detector are slightly shifted. Furthermore, the width of the (pek+pik) distribution

(0.06 a.u.) is just half of that in Fig. 3.9 (b), which is integrated over all xcoordinates.

As we discussed before (Fig.3.9) the width of (pek+pik) at zero electron momentum is

a measure of the ion momentum resolution ∆pik. Thus, the ion momentum resolution

can be amended by improving the electric field homogeneity over the spectrometer

volume traced out by the ion trajectories.

The electron momentum resolution in the transverse direction is determined by

the position uncertainty resulting from the detector resolution and the target size.

We can estimate it using Eq. (3.23) and taking into account that the position resolu-

tion in the detector plane is 0.36 mm (using a LeCroy TDC). The best momentum

resolution (better than 0.01 a.u.) is obtained for electrons with a time-of-flight,

which corresponds to sin ωTe

2= 1. However, this momentum resolution degrades

when the electron time-of-flight approaches an integer multiple of the cyclotron pe-

riod T= 2π/ω. At these times the resolution is lost completely.

The ion momentum resolution in the transverse direction acquired from the

SIMION simulations is about 0.18 atomic units. It is obtained without taking into

account the extension of the interaction volume. The real transverse momentum

resolution for ions is worse than this value.

3.7 The laser system

We used a Ti:Sapphire laser system in our experiment to irradiate the atoms and

molecules. It consists of a Ti:sapphire oscillator, a spatial light modulator (SLM), a

regenerative amplifier, and a prism compressor (Fig. 3.14). The detailed description

of the laser system can be found in Ref. [126]. Unlike the conventional chirped-pulse

amplification technique (pulse stretching-amplification-recompressing) used for high

peak-power ultrafast lasers [3, 127], the present system does not include a pulse

stretcher. Instead, pulse broadening occurs due to accumulation of a large amount

of dispersion during the amplification process.

The laser pulses are generated in a Kerr-lens mode-locked Ti:Sapphire oscillator

with a repetition rate of 78 MHz and a mean output power of 200 mW. This cor-

responds to a pulse energy of ∼3 nJ. The output pulse width of the oscillator is

about 30 fs.

A liquid-crystal phase modulator (SLM) serves for the pre-compensation of high-

order dispersion, while the main part of dispersion (second and third orders) is com-

pensated after amplification by the prism compressor. Without the SLM the pulse

width after the compressor is limited to 60 fs. Fig. 3.15 represents the non-collinear

autocorrelation curves of the amplified pulse with and without the SLM [126]. The

46 3. Experiment

Fig. 3.14: Diagram of the Ti:Sa laser system used in the experiment [128]. FR - Faraday

isolator; QS - Q switch; CD - cavity dumper; TOD - chirped mirrors to control

third-order dispersion; CM - chirped mirror; G - grating.

pulse width is two times shorter with the SLM used. The modulator consists of an

array of 128 pixels, each with a width of 97µm. Every pixel can independently affect

the phase of the corresponding spectral component of the beam passing through it

by means of an applied voltage. The phase distribution is controlled and optimized

by adjusting the voltages in a feedback loop using an evolutionary algorithm. In our

experiment the phase distribution was optimized on the ion rate, which served as a

feedback signal.

After passing the phase modulator the low-energy pulses from the oscillator are

amplified in a regenerative amplifier. The Q-switched cavity of the amplifier with a

Ti:sapphire crystal is CW pumped by an Ar+ion laser. The pulse repetition rate in

the amplifier is 100 kHz. After 18 round trips, the amplified pulses are ejected with

an energy of 8µJ and a pulse width of 15 ps.

In order to obtain short pulses the dispersion is compensated in a prism com-

pressor. A Proctor-Wise double prism configuration is used. Finally, the 35 fs laser

pulses at a wavelength of 800 nm and a repetition rate of 100 kHz are delivered to

the experiment. The maximum output energy is 7µJ.

3.8. Determination of light intensity 47

Fig. 3.15: Non-collinear autocorrelation traces of the amplified output pulses with SLM

(solid line) and without SLM (dashed line) [126].

3.8 Determination of light intensity

The light intensity of the on-axis focused laser beam cannot be measured directly.

It can be derived from the measurement of the integral ion yield dependence on

light intensity. Since single ionization of atoms in a strong laser field is well de-

scribed by the ADK tunneling theory, we can use the ADK single ionization rate

[10] for the light intensity calibration. Fig. 3.16 shows the result of this calibration

for krypton as a target. We measured the number of singly charged Kr+ions as

a function of the laser output power, which was determined by means of a power

meter. From the ADK curve which fits the experimental points we are able to derive

a relation between the laser output power and the light intensity peak in the focal

spot. However, the determination accuracy of the light intensity in this way is quite

low. Alternatively, it can be obtained from the differential ATI photoelectron energy

spectra (Fig. 3.17). Here, we can use the differential ADK formula for the electron

momentum distribution based on the tunneling theory [129]

ω(pk, p⊥) = ω(0) exp "−p2

kω2(2Ip)3/2

3E3−p2

⊥(2Ip)1/2

E#,(3.28)

where Ipis the ionization potential, ωis the laser frequency and Eis the electric

field strength of the light wave. The corresponding energy distribution (p2

e,k/(2me))

obtained using this formula fits very well to the experimentally observed one in

the low energy region (Ekin <2Up) of directly emitted electrons. Based on these

methods the light intensity is determined within an uncertainty of 20%. Furthermore,

48 3. Experiment

Fig. 3.16: Kr+ion yield as a function of light intensity: ADK theory (red curve) and

experiment (black squares).

we compared our electron energy spectra at different light intensities with those of

Paulus [130], which have been obtained using the same laser system.

3.8. Determination of light intensity 49

Fig. 3.17: ATI photoelectron kinetic energy distribution for single ionization of Ne (black

curve) at a light intensity of 3 ×1014 W/cm2; comparison with the kinetic

energy distribution calculated based on the tunneling theory (red curve).

50 3. Experiment

4. Non-sequential double ionization of

atoms: results and discussion

In this chapter we discuss the experimental results on non-sequential double ioniza-

tion of atoms (neon and argon) in linearly polarized high-intensity 35 fs laser pulses.

The laser system is considered in Chapter 3. As noted before, the ”CoboldPC” soft-

ware was employed to analyze the raw experimental data. The analysis program is

given in the Appendix A.

4.1 Electron sum-momentum distributions

We measured the electron sum-momentum distributions for Ar and Ne in order to

analyze in detail the NSDI mechanism and its atom specific features. For NSDI

via electron impact ionization one expects a double-hump structure of the e−sum-

momentum distribution parallel to the light polarization direction.

In the previous chapter we have discussed the momentum conservation for single

and double ionization by many photons. We have shown that in the direction of the

light polarization (z-axis, Fig. 3.3) and perpendicular to it (x-axis) the momentum

of a doubly charged ion is equal to the sum of the momenta of the two photoelec-

trons emerging from double ionization (3.12), (3.13). It follows that the momentum

distribution of doubly charged ions is equivalent to the sum-momentum distribution

of the two corresponding photoelectrons. We have to take into account that, after

ionization, electrons and ions are additionally accelerated by the oscillating electric

field of a light wave (2.14) along the z-axis during the laser pulse. The drift momen-

tum of the doubly charged ion gained during this acceleration is twice as large as

the drift momentum of each electron (2.18). In the experiment we actually measure

the final momenta of the photoelectrons and ions including the drift momentum

they get. The measured momentum distributions of ions and electrons in the trans-

verse direction (x-axis) are not affected by the light wave which is linearly polarized

along z.

Fig. 4.1 displays Ne2+ and Ar2+ ion momentum distributions (= sum-momentum

distributions of the two photoelectrons) measured in a focused linearly polarized

light beam at light intensities, where non-sequential double ionization is expected.

52 4. Non-sequential double ionization of atoms: results and discussion

The number of doubly charged ions per momentum bin1is plotted versus the corre-

sponding momentum measured in atomic units. The panels on the left show the mo-

mentum distributions projected onto the polarization axis f(pk), which is obtained

by integrating the momentum distribution function over all transverse momenta

f(pk) = Zd2~p⊥f(~p).(4.1)

The panels on the right show the corresponding distributions projected onto the

light beam propagation direction (x-axis) f(p⊥), where

f(p⊥,x) = Zdpkdp⊥,yf(~p).(4.2)

The Ne2+ momentum distribution (Fig. 4.1 (a)) exhibits a double-hump struc-

ture with a pronounced minimum at pk= 0 which has been observed earlier [17, 131].

Such a structure is an indication of instantaneous electron impact ionization (rescat-

tering) which occurs preferentially when the oscillating electric field of the light wave

E0(t) cos ωt is close to zero [17]. The final longitudinal drift momentum of the doubly

charged ion

pik=q

ωE0(tr) sin ωtr,(4.3)

reaches maxima at zero crossings of the electric field and it is equal to zero at

field maxima. For other ionization mechanisms (”shake-off”, collective tunneling)

both electrons are expected to be released near maxima of the oscillating electric

field. This should lead to small final longitudinal momenta with a maximum of the

momentum distribution at zero. Therefore these mechanisms can be ruled out to be

the main NSDI mechanism.

The Ar2+ momentum distribution exhibits a minimum at zero, which is much

less pronounced in comparison with Ne (Fig. 4.1 (a)) and is even absent at the lower

light intensity (Fig. 4.1 (c)). This may indicate that the ionization mechanism for

Ar differs from that for Ne. However, we supply evidence below that the main NSDI

mechanism for Ar is also instantaneous impact ionization by a rescattering electron.

First, we note that the longitudinal momentum distributions of Ar2+ and Ne2+ have

equal widths at the same light intensity (Fig. 4.1 (a)). Due to acceleration in the

light pulse after electron impact ionization, the doubly charged ion, and therefore

the electrons are able to gain a drift momentum. Based on the classical rescattering

model, the maximum drift momentum is 4pUp. This value determines the width of

the electron sum-momentum distribution along the light polarization direction. The

momentum |pk|= 4pUpis denoted by the arrows in Fig. 4.1 (a,c). It is close to the

1The bin size is determined in the ”CoboldPC” program; in our case it is 0.25 a.u.

4.1. Electron sum-momentum distributions 53

Fig. 4.1: Ar2+ (black curve) and Ne2+ (red curve with squares) momentum distributions

parallel (left) and perpendicular (right) to the light polarization direction: (a,b)

at 2.4×1014 W/cm2; (c,d) at 1.5×1014 W/cm2.

54 4. Non-sequential double ionization of atoms: results and discussion

half widths of the observed distributions for Ne and Ar. This is an indication that

the main double ionization mechanism of Ar and Ne is the same.

The transverse momentum distributions of Ar2+ and Ne2+ are practically identi-

cal at the same light intensity (Fig.4.1 (b)). This indicates that f(p⊥) is insensitive

to the initial state from which the electrons are removed, for Ne from the (2p) and

for Ar from the (3p) shell. Both momentum distributions are very narrow with a

maximum at p⊥= 0. In the semiclassical rescattering model, the kinetic energy of

the recolliding electron is expected to play a decisive role for electron impact ion-

ization. The latter occurs when Ekin,max is larger than the ionization potential of

the singly charged ion core I+

p(see Chapter 2). As can be seen in Fig. 4.1 (b), the

essential difference in the respective maximum electron excess energy (Ekin,max −I+

(4.5 eV for Ne and 18 eV for Ar) does not influence f(p⊥). The overall distribution

shape also does not change with light intensity (Fig.4.1 (d)). A detailed discussion

of the measured transverse momentum distributions f(p⊥) will be given in Sec. 4.4.

Summarizing, we observed a clear double-hump structure of the electron sum-

momentum distribution for Ne, that gives a strong evidence for electron impact

ionization as the main NSDI mechanism. Although for Ar these indications are not

pronounced, the width of the distribution suggests that electron impact ionization

contributes strongly to NSDI.

4.2 Electron momentum correlation

Our momentum spectrometer allows for highly resolved differential measurements of

the momentum correlation of the two photoelectrons emitted in double ionization.

We measured the e−-e−momentum correlation in order to clarify the origin of the

specific behavior of Ar, which we found in the electron sum-momentum measure-

ments.

Fig. 4.2 shows the e−-e−momentum correlation for Ar parallel to the light po-

larization axis at the same light intensities as in Fig. 4.1. In these density plots the

horizontal axis represents the momentum p1,kof the detected electron and the ver-

tical one corresponds to the momentum p2,kof the second photoelectron. The value

of p2,kis calculated from the momentum conservation (3.12) using the measured p1,k

and the measured momentum of the corresponding Ar2+ ion. The color scale denotes

the number of electron pairs detected per momentum bin.

At both light intensities two maxima along the main diagonal in the first and

third quadrants of the plots are observed. They have been found to be a characteris-

tic of instantaneous impact ionization [68, 69]. The maxima appear on the diagonal

p1,k=p2,kat ±1 a.u. (Fig. 4.2 (a)) and ±0.7 a.u. (Fig. 4.2 (b)) for the given light

intensities. This corresponds to the case where the two photoelectrons with equal

longitudinal momenta are emitted in the same half-space along the z-axis (positive

4.2. Electron momentum correlation 55

Fig. 4.2: The momentum correlation of the two photoelectrons after double ionization

of Ar in linearly polarized 35 fs laser pulses at two light intensities: (a) at

2.4×1014W/cm2; (b) at 1.5×1014 W/cm2. The momentum components parallel

to the light polarization direction are shown.

or negative direction). The two electrons ejected simultaneously after impact ioniza-

tion tend to have equal longitudinal final momentum vectors due to post-collision

acceleration in the light wave.

The projection of the plots onto the main diagonal p1,k=p2,kis equivalent to

the sum-momentum distributions we have already discussed (Fig. 4.1 (a,c)), where

pk=p1,k+p2,k. Evidently electrons from instantaneous impact ionization dominate

the sum-momentum distribution at large momenta |pk|. The reason for the shallow

minimum at pk= 0 at the light intensity 2.4×1014W/cm2and its disappearance

at 1.5×1014 W/cm2for Ar also becomes clear. In contrast to Ne, where the vast

majority of electron pairs is emitted with similar momenta into the same half-space

[72], in non-sequential double ionization of Ar many electron pairs are also emitted

into the opposite half-spaces with significant and similar |pi,k|(i= 1,2). This gives

rise to a large amount of events in the second and fourth quadrant of the plots in

Fig. 4.2. They contribute to the sum-momentum distribution near pk= 0.

