Ultrafast light sources and methods for attosecond pump-probe experiments

Ultrafast light sources and

methods for attosecond

pump-probe experiments

Davide Fabris

March 2015

Thesis submitted in partial fulfilment of the

requirements for the degree of Doctor of

Philosophy of Imperial College London and the

Diploma of Imperial College

Laser Consortium

Quantum Optics and Laser Science Group

Department of Physics

Imperial College London

London, SW7 2BW

United Kingdom

2

Declaration

I hereby certify that the work presented in this thesis is my own unless stated

otherwise.

Davide Fabris

The copyright of this thesis rests with the author and is made available un-

der a Creative Commons Attribution Non-Commercial No Derivatives licence.

Researchers are free to copy, distribute or transmit the thesis on the condition

that they attribute it, that they do not use it for commercial purposes and that

they do not alter, transform or build up on it. For any reuse or redistribution,

researchers must make clear to others the licence terms of this work.

3

Abstract

In this thesis I describe the development of novel light sources to be applied

in attosecond pump-probe experiments, together with new methods dedicated

to their characterisation and optimisation.

Femtosecond pulses are a necessary tool to enter the attosecond domain.

For this reason their development is a key element to unlock more capabili-

ties in pump-probe attosecond experiments. The dynamics of generation and

compression of few-cycle femtosecond pulses has been studied in a hollow core

fibre system. The carrier envelope phase stability performance under increas-

ing input power to the fibre system has been examined systematically, showing

the effects of ionisation on the carrier envelope phase stability.

Two characterisation techniques have been developed to measure ultrafast

femtosecond pulses. A version of the d-scan technique has been demonstrated

in the single shot regime for the first time, extending the utility of this diag-

nostic. An all optical technique (ARIES) for the characterisation of the full

waveform of a femtosecond pulse has been developed, exploiting the high har-

monics generation process and the sensitivity of the cut-off emission to the

instantaneous amplitude of the generating electric field.

The main results of the thesis are concerned with the generation of isolated

attosecond pulses in new spectral regions. Vacuum ultraviolet few-femtosecond

and attosecond pulses have been generated by filtering with metallic foils the

high harmonics emission driven by sub-4 fs pulses, and were characterised

with the attosecond streaking technique. When using indium as spectral filter

a pulse duration of 1.7±0.1 fs was measured at a photon energy of 15 eV. When

using tin as spectral filter a pulse duration of 585 ± 31 as was measured at a

photon energy of 20 eV. The experimental techniques developed in this thesis

allow these pulses to be generated simultaneously with a XUV pulse with a

measured duration of 270± 25 as. This work will open new opportunities for

pump-probe experiments, for example studies of ultrafast charge migration in

large molecules.

5

Contents

Abstract 3

List of Figures 12

List of Tables 13

1 Introduction 15

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Author’s contribution . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Author publications . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Theory and background 21

2.1 Ultrafast lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Mathematical description . . . . . . . . . . . . . . . . . 21

2.1.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.3 Non-linear optics . . . . . . . . . . . . . . . . . . . . . . 31

2.1.4 Short pulse production . . . . . . . . . . . . . . . . . . . 36

2.2 Characterisation techniques . . . . . . . . . . . . . . . . . . . . 39

2.2.1 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.2 Frequency Resolved Optical Gating . . . . . . . . . . . . 42

2.2.3 Spectral Phase Interferometry for Direct Electric field

Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.4 CEP measurement and stabilisation . . . . . . . . . . . . 48

2.3 Strong field physics . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 CONTENTS

2.3.1 Ionisation . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3.2 High Harmonics Generation . . . . . . . . . . . . . . . . 54

3 Ultrafast light sources development 63

3.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.1.2 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Hollow core fibre pulse compression . . . . . . . . . . . . . . . . 68

3.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . 75

3.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . 78

3.3 HHG for isolated VUV and XUV pulses production . . . . . . . 81

3.3.1 Gating the HHG emission . . . . . . . . . . . . . . . . . 81


3.3.3 Spectral results . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Plasma HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . 94

4 New methods for attosecond pump-probe experiments 99

4.1 Dispersion-Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.1.1 Standard d-scan . . . . . . . . . . . . . . . . . . . . . . . 100

4.1.2 Single shot implementation . . . . . . . . . . . . . . . . 101

4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 Attosecond Resolved Interferometric

Electric-field Sampling . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


4.3 Electron Velocity Map Imaging and ion time-of-flight . . . . . . 120

4.3.1 VMI background theory . . . . . . . . . . . . . . . . . . 120

4.3.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

CONTENTS 7

5 Attosecond streaking of XUV and VUV pulses 127

5.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . 128

5.1.1 Retrieval algorithms for streaking experiments . . . . . . 130

5.2 XUV and VUV streaking experiment . . . . . . . . . . . . . . . 134


5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3.1 XUV Results . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3.2 VUV Results . . . . . . . . . . . . . . . . . . . . . . . . 148

6 Conclusions and future work 161

Bibliography 165

Appendix A Retrieval algorithms 187

A.1 PCGPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

A.2 LSGPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Appendix B Copyright permissions 193

B.1 Item 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

B.2 Item 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

B.3 Item 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

B.4 Item 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

B.5 Item 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

B.6 Item 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

B.7 Item 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9

List of Figures

2.1 Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 CEP definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Dispersion effects . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Prism pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Chirped mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Non-linear optics . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 Second harmonic generation . . . . . . . . . . . . . . . . . . . . 34

2.8 Self-phase modulation . . . . . . . . . . . . . . . . . . . . . . . 35

2.9 Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10 Kerr-Lens Mode-Locking . . . . . . . . . . . . . . . . . . . . . . 37

2.11 CPA technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.13 Field autocorrelation setup . . . . . . . . . . . . . . . . . . . . . 40

2.14 Field autocorrelation trace . . . . . . . . . . . . . . . . . . . . . 40

2.15 Intensity autocorrelation setup . . . . . . . . . . . . . . . . . . . 41

2.16 Intensity autocorrelation trace . . . . . . . . . . . . . . . . . . . 42

2.17 Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.18 Projection algorithm . . . . . . . . . . . . . . . . . . . . . . . . 44

2.19 SPIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.20 SPIDER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.21 SEA-F-SPIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.22 Interferometry f − 2f . . . . . . . . . . . . . . . . . . . . . . . 48

10 LIST OF FIGURES

2.23 Ionisation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 51

2.24 ADK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.25 Three-step model . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.26 Typical HHG spectrum . . . . . . . . . . . . . . . . . . . . . . . 55

2.27 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.28 Long andShort trajectories . . . . . . . . . . . . . . . . . . . . . 58

2.29 HHG phase-matching . . . . . . . . . . . . . . . . . . . . . . . . 62

3.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 CEP fast loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Amplifier Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 CEP slow loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Fibre modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Fibre losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Fibre design parameters . . . . . . . . . . . . . . . . . . . . . . 74

3.8 Fibre setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.9 Focal spot size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.10 Fibre spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.11 Fibre CEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.12 Fibre transmission . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.13 Fibre gas pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.14 Fibre CEP stabilitye . . . . . . . . . . . . . . . . . . . . . . . . 80

3.15 Beamline layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.16 Gas jet schematics . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.17 Flat field spectrometer calibration . . . . . . . . . . . . . . . . . 86

3.18 XUV and VUV spectra . . . . . . . . . . . . . . . . . . . . . . . 87

3.19 Kr-Xe Low harmonics comparison . . . . . . . . . . . . . . . . . 87

3.20 Phase-matching conditions for the XUV and VUV spectra . . . 88

3.21 CEP scan for the XUV and VUV spectra . . . . . . . . . . . . . 89

3.22 Interaction effects on XUV and VUV spectra . . . . . . . . . . . 90

LIST OF FIGURES 11

3.23 Flat field XUV spectra CEP scan . . . . . . . . . . . . . . . . . 92

3.24 Four step model . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.25 Plasma plumes HHG . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1 Pulse measurement with d-scan . . . . . . . . . . . . . . . . . . 101

4.2 Prism for single shot d-scan . . . . . . . . . . . . . . . . . . . . 103

4.3 Single shot d-scan setup . . . . . . . . . . . . . . . . . . . . . . 104

4.4 Single shot d-scan calibration . . . . . . . . . . . . . . . . . . . 105

4.5 Single shot and scanning d-scan traces . . . . . . . . . . . . . . 106

4.6 Single shot and scanning d-scan results. . . . . . . . . . . . . . . 107

4.7 HHG CEP dependence . . . . . . . . . . . . . . . . . . . . . . . 109

4.8 ARIES concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.9 ARIES theoretical traces . . . . . . . . . . . . . . . . . . . . . . 112

4.10 ARIES setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.11 ARIES CEP measurement . . . . . . . . . . . . . . . . . . . . . 115

4.12 ARIES dispersion measurement . . . . . . . . . . . . . . . . . . 116

4.13 ARIES 2ω simulation . . . . . . . . . . . . . . . . . . . . . . . . 119

4.14 Electron VMI and ion TOF detectors . . . . . . . . . . . . . . . 120

4.15 Abel transform for VMI . . . . . . . . . . . . . . . . . . . . . . 121

4.16 VMI images for different β . . . . . . . . . . . . . . . . . . . . . 123

4.17 Electron VMI and ion TOF performance summary . . . . . . . 125

4.18 Electron VMI and ion TOF results . . . . . . . . . . . . . . . . 126

5.1 Theoretical streaking traces . . . . . . . . . . . . . . . . . . . . 130

5.2 Retrieval algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3 Retrieval algorithm comparison . . . . . . . . . . . . . . . . . . 133

5.4 TOF chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.5 Two part mirror assembly . . . . . . . . . . . . . . . . . . . . . 135

5.6 Filters for XUV and VUV pulse selection . . . . . . . . . . . . . 136

5.7 Combination of attosecond pulses . . . . . . . . . . . . . . . . . 137

5.8 Temporal overlap for straking experiment . . . . . . . . . . . . . 138

12 LIST OF FIGURES

5.9 TOF lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.10 TOF calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.11 CEP scan with the TOF spectrometer . . . . . . . . . . . . . . 143

5.12 XUV-NIR raw and smoothed streaking traces. . . . . . . . . . . 144

5.13 NIR pulse extraction from streaking traces . . . . . . . . . . . . 145

5.14 XUV pulse retrievals . . . . . . . . . . . . . . . . . . . . . . . . 147

5.15 Photon flux calibration . . . . . . . . . . . . . . . . . . . . . . . 150

5.16 Indium raw and smoothed traces . . . . . . . . . . . . . . . . . 153

5.17 Indium LSGPA results . . . . . . . . . . . . . . . . . . . . . . . 154

5.18 Calibration traces for the TOF nozzle . . . . . . . . . . . . . . . 156

5.19 Tin raw and smoothed traces . . . . . . . . . . . . . . . . . . . 157

5.20 Tin LSGPA results . . . . . . . . . . . . . . . . . . . . . . . . . 158

13

List of Tables

3.1 Visibility of harmonics generated in ablation plumes. . . . . . . 95

5.1 Photon flux estimate from count rate . . . . . . . . . . . . . . . 150

5.2 Photon flux measurement results . . . . . . . . . . . . . . . . . 151

5.3 Pulses properties . . . . . . . . . . . . . . . . . . . . . . . . . . 159

14

Acknowledgements

First of all many thanks to John for his great supervision. Not only he wasalways supportive and available for technical and scientific advice, but he wasalso always able to cheer us up when the morale was a bit down, either forlong and unfruitful scans in the lab, or just because it was another raining dayin London. Thanks to Jon for his relentless supervision and constant advicethroughout my PhD.