In Fig. 4.3 the sum-momentum distributions are obtained by projecting only a

narrow part (p1,k−p2,k∼0) of the e−-e−momentum distribution along the diag-

onal p1,k=p2,k. Constructed in this way, they include only the contribution from

the electron pairs gained in recollision ionization. A pronounced double-hump struc-

ture, which is not observed in the full sum-momentum distribution (Fig. 4.1) can

be clearly seen. Obviously the valley around pk= 0 is filled by the electron pairs

arising from a second ionization mechanism. A possible scenario for this mechanism

is electron impact excitation during recollision with subsequent electric field (tun-

56 4. Non-sequential double ionization of atoms: results and discussion

Fig. 4.3: Projections of a narrow part (p1,k−p2,k∼0) of the e−-e−momentum correlation

from Fig. 4.2 and Fig. 4.5 along the diagonal p1,k=p2,k. (a) at 2.4×1014 W/cm2;

(b) at 1.5×1014 W/cm2; (c) 0.9×1014 W/cm2.

neling) ionization [69]. The contribution of this ionization mechanism becomes more

prominent with decreasing light intensity (Fig. 4.2 (b)).

From our measurement of the electron momentum correlation we conclude that

for Ar in addition to electron impact ionization (two maxima on the main diagonal)

a second ionization mechanism contributes to NSDI. The latter is related to elec-

tron impact excitation with subsequent electric field ionization by the light wave.

The electron pairs from this mechanism conceal the double-hump structure in the

electron sum-momentum distribution for low light intensities.

4.3 Sub-threshold electron impact ionization

As discussed before, in the semiclassical rescattering picture an ionization threshold

for electron impact ionization is expected. If the kinetic energy of the recolliding

electron Ekin,max is smaller than the ionization potential of the singly charged ion

core I+

pelectron impact ionization should become impossible. However, preceding

experiments have shown that the dependence on light intensity of the integral ion

yield ratio of doubly charged to singly charged ions [A2+]/[A+] does not show any

abrupt change at light intensities where Ekin,max reaches the ionization threshold

I+

p[14, 67]. Since the rescattering scenario was found to be valid for non-sequential

double ionization of rare gases [17, 68], this fact has remained a puzzle and may

point to a gradual change of ionization mechanisms near threshold. We investigated

non-sequential double ionization of Ar at light intensities, where the kinetic energy

of the returning electron Ekin,max is close to or below the expected threshold for

instantaneous impact ionization of Ar+.

4.3. Sub-threshold electron impact ionization 57

Fig. 4.4: Ar2+ momentum distributions parallel (left) and perpendicular (right) to the

light polarization direction: (a,b) at 1.1×1014 W/cm2; (c,d) at 0.9×1014

W/cm2.

58 4. Non-sequential double ionization of atoms: results and discussion

Fig. 4.5: The momentum correlation of the two photoelectrons after double ionization of

Ar similar to Fig. 4.2 but at light intensities below the threshold for e−impact

ionization: (a) at 1.5×1014 W/cm2; (b) at 0.9×1014 W/cm2.

Fig. 4.4 shows the sum-momentum distributions of the two photoelectrons after

double ionization of Ar similar to those in Fig. 4.1, but this time at 1.1×1014 W/cm2

and 0.9×1014 W/cm2. At these light intensities Ekin,max is not sufficient for instan-

taneous impact ionization. The corresponding ”excess energies” (Ekin,max −I+

p) are

−7.2 eV and −10.5 eV. Comparing Fig. 4.4 with Fig. 4.1 we observe a gradual change

of the functional form of f(pk) (left panel) with decreasing intensity while crossing

the threshold Ekin,max =I+

p. It changes from a double-hump structure at 2.4×1014

W/cm2to a bell-shaped one at 0.9×1014 W/cm2. The distribution width becomes

smaller with decreasing intensity. In contrast, the transverse sum-momentum dis-

tribution f(p⊥) does not change its shape with decreasing light intensity whereas

its width slightly decreases. At first sight, a bell-shaped structure of f(pk) with a

maximum at pk= 0 seems to indicate the disappearance of impact ionization below

the threshold. But, as we have seen before, the sum-momentum distribution for Ar

near pk= 0 is dominated by a second ionization mechanism (impact excitation),

which seems to become more prominent with decreasing intensity (Fig. 4.2).

A closer look at the e−-e−momentum correlation at the lowest intensity 0.9×1014

W/cm2(Fig. 4.5 (b)) reveals that the characteristic maxima due to electron impact

ionization do not vanish, although Ekin,max I+

p. This becomes even more evi-

dent from Fig. 4.3 (c) where a narrow part of this e−-e−momentum distribution

along the diagonal p1,k=p2,kis projected. The double-hump structure does not

disappear in the below-threshold regime. It is just masked due to the electron pairs

from a second ionization mechanism. They completely fill the minimum at pk= 0

4.3. Sub-threshold electron impact ionization 59

Fig. 4.6: The dependence of the cutoff momentum of the Ar2+ and Ne2+ momentum

distributions on the light intensity. The data are acquired from Fig. 4.1 and

Fig. 4.4. The solid line is the function 4pUp.

of the full sum-momentum distribution in Fig. 4.4 (c). For convenience, in Fig. 4.5

the e−-e−momentum correlations at 0.9×1014 W/cm2(below threshold) and at

1.5×1014 W/cm2(above threshold) are directly compared. As can be seen, below

threshold instantaneous electron impact ionization still dominates the distribution

f(p1,k, p2,k), and therefore the corresponding sum-momentum distribution f(pk) at

large electron momenta where p1,k=p2,k. However, the overall contribution of elec-

tron pairs from recollision ionization becomes smaller in the intensity regime below

ionization threshold. In contrast, the second ionization mechanism, forming events

in the second and fourth quadrants of the e−-e−momentum correlation in Fig. 4.5,

seems to become more important below the threshold for impact ionization. At the

intensity 0.9×1014 W/cm2the maximum kinetic energy of the returning electron

just suffices to excite Ar+from the ground state 3s23p5to the lowest bound excited

states 3s23p4(3P)4s, that may facilitate electron impact excitation.

The importance of the e−impact ionization mechanism for the final electron

momentum distribution becomes more obvious in the dependence of the cutoff of the

sum-momentum distribution f(pk) on the light intensity (in terms of ponderomotive

energy) in Fig. 4.6. We here define the cutoff momentum of f(pk) as the momentum

where f(pk) reaches half of the maximum ion yield in Fig. 4.1 and Fig. 4.4. The

data points for Ar and Ne are closely following the function 4pUp(solid curve)

lying systematically slightly below. The error bars of Upreflect the uncertainty in

60 4. Non-sequential double ionization of atoms: results and discussion

determination of the light intensity. The deviation of the data points from 4pUpis

somewhat larger at small Updue to an increasing amount of events in the second

and fourth quadrants of Fig. 4.5. These events arise from the e−impact excitation

channel and give rise to the maximum of f(pk) at pk= 0 in Fig. 4.4. At low light

intensities they tend to decrease the cutoff momentum. Keeping in mind that 4pUp

is classically the largest momentum that the doubly charged ion can gain in post-

collision acceleration, we come to the conclusion that photoelectrons contributing to

large |pk|at all light intensities in experiment, including the sub-threshold regime,

are generated in instantaneous e−impact ionization. It should be noted that also at

large light intensity (1.3×1015W/cm2→Up= 2.8 a.u.) the data point for Ne from

Ref. [17] (not shown in Fig. 4.6) is in agreement with the tendency of our data lying

slightly below 4pUp.

The question arises how the observed instantaneous electron impact ionization

below the threshold Ekin,max =I+

pcan be explained within the semiclassical rescat-

tering model? In the quasi-static limit it may be understood by taking into account

that the electric field of the light wave is usually different from zero at the instant

of the electron recollision at tr. The actual threshold for e−impact ionization of a

singly charged ion in an external electric field of strength E0(tr) cos ωtrat the time

tris therefore lowered to the instantaneous saddle point energy of the combined

external and Coulomb potential, which is given by

I+

p(tr) = I+

p,0−2p2E0(tr) cos ωtr(a.u.),(4.4)

where I+

p,0is the unperturbed ionization potential. Fig. 4.7 displays schematically

the lowering of the ionization threshold. Here, the one-dimensional ionic Coulomb

potential modified by the electric field of the light wave is depicted at the instant

of recollision tr. The relation (4.4) is a good approximation of the real ionization

threshold as long as the Stark shift of the ionic ground state remains small and the

saddle point appears well outside of the electron charge cloud of the doubly charged

ion core. Such a field induced shift of the ionization threshold has been introduced

recently by van der Hart and Burnett in order to understand the missing threshold

behavior in the dependence of the total ion yield ratio [He++]/[He+] on the light

intensity for helium [94].

Fig. 4.8 (a) gives an idea of the instantaneous electron impact ionization below

threshold from the viewpoint of energy. The kinetic energy of the returning electron

Ekin(tr) (solid line) and the instantaneous ionization potential of the singly charged

argon ion Ip(Ar)+(tr) for two light intensities, below the impact ionization thresh-

old (0.9×1014 W/cm2- dashed line) and above (2.4×1014 W/cm2- dot-dashed

line), are plotted in terms of the ponderomotive energy Upversus the phase of the

external electric field at the instant of recollision ωtr. The oscillating electric field of

the light wave crosses zero at ωtr= 1.5π. At the instant of time 1.5π/ω the ioniza-

tion potential I+

p(tr) is unperturbed and reaches its maximum I+

p,0. At all ωtrwith

4.3. Sub-threshold electron impact ionization 61

Fig. 4.7: Schematic diagram of the 1D Coulomb potential of a singly charged ion core in

the presence of the external electric field of the laser at the instant of recollision

tr.I+

p,0is the unperturbed ionization potential of the singly charged ion, and

I+

p(tr) is the ionization potential in the non-zero electric field of the light wave

at the time of recollision tr.

Fig. 4.8: (a) The kinetic energy of the returning electron Ekin(tr) (solid curve) and the

instantaneous ionization potential Ip(Ar)+(tr) (dashed line at 0.9×1014 W/cm2

and dot-dashed line at 2.4×1014 W/cm2) as a function of ωtr, the electric field

phase at the instant of recollision (the energy scale is given in units of Up!). (b)

The corresponding excess energy Eexc after e−impact ionization.

62 4. Non-sequential double ionization of atoms: results and discussion

Ekin,max(tr)⩾I+

p(tr) the kinetic energy of the returning electron classically suffices

to kick out a second electron instantaneously in an inelastic collision. This may oc-

cur in a large interval of return times at the light intensity 2.4×1014 W/cm2, i.e.

above threshold. Below threshold, at 0.9×1014 W/cm2, electron impact ionization

may also happen in a restricted interval of ωtr, where the Ekin,max(tr) curve is lying

above the corresponding instantaneous ionization potential curve I+

p(tr), although

it is not possible with the unperturbed ionization potential I+

p,0. The dependence of

the instantaneous excess energy after collision Eexc =Ekin,max(tr)−I+

p(tr) on the

return time is plotted in Fig. 4.8 (b).

Concluding, we observed an electron momentum correlation characteristic of in-

stantaneous electron impact ionization below the corresponding ionization threshold.

We are able to explain this behavior within the semiclassical rescattering model by

taking into account the electric field of the light wave, which reduces the ionization

threshold at the instant of electron recollision. At all light intensities in our experi-

ment electron impact ionization was found to be the dominant NSDI mechanism.

4.4 The transverse electron sum-momentum distribution

We now draw our attention to the transverse electron sum-momentum distribution.

f(p⊥) is not affected by post-collision acceleration of ions, and thus of the photo-

electrons, in the electric field of the linearly polarized light wave. It, nevertheless,

is determined in the recollision event while the colliding electron interacts with the

singly charged ion core and with the second electron which is going to be kicked out.

Therefore f(p⊥) is expected to give more insight into the recollision process than

f(pk).

As we have seen in Fig. 4.1 (b,d) and Fig. 4.4 (b,d), at all light intensities in our

experiment f(p⊥) is a very narrow distribution with a maximum at zero momentum.

The distribution can be well fitted with a Gaussian function. Within the rescattering

model one would expect that the full width at half-maximum (FWHM) of f(p⊥)

depends on the square root of the available excess energy (Ekin,max −I+

p), and

therefore linearly on the external electric field strength E0. Surprisingly, we found

instead a scaling of the FWHM of f(p⊥) with √E0(Fig. 4.9). The experimental

data points for the FWHM of f(p⊥) of Ar and Ne follow closely the const ×√E0

curve. No abrupt change is found at the threshold for impact ionization of Ar+which

corresponds to the electric field strength E0= 0.06 a.u.. Also at large excess energy

(200 eV) for Ne the corresponding data point at E0= 0.19 a.u.2is lying on the

const ×√E0curve. A faster decrease in width is found only at the lowest E0, which

corresponds to the light intensity 0.9×1014 W/cm2. Here, a large amount of events

2the data point is taken from [131].

4.4. The transverse electron sum-momentum distribution 63

Fig. 4.9: The FWHM of the doubly charged ion momentum distribution f(p⊥) plotted

versus the peak electric field strength E0. Black squares and open triangles are

experimental data points of Ar and Ne, respectively. Dashed and dot-dashed

lines are corresponding results for Ne and Ar derived from the semiclassical

model calculations. The solid line along the experimental points is the function

of const ×√E0and the thin dotted line is proportional to E0.

not attributed to instantaneous impact ionization of Ar+contributes to f(p⊥). This

may be responsible for the deviation.

In Fig. 4.9 the experimentally observed dependence of the FWHM of f(p⊥) on E0

is directly compared with our semiclassical model calculations for Ne (dashed curve)

and Ar (dot-dashed curve). The calculation is based on the rescattering model with

taking into account the lowering of the ionization threshold I+

p3. The curves are

far from describing the experimental behavior. They are rising much faster than

√E0and even faster than E0, which is displayed by a thin dotted line. Therefore, it

seems that the width of the transverse electron sum-momentum distribution from the

experiment is not determined by the impact parameters of the recolliding electron.