I would like to thank all the people of the Atto Lab. I would like to thankTobi for teaching me a lot, from the very basics of laser science, to the morenerdy features of Matlab. Thanks to Tom, for bringing new and interestingknowledge in our group, and to Will, for trying to educate people about propervacuum procedures and jazz music. Thanks to Zara, Daniel and Paloma, forhelping a lot in the lab and making it a nice place to be.

I would like to thank all the Laser Consortium, where I had the luck to findmore than just colleagues. A big thank to a big guy, Steffen, for the long chatsand the many pints we had together. Thanks to Jon and Peter for makinggoing to the pub a true British experience. Thanks to all the others, thanksDane for having that big brain of yours, thanks Felicity for the cow pictures.Thanks Sid, Malte Richard, Marco, Martin, Emma, Hazel, Zsolt, Bridgette,Emilio, Nick, Konstantin, Allan, Alvaro and Lukas for making the office awelcoming place.

I would like to thank the people that saved the day more than once. ThanksJudith and Marcia for making things happening, from purchase orders to sem-inars. Thanks Peter, Andy and Susan to turn our ideas in the equipment weare so proud of.

I would like to thank all the friends that made London feel a bit more likehome. Thanks to Gabriele, for being a good friend and for dragging me to thegym. Thanks to the Andreas (aka the colleagues) , for sharing the never-a-joymoments with a good laugh.

A very big thank, and something more, to Laura, for being always therefor me, despite the distance and the difficulties. We made it so far and I lookforward for what the future holds for us.

Finally the most important thank-you of all, to my family. Thanks tomy mum, Margherita, for worrying just enough, thanks to my dad, Antonio,for always taking care of us, thanks to my sister Marta, for putting up witha brother like me. Thank you a lot, thanks for your support and for beingalways there for me.

I hope I did not forget anyone. In that case just forgive me, blame thethesis submission anxiety, and, if we end up in a pub together, the first pint ison me.

Davide

15

Chapter 1

Introduction

1.1 Introduction

Laser technology provides the foundation for attosecond science. The develop-

ment of intense femtosecond lasers has unlocked the generation of attosecond

pulses via the process of high harmonics generation [1, 2]. These attosecond

pulses are a powerful tool to investigate dynamics at the edge of known time

scales [3]. The experimental developments both in the generation and the char-

acterisation of attosecond pulses have allowed scientists to perform time re-

solved experiments that challenge the theoretical models developed to describe

light-matter interactions. Pump probe have been used to study photoionisation

and tunnelling dynamics [4] confirming existing models [5], but also discover-

ing new effects like a delay in photoemission from different electronic states [6,

7].

Currently, one of the driving aspects of research in attosecond science is to

apply the techniques and knowledge gained in atomic systems to more com-

plex systems, such as molecules [8–10] and even condensed phase matter [11,

12]. The challenge is both technical and theoretical. Performing experiments

in atoms allowed an understanding of both the experimental technique and

the physical system under investigation thanks to well developed theoretical

models such as the strong field approximation for a single atom [13].

16 CHAPTER 1. INTRODUCTION

When moving to molecular systems the theoretical modelling is more com-

plicated. To begin with the geometry of the molecule has to be taken into ac-

count [10] with challenges that are not present in the atomic case. The molecule

geometry and the delocalised electronic structure makes the electron-electron

interaction important especially when the molecule is in a excited state [14].

To help deconvolve all these dynamics the experiments should be designed

to isolate a specific feature of the process under investigation. The stability

requirements and specifications of the experimental setup are therefore more

demanding to allow the acquisition of a reliable data set of sufficiently high

quality for the comparison with theoretical models. For this reason there is

great interest both in the development of new attosecond sources, able to spec-

trally target specific dynamics, and of new methods and techniques to increase

the amount of information that can be gathered from an experiment.

Regarding the development of new methods for attosecond science, an im-

portant topic is the characterisation of ultrafast pulses [15]. The precise knowl-

edge of the pulses applied to the experiment is vital to fully understand the

dynamics studied. On one hand rapid and easy to use diagnostics are ideal

for regular monitoring of the pulses, but on the other hand more sophisticated

techniques are required for a detailed characterisation of the pulse. In fact,

with the advent of extremely short femtosecond pulses [16], the characterisa-

tion of the exact waveform is required since pulses with the same envelope can

be significantly different. For these reasons techniques such d-scan [17] aim

for simplicity, while others like the petahertz oscilloscope [18] aim for carrier

waveform precision.

Moving to the field of development of new attosecond sources, it can be said

that the production of isolated attosecond pulses using an intense near-infrared

(NIR) femtosecond pulse has in the last decade become a well established and

understood process [3]. The development of this method has extended many

limits of attosecond pulses, from the highest photon energy reached (1.4 keV

[19]), to the shortest attosecond pulse measured (67 as [20]). These impressive

CHAPTER 1. INTRODUCTION 17

parameters are a demonstration of how far attosecond science has developed

since its beginning, but much is left to do in order to make attosecond sources

more versatile and readily accessible to the scientific community.

One essential requirement to extend the capabilities of attosecond science is

the development of attosecond pulses across different photon energies [19, 21]

to be applied to attosecond-pump attosecond-probe experiments. Most of the

existing schemes cannot produce attosecond pulses with enough photons to be

split and used in a pump-probe experiment (with few exceptions [22]). For this

reason the driving NIR field is often used with the attosecond pulse for pump-

probe. However the high NIR intensity can distort the observed dynamics and

its longer time duration lowers the time resolution of the experiment. To reach

different photon energies, different methods are being studied, from the use of

monochromators [23], to new HHG sources like ablation plumes [24]. Pulses

at low photon energies (few eV) are good candidates for the excitation step of

small molecules due to their high cross section in this spectral range [25]. Such

pulses can be used at relatively low intensity to drive single photon transition,

allowing significant simplifications in the understanding and modelling of the

exciting step of the experiment. Low order non linearities (including non linear

processes in photonic crystal fibres) have been exploited to generate pulses

with 4-7 eV photon energy and 3-18 fs pulse duration [26–29]. High harmonics

generation is a promising strategy to shorten the pulse duration and attosecond

pulses have been measured at this photon energy [21, 30]. However, the task of

delivering two independent and stable attosecond pulses to an experiment is

still extremely challenging and has been the main reasearch goal of this thesis.


1.2 Author’s contribution

I played a key role in the work presented in this thesis, as a part of a team

with Dr. T. Witting, Dr. F. Frank, Dr. W. Okell, Dr. T. Barillot and with Z.

Abdelrahman, D. Walke and P. Matia-Hernando.

For the results on the VUV pulse measurements I was, together with Dr.

T. Witting, the main researcher responsible for the attosecond streaking mea-

surement of the simultaneously generated VUV and XUV pulses. I carefully

optimised the laser system for these experiment and obtained the streaking

traces. I developed the LSGPA algorithm code for the analysis of the traces.

With fellow PhD students P. Matia-Hernando and D. Walke I optimised the

procedure for the photon flux calibration of the VUV pulses and gathered the

relevant data-sets. With J. Henkel and Dr. M. Lein (Institut fur Theoretische

Physik and Centre for Quantum Engineering and Space-Time Research, Leib-

niz Universitat Hannover) we interpreted the results of the TDSE simulations

and compared them with the experimental data.

I was closely involved in the data collection of the experimental campaign

led by Dr. W. Okell, studying the performance of the hollow core fibre CEP

stability.

I operated the laser system and assisted with the data acquisition during

the experimental campaign, led by PhD student Z. Abdelrahman, dedicated to

the spatial coherence characterisation of high harmonics generated in ablation

plumes.

Together with Dr. M. Siano I simulated the electrodes geometry for the

design of the electron VMI and ion TOF in close collaboration with Dr. T.

Barillot and D. Walke.

I performed the calculations related to the design of the single shot d-scan

and I participated with Dr. T. Witting to the experimental campaign in Porto

in collaboration with W. Holgado (University of Salamanca), Dr. F. Silva

(ICFO) and Prof. H. Crespo (Porto University).

CHAPTER 1. INTRODUCTION 19

Together with Dr. T. Witting I was involved in the experimental campaign

that led to the development of the ARIES technique in collaboration with Dr.

A. Wyatt (Science and Technology Facilities Council Rutherford Appleton

Laboratory) who performed the data analysis and with A. Schiavi and Prof I.

Walmsley (Oxford University).

1.3 Author publications

• Prediction of attosecond light pulses in the VUV range in a

high-order-harmonic-generation regime

J. Henkel, T. Witting, D. Fabris, M. Lein, P. L. Knight, J. W. G. Tisch,

J. P. Marangos

Physical Review A 87.4 (Apr. 2013), p. 043818. [31]

• Carrier-envelope phase stability of hollow fibers used for high-

energy few-cycle pulse generations

W. A. Okell, T. Witting, D. Fabris, D. Austin, M. Bocoum, F. Frank,

A. Ricci, A. Jullien, D. Walke, J. P. Marangos, R. Lopez-Martens,

J. W. G. Tisch

Optics Letters 38.19 (Oct. 2013), pp. 3918-3921. [32]

• Spatial coherence measurements of non-resonant and resonant

high harmonics generated in laser ablation plumes

R. A. Ganeev, Z. Abdelrahman, F. Frank, T. Witting, W. A. Okell,

D. Fabris, C. Hutchison, J. P. Marangos, J. W. G. Tisch

Applied Physics Letters 104.2 (Jan. 2014), p. 021122. [33]

• Synchronised pulses generated at 20 eV and 90 eV for attosecond

pump probe experiments

D. Fabris, T. Witting, W. A. Okell, D. J. Walke, P. Matia-Hernando,

J. Henkel, T. R. Barillot, M. Lein, J. P. Marangos, J. W. G. Tisch

Accepted for Nature Photonics


• Attosecond streaking on metallic films

W. A. Okell, T. Witting, D. Fabris, C. A. Arrel, J. Hengster, D. Y. Lei,

Y. Sonnefraud, M. Rahmani, D. Walke, S. A. Maier, S. Stankov, Th.

Uphues, J. P. Marangos, J. W. G. Tisch

Accepted for Optica

• Ultra-fast pulse characterisation with the d-scan technique: a

single shot implementation

D. Fabris, W. Holgado, F. Silva, T. Witting, H. Crespo, J. W. G. Tisch

In preparation for Optics Express

• Attosecond sampling of arbitrary optical waveforms (ARIES)

A. Wyatt, A. Schiavi, I. A. Walmsley, D. Fabris, T. Witting, J. P.

Marangos, J. W. G. Tisch

In preparation

21

Chapter 2

Theory and background

In this chapter I will introduce basic concepts related to ultra-fast lasers, their

characterisation and their interaction with matter.

2.1 Ultrafast lasers

Time resolved measurements have entered the attosecond domain due to the

huge improvements in ultrafast laser technology. The unique properties of such

pulses allow very high intensities to be reached, where non-linear processes can

be exploited to generate even shorter pulses reaching the XUV spectral range,

and to probe dynamical systems on very short time scales. This section will

introduce some of the most important concepts to describe and understand

ultra-fast pulses. The mathematical notation to describe broadband pulses

will be presented, allowing us to describe the effect that dispersion has on

such broadband pulses and to introduce the first concepts of non-linear optics.

2.1.1 Mathematical description

When only the time dependence of the electric field is considered (i.e. the

pulse is linearly polarised and spatially uniform) equations (2.1) and (2.2) can

22 CHAPTER 2. THEORY AND BACKGROUND

be used to describe a laser pulse in the spectral and temporal domains.

E (ω) =∣∣∣E(ω)∣∣∣ · eiϕ(ω), (2.1)

E(t) = |E(t)| · eiϕ(t). (2.2)

Assuming a linearly polarised pulse, only one component is necessary to de-

scribe the temporal profile of the pulse, while for an arbitrary polarised pulse

the full 2D description (Ex, Ey) is necessary. The two definitions provided in

the time and spectral domain are related by the Fourier transform:

E (ω) = F [E (t)] =1√2

∫ ∞

−∞E (t) eiωtdt

E(t) = F−1[E (ω)

]=

1√2

∫ ∞

−∞E (ω) e−iωtdω

(2.3)

This Fourier relation between time and frequency prescribes that a short

pulse in the time domain has a broad spectrum. For a given pulse shape

the spectral width Δν and the corresponding temporal width Δτ , are related

according the time bandwidth product (TBP):

ΔνΔτ ≥ a (2.4)

The constant a is specific for a given pulse shape assumed and the equality in

equation (2.4) is obtained only for a flat spectral phase ϕ(ω). In the case of

a Gaussian intensity profile ΔνΔτ = ln(4)/π ≈ 0.441. So for a given spectral

bandwidth the TBP can be used as an initial estimate for the possible pulse

duration. The assumption of a flat ϕ(ω) is however a strong one, especially

for broadband spectra such as those involved in the generation of ultra-short

pulses. For the characterisation of such pulses, therefore, both spectral am-

plitude and phase should be measured. Usually the spectral intensity I(ω)

of a visible or NIR pulse can be easily measured and related to the spectral

amplitude as I(ω) =∣∣∣E (ω)

∣∣∣2. The measurement of the spectral phase ϕ(ω)

CHAPTER 2. THEORY AND BACKGROUND 23

is, on the other hand, a challenging task and will be covered in more detail in

section 2.2.

A full description of a laser beam includes the spatial properties as well.

As a first approximation it is possible to decouple the temporal structure of

the pulse from the spatial properties:

E(x, y, z, t) = f(x, y, z)g(t) (2.5)

Here f(x, y, z) represents the spatial properties of the pulse, while g(t) the

temporal ones. Considering the spatial properties of an electromagnetic wave it

is possible to find a solution to Maxwell’s equations that corresponds to a beam

with a Gaussian transverse field distribution that, in cylindrical coordinates

(z, r), with z being the propagation axis, can be expressed as [34]:

E (r, z) =E0w0

w(z)exp

(− r2

w(z)2

)exp

{−i[kz − tan−1

(z

zR

)+

kr2

2R(z)

]}. (2.6)

Figure 2.1 shows the properties of such a beam.

The beam radius is defined as w(z) = w0

√1 + (z/zR)

2, with the Rayleigh

length zR = πw20/λ or the confocal parameter b = 2zR defining the beam

collimation. In fact the Rayleigh range represents the distance from the focus

where the beam area is twice the beam area at the focus. Therefore, considering

a beam path of length L, if b � L the beam can be considered collimated.

The beam radius at focus (z = 0) is w0, while k is the wave-vector k = 2π/λ

and R(z) is the radius of curvature of the wave-fronts R(z) = z(1 + z2r/z2).

A Gaussian beam can be completely described with the complex factor q(z)

[34] as follow:

E(z, r) ∝ exp

[−i πr

2

λq(z)

]with

1

q(z)=

1

R(z)− i

λ

πw2(z)(2.7)


-3 -2 -1 0 1 2 3

-4

-2

0

2

4

-0.5

0

0.5

1

Figure 2.1: Properties of a Gaussian beam. The black line shows the beam radiusw(z) while the dashed lines show the wave front curvature.

This parameter is useful as it allows, following the matrix treatment of stan-

dard geometrical optics, the propagation of a Gaussian beam in space applying

the transformation matrices:

qout =Aqin +B

Cqin +D. (2.8)

It is then possible to compute, from the input Gaussian beam qin, the modified

Gaussian beam described by qout, once the matrix elements(A BC D

)of the optical

path the beam is going through are known.

In general, the temporal and spatial properties of a pulse can be coupled and

therefore the assumption made in equation (2.5) may not be valid. For example

different pulse durations can be measured at different spatial positions, due to

a non spatially uniform distribution of the frequencies of the pulse spectrum.


2.1.2 Dispersion

A laser pulse will be now described from the spectral point of view, dropping

the tilde notation for convenience. From the definition in (2.1) the complex

phase ϕ(ω) can be expanded around the central frequency ω0 of the spectrum

|E(ω)| in a Taylor series:

ϕ(ω) =∞∑n=0

1

n!ϕn (ω − ω0)

n =

=ϕ0 + ϕ1 (ω − ω0) +ϕ2

2!(ω − ω0)

2 +ϕ3

3!(ω − ω0)

3 + . . .

(2.9)

The ϕn terms describe the effect of dispersion on a pulse, i.e. the fact

that waves of different frequencies travel at different velocities in a medium.

The first term is known as the carrier envelope phase (CEP). It is of particular

interest in few-cycle pulses, where the field envelope has a duration comparable

to the optical period (T = 2π/ω0). It gives the phase offset between the

maximum of the envelope and the maximum of the carrier as shown in figure

2.2.

Fiel

d S

treng

th (a

rb. u

.)

-1

-0.5

0

0.5

1 (a)

Time (fs)-5 -4 -3 -2 -1 0 1 2 3 4 5

Inte

nsity

(arb

u.)

0

0.2

0.4

0.6

0.8

1 (b)

Figure 2.2: Effect of different CEP’s in the waveform of a few cycle pulse. Goingfrom a cos-like pulse (ϕ0 = 0 dashed blue), to a sin-like pulse ϕ0 = π/2 changes themaximum field strength reached (a) and instantaneous intensity (b).


The peak of the carrier slips with respect the peak of the field envelope by

a quarter of the fundamental period when the CEP is changed from ϕ0 = 0 to

ϕ0 = π/2. This not only changes the maximum instantaneous field strength

and intensity, with a drop in field strength of 4.2% and in intensity of 8.2%

(assuming a pulse duration of 1.45 cycles), but also changes the temporal

symmetry of the pulse, going from an isolated peak in the case of ϕ0 = 0,

where the secondary peaks are 29% lower in intensity, to the situation of two

equal peaks when ϕ0 = π/2. These differences have a great impact in the

highly non-linear processes discussed in this thesis.

The second term in the spectral phase Taylor expansion is ϕ1 = dϕdω

∣∣ω=ω0

also known as group delay (GD). This term describes a translation in time

of the pulse, as it adds a pure phase factor that is linear in frequency. This

does not affect the pulse duration, but should be taken into account when

temporal overlap between two pulses is required, for example in pump-probe

experiments.

The next term in the expansion is ϕ2 = d2ϕdω2

∣∣∣ω=ω0

, called the group delay

dispersion (GDD). Being the derivative of the GD with respect to frequency, it

represents how much the arrival time of a given frequency component changes

with respect the others. Most materials have a positive GDD in the wave-

length region of interest (visible to NIR), therefore a positive GDD is defined

as “normal”: red components of the spectrum travel faster than the blue ones.

A negative dispersion is defined as “anomalous”. The effect of GDD is usually

defined for a given material via the group velocity dispersion (GVD). The spec-

tral phase accumulated by the pulse passing through a length L of a medium

is ϕM(ω) = k(ω)L. Taking the second derivative with respect frequency to

compute the GDD gives:

GDD =d2ϕM(ω)

dω2

∣∣∣∣ω=ω0

=

(d2k(ω)

dω2

∣∣∣∣ω=ω0

)L = (GVD)L (2.10)


In terms of the refractive index of the medium, the GVD and the following

term known as third order dispersion (TOD) can be written as:

GVD =λ3

2πc2d2n

dλ2

TOD = − λ2

4π2c3

(3λ2

d2n

dλ2+ λ3

d3n

dλ3

) (2.11)

The phase accumulated, however, cannot be solely described in terms of

GVD and TOD, and the higher order terms are generally summarised with

the Sellmeier equation:

n(λ) = 1 +∑i

Biλ2

λ2 − Ci

(2.12)

The effects of dispersion on a few cycle (3.6 fs at 800 nm) pulse are shown

in figure 2.3.

1 1.5 2 2.5 3 3.5 40

5

10

15

20(a)

Time (fs)-20 -15 -10 -5 0 5 10 15 20

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1 (b)

Figure 2.3: The effects of different amount of dispersion in a 3.6 fs pulse. Thespectrum (solid black line) and different phases are shown in (a). the respectivetemporal profiles are shown in (b). Note how the 1 m path in air can broadensignificantly the pulse (magenta dashed line).


When only ϕ0 and ϕ1 are present, the pulse is defined as Fourier Transform

Limited (FTL), because the shortest pulse duration possible is achieved with

the given spectrum. Any additional phase term broadens and modifies the

temporal profile. Note how a beam path of 1 meter in air broadens a 3.6 fs

pulse (with spectrum centred at 800 nm) to 13.6 fs.

It becomes evident that phase compensation of air path, vacuum windows

and optics, is required to deliver a short pulse to an experiment. Different

pulse compressor and pulse stretcher designs have been developed to remove or

introduce phase to a pulse. Most common are prism [35] and grating [36] based

stretchers and compressors, where spatially separating the spectral components

of a pulse allows a frequency dependent path length through the system. Figure

2.4 shows how these elements can be set up to introduce negative and positive

dispersion.