The scaling of f(p⊥) with √E0reminds one of the scaling found for the FWHM

of the momentum distribution of the photoelectron in strong field single ionization

in the quasi-static limit (see, for example, [132]). The corresponding momentum

distribution is

f(~p⊥) = ω0exp −p2Ip~p2

⊥

,(4.5)

3the details of the model calculation will be discussed in the next section.

64 4. Non-sequential double ionization of atoms: results and discussion

with Ipthe ionization potential of the atom. The FWHM of this Gaussian function

FWHM =2√ln 2

(2Ip)1/4pE0.(4.6)

Here, an electron tunnels through an effective potential barrier formed by the com-

bined Coulomb and external electric field. The similar behavior of f(~p⊥) for single

ionization and for the electron sum-momentum distribution in double ionization is

not accidental. This implies that in electron impact ionization of the singly charged

ion, at the instant of recollision, the electric field of the light wave plays an important

role to free the second electron. Such a behavior can be explained if one assumes

that the returning electron does not transfer enough energy to the second one so

that either the second electron or the whole collision complex gets ionized by tun-

neling or above-barrier ionization in the presence of the external field. This scenario

has been discussed by Sacha and Eckhardt [101]. The light electric field in that case

would have a significant influence on the transverse momentum distribution of the

two electrons similar to single ionization. Barrier suppression or tunneling would

favor small transverse momenta and a scaling of the width of f(~p⊥) with √E0, as

found in the experiment, may result. For this scenario the collisions which most

efficiently lead to double ionization are expected to happen neither at zero crossings

of the electric field of the light wave nor at the highest kinetic energy possible for

the returning electron.

The importance of recollision events at non-zero electric field which do not result

from the most energetic electrons (Ekin <3.17Up) but contribute efficiently to non-

sequential double ionization, has also been discussed by Panfili et al. [133] within a

one-dimensional classical simulation. They found that the most effective collisions

happen at 1/2π < ωtr< π and 3/2π < ωtr<2π, where the recolliding electron is

moving against the external light electric field back to the ion core. After the energy

transfer during the collision, even if small, the bound electron is able to escape the

Coulomb potential while the barrier suppression is maximum. In this case one would

also expect that the final electron momentum distribution is mainly determined by

the field and not by the excess energy of the recolliding electron. However, the

relevance of this one-dimensional simulation in the full three-dimensional real world

is presently not assured. It is also not clear whether the recollision scenario outlined

above allows one to reproduce the electron sum-momentum distribution parallel to

the polarization direction f(pk) and the e-e momentum correlation found in the

experiment.

To conclude, our analysis of the transverse electron sum-momentum distribution

indicates that the distribution width is not determined by the impact parameters of

the recolliding electron, as expected from the rescattering model, but by the electric

field of the light wave, which was found to play an important role at the instant of

4.5. Comparison with theoretical results 65

electron recollision in NSDI.

4.5 Comparison with theoretical results

Let us compare our experimental results to several theoretical calculations for NSDI.

Most of them assume instantaneous electron impact ionization. Therefore we first

concentrate on this NSDI mechanism.

The quantum-mechanical S-matrix calculations [87, 85, 90] yield a good qual-

itative agreement with our experimentally observed momentum correlation of the

two photoelectrons. In this S-matrix approach the electron interaction at the instant

of recollision is described by a three-body contact interaction potential of the two

electrons and the ion core. Figs. 4.10 (b) and 4.11 (b) show the results of such a

calculation made by Schomerus [134] for Ar. A strong field approximation has been

used in this quantum-mechanical calculation. The two maxima in the first and third

quadrants of the momentum correlation are well reproduced. This result is quite

remarkable since the contact interaction of the electrons at the ion core seems a

crude approximation.

Replacing the three-body contact interaction in the S-matrix calculations by the

e−-e−Coulomb potential [82, 84, 90, 91] and applying first-order Born approximation

leads to a momentum distribution that differs substantially from the experimental

one. This approach favors different longitudinal momenta of the two photoelectrons

in the final state (|p1,k| 6=|p2,k|), especially with increasing light intensity. The

plausible reason for this behavior is that the potential of the ion core is neglected.

The correct Coulomb interaction has been used by L.-B. Fu and co-workers

[100] in a semiclassical trajectory calculation to model NSDI of helium. It included

e−-e−repulsion as well as the attractive Coulomb interaction of both electrons to

the ion core. The momentum correlation f(p1,k, p2,k) for He obtained within this

model qualitatively reproduces the experimentally expected (f(p1,k, p2,k) has not

been measured for He but it is expected to behave as Ar). Both electrons tend to have

large and equal momenta and thus the distribution f(p1,k, p2,k) peaks at the main

diagonal where |p1,k|=|p2,k|= 2pUp. No such calculation has been performed so far

for Ar or Ne since the inclusion of further electrons turns the problem intractable.

At the time of our experiment no calculations have been available at the relevant

light intensities that could be compared to our experimental results. We performed

a semiclassical model calculation which allows one to obtain the momentum cor-

relation of the two photoelectrons in double ionization for a comparison with the

experimental results. Our calculation is based on the replacement of the quantum-

mechanical S-matrix for recollision double ionization by its classical analog [87]. The

distribution of the momenta ~p1and ~p2of the two photoelectrons is then determined

66 4. Non-sequential double ionization of atoms: results and discussion

|S(~p1, ~p2)|2= (4.7)

Zdt0R(t0)δ



Ekin(tr)−~p1−~

A(tr)/c2

2−~p2−~

A(tr)/c2

2−I+

p(tr)



|V~p1,~p2,~

k|2,

where ~

Ais the vector potential for the electric field of the light wave4and |V~p1,~p2,~

k|2

is the collision form factor, which is determined by the e−-e−interaction at the in-

stant of recollision. The δ-function expresses energy conservation for electron impact

ionization of the singly charged ion. The ionization probability of the first electron is

calculated using the tunneling ADK rate R(t0) [10]. We assume that the recolliding

electron starts its motion in the external electric field of the light wave E=E0sin ωt

at a time t0with zero-velocity. The electron trajectories are calculated purely clas-

sically. In the next step we take into account only those trajectories, for which the

electron returns back to the singly charged ion core. This happens at a time tr.

Ionization of the bound electron from the ground state of the ion occurs, provided

that the kinetic energy of the returning electron Ekin(tr) is larger than the instan-

taneous ionization potential I+

p(tr). The latter is changing with the electric field of

the light wave (see Eq. (4.4)). We investigated the influence of two different types

of e−-e−interactions at the instant of recollision: the three-body contact interaction

of the two electrons and the ion core, analogously to that in the S-matrix calcula-

tions in Ref. [86, 91], and the electron-electron contact interaction. Finally, we take

into account the acceleration of the two free photoelectrons in the external field

after electron impact ionization. In Eq. (4.7) this is incorporated by the argument

~pi−~

A(tr)/c2,(i= 1,2) in the δ-function. The main code of the simulation can

be found in Appendix B.

Let us consider our theoretical results for the three-body contact interaction. In

our model the three-body contact interaction results in a form factor |V~p1,~p2,~

k|2=

const in Eq. (4.7). It neither depends on the intermediate state momentum ~

kof the

recolliding electron nor on the final state electron momenta ~p1,~p2. In Fig. 4.10 we

compare our calculation (left panel) for the electron-electron momentum correlation

for Ar at the light intensity 1.5×1014 W/cm2to the quantum-mechanical S-matrix

calculation (right panel) by Schomerus [134]. In Fig. 4.11 the same comparison is

done for the light intensity 0.9×1014 W/cm2. The distributions have been integrated

over all final transverse momentum components. As can be seen, our simple semiclas-

sical model yields similar results as the quantum-mechanical S-matrix calculation at

the given light intensities. In both calculations the two maxima appearing on the

E(t) = −1

∂

∂t ~

A(t)

4.5. Comparison with theoretical results 67

Fig. 4.10: e−-e−momentum correlation for the longitudinal momentum components at

a light intensity of 1.5×1014 W/cm2derived from (a) the semiclassical cal-

culation; (b) the quantum-mechanical S-matrix calculation [134].

Fig. 4.11: The same as Fig. 4.10, but at 0.9×1014 W/cm2. (a) the semiclassical calcu-

lation; (b) the quantum-mechanical S-matrix calculation [134].

68 4. Non-sequential double ionization of atoms: results and discussion

main diagonal are qualitatively similar to those observed in the experiment. At the

intensity 1.5×1014 W/cm2the semiclassical momentum distribution fills a larger

phase space area than the quantum-mechanical one. The opposite is found at the

lower light intensity. This tendency is a consequence of the shifting ionization poten-

tial I+

pin the presence of the external electric field incorporated in our semiclassical

model. At high light intensity this leads to a larger available phase space than the

constant I+

p,0. Without taking into account this effect we obtain a very restricted

electron momentum distribution near the classical ionization threshold and none at

all below the threshold. At higher intensities, far above the threshold, the effect due

to the decrease of I+

pis less prominent. The quantum-mechanical description, in

contrast to the classical one, avoids the problem of the ionization threshold, since

the absorption of a corresponding number of photons by the first electron in the con-

tinuum provides the energy needed for e−impact ionization of the singly charged

ion. This can be seen in the electron sum-momentum distribution obtained by the

strong-field S-matrix calculation by Kopold et al. [85]. At light intensities below the

classical threshold the distribution exhibits a sawtooth-like structure which reflects

the discrete energy transfer from the light field.

The two maxima of the distributions in Fig. 4.10 (b) and Fig. 4.11 (b) appear

at p1,k=p2,k=±2pUp, the value of the maximum classically allowed momentum.

This is in a good agreement with previous results, e.g. [90]. A slightly smaller value is

found for the peak of the classical distribution at low intensity (Fig. 4.11 (a)). How-

ever, both calculations predict a peak position at slightly higher momenta compared

to the experiment (Fig. 4.5).

In the case of electron-electron contact interaction the form factor in Eq. (4.7)

depends on the momentum ~

kof the returning electron:

|V~p1,~p2,~

k|2∼1

[2I+

p+ (~

k−~p1−~p2)2]2.(4.8)

In a linearly polarized electric field ~

khas only one component parallel to the field.

In the length gauge ~

kis the instantaneous electron momentum at recollision

k= 2pUp(cos ωtr−cos ωt0)~e, (4.9)

where ~e is the unit vector along the electric field, while in the velocity gauge ~

kis

the electron drift momentum at ωtr:

k=−2pUpcos ωt0~e. (4.10)

This interaction potential thus results in a dependence of the final electron momen-

tum correlation on the gauge chosen. Fig. 4.12 displays the results on the e−-e−

momentum correlation obtained using velocity (on the left) and length gauge (on

4.5. Comparison with theoretical results 69

the right) for several light intensities. In velocity gauge the two peaks of the distri-

bution are lying closer to the center compared to that in length gauge. The gauge

dependence of |S(~p1, ~p2)|2is certainly not physical. It appears due to a number of

approximations which are not gauge invariant. However, the tendency to shift the

maxima of f(p1,k, p2,k) on the diagonal p1,k=p2,kcloser to zero is a general feature

of the e−-e−contact interaction potential. The experimentally found positions of the

maxima of f(p1,k, p2,k) are described better by the model with e−-e−contact interac-

tion than with the three-body contact one. A dramatic change happens for the e−-e−

contact interaction potential in both gauges with increasing intensity. The peaks on

the main diagonal are spreading out along the direction p1,k=−p2,k. Consequently

the almost round distribution found at 0.9×1014 W/cm2turns into a distribution

stretched along the direction perpendicular to the main diagonal at higher intensi-

ties. In this case the calculated final momentum distribution is dominated by the

photoelectron pairs emitted with different momenta pi,k. Such a behavior is, how-

ever, not confirmed experimentally. It should be noted here that in the case of the

three-body contact interaction, discussed above, the round shape of the distribution

does not change essentially with the light intensity.

So far we have discussed the theoretical results for the main NSDI mechanism,

i.e. electron impact ionization. There is a well-known second ionization mechanism

contributing to NSDI of Ar which is associated with instantaneous impact excitation

followed by electric field ionization of the exited singly charged ion. This mechanism

gives rise to the large amount of electrons emitted into opposite half-spaces along

the light polarization direction. These electrons contribute to the second and forth

quadrants of the e−-e−momentum correlation (Fig. 4.2) and thus lead to the dis-

appearance of the minimum at pk= 0 in the sum-momentum distribution for Ar at

low intensities. A similar experimental behavior was observed for NSDI of He [16]

but no such features were found for Ne [72].

Up to now just a limited number of NSDI calculations have taken into account the

electron impact excitation mechanism. So far such calculations have only been done

for He. The main problem of performing these calculations for other atomic systems

is the deficiency of the theoretical and experimental data on total inelastic cross

section for electron impact excitation. Moreover, excited states may be perturbed in

the external laser field, that turns accurate calculations of the ionization probability

from an excited state difficult.

The quantum-mechanical S-matrix calculations for He by Kopold et al. [85] and

Goreslavskii et al. [86] showed that including electron impact excitation mechanism

as a superposition of contributions from different excited states may yield momen-

tum distributions similar to that found in experiments. It has been found that for

the lowest excited states the electron sum-momentum distribution f(pk) does not

change its double-hump shape, while for higher excited states the minimum at zero

momentum fills up quickly and the absolute electron yield rises.

70 4. Non-sequential double ionization of atoms: results and discussion

Fig. 4.12: e−-e−momentum correlation for the longitudinal momentum components ob-

tained by our semiclassical model calculation using the velocity gauge (left

panel) and the length gauge (right panel). The light intensities are: 0.9×1014

W/cm2(a,b); 1.5×1014 W/cm2(c,d); 2.4×1014 W/cm2(e,f).

4.5. Comparison with theoretical results 71

More recently it has been argued, that the difference in the shape of the electron

sum-momentum distributions, for Ar and He on one hand and for Ne on the other

hand, appears due to the atom specific relative contribution of impact excitation

and impact ionization to NSDI [97]. Indeed, the dependence of the total excitation

and ionization cross sections on the kinetic energy of the recolliding electron shows,

that the first dominates over the second for Ar+and He+, while the opposite is the

case for Ne+in a large range of energies. One may expect that this gives rise to

different contributions to NSDI from the two ionization mechanisms. As a result,

different sum-momentum distributions are observed. Although this model explains

well the experimental results, its relevance is questionable since it is based on total

instead of differential cross sections. The latter are more realistic but difficult to

include.

The e−impact excitation-electric field ionization mechanism for NSDI is strictly

speaking sequential. The second electron is ionized by the electric field of the light

wave from an excited state at a later time and not simultaneously as in the impact

ionization mechanism. As we have seen in Figs. 4.2 and 4.5, the photoelectron pairs

arising from impact excitation-field ionization start to dominate the e−-e−momen-

tum distribution with decreasing light intensity. At low intensity, where the kinetic

energy of the returning electron is not high enough for direct impact ionization, only

impact excitation followed by field ionization should contribute to NSDI. With in-

creasing intensity the probability for electron impact ionization grows. On the other

hand the lifetimes of excited states of Ar+should decrease in a stronger field. As

a result both electrons in the excitation-field ionization pathway are also emitted

almost simultaneously giving rise to the similar momenta. This idea has been drawn

from the calculations for He by van der Hart [135]. It allows one to explain the

intensity dependent features of the impact excitation-field ionization we found for

the Ar data.

In conclusion of this chapter it can be stated that our semiclassical calculations

based on the rescattering model agree qualitatively well with our experimental re-

sults for Ar and Ne. Our model is able to reproduce the momentum correlation

spectra with the distinctive features of the instantaneous electron impact ionization.

Among other theoretical models the most successful are the quantum-mechanical S-

matrix calculations applying three-body contact interaction [87, 85, 90] as well as the

most complete semiclassical trajectory calculation with the full three-body Coulomb

interaction [99, 100]. A quantitative disagreement of the existing theoretical results

with experiment is not surprising in view of the approximations made. Till now

no complete quantum-mechanical calculation has been performed at the relevant

wavelength and for complex atomic systems like Ar and Ne.

We observed the footprint of instantaneous electron impact ionization in the

72 4. Non-sequential double ionization of atoms: results and discussion

light intensity region, where it is forbidden within the classical rescattering model.

This can be explained by taking a lowering of the unperturbed ionization potential

of the singly charged ion in the external electric field of the light wave at the instant

of recollision into account. The light field thus essentially influences the rescatter-

ing process. The dependence of the transverse electron sum-momentum distribution

f(p⊥) on light intensity we found experimentally may indicate that this is also the

case at high light intensities, where Ekin,max I+

p. An alternative way of under-

standing the observed impact ionization ”below” threshold is to assume that the

collision complex which forms when the electron returns to the ion core absorbs

photons. If a sufficient number is absorbed the internal energy suffices for instanta-

neous double ionization of the atom.

Comparing Ar to Ne data atom specific features appear. For Ne the instantaneous

electron impact ionization prevails in a large range of light intensities, from the

classical threshold I+

pto far above, as a comparison with previous results shows

[131]. In the case of Ar, a second ionization mechanism, which is believed to be

electron impact excitation with subsequent electric field ionization, competes with

the first one and dominates with decreasing light intensity.

5. Molecules in high-intensity ultra-short

laser pulses

In the previous chapter the experimental results on non-sequential double ionization

of atoms in high-intensity ultra-short laser pulses have been presented and discussed.

The present chapter is dedicated to molecules, in particular to N2and O2.

There are several differences between molecules and atoms that might be relevant

for the ionization dynamics in a strong laser field. The presence of several nuclear

centers may lead to new interference effects between electron wave packets emitted

at different nuclei. Moreover, the symmetry of the ground state orbitals occupied

by the valence electron (e.g. bonding or antibonding) may have an effect on the

ionization process. Finally, the orientation of the molecular axis with respect to the

electric field of the light wave changes the effective potential that the valence electron

experiences during photoionization. So far, only a few experiments have addressed

these questions and the theory is not as developed as it is for atoms.

5.1 Introduction

Similar to atoms, several homonuclear and heteronuclear molecules show an exces-

sive double ionization probability observed in integral measurements of the doubly

charged ion yield [136, 58, 59, 137, 138]. This fact indicates that double ionization

of molecules may also proceed non-sequentially. However, the total ion yield data

are not sufficient to identify the ionization mechanism in detail.

A study of the dependence of double ionization on the ellipticity of the light

polarization may provide a clue on the NSDI mechanism, such as the suppression

of the doubly charged ion yield in elliptically polarized light found for atoms. For

example, for the benzene molecule (C6H6) the dependence of the doubly to singly

charged ion yield ratio on the ellipticity indicated that NSDI is due to electron

recollision [139].

Electron rescattering, which is the main mechanism of atomic NSDI, was recently

reported for molecular hydrogen [140] and also for D2[141, 142]. The high-energy

protons (and D+ions) observed in these experiments were interpreted either as the

outcome of rescattering excitation of the singly charged molecular ion followed by

74 5. Molecules in high-intensity ultra-short laser pulses

its dissociation or as coming from rescattering-induced ionization. Based on these

experimental findings one may thus expect that electron rescattering is the general

mechanism of molecular NSDI.

In this chapter we study double ionization of N2and O2by means of momentum

spectroscopy in order to identify the non-sequential mechanism. We also investigate

molecule specific effects on the final electron momentum correlation. In a recollision

double ionization process the probability of ionization depends primarily on the ex-

ternal electric field strength and on the ionization energy of the system. Since N2

and Ar have the same ionization potentials, and O2has the same ionization poten-

tial as Xe, one would expect similar double ionization behavior of the corresponding

molecular and atomic systems. A direct comparison of molecules with their atomic

”companions” allows one to clarify the influence of the molecular structure on the

ionization dynamics and to resolve the question whether the molecular double ion-

ization mechanism is the same as that for atoms.

Recent single ionization experiments with N2and O2have shown that N2behaves

like Ar, while O2and Xe, despite their similar ionization potentials, clearly differ.

The difference has been observed in the total ion yield measurement as a function

of light intensity, where the signal for the O2molecule was strongly suppressed

compared to that for Xe [59, 143]. In above-threshold ionization (ATI) spectra [144]

a reduced electron yield has been found for O2at low and high electron kinetic energy

compared to corresponding ATI spectra of Xe, whereas the spectra of N2and Ar

looked similar. These effects have been attributed to the different symmetry of the

ground state orbitals occupied by the valence electron (σg, bonding for N2and πg,

antibonding for O2) [144, 145]. In a diatomic molecule, quantum interference effects

of electron wave packets emitted at the two nuclear centers may occur. A destructive

interference appears in case of antibonding symmetry of the active molecular orbital.

It results in a suppression of molecular ionization and a reduced production of low-

energy electrons in the ATI spectrum compared to an atomic system with the same

ionization potential.

A simple semiclassical model for double ionization of molecules based on electron

recollision will be developed in this chapter. It takes into account the symmetry of

the initial state orbitals occupied by the valence electron. Thus, we can directly

compare our experimental results with model calculations in order to understand

the influence of the initial state symmetry on molecular double ionization and the

role of interference effects.

5.2. The electron sum-momentum distributions for N2and O275

5.2 The electron sum-momentum distributions

for N2and O2

As we have seen in the previous chapter about NSDI of atoms, the sum-momentum

distribution of the two photoelectrons for the momentum component parallel to

the polarization direction of the laser beam provides an obvious indication of the

ionization mechanism. We performed these measurements for N2and O2in order to

identify the NSDI mechanism for diatomic molecules.

Fig. 5.1 shows these distributions for N2(left panel) and O2(right panel) at

different light intensities. The laser field strengths were chosen in a regime where

NSDI has been reported to predominate [59, 137]. The upper two distributions were

obtained at light intensities where, within the rescattering model, the maximum ki-

netic energy of the recolliding electron is not enough for direct e−impact ionization

of the singly charged ion. As can be seen, at all intensities used in the experiment,

both N2as well as O2do not exhibit the double hump structure in the momentum

distribution which is considered to be characteristic for NSDI due to recollision.

However, we have already seen for Ar that the absence of the pronounced double

hump structure in the f(pk) distribution does not necessarily mean the absence of

NSDI by electron recollision. The Gauss-like functional form of the O2distribu-

tions does not change with increasing laser field strength from below the ionization

threshold (1.3×1014 W/cm2in Fig. 5.1 (b)) to far above (2.2×1014 W/cm2in

Fig. 5.1 (h)). The width of the distribution just becomes broader. In contrast, the

N2distributions exhibit a narrow plateau which develops to a Gaussian distribution

only at the highest light intensity.

A further characteristic of NSDI via rescattering is a dependence of the cutoff

momentum of f(pk) on the ponderomotive potential Upwhich follows a 4pUprule.

A relation like this is expected for instantaneous electron impact ionization of atoms

(see Sec. 4.3). Fig. 5.2 shows the experimentally observed cutoff momenta for N2

(filled squares) and O2(open circles) and also the function 4pUpfor comparison.

Three experimental points for N2are located close to the curve 4pUpand have the

same trend with light intensity. Only at the highest intensity 2.5×1014 W/cm2the

data point deviates from the common tendency being located far below the curve.

At this laser intensity the corresponding momentum distribution f(pk) is dominated

by electrons with momenta close to zero. Thus, from the dependence of the cutoff

momentum on the laser intensity we find a first signature of instantaneous e−impact

ionization for N2, except for the highest intensity. A full match of the data points with

the curve 4pUpcannot be expected since the sum-momentum distributions in Fig.

5.1 include contributions from electrons which are not coming from the e−impact

ionization channel. These electrons may result in a sum-momentum distribution

with maximum at pk= 0. Consequently, the cutoff momentum determined from the

76 5. Molecules in high-intensity ultra-short laser pulses

Fig. 5.1: Electron sum-momentum distributions after double ionization projected onto

an axis parallel to the laser beam polarization vector. Left panel: N2at light

intensities 1.3×1014 W/cm2(a), 1.5×1014 W/cm2(c), 1.8×1014 W/cm2(e),

2.5×1014 W/cm2(g); Right panel: O2at 1.1×1014 W/cm2(b), 1.7×1014

W/cm2(d), 1.9×1014 W/cm2(f), 2.2×1014 W/cm2(h).

5.3. e−- e−momentum correlation for N2and O277

Fig. 5.2: The dependence of the cutoff momentum of the N2and O2momentum distrib-

utions on the light intensity. The data have been extracted from Fig. 5.1. The

error bars denote the uncertainty in the determination of the light intensity.

The solid line represents the function 4pUp.

full sum-momentum distribution is smaller than 4pUp.

For O2, all data points are far below the 4pUpcurve. The cutoff momenta

for O2are about half of the corresponding cutoff momenta for N2. This indicates,

at first sight, that electrons which determine the cutoff momentum do not result

from instantaneous e−impact ionization. In Sec. 5.5 we will show, however, that e−

impact ionization of a molecule like O2may give rise to a narrow sum-momentum

distribution with a small cutoff momentum.

Summarizing our measurements of the electron sum-momentum distribution for

N2and O2, the double-hump structure found for atoms has not been observed for

molecules. However, for N2the width of the distribution indicates a significant con-

tribution of instantaneous electron impact ionization.

5.3 e−- e−momentum correlation for N2and O2

Since the momentum correlation spectra of the two photoelectrons provide the most

complete information about the NSDI dynamics, we expect to find other signatures

of electron impact ionization for N2and to understand the reason for the observed

e−sum-momentum distributions for both, N2and O2.

Fig. 5.3 shows these spectra for N2at 1.5×1014 W/cm2and for O2at three

78 5. Molecules in high-intensity ultra-short laser pulses

different laser intensities. The spectra are shown for the momentum component

parallel to the laser polarization direction.

In the previous chapter we have discussed that electrons emitted from atoms

during NSDI to the same half-space along the polarization direction give rise to

the two maxima on the main diagonal in the momentum correlation spectrum (first

and third quadrants). Moreover this is the main signature of instantaneous electron

impact ionization [68, 69].

In Fig. 5.3 (b) we observe such maxima for N2, where the two photoelectrons are

emitted preferentially with similar momentum components (p1,k=p2,k). However,

significantly more electron pairs with small momenta (pi,k≈0, i = 1,2) are found

for N2in comparison with its atomic companion Ar at the same light intensity (see

Fig. 4.2 (b)). We will show in the next section that these pairs may be attributed to

rescattering impact ionization of N+

2. The final momentum correlation of this NSDI

mechanism is thus decisively influenced by the molecular structure. Similar to Ar,

those electron pairs in the 2nd and 4th quadrants of Fig. 5.3 (b) found significantly

off the main diagonal have to be attributed to rescattering electron impact excitation

of N+

2with subsequent electric field ionization of the excited electron [69].

In contrast, the momentum correlation spectra for O2do not show any charac-

teristics expected for rescattering impact ionization (Fig. 5.3 (a,c,d)). At all laser

intensities in our experiment the momentum correlation for O2has a homogeneous

distribution with a maximum close to zero momentum. Such a distribution is usu-

ally characteristic for sequential double ionization. However, we have noted earlier

that the light intensities in our experiment were chosen in the regime where NSDI

is expected. Therefore a possible conclusion is that the molecular structure modifies

the final state momentum correlation of NSDI significantly.

To give the main points of our measurements of the e−-e−momentum correlation

for N2and O2, we observed the typical structures of the electron impact ionization

for N2but not for O2. Moreover, the significant contribution of electron pairs at zero

momentum found for both molecules is an evidence of the strong influence of the

molecular structure on the final electron momentum correlation.

In the next section we will develop simple models which allow us to test both,

the possibility of rescattering NSDI for N2and O2molecules as well as the sequential

double ionization scenario for O2.

5.4 Modeling of sequential ionization

In order to clarify whether the reason for the observed momentum distributions for

O2is sequential ionization, we first calculate electron sum-momentum distributions

expected from this mechanism.

Our semiclassical model is based on the KFR approximation as applied by

5.4. Modeling of sequential ionization 79

Fig. 5.3: e−-e−momentum correlation (longitudinal projection on the light polarization

axis) for N2at 1.5×1014 W/cm2(b) and for O2at 1.1×1014 W/cm2(a), at

1.7×1014 W/cm2(c) and at 2.2×1014 W/cm2(d).