Lp

Lg

(a)

(d)(c)

(b)

Figure 2.4: Example of prism and grating stretchers (a), (c) and prism and gratingcompressors (b), (d). See equation (2.13) for an expression of the GDD provided bythe compressors shown.


The tuning of the dispersion can be performed by adjusting the separation

of the prisms or the gratings. Usually gratings set-ups are able to provide

more dispersion than prisms set-ups of similar size, but the two have different

signs for the TOD that is positive for a grating compressor, and negative for a

prism compressor. For this reason they can be used in series to achieve close to

zero GDV and TOD at the same time. When considering compressors, other

than their phase range and dimensions, a critical constraint is related to the

maximum intensity they have to withstand. In this aspect, a reflective grating

compressor can perform better than a prism compressor because the pulse does

not travel through the material of the compressor but it is simply reflected.

The full characterisation and analytical description of such devices is difficult,

as the temporal and spatial properties of the pulse are now coupled. For beams

with small beam size and symmetric (perfectly aligned) compressors set-ups

the following equations can be used [35, 36].

GDDprism =λ3

2πc24Lp{[

d2n

dλ2+(2n− n−3

)(dndλ

)2]sin β − 2

(dn

dλ

)2

cos β

}

GDDgrating = − λ3Lg

2πc2d2

[1−(λ

d− sin θi

)2]−3/2

,

(2.13)

where grating and prism separation is Lg and Lp respectively. The parameter

β is the spread angle after the first prism between the highest and lowest

frequency in the pulse, usually small enough that the approximation sin β ≈ 0

and cos β ≈ 1 is valid. The parameter d is the grating period and θi is the

angle of incidence of the incoming beam with respect the grating normal.

Other devices exist for the phase manipulation of a pulse that do not rely on

spatially dispersing the beam. It is possible to design chirped mirrors [37, 38],

a stack of dielectric materials of varying thickness such that the penetration

depth of different wavelengths varies introducing negative dispersion as shown

in figure 2.5. Another approach for phase manipulation is to use the coupling


between acoustic and optical waves in a birefringent medium. If different

wavelengths travel for different lengths on each axis, they will accumulate

different phases at the end of the birefringent medium. An acoustic wave

launched into the birefringent crystal creates a density-induced grating able

to diffract, at different spatial positions, different wavelengths onto the other

optical axis of the birefringent crystal [39]. In this way, as shown in figure 2.5,

it is possible to slow down the red components (fast) on the slow axis while

the blue components (slow) still travel on the fast axis.

fast

slow

acousticwave

(a)

(b)

birefringent medium

Figure 2.5: Chirped mirror (a), and Dazzler (b) concepts for inducing negativedispersion. The chirped mirrors are manufactured such that different wavelengthsare reflected at different depths. In a Dazzler an acoustic wave is used to coupledifferent wavelengths onto the fast/slow axis of a birefringent medium at differenttimes. In this way the optical path length of different frequencies can be manipu-lated. In figure is shown how the red components of a pulse can be slowed downwhen allowed to travel on the slow axis longer than the blue components.


2.1.3 Non-linear optics

With the strong optical fields available from lasers it is possible to induce a

non-linear response in the polarisation of a medium. Maxwell equations for

the electric and magnetic fields E and B with the only current component due

to the polarisation of the medium P can be written as:

∇ · E = 0 (2.14a)

∇× E = −∂B∂t

(2.14b)

∇ ·B = 0 (2.14c)

∇×B =1

c2∂E

∂t+ μ0

∂P

∂t(2.14d)

From these equation the wave equation can be obtained:

∇2E− 1

c2∂2E

∂t2= μ0

∂2P

∂t2(2.15)

So far no specific assumptions have been made, however to find the solutions

of equation (2.15) the relation between E and P has to be established. In the

usual case of a linear relation:

P = ε0χ(1)E, (2.16)

the usual plane wave solution is found together with the introduction of the

refractive index n =√1 + χ(1) = c/v, where v is the speed of the wave in

the medium and χ(1) is the linear susceptibility of the medium. In this linear

regime no new frequencies are generated, so when the system is driven at

frequency ω0, the polarisation field P will still be at frequency ω0 with only a

difference in speed. However this is just a first approximation, and due to the

high intensities achievable with lasers, a non-linear response of the medium can


be expected and therefore (2.16) should be expanded to higher order terms:

P = ε0[χ(1)E+ χ(2)E2 + χ(3)E3 + · · · ] (2.17)

When non-linear terms are introduced in the response of a system, new fre-

quencies can be created. In fact assuming the driving electric field to be:

E(t) = E1 exp(iω1t) + E2 exp(iω2t) + c.c. (2.18)

the second order response of the medium can be written as:

E2(t) =E21 exp(i2ω1t) + c.c.

+ E22 exp(i2ω2t) + c.c.

+ 2E1E2 exp[i(ω1 + ω2)t] + c.c.

+ 2E1E∗2 exp[i(ω1 − ω2)t] + c.c.

+ 2|E1|2 + 2|E2|2

(2.19)

In equation (2.19) are present frequencies that were absent in equation (2.18).

The new terms corresponds to the second harmonics of the fundamental fre-

quencies (2ω1 and 2ω2), and to their sum and difference frequencies (ω1 ±ω2).

It is possible to describe such phenomena in the photon picture as shown in

figure 2.6, where arrows pointing upwards are photons absorbed, and arrows

pointing downwards are photons emitted. The higher the order considered,

the more photons are involved in the process, and the less likely is the process

to happen.

Figure 2.6: Non linear optics processes depicted as multiple photon interactions.


Let us focus on second harmonic generation (SHG), that allows me to in-

troduce concepts useful in general in non-linear optics. The non-linear relation

between E and P makes finding the solution to equation (2.15) more difficult.

The first approximation usually introduced is the low depletion approxima-

tion where the input field E is assumed not to be perturbed by the non-linear

process occurring. In the case of SHG this means that the energy transfer be-

tween the fundamental frequency and the second harmonic is negligible, and

only equation (2.15) for the field at 2ω0 needs to be considered. The slowly

varying envelope approximation is usually made as well, where the field enve-

lope is assumed to be varying slowly both in time and space:

∂2E2ω0

∂t2� 2ω0

∂E2ω0

∂t

∂2E2ω0

∂z2� k2ω0

∂E2ω0

∂z(2.20)

With these assumptions, equation (2.15) can be written, in a co-moving frame,

propagating along z, in the form:

∂E

∂z= −iμ0ω

2

2kP exp(iΔkz) (2.21)

The Δk factor, called the phase mismatch factor, represents the difference

between the wave-vector at the frequency under investigation k2ω0 = 2ω0/v =

2ω0n(2ω0)/c and the wave-vector of the polarisation induced kpol = kω0+kω0 =

2ω0n(ω0)/c. The solution to equation (2.21) integrating over a length L is:

E2ω0 = −iμ0(2ω0)2

2k2ω0

P2ω0L exp

(iΔkL

2

)sinc

(ΔkL

2

)(2.22)

from which the intensity can be written:

I2ω0 ∝ |P2ω0 |2L2 sinc2(ΔkL/2) (2.23)

The coherent length, Lcoh = π/Δk, can be now introduced, and the meaning

of phase-matching understood. The production of the non linear signal in fact


strongly depends on Δk. When Δk = 0 the intensity of the second harmonic

grows quadratically with length. When Δk = 0 the output signal is reduced

and has maxima and minima with periodicity Lcoh as shown in figure 2.7.

0 1 2 3 4 5 6

Figure 2.7: Second harmonic generation phase-matching happens when Δk = 0,while the bigger Δk, the lower and more oscillating the signal is.

In the case of SHG the phase-matching condition can be written as

k2ω0 = 2kω0 i.e. n(ω0) = n(2ω0) (2.24)

This condition is usually prohibited by dispersion. However phase-matching

conditions can be met, for example with birefringent material using no(2ω0) =

ne(ω0), or other phase matching techniques [40].

Another important effect is the Pockels effect (a second order or χ(2) effect),

where a DC field can induce birefringence in an otherwise non-birefringent

crystal. In this way it is possible to switch on and off a quarter wave plate

rapidly, using pulsed voltages, in a device known as “Pockels cell”.

Important χ(3) processes include self phase modulation (SPM) and Kerr-

Lensing, that can be described as the effect of a non-linear refractive index in

time (SPM) and space (Kerr-lensing):

n(r) = n0 + I(r)n2 (Kerr-lens)

n(t) = n0 + I(t)n2 (SPM)(2.25)


In the case of Kerr-lensing, the effect for an intense beam with a Gaussian

profile is self-focusing. In fact the the intensity profile curvature in space, I(r),

will result in an effective refractive index curvature, with the central region of

the beam experiencing a stronger refractive index than the outer part and

therefore experiencing the same effects of a focusing optic. This effect can

lead to a self-sustaining feedback as the intensity will increase due to the self-

focusing, and may lead to damage to the non-linear medium. On the other

hand this mechanism can be usefully applied to achieve mode-locking.

In the case of SPM, the temporal phase accumulated through a non-linear

medium of length z can be computed as ϕ(t, z)SPM = k0n2I(t)z. In the same

way that a spectral phase can broaden the temporal profile of a pulse without

changing the spectral amplitude, a temporal phase can broaden the spectral

profile without changing the temporal shape. This can be exploited to spec-

trally broaden a pulse in order to support shorter pulse durations, as shown

in figure 2.8.

(a)

0 0.5 1

1.52

2.53

3.51.5 2 2.5 3 3.5

Inte

nsity

(arb

. u.)

0

0.5

1(b)

(c)

0 0.5 1

-50

0

50-50 0 50

Inte

nsity

(arb

. u.)

0

0.5

1(d)

Figure 2.8: Spectral broadening due to self-phase modulation. In (a) the spectralprofile is plotted as a function of propagation. In (c) the corresponding FTL tem-poral profile of (a) is shown. In (b) and (d) the detail of the computed profiles atbeginning (red) and at the end (dashed black) of the propagation through he nonlinear medium are shown.


In this case, a 30 fs pulse intensity profile has been used, and it can be noted

how, with propagation along L, new frequencies are generated by comparing

the line-out at the beginning (solid red) with the line-out at the end (dashed

black) in panel (a) and (b). If these frequencies are compressed appropriately

they can result in short pulses as shown in panels (c) and (d), where the FTL

temporal profile has been computed. This process is harnessed in hollow fibre

pulse compression that will be discussed in 3.2 which is the main method to

produce intense few cycle pulses.