Smirnov and Krainov [146, 147] to the H+

2molecular ion. Quasistatic electric field

ionization is assumed for both ionization steps. The symmetry of the initial state

orbital occupied by the valence electron is included in the calculation, however in

a simplified way. Averaging over the randomly oriented internuclear axis was per-

formed. This is allowed since the 35 fs laser pulses do not align the molecule.

Generally, molecules can be aligned by a linearly polarized low-frequency intense

laser field due to the torque on the laser-induced molecular dipole moment [148, 149,

150]. For short laser pulses, alignment will be important for the ionization process

if it occurs on a time scale comparable with the laser pulse duration. As has been

shown in [151], for a given laser field strength E0the alignment time τof N2and O2

can be estimated based on the scaling law τ∝pµ/R/E0found by Bandrauk and

Ruel [152] in their calculations for H+

2. Here, Ris the internuclear separation and µ

80 5. Molecules in high-intensity ultra-short laser pulses

is the molecular reduced mass. In our case the estimated shortest alignment times

for N2and O2are assured to be longer than the pulse duration of 35 fs.

In our models we also assume a fixed internuclear separation. This simplification

seems reasonable since for O2and N2as well as for the respective singly charged ions

the periods of molecular vibration are several times longer then the optical cycle of

the laser field (2.7 fs).

Let f1(~p1) and f2(~p2) be the momentum distribution functions of the two pho-

toelectrons after stepwise double ionization. They are independent for sequential

ionization. Therefore the sum-momentum distribution can be written as a convolu-

tion integral of the two distribution functions

g(~p) = Zf1(~p1)·f2(~p2)·δ3(~p −~p1−~p2)d3~p1d3~p2

=Zf1(~p1)·f2(~p −~p1)d3~p1,(5.1)

where ~p =~p1+~p2. We are interested in the distribution of the sum-momentum

component pkparallel to the light polarization direction for a direct comparison

with the experimentally observed distributions in Fig. 5.1

g(pk) = Zg(~p)d2~p⊥=ZZ f1(~p1)·f2(~p −~p1)d2~p⊥d3~p1.(5.2)

The momentum distribution fi(~pi) of each electron is given by

fi(~pi) = ω0sin 2 ~pi~

2!e−αikp2

ik−αi⊥p2

i⊥,(5.3)

with

αik=ω2(2Ip,i)2/3

3E3;αi⊥=(2Ip,i)1/2

E[129].(5.4)

Here, ωis the laser field frequency, Ip,i is the ionization potential of O2(i= 1) and

O+

2(i= 2) respectively, Eis the amplitude of the electric field of the light wave,

and ~

Ris the vector of the internuclear separation. The sin term in Eq. (5.3) takes

the antibonding symmetry of the valence molecular orbital into account [144, 145].

In Eq. (5.3) ω0only determines the maximum value of the distribution while it has

no influence on its shape.

5.4. Modeling of sequential ionization 81

After integration in Eq. (5.2) we obtain:

g(pk) = 1 + 1

2cos pkRke−1

4(1

α1⊥+1

α2⊥)R2

⊥

−cos α1k

α1k+α2k

pkRke−R2

⊥

4α2⊥−

4(α1k+α2k)

−cos α2k

α1k+α2k

pkRke−R2

⊥

4α1⊥−

4(α1k+α2k)

2cos α1k−α2k

α1k+α2k

pkRke−1

4(1

α1⊥+1

α2⊥)R2

⊥−

α1k+α2k.(5.5)

Here Rkand R⊥are projections of ~

Ronto the axis parallel and perpendicular to the

electric field direction respectively.

Finally we average over all possible orientations of the internuclear axis with

respect to the external electric field:

¯g(pk) = Z2π

0Zπ

g(pk)sinθ dθ dϕ. (5.6)

Fig. 5.4: Electron sum-momentum distribution calculated for sequential double ioniza-

tion of O2(solid line) compared with the corresponding experimental results

(dashed line): (a) at 1.7×1014 W/cm2, (b) at 2.2×1014 W/cm2.

Fig. 5.4 shows the results of our calculation for sequential double ionization of

O2at two light intensities (solid lines). The dashed lines are the sum-momentum

82 5. Molecules in high-intensity ultra-short laser pulses

distributions obtained in the experiment (Fig. 5.1). Obviously, the theoretical dis-

tributions are narrower than the experimental ones.

The theoretical approach stated above is known to overestimate the width of

the momentum distribution when applied to sequential ionization of atoms. If the

O2molecule in our experiment is ionized sequentially one would therefore expect

that the experimental width of the electron sum-momentum distribution is narrower

than the theoretical one. From this we conclude that the double ionization of O2does

not proceed sequentially. Despite the absence of any characteristics of NSDI, double

ionization of the O2molecule is thus expected to be non-sequential! The question,

however, remains whether recollision is the underlying ionization mechanism.

5.5 NSDI model calculation for molecules

Let us now assume that the main mechanism for molecular NSDI is recollision.

We will test this hypothesis by comparing the theoretical and experimental data.

The theoretical model includes the recollision scenario as well as the symmetry

of the initial state orbitals of the valence electrons in the molecule. If correct, the

model should be able to reproduce the qualitative differences in the sum-momentum

distributions and final electron momentum correlations for N2and O2.

The calculation is based on a classical analog of the quantum mechanical S-

matrix for recollision double ionization [91, 87]. The model has been described in

detail in the previous chapter on atoms (Sec. 4.5). According to this, the differential

probability for double ionization |S(~p1, ~p2)|2is proportional to a form factor |V~p1,~p2,~

k|2

[90]. In case of molecules only the form factor in Eq. (4.7) will change, depending

on the molecular structure and the type of electron interaction at recollision.

We factorize the form factor into a contribution g1(~

k) describing the transition of

the first electron to the continuum and a second contribution g2(~p1, ~p2,~

k) describing

the inelastic recollision

|V~p1,~p2,~

k|2=|g1(~

k)·g2(~p1, ~p2,~

k)|2.(5.7)

~p1and ~p2are the final state momenta of the two photoelectrons and ~

kis the inter-

mediate state momentum of the recolliding electron.

The inelastic scattering matrix element g2is calculated in Born approximation

for the transition of the second electron bound in the single particle orbital ψ2(~x)

to the continuum upon recollision of the first electron

g2(~p1, ~p2,~

k) = Zd3~x d3~x0e−i~p1~x−i~p2~x0Ve(~x, ~x0)ei~

k~x ψ2(~x0).(5.8)

The wave function of the bound electron ψ2(~x) is constructed by the method of

linear combination of atomic orbitals (LCAO) for bonding (g) and antibonding (u)

5.5. NSDI model calculation for molecules 83

molecular orbitals

ψ2,g =c1

e−κr1

+c2

e−κr2

ψ2,u =c1

e−κr1

r1−c2

e−κr2

.(5.9)

Here, c1and c2are normalization constants which are determined by the internuclear

separation |~

R|. The binding energy κ2

2=Ebdefines κ. The interaction of the two

nuclei with the bound electron is described by means of a δ-potential

V(~r) = V0δ(~r1)∂

∂r1

r1+δ(~r2)∂

∂r2

r2.(5.10)

Here, ~r1and ~r2are vectors pointing from the respective nuclear centers to the

electron.

In this way we have constructed a simple model molecule which incorporates the

main features of a real homonuclear molecule, in particular the two-center effect. As

a consequence, the molecular orbital of our model molecule is either σg(bonding) or

σu(antibonding). Although the relevant O2orbital is not a σuone, it nevertheless

reflects main features of the initial state symmetry, and thus allows one to test the

statement that the symmetry of the initial valence orbital is decisive. The electron-

electron interaction Ve(~x, ~x0) at the instant of recollision is assumed to be a contact

interaction

Ve(~x, ~x0) = V0δ(~x −~

x0)∂

∂|~x −~

x0||~x −~

x0|.(5.11)

With these approximations, g2reads explicitly

g2,g/u(~p1, ~p2,~

k) = h2,g/u(|~

R|)

2I+

p(t) + (~

k−~p1−~p2)2(cos (~

k−~p1−~p2)~

2(g)

sin (~

k−~p1−~p2)~

2(u),(5.12)

where h2,g/u is a function of the internuclear separation |~

R|alone. It has a constant

value, since |~

R|is assumed fixed in our model, and thus does not influence the

properties of the momentum distribution . I+

p(t) is the time dependent ionization

potential of the singly charged ion in the external electric field of the light wave (see

Eq. (4.4)).

The transition matrix element of the first electron from the ground state of the

molecule to the ionization continuum is split into an ionization rate part R(t) and

a geometry part g1. The ionization rate R(t) is calculated similar to that for atoms

using the ADK tunneling rate [10]. The geometry part g1is incorporated in the form

factor and is given by

g1(~

k) = Zd3~x e−i~

k~x V(~x)ψ1(~x),(5.13)

84 5. Molecules in high-intensity ultra-short laser pulses

Fig. 5.5: Calculated e−-e−momentum correlation for the model molecules: (a) N2at

1.5×1014 W/cm2; (b) O2at 1.7×1014 W/cm2.

where V(~x) is the interaction potential of the active electron with the singly charged

ion core (see Eq. (5.5)). ψ1(~x) is the wave function of the valence orbital occupied

by the electron removed first to a plane wave intermediate Volkov state. ψ1(~x) is

constructed in the same way as ψ2(~x) using the LCAO method. The term g1then

appears as an intermediate state interference term

g1,g/u(~

k) = h1,g/u(|~

R|)(cos ~

2(g)

sin ~

2(u).(5.14)

In Eqs. (5.12) and (5.14), the subscript (g) is for the σgand (u) for the σuvalence

orbital. Finally, we average again over all orientations of the internuclear axis of the

molecule.

The calculated electron momentum correlation after NSDI is shown in Fig. 5.5

for N2at the light intensity of 1.5×1014 W/cm2(a) and for O2at 1.7×1014 W/cm2

(b). These theoretical results can be compared to the corresponding experimental

data at the same light intensities in Fig. 5.3 (b,c). For N2, we find a correlation

similar to that observed in the experiment with a significant amount of electron

pairs with small and similar momenta pei,k, i = 1,2. The calculation for N2clearly

shows an electron momentum correlation that is different from that of Ar at the

same light intensity (Fig. 4.10 (a)). Thus, compared to the corresponding atom the

presence of the two atomic centers in a molecule leads to a significantly different

electron momentum correlation in the final state after NSDI at the same light in-

tensity. The calculation also indicates that the specific ”sausage” shape of the N2

distribution is mainly determined by final state interference through the matrix

5.5. NSDI model calculation for molecules 85

Fig. 5.6: e−-e−momentum correlation calculated with g1(~

k) = 1 in Eq. (5.7) for model

molecules: (a) N2at 1.5×1014 W/cm2; (b) O2at 1.7×1014 W/cm2.

element g2,g(~p1, ~p2,~

k) (5.12). In contrast, g1,g(~

k) has only little effect on the final

momentum correlation. To demonstrate this we assume g1= 1. Fig. 5.6 (a) shows

the e−-e−momentum correlation calculated at 1.5×1014 W/cm2for g1= 1. The

specific shape of the distribution is quite similar to that in Fig. 5.5 (a).

For the model molecule corresponding to O2an electron momentum correlation

significantly different from N2is found (Fig. 5.5 (b)). Despite the slightly higher

light intensity, it is concentrated near pe1,k=pe2,k= 0. This is in agreement with

the experimental finding at the same light intensity (Fig. 5.3 (c)). The g1,u(~

k) matrix

element is found to be responsible for the electron momentum correlation which is

localized at zero momentum for O2. The sin term in Eq. (5.14) leads to a reduction

of recollision electrons with small ~

k. They are emitted near the extrema of the

oscillating electric field, where the tunnel ionization probability is maximum. These

electrons with small momenta ~

kin the intermediate state are mainly responsible for

the emission of final state electrons with large pei,k. The predominant influence of

the sin term in Eq. (5.14) on the final state electron momenta can easily be seen

if we compare the e−-e−momentum correlation in Fig. 5.5 (b) with that in Fig.

5.6 (b), where g1= 1 is assumed. The distribution with g1= 1 becomes stretched

along the diagonal pe1,k=pe2,k. Thus, different from N2, the antibonding initial state

symmetry of the model molecule corresponding to O2shows up mainly through the

g1,u(~

k) matrix element.

A full quantitative agreement between the experiment and the model calculation

is not expected because of the simplicity of the model. In a certain range of laser

intensities our model correctly reproduces the qualitative difference between the two

molecular systems with bonding and antibonding initial state symmetry. Fig. 5.7

86 5. Molecules in high-intensity ultra-short laser pulses

Fig. 5.7: Calculated electron sum-momentum distributions for N2(a,c,e,g) and O2

(b,d,f,h) model molecules at the same light intensities as in Fig. 5.1. The arrows

are located at pk= 4pUp.

5.5. NSDI model calculation for molecules 87

shows the results of the model calculation for the sum-momentum distributions of

the two photoelectrons produced in NSDI. Similar to real N2and O2molecules,

the width of the momentum distribution for the σginitial state molecular orbital

is about 4pUpand about twice as large as that for the σuorbital. Moreover, our

model calculation reproduces qualitatively the main tendency of the sum-momentum

distribution for N2with increasing light intensity, including its strong change in

shape at 2.5×1014 W/cm2(compare Fig. 5.7 and Fig. 5.1).

In case of O2, the deviations between our calculations and the experimental re-

sults become stronger with increasing light intensity. The calculated sum-momentum

distribution narrows rather than broadens. The main reason for this is suspected in

the initial state symmetry of the valence molecular orbital, which is taken as σuand

not the correct πg. A further reason is probably the approximation for the instant

Fig. 5.8: Calculated e−-e−momentum correlation for the model molecules: O2at 1.9×

1014 W/cm2(a) and at 2.2×1014 W/cm2(b); N2at 1.8×1014 W/cm2(c) and

at 2.5×1014 W/cm2(d).