2.1.4 Short pulse production

To obtain a pulse duration in the femtosecond range the required bandwidth

needs to span a range of tens of nanometres. For example a 30 fs pulse at

800nm requires a bandwidth of ≈30 nm. A gain medium capable of providing

such bandwidth is Titanium doped Sapphire (Ti:Sapphire) [41]. The modes

inside a laser cavity are the longitudinal modes with wavelength:

λn =2Lc

n(2.26)

where Lc is the laser cavity length and the integer n is the mode number. To

achieve short pulse durations the intra-cavity dispersion has to be compensated

so that the cavity modes can interfere to generate a short pulse once their

phases are locked, as explained in figure 2.9.

This procedure is called mode-locking [42, 43] and is usually achieved ex-

ploiting the Kerr-lensing effect [44]. Self-focusing can be exploited to spatially

select only intense pulses as shown in figure 2.10. When soft Kerr-lensing is

applied there is no physical aperture, but the focusing geometry and the os-

cillator cavity are designed so that only the more intense part of the beam,

undergoing self focusing, is overlapped with the pump beam. In this way the

mode locked, shorter and therefore more intense pulses are selected and am-

plified. This process can usually be initiated by a noise spike [45, 46] obtained


Mode number-25 -20 -15 -10 -5 0 5 10 15 20 25

Pha

se (r

ad)

0

2

4

6

8Modes Amplitude=1Random phaseLocked phase

Time (fs)-60 -40 -20 0 20 40 60

Inte

nsity

(arb

. u.)

0

0.5

1Random PhaseLocked Phase

Figure 2.9: Mode locking effect. If the phase of a set of modes is random (bluecircles), the overall signal will be spread in time and will result in a CW emission. Ifthe phase of the modes is locked (black crosses), constructive interference producesan isolated strong pulse.

by a quick movement of one of the optical components in the laser cavity (e.g.

by tapping it). This spike starts a positive feedback process that leads to the

mode-locked operation of the oscillator. With this technique oscillator pulses

as short as 4.4 fs have been produced [47].

The typical oscillator output, with only few nJ per pulse, can be amplified

to the millijoule level via chirped pulse amplification [48]. A direct amplification

of the short oscillator pulses will lead to damage of the optical components due

to self focusing. To prevent this occurrence the pulses are dispersed, or chirped,

non-linear medium

Intense pulseamplified

Figure 2.10: Kerr-Lens Mode-Locking concept: the intense part of the beam isfocused (red profile) and the optical designof the oscillator naturally favours theintense pulses with optimal overlap with the pump beam represented by the greenarea.


before amplification. In this way the pulses are stretched in time and the peak

power is lowered allowing the amplification step to happen without damaging

the system. Once amplification has occurred, the pulse can be compressed once

again and delivered to the experiment. Figure 2.11 shows the CPA technique.

stretcher

amplifier

compressor

Figure 2.11: Chirped pulse amplification. The pulse to be amplified is stretchedin time in order to lower the peak power. Once amplification has occurred the pulsecan be compressed once again.

The mechanism of chirped pulse amplification heavily relies on the accurate

phase dispersion compensation after amplification, from which the importance

of stretchers and compressors introduced in figure 2.4 appears clear. During

amplification other effects have to be taken in account, for example gain nar-

rowing can limit the amplification of the full bandwidth available from the

oscillator pulses. In fact the amplification process is not homogeneous for

different wavelengths [41], and the gain curve of the amplifying medium is

imprinted onto the spectrum of the output pulse.

Ti:Sapphire CPA systems are nowadays commercially available and can

provide pulses with a duration of ≈ 30 fs and pulse energy in the multi-

millijoule level. With these parameters, peak powers in the TW (1012 W) level

are easily achievable, giving access to the high harmonics generation (HHG)

process (see section 2.3), at the base of attosecond science.


2.2 Characterisation techniques

Pulse characterisation in the ultra-fast domain can be challenging since both

spectral amplitude and phase of the pulse have to be measured accurately.

A direct measurement in the time domain is not possible because electronics

detectors (i.e. photodiodes) usually have a response time in the nanosecond

range, so are not suitable for femtosecond pulses. In the spectral domain it

is relatively easy to obtain the spectrum of a pulse with a spectrometer, as

shown in figure 2.12, but no spectral phase is recorded. Most ultrafast pulse

characterisation techniques rely on using the pulse to measure itself. In the

following sections some characterisation techniques will be discussed.

dete

ctor

input beam

grating

M

M

Figure 2.12: Typical spectrometer device, providing I(ω). A dispersive element,usually a grating, spatially separates the wavelengths components of the pulse thatare then imaged onto a detector, for example a CCD array.

2.2.1 Autocorrelation

The signal measured from an ideal slow detector can be written as:

Vsig =

∫ ∞

−∞|E(t)|2dt (2.27)

The simplest idea to measure the pulse with itself is to send two replica

of the pulse onto the detector and measure the signal as a function of delay

between the two, as shown in figure 2.13.


BS

delaystage

detector

Figure 2.13: Field autocorrelation set-up. Two replica of the pulse are createdwith a beam splitter. A simple translation stage provides the delay between the twopulses that are then recombined on the detector.

Vsig(τ) =

∫|E(t) +E(t− τ)|2dt = const. + 2

{∫E(t)E∗(t− τ)dt

}(2.28)

This method is known as field autocorrelation and the measured signal can

be expressed as in equation (2.28). The measured constant level is proportional

to the pulse energy, while the remaining delay dependent term is oscillatory,

but contains only the spectral amplitude information as shown in figure 2.14.

Inte

nsity

(arb

. u.)

0

0.5

1(a) FTL

40 fs3

time (fs)-20 -15 -10 -5 0 5 10 15 20

Vsi

g (arb

. u.)

-1

-0.5

0

0.5

1(b)

Figure 2.14: Field autocorrelation trace, no phase information is retrieved, aswhen the two pulses share the same amplitude but different phase, as shown in plot(a), the autocorrelation trace, shown in (b), is identical.


The field autocorrelation has been computed for a FTL pulse and for a pulse

with identical spectral amplitude but with 40 fs3 chirp. Note how the two very

different temporal profiles in 2.14(a) provide the same field autocorrelation

trace in 2.14(b) due to the fact they share the same |E(ω)|.It is possible to go from field autocorrelation to intensity autocorrelation

by introducing a non-linear crystal and measuring the second harmonic rather

than the direct beam, as shown in figure 2.15.

SHG

BS

detector

delaystage

Figure 2.15: Intensity autocorrelation setup. The spatial selection of the middlebeam after SHG allows to measure the required signal.

Vsig(τ) =

∫|E(t)|2 · |E(t− τ)|2dt (2.29)

In this case the measured signal will be expressed as in equation (2.29).

The signal is not oscillating, and provides an envelope that can be related to

the pulse intensity envelope I(t) = |E(t)|2. If a Gaussian intensity profile is

assumed, it is possible to compute the factor between the trace width and

the pulse width as Δτautocorr/Δτpulse = 1.41. The trace is always symmetric

in time, so the time direction has to be deduced from the geometry of the

experimental setup. However little information about the spectral phase can

be retrieved with this method, as is shown in figure 2.16.

It can be seen how the intensity autocorrelation traces in 2.16(b) are very

similar, despite the chirp of the pulses being very different as shown in 2.16(a).

This technique can be used as a first estimate of the pulse duration for sim-

ple intensity profiles, for example when a Gaussian profile can be assumed,

and when dispersion is not affecting the pulse duration too much. For more


Inte

nsity

(arb

. u.)

0

0.5

1(a) 6 fs2

40 fs3

time (fs)-20 -15 -10 -5 0 5 10 15 20

Vsi

g (arb

. u.)

0

0.5

1(b)

Figure 2.16: Intensity autocorrelation for two pulses sharing the same spectrumbut with different phase as shown in (a). Note how the autocorrelation trace, shownin (b), is very similar in the two cases.

complicated pulses, intensity autocorrelation can lead to ambiguous results

[49].

It is possible to obtain an interferometric autocorrelation trace [50] by

combining two collinear replica of the pulse on the SHG crystal. The trace

measured with this method can distinguish between almost identical intensity

autocorrelation traces, especially if appropriate filtering and analysis of the

trace is applied [51, 52]. However a precise measurement of the spectral phase

of the pulse, in the few-cycle regime, is not possible with these techniques [53],

and even increasing the order of the non linear process exploited to obtain the

measured trace does not yield substantial improvements to this approach [15].

2.2.2 Frequency Resolved Optical Gating

Frequency resolved optical gating (FROG) [53, 54] extends the idea of auto-

correlation, allowing for the retrieval of the spectral phase. The output of a

FROG device is a spectrogram, which is a 2-D map of frequencies (spectra) as


a function of time (delay). For a signal E(t) and a gating function g(t) it can

be expressed mathematically as:

S(ω, τ) =

∣∣∣∣∫ ∞

−∞E(t)g(t− τ)eiωtdt

∣∣∣∣2 (2.30)

Figure 2.17 shows the spectrogram of a pulse with three different phases, using

a Gaussian gating function with 10 fs FWHM.

(d)(d)

time (fs)-40 -20 0 20 40

Freq

uenc

y (r

ad/fs

)

0

1

2

3

4

5

6(e)(e)

time (fs)-40 -20 0 20 400

1

2

3

4

5

6(f)(f)

time (fs)-40 -20 0 20 400

1

2

3

4

5

6

Am

plitu

de (a

rb. u

.)

-1

-0.5

0

0.5

1(a)

-1

-0.5

0

0.5

1(b)

-1

-0.5

0

0.5

1(c)

Figure 2.17: Spectrogram of three pulses with same spectrum and positive (a),negative (b) and without (b) chirp. The corresponding spectrograms were computedwith a 10 fs Gaussian gate, and are shown in (d), (e) and (f).

The differences of the three spectrograms (d),(e) and (f) for a positively

chirped pulse (a), a FTL pulse (b), and negatively chirped pulse (c), are clear.

Even if the gate is not much shorter than the actual pulse it is possible, for

example, to track the instantaneous frequency ω(t) as a function of time, show-

ing how the spectrograms yield useful information about the signal phase. The

unknown pulse can be used as the gate in equation (2.30), using the same op-

tical set-up as in figure 2.15, simply measuring the second harmonic spectrum


generated at each delay. The measured signal can be expressed as [53]:

IFROG(ω, τ) =

∣∣∣∣∣∣∣∫ ∞

−∞

SHG︷︸︸︷E(t)E(t− τ)︸︷︷︸

g(t)

eiωtdt

∣∣∣∣∣∣∣2

(2.31)

In this case g(t) is an amplitude and phase gate at the same time and it is

unknown. While this seems like a dead end, having now a 2-D map allows

one to fully determine both the signal and gate function using a retrieval al-

gorithm. This is possible when at least the integrand of equation (2.31) is

known, i.e. when the interaction between g(t− τ) and E(t) is known. If this isthe case, a generalised projection algorithm (GPA) [55] can be used to apply

two constraints, one is the measured spectrogram, the other is the assumed

interaction between signal and gate. The algorithm modifies, in an iterative

fashion, amplitude and phase of both E(t) and g(t) to fulfil the imposed con-

straints. In this way, if the signal and gate are well behaved, the algorithm

will converge to a solution as shown in figure 2.18.

apply

apply

solutioninitial guess

Figure 2.18: Generalised Projection Algorithm (GPA). From an initial guess ofgate and signal, the measured intensity of the spectrogram, and the required gat-ing action are applied to iteratively improve the agreement between the measuredspectrogram and the retrieved one.


2.2.3 Spectral Phase Interferometry for Direct Electric

field Reconstruction

Another approach to characterise ultra-short pulses comes from interferome-

try. Interferometry can be very sensitive to phase variation in signal and has

been applied to ultra-short pulses to gather information about the spectral

phase [15]. The idea is to make the ultra-short pulse interfere with a spectrally

sheared replica of itself, and from the interference pattern recorded extract

the spectral phase. This technique is called spectral phase interferometry for

direct electric field reconstruction (SPIDER) [56]. It is possible to achieve a

pulse replica shifted in frequency with a chirped pulse and SHG. A chirped

pulse has its frequency components smeared in time, so that, if a shorter pulse

is combined at different times in a non linear crystal, the second harmonic

will be produced at slightly different frequencies, as shown in figure 2.19 and

expressed in equation (2.32)

E1 = E(ω − ω0)eiϕ(ω−ω0) E2 = E(ω − ω0 − Ω)ei[ϕ(ω−ω0−Ω)+ωτ ] (2.32)

stretcher

SHG

Spectrom

eter

Figure 2.19: Standard SPIDER set-up. After splitting the beam, one arm of theinterferometer includes a stretcher, to provide the ancilla. In the other arm a delayedreplica of the pulse is produced, for example with a split mirror or an etalon. Suchpulses are then recombined for SHG resulting in two up-converted pulses at differentcentral frequencies. The resulting spectrum is measured on a standard spectrometer.


The chirped pulse is generally referred to as the “ancilla” as it just serves

for shearing the two replica of the pulse under investigation. With such ar-

rangement the SHG measured on axis can be written as:

S(ω) = |E1 + E2|2 == |E(ω − ω0)|2 + |E(ω − ω0 − Ω)|2 + (DC term)

+ 2E(ω − ω0)E(ω − ω0 − Ω)· (2.33)

· exp [φ(ω − ω0)− φ(ω − ω0 − Ω) + ωτ ] . (AC term)

The phase term in equation (2.33) is an approximation of the derivative of

the wanted ϕ(ω), as it expresses the difference of phase in steps of width Ω. It

can be retrieved via Fourier filtering as shown in figure 2.20.

1 2 3 4 5 6 7 8

Inte

nsity

(arb

. u.)

0

0.5

1

1.5

2

2.5

0

(rad/ fs)

(a)

0.5 1 1.5 2 2.5 3 3.5 4

Inte

nsity

(arb

. u.)

0

0.5

Pha

se (r

ad)

-20

-15

-10

-5

0

5

10

15

20

(d)

FT

filteringand FT-1

integration&

shift

(rad/ fs)

-60 -40 -20 0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

filter

(b)

Inte

nsity

(arb

. u.)

time (fs)

-100

0

100

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 80

0.5

1

1.5

2

2.5

-1

-1

ang

(c)

phas

e (r

ad)

Inte

nsity

(arb

. u.)

(rad/ fs)

Figure 2.20: SPIDER analysis involving the Fourier transform of the interferencepattern measured in (a), to select the AC component in (b). Once the selection isperformed the inverse Fourier transform (c) provides the phase values from whichthe spectral phase can be reconstructed (d).

The interference pattern measured in the frequency domain (2.20(a)) is first

Fourier transformed. A peak around τ can be isolated from the DC component

(2.20(b)) and then Fourier transformed back in the frequency domain (2.20(c)).


The obtained phase is the derivative of the spectral phase of the pulse, that

can be calculated by integration [57]. After shifting the measured phase and

spectrum to the original ω0 position the full amplitude and spectrum of the

pulse can be retrieved.

In our laboratory the optical characterisation of the few cycle pulse is usu-

ally performed either with the standard 1-D implementation of SPIDER, with

the stretcher consisting of a rod of BK7 and the two replica created on a split

mirror, or with the more sophisticated spatially-encoded arrangement with fil-

ters SPIDER (SEA-F-SPIDER) [58]. In this case, two ancillas are prepared

with narrow-band spectral filters and the two out-coming sheared replicas in-

terfere spatially as they are recombined at an angle. To record the spatial

fringes an imaging spectrometer is required, allowing the characterisation of

the pulse both in the temporal and spatial domains as shown in figure 2.21.

Figure 2.21: SEA-F-SPIDER. Interference of the two pulse replica is obtain bycombining them in the BBO crystal at an angle. The ancilla preparation is per-formed with angle tunable narrow bandwidth spectral filters. The signal treatmentis similar to the 1D SPIDER case. Figure reproduced from [58]. c©IOP Publishing.Reproduced by permission of IOP Publishing. All rights reserved.


2.2.4 CEP measurement and stabilisation

The CEP can be extremely important in experiments involving intense few-

cycle lasers such as HHG. For this reason it is an important parameter to

measure and stabilise. The locking and stabilisation of the CEP begins in

the oscillator. The CEP would change on a shot to shot basis, even in a

perfect environment, without external contributions like temperature drifts or

mechanical vibrations, due to the difference between the phase velocity of the

carrier, and the group velocity of the pulse envelope. Each consecutive pulse

would therefore experience a CEP slip Δϕ0 that should be taken into account

and controlled to obtain a pulse train of CEP stable pulses. In fact if Δϕ0 =

2π/n each nth pulse will have the same CEP. While different experimental

methods are available to determine the CEP value, including stereo ATI [59]

and solid state detectors [60], the most common way to measure the CEP is

interferometry [61]. In figure 2.22 is shown the frequency comb corresponding

to the mode-locked pulse train. Each line is a longitudinal mode frequency,

and only those overlapping with the Ti:Sapphire gain curve are amplified.

Frequency

Inte

nsity

Gain BandwidthGain BandwidthGain Bandwidth

Figure 2.22: Spectral structure of a mode locked pulse train. Each line is alongitudinal cavity mode, and those corresponding to the gain medium gain curveare amplified. The separation between the modes is frep, while the offset f0 can bewritten as f0 = frepΔϕ0/(2π)


Each frequency component can be written as [62] fn = nfrep + f0, with

f0 = frepΔϕ0/2π. If the frequency doubled part of the low frequency spectrum

interferes with the corresponding high frequency part one can obtain:

2fn − f2n = 2(nfrep + f0)− (2nfrep + f0) = f0 = frepΔϕ0/2π (2.34)

Therefore, measuring the beat note 2fn − f2n = f0 and frep allows one to

measure the CEP slip Δϕ0.

In order to stabilise the CEP of a pulse train the beat note obtained from

the f − 2f interferometer has to be locked to f0 = frepn. If this condition is

met, every nth pulse of the train has the same CEP. The locking procedure is

usually obtained with a feedback loop. Once f0 is measured it can be compared

to the wanted frequency, and the difference between the target frequency and

the measured one can be used as error signal in a PID controller. To act on

f0 it is necessary to have a fast actuator, able to modify f0 at a rate as fast

as the repetition rate of the laser system. An efficient method is to modulate

the power of the pump laser of the oscillator with an acousto-optic modulator

(AOM) [62] providing a satisfactory dynamic range and response time.

With this system only one every nth pulses can be used for CEP sensitive

experiments. Locking the beat note to frep would allow one to have a pulse

train with identical CEP at each pulse, but it is not experimentally practical

due to the difficulty in separating the signal coming from the beat note f0 and

the one coming from the input laser [63]. An elegant solution to this problem

has been proposed by Koke et al. [64] with a feed-forward scheme. In this

case the error signal is not fed back to the laser system, but is used to drive

an acousto-optic shifter to correct the output pulse, before delivery to the

experiment. This method allows to leave the laser system free-running, with

less stringent constraint on its stability, and provided a residual time jitter as

small as 12 as.


2.3 Strong field physics

2.3.1 Ionisation

Einstein showed how, when light with sufficient photon energy (�ωL > W )

interacts with solids, photo ionisation occurs [65]. The photoelectric effect is

explained as the exchange of energy between light (�ωL) and the system (work-

function W ), so that when the potential barrier can be overcome, an electron

is created from a photon. If an atom is considered, rather than a solid, as the

target system, to a first approximation the work-function W will be replaced

by the ionisation potential Ip.

At low intensities, no electrons are emitted when �ωL < Ip, but with nowa-

days intense lasers, photoionisation can still occur through different processes.

In all of them the laser intensity has to be relatively high, as single photon

ionisation is not possible any more. The first effect encountered increasing the

laser intensity is multi photon ionisation (MPI) [66]. This is a non-linear pro-

cess where the minimum number of photons from the laser field is absorbed

at once to overcome the Ip. If the intensity is increased even further, more

photons can be absorbed at once and therefore electrons at different energies

will be produced in what is called above threshold ionisation (ATI) [67]. The

ionisation rate (Γ) of these two processes can be written as:

Γn = σnIn (2.35)

The order of the process n refers to the number of photons involved in the

photoionisation process, and σn is the cross section for the absorption of n

photons. In the case of ATI the photoelectrons spectrum is a comb of peaks

spaced by one photon energy, signature of the quantum nature of the energy

transfer [68]. If the intensity is increased even more (I > 1013 Wcm−2), the field

amplitude can become significant compared to the atomic Coulomb potential.

Photoionisation is now possible due to tunnelling [69]. An electron can tunnel


ionise through the distorted potential and be set free in the continuum. This

is the first step in the process of high harmonics generation (see section 2.3.2).

With the intensity even higher, the distortion of the potential due to the electric

field can be so strong that no potential is seen by the electron that is therefore

free to leave the parent atom, in a process called over the barrier ionisation

(OTBI) [69]. A parameter to predict what will be the dominant mechanism is

the Keldish parameter [70]:

γ =

√Ip2Up

(2.36)

In this equation the ionisation potential, Ip is compared to the pondero-

motive energy, Up, the time averaged energy of the oscillating electron in the

field:

Up =e2E2

4meω2L

(2.37)

Another way to interpret the Keldish parameter is via the tunnelling rate

(ωt), and the frequency of the laser field (ω0). In fact equation (2.36) is equiv-

alent to γ = ω0/ωt.

Ip

MPI ATI TunnelIonisation OTBI

Figure 2.23: Ionisation dynamics. In the perturbative regime γ � 1 the Coulombpotential is not significantly modified by the laser field, and MPI or ATI lead tophoto-ionisation through absorption of multiple photons. When γ � 1 the Coulombpotential is lowered and tunnel ionisation or OTBI can happen.