88 5. Molecules in high-intensity ultra-short laser pulses

of recollision, where an e−-e−contact interaction potential in velocity gauge was

assumed. Our calculations for atom have revealed the gauge dependence of the final

electron momentum distribution, especially at higher light intensities. As can be seen

in Fig. 5.8, for molecules f(pe1,k, pe2,k) stretches along the diagonal pe1,k=−pe2,kin

a similar way as for atoms.

At the lowest light intensities the agreement between the model and the exper-

iment is lost (compare Fig. 5.7 (a,b) with Fig. 5.1 (a,b) respectively). In contrast

to the experiment, the calculated electron sum-momentum distribution develops a

double hump structure. This discrepancy has probably its origin in our classical

approach. The classically accessible phase space for the recolliding electron where

Ekin,max(t)≥I+

p(t) gets very small around pei,k= 0. A quantum mechanical calcu-

lation may improve the situation for low light intensities.

So far we have discussed momentum distributions that are averaged over the

orientation of the internuclear axis. This corresponds to our experimental conditions.

For a fixed molecular orientation our model calculation shows that the final electron

momentum distribution changes strongly with the alignment of the molecule with

respect to the polarization direction of the light wave. Fig. 5.9 (a,c) displays the

electron-electron momentum correlation with the molecular axis ~

Raligned parallel

with respect to the polarization direction ~

Eof the light wave. The same distributions

are shown in Fig. 5.9 (b,d) for molecules aligned perpendicular to ~

E. The upper panel

shows the results for N2and the lower one for O2. Obviously, for N2the momentum

correlation averaged over all orientations (Fig. 5.5 (a)) is mainly determined by the

events where ~

R⊥~

E. The contribution of electrons emitted at ~

Rk~

Eis orders of

magnitude smaller. This is due to sin θbeing small in the orientation average integral

in Eq. (5.6) for θ(~

R, ~

E) close to zero. The same is true for O2if ~

Rk~

E. Additionally,

for ~

R⊥~

E, due to the antibonding symmetry of O2, electron waves emitted from the

two nuclear centers cancel each other on the observation axis along the electric field

direction. This destructive interference results in a negligibly small contribution to

the momentum correlation from ~

R⊥~

Ein Fig. 5.9 (d). In this case intermediate

angles 0 < θ(~

R, ~

E)< π/2 determine the main contribution to the e−-e−momentum

distribution after averaging.

Fig. 5.10 shows the relative contribution of the photoelectron pairs at different

θ(~

R, ~

E) to the full orientation-averaged e−-e−momentum correlation N(θ)



N(θ)dθ . The

maximum contribution for the O2model molecule (left panel) is found at an angle

of about 35 deg. For N2(right panel), the dependence on the molecular orientation

changes from a minimum at 0 deg, i.e. along the laser field direction, to a maximum

for the perpendicular orientation of the molecule with respect to the field axis. It is

important to note that the molecular orientation dependence in our model is only

taken into account in the form factor |V~p1,~p2,~

k|2in Eq. (4.7). The ionization rate R(t)

for a molecule is assumed to be the same as for an atom with the corresponding

5.5. NSDI model calculation for molecules 89

ionization potential, i.e. independent of the molecular orientation in the laser field.

Fig. 5.9: Calculated e−-e−momentum correlation similar to that in Fig. 5.5 but for the

fixed alignment of the molecular axis with respect to the light beam polarization

axis. ~

Rk~

E(right panel), ~

R⊥~

E(left panel). N2at 1.5×1014 W/cm2(a,b); O2

at 1.7×1014 W/cm2(c,d).

Actually, several theoretical models indicate that the strong field single electron

ionization rate of molecules depends on the molecular orientation with respect to the

laser field [151, 153, 154, 155, 156]. Recently, it has been found experimentally that

N2molecules aligned parallel to the ionizing electric field of the linearly polarized

light wave get ionized four times more likely than molecules aligned perpendicular

to the field [157]. As has been discussed above, ultra-short laser pulses ⩽50 fs

do not align molecules like N2and O2. In this case, orientation effects cannot be

observed directly in the experiment because of the isotropic distribution of molecular

orientations. Nevertheless, it has been shown by DeWitt et al. within the so-called

structure-based tunneling model [151], as well as in calculations by Tong et al. based

90 5. Molecules in high-intensity ultra-short laser pulses

Fig. 5.10: Relative contribution to the e−-e−momentum correlation after NSDI of the

photoelectron pairs ejected at different molecular orientation with respect to

the laser field direction θ(~

R, ~

E0).

on molecular ADK theory [154], that molecular orientation has a significant effect

on the tunneling ionization rate. Molecules with anisotropic electron distribution

like O2have a small ionization probability if they are aligned along the laser field

direction [154]. As a result, the molecular orientation effects contribute to the overall

ionization rate averaged over randomly oriented molecules in an intense laser field. In

particular, this may lead to a suppressed ionization rate of molecules with anisotropic

electron distribution like O2.

Summarizing, our NSDI calculations for the model molecules corresponding to N2

and O2are able to reproduce the main features of the electron momentum distribu-

tions found experimentally. They also indicate that the e−-e−momentum correlation

is mainly determined during tunneling of the first electron for O2and during the

inelastic recollision for N2. The initial state symmetry of the molecular orbital is

found to be responsible for the differences between N2and O2in the final state after

NSDI. We have also shown the influence of the molecular orientation on the electron

momentum correlation, which can not be resolved in our experiment.

5.6 Other theoretical models and their relevance in

experiments

We have seen that the differences between N2and O2found in our experiment

on double ionization by ultrashort high-intensity laser pulses can be successfully ex-

5.6. Other theoretical models and their relevance in experiments 91

plained by an initial state symmetry effect. Molecules with bonding and antibonding

symmetry of the initial state valence orbitals can give rise to entirely different elec-

tron momentum correlations in the final state after NSDI. The same symmetry effect

has been found to be responsible for suppressed single ionization and the reduced

yield of low-energy electrons in ATI spectra for O2with respect to Xe, which has

the same ionization potential [144, 145]. However, destructive interference due to

the antibonding symmetry of the electronic ground state is not the only mechanism

which explains the suppressed single ionization of O2.

An alternative explanation has been suggested by Guo in [158]. He introduced

a so-called charge-screening correction to the tunneling theory, that allows one to

describe correctly the observed suppressed yield of O+

2ions. This model assumes

that the detailed electronic structure of the molecule plays the key role in strong

field tunneling ionization. In particular, a molecule with the spatially symmetric

two-electron wave function (a singlet state) is expected to behave in a strong laser

field like a structureless atom with the same ionization potential. In contrast, if

an electron is removed from a molecule with an antisymmetric two-electron wave

function (a triplet state) via strong field tunneling ionization, the remaining one has

no time to adjust its position and to minimize the energy. As a consequence, the ion

core charge seen by the ionizing electron effectively increases and ionization becomes

less probable than in a weak field. The ADK calculation for O2, which has a triplet

ground state, with the corrected Ipand with an effective charge well reproduces the

experimentally observed suppressed single ionization. This effect does not appear for

N2, which has a closed shell electronic structure (singlet state), as well as for rare

gas atoms where the outermost electrons are assumed to be uniformly distributed

around the ion core [158].

A drawback of the charge-screening model as it is introduced in [158] is that

the presence of the two nuclei in the molecule is ignored. The molecule is simply

modeled by a single ion core and the valence electrons. The applicability of the

model to realistic molecular systems is questionable, not to mention its extension

to NSDI of molecules. So far only one ab initio 1D quantum-mechanical calculation

has been performed for a model atom with a spatially symmetric (singlet state)

and antisymmetric (triplet state) two-electron wave function [159]. The calculation

has shown that the electronic structure (singlet or triplet) may influence the NSDI

process.

A further theoretical approach has investigated the possible influence of vibra-

tional motion on the single ionization rate of diatomic molecules [160]. The calcula-

tion predicts a reduced ionization rate for O2and H2molecules but the effect due to

the vibrational motion seems to be too small to explain the experimental data. The

relevance of the model for NSDI is even more doubtful since the typical vibrational

period of the molecule is much longer than the optical cycle of 800 nm laser radiation

which sets the scale for NSDI.

92 5. Molecules in high-intensity ultra-short laser pulses

Recently, a simple classical analysis for NSDI of N2and O2by high-intensity

short laser pulses [161] has been performed. In this calculation only the final stage

of the ionization process is analyzed, i.e. the evolution of the two photoelectrons in

combined Coulomb and external fields after a highly excited molecular complex has

formed during electron rescattering. This analysis does not indicate any difference in

the final e−-e−momentum correlation for N2and O2with similar initial conditions.

The result of this calculation strongly suggests that the difference between N2and

O2observed in our experiment has its origin in an earlier stage of the ionization

process.

Both mechanisms, the interference due to bonding or antibonding symmetry

of the molecular orbital and the charge-screening correction, are quite general and

thus should apply at least to every diatomic molecule. However, a recent experiment

on single ionization of diatomic molecules in a strong laser field seems to indicate

that neither the first nor the second mechanism alone is sufficient to quantitatively

predict integral ion yields in single ionization of diatomics in general [162]. Besides

N2and O2, single ionization of S2and F2molecules has been investigated in this

study. F2has a singlet electronic ground state and an ionization potential similar to

N2, but the symmetry of the ground state molecular valence orbital is antibonding

(πg) as for O2. According to the interference mechanism, single ionization of F2

is thus expected to be suppressed with respect to N2[145]. This, however, is not

observed in the experiment [162]. The S2molecule has a triplet electronic ground

state with antibonding symmetry of the active molecular valence orbital similar to

O2, but with significantly lower ionization potential Ip= 9.26 eV. The ionization

suppression effect in S2, which is expected to be less pronounced compared to O2

due to a larger internuclear separation, is also not confirmed experimentally. On

the other hand, also the charge-screening model with its prediction of suppressed

ionization for F2fails in describing the experimental data.

Further experimental investigation of diatomics, including heteronuclear, has

shown that molecules with singlet electron configuration exhibit more atomlike single

ionization properties, while molecules with doublet and triplet electronic structure

appear to have suppressed ionization relative to atoms with the same Ip[163].

Contrary, the most recent experimental data for Cl2, a molecule with the closed-

shell antibonding valence orbital and Ipsimilar to Xe, seem to confirm the hypothesis

that suppressed single ionization in a strong laser field appears due to the antibond-

ing symmetry effect of the molecular valence orbital [164]. This result may indicate

that the absence of ionization suppression for F2could be an exception from the

main tendency.

A tunneling ionization theory developed for molecules [154, 155] was able to

reproduce several experimental results for the ratio of ionization signals of molecules

with respect to the corresponding atoms with similar Ip. Based on the ADK model

for tunneling ionization of atoms, this theory has been modified to account for

5.6. Other theoretical models and their relevance in experiments 93

the symmetry differences in the initial-state wave function of a valence electron in

the molecular orbital. The molecular tunneling ionization theory fails, however, to

predict the observed single ionization ratio for F2:Ar [154].

Just recently, the quantum-mechanical strong-field S-matrix calculations per-

formed for N2and O2in [144, 145] have been extended to other linear molecules

[156] in order to study the effect of molecular orbital symmetry and molecular ori-

entation on the photoelectron energy spectra and angular distributions as well as on

the total ionization rates. For a molecule with πsymmetry of its valence molecular

orbital a minimum in the angular distribution after single electron ionization has

been found at 0 deg and a maximum at about 40 deg with respect to the polariza-

tion direction. Such an angular distribution occurs if the molecule is aligned along

the polarization axis but not for randomly oriented π-symmetric molecules and not

for σ-symmetric molecules. These symmetry-induced effects predicted for angular

distributions [156] are remarkable in light of recent experimental observations by

Alnaser at al. [165]. In this experiment signatures of symmetry effects have been

observed in angular distributions for the decay of excited electronic states of N2+

and O2+

2produced in NSDI of N2and O2via rescattering. The observed angular

distribution spectra exhibit a similar structure as the theoretically predicted one

for the single electron ionization of N2and O2[155, 156]. This result suggests that

the strongest symmetry-induced anisotropic angular effects appear at the first step

of NSDI, i.e. when the outermost valence πor σelectron is removed via tunneling

ionization[165].

Finally, a strong-field approximation calculation by Kjeldsen and Madsen [166],

similar to that in [156], for the N2molecule should be mentioned. It has been per-

formed in both, the length and velocity gauge. The results of this calculation for

the photoelectron angular distribution as well as for the dependence of the single

ionization rate on the molecular orientation are different for the two gauges. In the

length gauge the ionization rate is maximum when the molecule is aligned along

the laser field. In contrast, in the velocity gauge the maximum ionization rate is

found at the perpendicular orientation of the molecule. Our model calculations have

also been performed in velocity gauge. This may be the reason for the maximum at

θ(~

R, ~

E) = 90 deg found for N2in the dependence on θ(~

R, ~

E) of the relative contri-

bution of the photoelectron pairs to the e−-e−momentum correlation averaged over

molecular orientations (see Fig. 5.10).

Summarizing, in this chapter we have presented and discussed our results on

NSDI of diatomic molecules. The main features found in our experiment can be

understood within the framework of the rescattering model with instantaneous elec-

tron impact ionization of the singly charged molecular ion. The molecular structure

decisively influences the final e−-e−momentum correlation. The differences found

94 5. Molecules in high-intensity ultra-short laser pulses

between N2and Ar momentum correlation spectra indicates that this influence is

more pronounced in double ionization than in single ionization. The N2molecu-

lar structure strongly affects the inelastic recollision. On the other hand, the e−-e−

momentum correlation for O2is mainly determined in the first stage of the ioniza-

tion process during the transition of the first electron to the ionization continuum.

Our model calculation shows that the final electron correlation can be significantly

different depending on the alignment of the molecule in the external field of the

light wave. Averaging over randomly oriented molecular axes washes out much of

the alignment-induced effects. The hypothesis that the symmetry of the initial state

orbitals occupied by the active valence electrons plays a decisive role for NSDI of

molecules seems to be confirmed by our experiment for N2and O2. However, its rel-

evance for other diatomic molecules is still not clear and has to be proven. Finally,

for a correct description of molecular NSDI, which is limited in our semiclassical

model, a full quantum-mechanical calculation is strongly desired.