When γ � 1 the laser field is not significantly perturbing the Coulomb po-

tential, leaving MPI and ATI as the principal mechanisms of photoionisation.

On the other hand when γ � 1, given the strong perturbation induced by the

electric field, tunnel ionisation and OTBI are the most likely dynamics leading

to photoionisation, as shown in figure 2.23.

A model to predict the instantanous photoionisation rate is provided by

the Ammosov-Delone-Krainov (ADK) theory [71]. According to this model

the instantaneous ionisation rate can be written as:

W (t) = ΩP |Cn∗ |2(4ΩP

ωt

)2n∗−1

e− 4ΩP

3ωt , (2.38)

where the following parameters are defined

Ωp =Ip�

ωt =e |E(t)|√2meIp

n∗ = Z(IH/Ip)1/2

|Cn∗ |2 = 22n∗

n∗Γ(n∗ + l∗ + 1)Γ(n∗ − l∗)

(2.39)

In this expression IH is the hydrogen ionisation potential, while Ip is the ioni-

sation potential of the atom under consideration. Z is the final charged state

of the ion. The effective quantum number l∗ is defined as n∗ − 1 and Γ(x) is

the gamma function (extension of the factorial function). E(t) is the electric

field and me the electron mass. Once the ionisation rate is known, the overall

population of neutrals can be computed as:

N(t) = N0e− ∫ t

−∞ W (τ)dτ , (2.40)

where N0 is the initial population. In this way it is possible to calculate the

instantaneous population of neutrals and free electrons as shown in figure 2.24.

Such information will be relevant both for the broadening of the pulse in the

hollow core fibre, and for the high harmonic phase-matching conditions, as will

be seen in section 2.3.2.


Fiel

d (V

/nm

)

-100

0

100(a)

Ioni

satio

n ra

te (f

s-1)

0

0.2

0.4

0.6(b)

time (fs)-10 -5 0 5 10

Pop

ulat

ion

(%)

0

50

100(c) free electrons

neutrals

Figure 2.24: ADK calculations in Ne for an intensity I = 1015Wcm−2, λ = 800nm, duration 5 fs. Panel (a) shows the pulse, in (b) is the instantaneous ionisationrate peaked around the field maxima, from which one can compute the neutral andfree electron population in (c).


2.3.2 High Harmonics Generation

Now I will give an overview of the process underpinning the generation of

attosecond pulses. High harmonics generation happens when, after tunnel

ionisation, the electron manages to recombine with the parent atom in the

presence of an intense oscillating field. After tunnel ionisation the electron is

accelerated in the continuum by the presence of the oscillating electric field,

gaining energy. If the electron is brought back close to the parent atom it can

recombine with it, releasing the energy gained in the continuum in the form

of a photon. The features of this process allow for the production of short

pulses in the spectral range from UV to soft X-ray in a relatively compact

experimental apparatus. For this reason HHG is an attractive method to

develop table-top sources of short and coherent XUV pulses and attosecond

pulses. Experimentally, HHG is usually achieved by focusing an intense laser

pulse onto an atomic or molecular target. As shown in the previous section,

the Keldish parameter should be bigger than one to efficiently tunnel ionise the

target. However, many other aspects of the experiment influence the harmonics

generation including the laser wavelength, the focusing geometry and the pulse

duration. In the following sections the aspects of HHG more relevant to the

present thesis will be discussed.

Three step model

The HHG process can be described by the three-step model [13, 72, 73]. Figure

2.25 depicts these steps. In the first step, the electron tunnel ionises through

the distorted coulomb potential. In the second step, reaching the continuum,

the wave-packet can gain energy from the laser field. For certain ionisation

times, the wave-packet recombines with the parent atom in step three, emitting

the kinetic energy gained in the field plus the ionisation potential as a photon.

The HHG spectrum has very distinctive features. Assuming a linearly

polarised pulse in the NIR, the three step process happens every half cycle of


TunnelIonisation Acceleration Recombination

(1) (2) (3)

Figure 2.25: Three-step model, The laser field distorts the Coulomb potential in(1) allowing tunnel ionisation to happen. The electron gains kinetic energy in thecontinuum in (2). Upon recombination a photon of energy Ip + KE is emitted in(3).

the electric field. Hence the spectrum shows the interference pattern of a signal

repeated every half cycle, giving rise to the odd harmonics comb separated by

2�ωL. Another notable feature is the overall shape of the spectral emission.

For low order harmonics, where the perturbative regime can be used to

describe the harmonic generation, the conversion efficiency drops rapidly, until

a plateau region is reached, where the conversion efficiency is roughly constant.

At the high energy end of the spectrum we find the cut-off region, where the

harmonics production drops to zero. Figure 2.26 shows the main features of a

HHG spectrum. In the next section I will show how a simple classical model

can reproduce many characteristics of this spectrum

Plateau

Cut-off

Energy

Drop-off

Figure 2.26: Typical HHG spectrum, showing how the conversion efficiency dropsfor low order harmonics, the plateau region, and the cut-off.


Classical trajectories

The measured HHG spectrum, for a given laser pulse and target, depends on

both the single-atom response and the macroscopic, or phase-matching, re-

sponse. The easiest model for the single-atom response is the classical one,

and despite ignoring the quantum nature of the process, it is able to repro-

duce many features of the experiments. The classical model involves solving

the equation of motion of an electron in an oscillating field. Considering the

case of a linearly polarised field along x, the resulting force on an electron is

F (t) = eE0 cos(ωLt). The equations of acceleration, velocity and position can

be written as:

a(t) =eE0

me

cos(ωLt),

v(t) =eE0

meωL

· [sin (ωLt)− sin (ωLti)] ,

x(t) = − eE0

meω2L

· {[cos (ωLt)− cos (ωLti)] + ωL(t− ti) sin(ωLti)} ,

(2.41)

where ti is the ionisation time when the electron starts moving under the

effect of the electric field only. The initial conditions are set to be v(ti) =

x(ti) = 0. In general, the form of the electric field can be simply integrated to

find the corresponding v(t) and x(t). The extension to the full 3 dimensional

classical description, as required for an arbitrary laser polarisation, is trivial,

each dimension being independent from the others. The interesting parameters

are the boundary conditions. In fact these conditions select the recombining

electrons, which are the ones responsible for the HHG spectrum. Only for some

ti the electron trajectory will come back (re-collide) to the parent atom at a

later time tr. For the simulations I performed with this model the returning

electrons were selected if they crossed the origin within one cycle. However, in

2D this aspect is more complicated, and from a classical point of view a small

region around the core atom has to be defined as the target for recollision. In

figure 2.27 are shown the trajectories computed for a field λL = 800 nm and

different ti.


time (fs)0 0.5 1 1.5 2 2.5 3

Pos

ition

(nm

)

-15

-10

-5

0

5

10

15

0

1.6

3.2Up

Figure 2.27: Electron trajectories in a oscillating field. The color scale refersto the recombination energy. Black trajectories do not return to the origin. Theprocess repeats every half-cycle but only the first half cycle of trajectories is shownfor clarity. The electric field (red-dashed line) is shown for reference.

As already mentioned, only certain ti allow the electron to recombine

(coloured trajectories), while for others the electron does not return (black

trajectories). The maximum kinetic energy of the returning electrons is 3.2Up.

This corresponds to the cut-off energy of the HHG spectrum. Recalling the

definition of Up given in equation (2.37), to increase the energy cut-off one can

either increase the laser intensity, or increase the driving wavelength. In the

first case the limit is the ionisation saturation intensity, where the target is

almost fully ionised, in the second case the conversion efficiency of the HHG

process decreases with wavelength (scaling as ≈ λ−6.5L ) [74], making it difficult

to measure and use the weak HHG spectra produced with long wavelength

laser fields. This is due to the longer time an electrons spends in the con-

tinuum when driven by a long wavelength field. The longer the electron is

in the continuum the more the wavepacket is spread in space, decreasing the

probability of recombination with the localised parent atom.


Another feature of the HHG spectrum is that different trajectories have

different excursion times in the continuum. In figure 2.28 the final kinetic en-

ergy of different trajectories is plotted as a function of ti (on the left side of the

plot), and of tr (on the right side of the plot). The trajectories between the two

peaks are defined as “short”, while the ones on the outside are defined “long”.

Moreover, it can be seen that the distribution of ti on the left is narrower than

the distribution of tr on the right. This spread is called “attochirp”, and is the

natural dispersion of the HHG process, with different photon energies being

emitted at different times. This chirp can either be compensated to obtain the

shortest possible attosecond pulse for a given spectrum [20], or can be used as

a natural delay scan to investigate ultrafast dynamics [8]

time (fs)0 0.5 1 1.5 2 2.5 3

Ene

rgy

(Up

units

)

0

0.5

1

1.5

2

2.5

3

3.5 ti tr

Short

Long

0

1.6

3.2Up

Figure 2.28: Long and short trajectories. The left part of the plot represents therecombination energy as a function of ionisation time ti. The right part o the plotshows the same as a function of returning time tr.


Phase-matching in HHG

In the previous section the single atom response was analysed, and the semi-

classical treatment shown to reproduce the basics features of the HHG spec-

trum. The collective behaviour of the target atoms can now be considered.

The key aspect of HHG generation is coherence, as otherwise much brighter

X-ray sources would be preferable (for example X-ray tubes). For this reason

the coherent emission of all the emitters in the target has to be considered

and the experimental conditions where they constructively interfere has to be

found. The approach is similar to the phase-matching problem of low-order

non-linear optics, with the definition of a phase mismatch parameter Δk, valid

for HHG as well, and the relative coherence length Lcoh = π/Δk over which

constructive interfere happens. In HHG the contributions to the phase mis-

match factor come from different sources. One term is the phase ϕfocus present

due to the focusing of the beam. Another term is provided by the atomic re-

sponse of the atom ϕdip,j, that is related to the phase the electron following a

trajectory j accumulates while in the continuum. Following the notation pro-

vided in [75] the phase mismatch factor for these two factors can be written

as:

Δkωq(r, z) =ωq

c−∣∣∣∣�kdip,j(r, z) + ωq

ωL

(�kfocus(r, z) + �kL)

∣∣∣∣ (2.42)

where �kL is the laser fundamental wave-vector with frequency ωL. The har-

monic order q has frequency ωq. The Gouy phase can be written as:

ϕfocus(z, r) = − tan−1

(2z

b

)+

2kLr2z

b2 + 4z2(2.43)

From equation (2.43) it can be seen that, on axis (r = 0) the wave-vector

�kfocus points always in the same direction, since tan−1 is a monotonic function

and �kGouy = ∇ϕfocus. The other phase factor is ϕdip,j(r, z). In this case j

labels a specific trajectory. In general it can be written as the classical action

on a trajectory j with ionisation time ti and return time tr:


ϕ(ti, tr) = −∫ tr

ti

S(t′′)dt′′ (2.44)

The integrand of equation (2.44) is defined as S(t′′) = Ekin(t′′) − Epot(t

′′).