6. Summary and outlook

In the present work, the ionization dynamics of the atoms Ar and Ne and the

diatomic molecules N2and O2in an intense laser field has been investigated ex-

perimentally by means of correlated electron-ion momentum spectroscopy based on

COLTRIMS. The main interest of this experimental study is non-sequential double

ionization (NSDI), a nonlinear process which occurs in an electromagnetic field of

highly intense laser radiation. The characteristic features of NSDI are most promi-

nent for linearly polarized light. The typical light intensities for experimental obser-

vation of NSDI are of the order of 1014 W/cm2.

In the present experiment 35 fs laser pulses with a central wave length of 800

nm and a repetition rate of 100 kHz have been used as a source of laser radiation.

To the present knowledge, the underlying ionization mechanism of NSDI is rescat-

tering [62, 63]. In this mechanism an electron, ionized by the electric field of a light

wave, is driven back by this field to its singly charged parent ion core, which is then

ionized by the electron impact. In this way the two photoelectrons get correlated.

In previous experiments spectra for the momentum correlation of the two photo-

electrons after NSDI were observed for several atoms [16, 68, 72]. They revealed

differences in NSDI of Ar and Ne. These atom specific features of NSDI are cor-

roborated in this experiment. Whereas the results for Ne can be well understood

within the electron impact ionization mechanism, the assumption of an additional

ionization mechanism seems to be necessary to interpret the Ar data [68, 69, 167].

A possible scenario for this mechanism is instantaneous electron impact excitation

of the singly charged ion with subsequent electric field ionization of the excited ion.

The experiment shows that for Ar this second ionization mechanism becomes more

significant with decreasing light intensity. In contrast, for Ne the instantaneous elec-

tron impact ionization prevails in a large range of light intensities, from the classical

threshold I+

pto far above.

Within the classical rescattering scenario a threshold for instantaneous electron

impact ionization is expected when the maximum kinetic energy of the recolliding

electron Ekin,max is comparable with the ionization potential of the singly charged

ion I+

p. It becomes energetically forbidden at Ekin,max < I+

p. Nevertheless, in this

experiment, structures characteristic of NSDI by instantaneous electron impact have

been discovered for Ar at light intensities where Ekin,max < I+

p, i.e. below threshold.

It has been shown that sub-threshold e−impact ionization may be understood in

96 6. Summary and outlook

the framework of the rescattering model by taking into account the shifting ioniza-

tion threshold in the presence of the external electric field of the light wave at the

instant of recollision. In the quasistatic approach this field instantaneously reduces

the ionization potential of the singly charged ion, thus facilitating e−impact ioniza-

tion below threshold. A semiclassical calculation based on this improved rescattering

model yields e−-e−momentum correlations which are in good qualitative agreement

with the experimental results including those at light intensities below NSDI thresh-

old (see Chapter 4). A threshold lowering also explains why no abrupt decrease in

the total ion yield ratio [A++]/[A+] was found at Ekin,max ∼I+

pfor various noble

gases. Consequently, one can conclude that the electric field of the light wave es-

sentially influences the electron scattering event in NSDI. The dependence on the

light intensity of the electron sum-momentum distribution f(p⊥) in the plane per-

pendicular to the light polarization direction seems to confirm this conclusion also

for Ekin,max I+

The comparison of this experiment with several theoretical calculations shows

that specific discrepancies exist. For the time being, the most successful model calcu-

lations yielding results qualitatively similar to the experiment are the semiclassical

and quantum-mechanical calculations applying the three-body contact interaction

[87, 85, 90]. Also the most complete semiclassical trajectory calculation with the

full three-body Coulomb interaction [99, 100] is able to reproduce the experimental

findings. The quantitative disagreement with experiment of the available theoretical

results is not surprising in view of the approximations made.

Another interesting problem addressed in this thesis is non-sequential double

ionization of molecules. At present it is not sufficiently explored experimentally as

well as theoretically. Previous experiments revealed footprints of NSDI for several

molecules in the dependence on light intensity of the total doubly charged ion yield

[58, 59, 136, 137, 138]. However, the ionization mechanism could not be identified

completely. The main goals of this experiment have been to identify the molecular

NSDI mechanism and to investigate the influence of molecule specific effects on the

ionization process, and thus on the final electron momentum correlation. Homonu-

clear diatomics like N2and O2are good candidates for this investigation. Their

relatively simple structure with two identical nuclear centers allows one to model

them theoretically more accurate and to understand the influence of the molecular

structure in comparison with atoms.

The e−-e−momentum correlation found in the experiment differs significantly for

N2and O2. The distribution f(pe1,k, pe2,k) for N2exhibits structures characteristic

of instantaneous e−impact ionization while this is not the case for O2. Both, N2

and O2show new features in the final electron momentum correlation which are not

observed for atoms. The comparison of N2molecule with the Ar atom, which has

a similar ionization potential I+

p, at the same light intensity reveals most obviously

the effect of two nuclei (see Chapter 5).

A semiclassical analysis based on the rescattering model has been applied to

the N2and O2molecules in order to test the applicability of the electron impact

ionization mechanism. The model calculation includes the symmetry of the initial

molecular state occupied by the active valence electrons. The results of this calcula-

tion qualitatively reproduce the main features found in the experiment. Thus, two

important conclusions can be drawn. First, instantaneous electron impact ionization

seems to be the general mechanism for NSDI of atoms and molecules. Second, the

molecular structure decisively influences the final e−-e−momentum correlation after

NSDI. It has been found that the bonding symmetry of the initial state of the mole-

cule strongly modifies the inelastic electron recollision. In contrast, the antibonding

symmetry mainly affects the first stage of NSDI during the electron transition to

the ionization continuum via tunneling.

Although this experiment on N2and O2indicates that the symmetry of the

initial state orbital occupied by the valence electron plays a decisive role for NSDI,

further experimental investigations are necessary to generalize this statement for all

diatomic molecules.

From the theoretical point of view, complete NSDI quantum-mechanical calcu-

lations which correctly describe the interaction of the recolliding electron with the

bound e−and with the ion core are strongly desirable for atoms as well as for di-

atomic molecules. Also instantaneous electron excitation followed by electric field

ionization has to be taken into account for a correct description of NSDI of Ar and

He.

98 6. Summary and outlook

Appendix A

C++ code for the data analysis

The analysis of the measured data has been performed by means of the ”CoboldPC”

software (ReontDek Handles GmbH). We have modified the main C++ code and

adjusted it for our experiment. It allows to generate a dynamic link library (dll) file

which, being recalled from the ”CoboldPC”, determines the data analysis.

///////////////////////////////////////////////////////////////////////////

//USER DEFINED ANALYSIS PART CALLED FROM COBOLD MAIN PROGRAM

///////////////////////////////////////////////////////////////////////////

//COORDINATES//

double x0e, y0e;

double Ua, L_accel, L_free, T0e, Tc, e, Wc, L_accel_i;

double /*Xi0, Yi0, */Ti1, Ti2;

int Start_Sort,Stop_Sort;

//FUNCTIONS FOR THE ANALYSIS//

//FUNCTION TO EXTRACT DATA FROM LeCroy TDC//

int merge(int high_byte,int low_byte)

{

return ((high_byte & 0x000000FF)*256 + (low_byte & 0x000000FF));

}

//FUNCTION TO EXTRACT INFORMATION OF THE TDC CHANNEL//

int extr_channel(int in)

{

return ((in & 0x00007C00)/0x00000400); //extract channel

}

//TEST WHETHER THE RAW DATA MAKES SENSE//

int bit(int x)

100 Appendix A. C++ code for the data analysis

{

return ((x & 0x00000100)/0x00000100);

}

double position(double t1,double t2)

{

return (t2-t1);

}

double time_test(double time_x1, double time_x2, double time_of_flight)

{

return (time_x1+time_x2-2*time_of_flight);

}

//EXTRACT THE RAW DATA FROM THE PHILLIPS TDC//

int merge_ion(int data)

{

return (data & 0x00000FFF);

}

//EXTRACT INFORMATION OF THE TDC CHANNEL

int extr_channel_ion(int in)

{

return ((in & 0x0000F000)/0x00001000);

}

double time_test_ion(double time_x1, double time_x2)

{

return (time_x1+time_x2);

}

double F (double p, double E, double la, double lf, double t0, double t)

{

return (p/E + sqrt(p*p + 2*E*la)/E + lf/sqrt(p*p + 2*E*la) + t0 - t);

}

double DF (double p, double E, double la, double lf)

{

return (1/E + (p/sqrt(p*p + 2*E*la))*(1/E - lf/(p*p + 2*E*la)));

}

double P_transver (double w, double r, double t)

101

{

return (w*r/fabs(2*sin(w*t/2)));

}

///////////////////////////////////////////////////////////////////////////

CDAN_API LPCTSTR AnalysisGetInformationString()

{

return LPCTSTR("DAN for CoboldPC 2002");

}

///////////////////////////////////////////////////////////////////////////

//PARAMETERS FOR THE ANALYSIS//

CDAN_API BOOL AnalysisInitialize(CDoubleArray *pEventData,

CDoubleArray *pParameters, CDoubleArray *pWeighParameter)

{

x0e = (pParameters->GetAt(200));

y0e= (pParameters->GetAt(201));

Ua = (pParameters->GetAt(202));

L_accel = (pParameters->GetAt(203))/0.529E-10;

L_free = (pParameters->GetAt(204))/0.529E-10;

T0e = 2.07E+7*(pParameters->GetAt(205));

Tc = 2.07E+7*(pParameters->GetAt(206));

e = (pParameters->GetAt(207));

Wc = 6.283185/(2.07E+7*(pParameters->GetAt(208)));

Ti1 = (pParameters->GetAt(209));

Ti2 = (pParameters->GetAt(210));

L_accel_i = (pParameters->GetAt(211))/0.529E-10;

//Xi0 = (pParameters->GetAt(212));

//Yi0 = (pParameters->GetAt(213));

// First event to be taken into account//

Start_Sort = (int) (pParameters->GetAt(212) + 0.1);

Stop_Sort = (int) (pParameters->GetAt(213) + 0.1);

srand( (unsigned)time( NULL ) );

return TRUE;

}

//COORDINATES USED IN THE ANALYSIS//

CDAN_API void AnalysisProcessEvent(CDoubleArray *pEventData,

CDoubleArray *pParameters, CDoubleArray *pWeighParameter)

{

102 Appendix A. C++ code for the data analysis

int i, n, m, channel = 0;

int Event_Counter;

double hit_1[32], hit[32], pos_e[2], re, pzi1, pzi2;

double x = 0;

double Xi, Yi/*, TOFi,Ri, pxyi*/;

double pe, pe0, E_accel, T, pxye, R, T1, Q, Ekin, p_calc;

bool good_ev;

double trans = 3.7996e5;

//EVENT COUNTER//

Event_Counter = (int) (pParameters->GetAt(214) + 1.1);

pParameters->SetAt(214, Event_Counter);

for (n = 0; n < 32; n++)

{

hit_1[n] = (double) (-1000.0);

hit[n] = (double) (-1000.0);

}

for (n = 0; n < 43; n++)

{

pEventData->SetAt(30 + n,(double)(-1000.0));

}

//EXTRACTION OF TIME SIGNALS FROM THE LeCroy TDC (ELECTRONS AND IONS)//

for (n = 0; n < 6; n++)

{

if (bit((int)(pEventData->GetAt(2*n+1)+0.1)) == 1)

{

channel = extr_channel((int)(pEventData->GetAt(2*n+2)+0.1));

if ((0 <= channel) && (channel < 5))

{

hit_1[channel] = (double)merge((int)(pEventData->GetAt(2*n+1)+0.1),

(int)(pEventData->GetAt(2*n+2)+0.1));

//set a real value

pEventData->SetAt(30 + channel, hit_1[channel]);

}

if (channel == 10)

{

103

hit_1[5] = (double)merge((int)(pEventData->GetAt(2*n+1)+0.1),

(int)(pEventData->GetAt(2*n+2)+0.1));

//set a real value

pEventData->SetAt(35, hit_1[5]);

}

else break;

}

for (n = 7; n < 12; n++)

{

if (bit((int)(pEventData->GetAt(2*n)+0.1)) == 1)

{

channel = extr_channel((int)(pEventData->GetAt(2*n+1)+0.1));

if ((0 <= channel) && (channel < 5))

{

hit_1[channel+6] = (double)merge((int)(pEventData->GetAt(2*n)+0.1),

(int)(pEventData->GetAt(2*n+1)+0.1));

//set a real value

pEventData->SetAt(36 + channel , hit_1[channel+6]);

}

else break;

}

//EXTRACTION OF TIME SIGNALS FROM THE Phillips TDC (IONS)//

for (n = 24; n < 28; n++)

{

channel = extr_channel_ion((int)(pEventData->GetAt(n)+0.1));

if ((channel >= 0) && (channel <= 3))

{

hit[channel] = (double)merge_ion((int)(pEventData->GetAt(n)+0.1));

//set a real value

if (hit[channel] < 4095)

pEventData->SetAt(41+channel, hit[channel]);

}

104 Appendix A. C++ code for the data analysis

// TEST WHETHER ALL DELAY-LINE SIGNALS ARE PRESENT//

//presence of laser pulse and time-of-flight is garanteed by hardware//

m = 0;

for (channel = 1; channel < 5; channel++)

{

if (0.0 < hit_1[channel]) m = m+1;

}

for (channel = 7; channel < 11; channel++)

{

if (0.0 < hit_1[channel]) m = m+1;

}

pEventData->SetAt(45, (double) (m));

// all signals are present!

if ((Event_Counter > Start_Sort) && (Event_Counter < Stop_Sort))

{

if (m == 8)

{

for (channel = 0; channel < 2; channel++)

{

// position X,Y of e

pos_e[channel] = position(hit_1[2*channel+1], hit_1[2*channel+2]);

pEventData->SetAt(46+channel,position(hit_1[2*channel+1],

hit_1[2*channel+2]));

// T2+T1-2*Tf of e

pEventData->SetAt(48+channel,time_test(hit_1[2*channel+1],

hit_1[2*channel+2], hit_1[0]));

// position X,Y of i

pEventData->SetAt(50+channel,position(hit_1[2*channel+8],

hit_1[2*channel+7]));

// position X,Y of i

Xi = ((rand() % 100)*0.01 - 0.5) + position(hit_1[8],hit_1[7]);

Yi = ((rand() % 100)*0.01 - 0.5) + position(hit_1[10],hit_1[9]);

// T2+T1-2*Tf of i

pEventData->SetAt(52+channel,time_test(hit_1[2*channel+8],

hit_1[2*channel+7], hit_1[6]));

}

105

//CALCULATION OF THE ELECTRON TIME-OF-FLIGHT//

if ((hit_1[5] > 0.0) && (hit_1[0] > 0.0))

pEventData->SetAt(54,hit_1[5] - hit_1[0]);

if ((hit_1[5] > 0.0) && (hit_1[6] > 0.0) && (hit_1[0] > 0.0))

{

// ELECTRON CYCLOTRON MOTION//

re = sqrt((pos_e[0]+((rand() % 100)*0.01 - 0.5)-x0e)*(pos_e[0]+

((rand() % 100)*0.01 - 0.5)-x0e) +

(pos_e[1]+((rand() % 100)*0.01 - 0.5)-y0e)*(pos_e[1]+

((rand() % 100)*0.01 - 0.5)-y0e));

pEventData->SetAt(55,re);

//CALCULATION OF THE ION TIME-OF-FLIGHT//

pEventData->SetAt(56,hit_1[6] + hit_1[5] - hit_1[0]);// TOF of i

}

n = 0;

if (0.0 < hit_1[0]) n = n+1;

if (0.0 < hit_1[6]) n = n+1;

pEventData->SetAt(57, (double) (n));

//CALCULATION OF THE ION POSITION FROM THE Phillips TDC//

n = 0;

for (channel = 0; channel < 4; channel++)

{

if (hit[channel] < 4095) n = n+1;

}

if (n == 4)

{

for (channel = 0; channel < 2; channel++)

{

pEventData->SetAt(58+channel,position(hit[2*channel+1],

hit[2*channel]));

// T2+T1-2*Tf of i//

pEventData->SetAt(60+channel,time_test_ion(hit[2*channel],

hit[2*channel+1]));

}

106 Appendix A. C++ code for the data analysis

good_ev = true;

for (i = 0; i < 6; i++)

{

if (hit_1[i] <= 0.0) good_ev = false;

}

//MOMENTUM COMPONMENT Pz FOR THE ELECTRON//

E_accel = Ua/(54.4*L_accel);

pe = Tc*E_accel;

T = 2.07E+7*(hit_1[5] - hit_1[0] + ((rand() % 100)*0.01 - 0.5));

T1 = T - T0e;

{

pe0 = pe;

pe = pe0 - F(pe0, E_accel, L_accel, L_free, T0e, T)/DF(pe0, E_accel,

L_accel, L_free);

}

while (fabs(pe - pe0) >= e);

if ((hit_1[5] > 0.0) && (hit_1[0] > 0.0))

{

pEventData->SetAt(62, (double) (pe));

//MOMENTUM COMPONMENT Pxy FOR THE ELECTRON//

R = 5.4E+6*re;

pxye = P_transver (Wc, R, T1);

pEventData->SetAt(63, (double)(pxye));

//ELECTRON EMITION ANGLE theta//

Q = acos(pe/sqrt(pe*pe + pxye*pxye));

pEventData->SetAt(64, (double)(Q));

//KINETIC ENERGY OF THE ELECTRON//

Ekin = 0.5*(pe*pe + pxye*pxye)*0.77;

pEventData->SetAt(65, (double)(Ekin));

}

//MOMENTUM COMPONMENT Pz FOR i+, i++//

if ((hit_1[5] > 0.0) && (hit_1[6] > 0.0) && (hit_1[0] > 0.0))

{

107

//i+

pzi1 = (-hit_1[6] - hit_1[5] + hit_1[0] + Ti1 +

((rand() % 100)*0.01 - 0.5)) * trans * Ua / L_accel_i;

pEventData->SetAt(66, (double) (pzi1));

//i++

pzi2 = (-hit_1[6] - hit_1[5] + hit_1[0] + Ti2 +

((rand() % 100)*0.01 - 0.5)) * 2.0 * trans * Ua / L_accel_i;

pEventData->SetAt(67, (double) (pzi2));

//MOMENTUM Pz FOR THE SECOND ELECTRON//

p_calc = -pzi2 - pe;

pEventData->SetAt(68, (double) (-pzi2 - pe));

//MOMENTUM CONSERVATION FOR SINGLE ELECTRON IONIZATION (SHOULD BE ZERO)//

pEventData->SetAt(69, (double) (-pzi1 - pe));

//MOMENTUM DIFFERENCE FOR DOUBLE IONIZATION//

pEventData->SetAt(70, (double) (-pzi2 - 2*pe));

//CONDITRION THAT THE DETECTED ELECTRON IS COMING AT THE DETECTOR FIRST//

if (pe <= p_calc)

{

pEventData->SetAt(71, (double)(Q));

pEventData->SetAt(72, (double)(Ekin));

}

return;

}

CDAN_API void AnalysisFinalize(CDoubleArray *pEventData,

CDoubleArray *pParameters, CDoubleArray *pWeighParameter)

{

}

108 Appendix A. C++ code for the data analysis

Appendix B

C++ code for NSDI model calculation

This program has been written in the course of the present work and used for the

NSDI model calculations for Ar, Ne, N2, and O2.

//Program for a corr-Spectrum

//written by E. Eremina

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

#include <float.h>

#include <iostream>

#include <conio.h>

#include <fstream>

#include <string>

#include <time.h>

using namespace std;

double Ip1, Ip2, Up;

double w;//laser frequency

double F, F0;//laser field strengt

////////////functions///////////////////////////////////////////////////

//momentum function

double mom (double U, double x0,double x)

{

return (2*sqrt(U)*(cos(x)-cos(x0)));

}

/*position function

double pos (double U, double y0,double y)

{

110 Appendix B. C++ code for NSDI model calculation

return (2*sqrt(Up)/w*((sin(y)-sin(y0))-cos(y0)*(y-y0)));

}

double Fu (double y0, double y)

{

return ((sin(y)-sin(y0))-cos(y0)*(y-y0));

}

double delta (double p1, double p2, double I, double Er)

{

return (Er - I - p1*p1/2 - p2*p2/2);

}

double Ip (double I, double s, double F0)

{

return (I - 2*sqrt(2*fabs(F0*sin(s))));

}

////////////////////////////////////////////////////////////////////////

void main()

{

string dataname_new;

double Ip1, Ip2, Up;

double w;//laser frequency

double F, F0;//laser field strengt

int l, i;

long double Pi = 3.1415926535;

double v;

double b=1.5;

double a[3][12100];

double wt,wt0, wt1, wt_r;

double rate;//ADK-ionization rate

double Ilas, e;

double Eret, Exess;

double p1, p2;

double Nu;

cout << "Matrix: " << endl;

////////////////////////////////////////////////////////////////////////

for (l = 0; l < 12100; l++)

111

{

a[0][l] = (abs(l/110)-50)*0.1;

if (l%110 == 0)

{

i=0;

{

a[1][l+i] = (i-50)*0.1;

i++;

}

while (i<110);

}

a[2][l] = 0;

}

///////////////////////////////////////////////////////////////////////

//laser intensity in units of W/cm^2

cout << "Laser intensity in W/cm^2: " << endl;

cin >> Ilas;

F0 = sqrt(Ilas/351E14);

w = 5.7;//in a.u. for lambda = 800nm

//ionization potential of Ar+ and Ar2+ in a.u.

Ip1 = 15.76/27.21;

Ip2 = 27.63/27.21;

v = 2*Ip1;

e = 1E-4;//accuracy

//corresponding ponderomotive potential in a.u.

Up = 5.9712*(Ilas/1E14)/27.21;

////////////////////////////////////////////////////////////////////////

wt0 = Pi/2;

{

wt = wt0 + Pi/40;

F = F0*sin(wt0);

rate = (4/F)*exp(-(2*sqrt(8*Ip1*Ip1*Ip1))/(3*F));

112 Appendix B. C++ code for NSDI model calculation

// cout << "wt0 = " << wt0 << " Rate = " << rate << endl;

// condition for the return is

// 2*sqrt(Up)/w*((sin(y)-sin(y0))-cos(y0)*(y-y0)) = 0

//=> Fu(wt0,wt_r)->0

wt_r = wt0+0.1;

{

wt1 = wt_r + 0.00001*Pi;

wt_r = wt1;

}

while (fabs(Fu(wt0,wt_r/*-0.00001*Pi*/)) >= e);

Eret = 0.5*mom(Up,wt0,wt_r)*mom(Up,wt0,wt_r);

cout << "wt0 = " << wt0 << " wt_r = " << wt_r << " Eret = "

<< Eret << " Ip= " << Ip(Ip2,wt_r,F0) << endl;

////////////////////////////////////////////////////////////////////////

// drift after rescattering ionization if (Pi/2 <= wt0 <= Pi)

for (l = 0; l < 12100; l++)

{

p1 = a[0][l] - 2*sqrt(Up)*cos(wt_r);

p2 = a[1][l] - 2*sqrt(Up)*cos(wt_r);

if (delta(p1,p2,Ip(Ip2,wt_r,F0),Eret)>0)

{

Exess = delta(p1,p2,Ip(Ip2,wt_r,F0),Eret);

Nu = a[2][l]+ rate*Exess;// rate*Exess;

// cout << "wt0 = " << wt0 << " wt_r = " << wt_r

<< " Ip = " << Ip2 << endl;

a[2][l] = Nu;

}

// symmetry part due to drift after rescattering ionization

//if (3*Pi/2 <= wt0 <= 2*Pi)

for (l = 0; l < 12100; l++)

{

113

p1 = a[0][l] + 2*sqrt(Up)*cos(wt_r);

p2 = a[1][l] + 2*sqrt(Up)*cos(wt_r);

if (delta(p1,p2,Ip(Ip2,(wt_r+Pi),F0),Eret)>0)

{

Exess = delta(p1,p2,Ip(Ip2,(wt_r+Pi),F0),Eret);

Nu = a[2][l]+ rate*Exess;// rate*Exess;

a[2][l] = Nu;

}

wt0 = wt;

}

while (wt0 < (Pi-0.001*Pi));//

////////////////////////////////////////////////////////////////////////

cout << "Name of the new file: " << endl;

cin >> dataname_new;

ofstream newdata(dataname_new.c_str(),ios::out);

if (!newdata)

{

cerr << "could not be written!" << endl;

exit(1);

}

newdata.setf(ios_base::scientific);

for (l = 0; l < 12100; l++)

{

newdata << a[0][l] << "\t" << a[1][l] << "\t" << a[2][l] << "\n";

}

cout << "Ready :)" << endl;

////////////////////////////////////////////////////////////////////////

getch();

return;

}

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Publications

The following papers have been published in the course of the present work:

•E. Eremina, X. Liu, H. Rottke, W. Sandner, M.G. Sch¨atzel, A. Dreischuh,

G.G. Paulus, H. Walther, R. Moshammer, and J. Ullrich,

Influence of molecular structure on double ionisation of N2and O2by high

intensity ultra-short laser pulses,

Phys. Rev. Lett. 92, 173001 (2004);

•E. Eremina, X. Liu, H. Rottke, W. Sandner, A. Dreischuh, F. Lindner, F. Gras-

bon, G.G. Paulus, H. Walther, R. Moshammer, B. Feuerstein and J. Ullrich,

Laser-induced non-sequential double ionisation investigated at and below the

threshold for electron impact ionization,

J. Phys. B 36, 3269-3280 (2003);

•X. Liu, H. Rottke, E. Eremina, W. Sandner, E. Goulielmakis, K. O´Keeffe,

M. Lezius, F. Krausz, F. Lindner, M.G. Sch¨atzel, G.G. Paulus, H. Walther,

Nonsequential double ionisation at the single-optical-cycle limit,

Phys. Rev. Lett. 93, 263001 (2004).

Acknowledgements

At the end of my thesis I would like to thank all those people who have contributed

to the success of this work.

First of all I want to express my deepest gratitude to Prof. Dr. Wolfgang Sandner

for giving me the opportunity to carry out this exciting research work in his group at

the Max Born Institute, for encouraging and supporting me throughout all my PhD,

for promoting the presentation of the experimental results at numereous conferences

and meetings.

I am most indebted to Dr. Horst Rottke for guiding this project and my work,

for providing most of the ideas realized in this experiment, for being always open

for discussions, giving me advice and helping in all questions. The success of our

experiments and my thesis in particular would not be possible without his enormous

contribution and support. I also thank Horst for thoroughly proofreading of my

thesis.

I am very grateful to my colleagues from the Max Plank Institute for Quantum

Optics, Gerhard Paulus, Michael Sch¨atzel, Alexander Dreischuh, and Fabrizio Lind-

ner for providing the laser system for our experiments, for the collaboration, for the

support and hospitality during the time of our experiment in Garching.

I also thank my colleagues from the Max Plank Institute f¨ur Kernphysik for

collaboration, especially Robert Moshammer and Bernhold Feuerstein for helping

during the adjustment of our first coincident electron-ion measurements.

I would especially like to thank my colleagues at the Division B2 of the Max

Born Institute:

Elena Gubbini for the friendly atmosphere and heartiness, her aid during all the

time and especially at writing my thesis;

Dr. Xiaojun Liu for the excellent cooperation, assistance in many problems, and

giving an incentive for our semiclassical model calculations;

Rainer Hoffmann for helping with electrical equipment during the build-up phase,

as well as Ralph Ewers for the mechanical part;

Stefan Gerlach for helping with the computer problems;

Silvia Szlapka and Bettina Haase for the perfect organizing work, mutual under-

standing and cordiality;

I would like to express my sincere thanks to my colleagues and friends Michael

Scht¨atzel, Dr. Artem Rudenko, Dr. Sven Gnutzmann for their countenance and

making me confident with my abilities during the difficult stages of my thesis. I am

also indebted to Sven for his help with the correction of my thesis.

Moreover, many thanks to the people, not mentioned by name, who contributed

to the successful completion of this thesis.

Finally, I heartily thank my family for the emotional support and encouragement.