In the long pulse approximation (the envelope shape is assumed not relevant

to the intra-cycle dynamics), the energy is proportional to the ponderomotive

energy Up, and therefore to the laser intensity, since Up ∝ I/ω2L. For this

reason equation (2.44) can be rewritten as:

ϕdip,j = −αjUp/ωL (2.45)

As shown in [76], the number of trajectories that has to be taken into

account can be limited to two paths, the long and short trajectories.

Neglecting dispersion effects (kq = qkL = qωL/c), the expression for the

on-axis phase mismatch factor can be written as:

Δkωq = αjdUp/ωL

dz+ωq

ωL

2

b(1 + 2z2/b2)(2.46)

The factor representing focusing is always positive, while the dipole factor,

being proportional to the laser intensity, is positive before the focus (dI/dz >

0) and negative after the focus (dI/dz < 0), giving the well-known result of

phase-matching of HHG after the focus. Figure 2.29 is a representation of

different phase-matching conditions, for different trajectories, short in the left

column, and long in the right column, and for different harmonics, 21st, 35th

and 45th from top to bottom. The color scale represents the coherent length

Lcoh. Note how phase-matching conditions are obtained in different positions

for long and short trajectories. The resulting divergence of these trajectories

is different as well, as shown by the different direction of the generated wave

indicated with the red arrows.

So far dispersion has not been included, together with the more dynamic

aspects for phase-matching. The reason is that such effects are not adiabatic,


i.e. they cannot be considered constant during the HHG process, as was for

the Gouy and the dipole phase [75]. However, if considered for small spatial

intervals the local refractive index can be expressed as a function of other

parameters like density and wavelength, and terms including the neutral and

electron dispersion can be added to the phase-matching equation (2.42), for

example computing the instantaneous free electron population with the ADK

model.

By now the complexity of phase-matching should be clear, it involves both

the understanding of the microscopic aspects of HHG, and of the macroscopic

aspects. This sensitivity can be sometimes an obstacle in the path of scaling

high harmonics yield and energies, but it can be used to tailor the attosecond

emission, for example selecting or gating an isolated attosecond pulse (see

3.3), or as a diagnostic tool as well, for example retrieving the dipole phase of

a target of interest as it has been done in [78].


Figure 2.29: HHG phase-matching. Efficient generation (white parts of the col-ormap), are achieved at different positions for short (left) and long (right) trajecto-ries. From top to bottom harmonic 21, 35 and 45 are considered. The red arrowspoint in the direction of the harmonic wave-vector, showing how the divergence ofthe radiation is different for the two paths. Image adapted from [77].

63

Chapter 3

Ultrafast light sources

development

In this chapter I will present the main results obtained during my PhD. Some

more background details will be provided, focusing on the aspects more rel-

evant to the experiments performed. In general the experiments performed

aimed at developing new light sources, or improving the performance of some

others already well established, to allow new attosecond experiments.

3.1 Laser System

In the following sections the features of the laser system in the Attosecond

Laboratory at Imperial College will be described [79]. The commercial CPA

system is based on Ti:Sapphire oscillator and amplifier (Femtopower HE CEP,

Femtolasers GmbH). The output is 2.5 mJ, 28 fs at 1 kHz repetition rate.

The pulses are spectrally broadened in a hollow core fibre to achieve, after

compression with chirped mirrors, a pulse duration as short as 3.5 fs and up

to 1 mJ pulse energy, while maintaining CEP stability.

64 CHAPTER 3. ULTRAFAST LIGHT SOURCES DEVELOPMENT

3.1.1 Oscillator

The Ti:Sapphire oscillator (Femtosource, Femtolasers GmbH) is shown in fig-

ure 3.1. The pump laser (Verdi, Coherent) is a 532 nm, 4.4 W frequency

doubled Nd:YVO4 CW laser. Soft Kerr-lens mode-locking, initiated by mov-

ing one of the intracavity mirrors (S) mounted on translation stage, produces

a pulse train at 78 MHz with a duration of 10 fs and 400 mW average power

(5 nJ/pulse). Intracavity dispersion compensation is provided by two chirped

mirrors (CM), further dispersion control is given by a pair of wedges (W). An

acousto-optic modulator (AOM) is installed in the pump beam to allow fast

modulations of the pump power for CEP control purposes.

A 50% beam splitter (BS) delivers half of the oscillator energy to an f−2f

for the measurement of the CEP, and the remaining half is delivered to the

multi-stage amplifier through a Faraday isolator (F).

Pump Laser: 4.4W 532nm

to f-to-2f

PD

Ti:S

W

to streatcher andamplifier

BS CMCM

Figure 3.1: Oscillator layout. A Ti:S crystal is pumped at 532 nm by 4.4 W of afrequency doubled Nd:YVO4. Two chirped mirrors (CM) provide intra-cavity neg-ative dispersion. Intra-cavity wedges (W) and an acousto-optic modulator (AOM)in the pump beam allow manipulation of the CEP and intra-cavity dispersion. Theoscillator output is delivered to the amplifier through a Faraday isolator (F). theoscillator output is monitored by a photodiode (PD).

The CEP of the oscillator is measured in the so called “fast loop” interfer-

ometer (XPS800, Menlo Systems) shown in figure 3.2. A photonic crystal fibre

(PCF) is used to obtain an octave spanning spectrum from the oscillator sig-

nal. A dichroic beam splitter (DBS) separates the green components (around

532 nm) from the red components (around 1064 nm). The red components are

CHAPTER 3. ULTRAFAST LIGHT SOURCES DEVELOPMENT 65

focused onto a periodically poled potassium niobate crystal (KNbO3, PPL)

for efficient SHG. The two arms of the interferometer are recombined with a

polarising beam splitter. The signal obtained is sent to a grating, dispersed

and the beat note f0 is recorded on an avalanche photodiode. The transmitted

part of the green arm of the interferometer is sent to a second avalanche pho-

todiode for the measurement of the repetition rate frep. These two signals are

fed into fast analogue electronics to provide a correction signal to the AOM

in the oscillator pump beam. This interferometer stabilises the pulse to pulse

fluctuations of the CEP by locking the f − 2f beat note at frep/4. In this way

the waveform of the pulse is the same every four pulses. The signal manip-

ulation is done with fast analogue electronics so that the feedback loop can

act on a single shot basis. The actuator of the feedback loop is implemented

with the acoustic optic modulator on the pump beam, so that a difference in

pump power can correct the CEP. When a coarser manipulation of the CEP

is necessary, the thin wedges pair can be translated to provide further control

of the intra cavity dispersion and CEP.

fromoscillatorPCF

DBS

PPL

PID AOM

Figure 3.2: CEP fast loop. The oscillator pulse is spectrally broadened in a PCFbefore a dichroic mirror (DBS) selects different wavelengths components sending thelong one (red) to a doubling crystal. Recombination of the two arms happens at apolarised beam splitter and the beat note is measured in parallel to the repetitionrate.


3.1.2 Amplifier

The amplifier consists of a Ti:Sapphire crystal pumped by a 12 mJ, 527 nm

(frequency doubled Nd:YLF) Q-switched laser pulse (DM30, Photonics Indus-

tries). The crystal is cryo-cooled to 170 K in a vacuum chamber at ≈10−2

mbar. The oscillator pulse is stretched to a pulse duration of ≈20 ps with a

double pass through 4 rods (5 cm each) of SF57 glass. The stretched pulse

goes through the Ti:S crystal 10 times. After pass four the repetition rate is

reduced from 80 MHz to 1 kHz through a Pockel cell unit. The pulse is then

shaped with a Dazzler [39] (Dazzler, Fastlite) to compensate for dispersion and

gain narrowing. After pass eight a telescope recollimates the beam to reduce

the intensity in the crystal. After the tenth pass the beam goes through a

transmission grating compressor, delivering an output of 2.5 mJ, 28 fs, at 1

kHz repetition rate. Figure 3.3, shows the amplifier layout.

PC

PD

PBS1

PBS2

Ti:S

From pump laser

stretcherglass

Dazzler

Telescope

2.5mJ, 28fs, 1kHz transmission grating compressor

fromoscillator

Figure 3.3: Amplifier layout. The oscillator output is stretched in a glass stretcherbefore performing 10 passes through the amplifier crystal. Compression is achievedin a transmission grating compressor. See text for details.

While the fast-loop locking system stabilises the oscillator CEP on a shot

to shot basis, a second feedback loop is necessary to stabilise CEP fluctuations

on a longer time scale that are caused by effects beyond the oscillator stability.

For example, vibration of the compressor, energy fluctuations in the pump


power of the amplifier or thermal drifts of the optical system can change the

CEP value of the output pulse. For this reason a CEP monitor and feedback

loop are used on a portion of the CPA output beam. While the low pulse

energies of the oscillator require a photonic crystal fibre to obtain an octave

spanning spectrum, the output of the amplifier is strong enough to produce

octave spanning white light in a sapphire plate. Figure 3.4 shows how the slow

locking loop measures the CEP.

sapphireplate

BBOcrystal

Spectrometer

polariser

SHGWLG

PIDAOM FFT

from fast loop

Figure 3.4: Slow locking CEP loop. The power available after the amplifier allowsto produce an octave spanning spectrum by focusing the beam in a sapphire plate.The feedback loop runs through a computer at 20-30Hz .

The fringe pattern obtained interfering the short wavelengths and the

frequency-doubled long wavelength components of the white light generated

in the sapphire plate, provides direct access to Δϕ0, that can be measured on

a computer at a repetition rate of 20-30 Hz. Given that the fast fluctuations

of the CEP are already stabilised by the fast loop, this bandwidth is enough

to compensate for the slower drifts of the system. The measured drifts are

converted to a voltage signal acting on the AOM in the pump beam of the

oscillator, together with the fast loop corrections.


3.2 Hollow core fibre pulse compression

As introduced in chapter 2, the main way to obtain attosecond pulses is HHG.

In our laboratory, focused on developing techniques for attosecond pump-probe

studies [3], we adopted amplitude gating to produce isolated attosecond pulses

[58]. More about gating and shaping techniques of attosecond radiation will be

discussed in section 3.3. In this section the production of the few cycle pulse

used to implement amplitude gating will be presented.

To obtain a few cycle pulse starting from the CPA laser output, spectral

broadening was achieved through SPM (see figure 2.8) in a hollow core fibre,

and pulse compression through chirped mirrors (see figure 2.5). This technique

is well established [80] with the capability of delivering up to a 1TW of peak

power in a 5 fs, 5 mJ pulse [81] or providing sub-4 fs pulses at slightly lower

energies [82–84], and it has been possible to reach even the single-cycle duration

[16]. The most challenging aspect in developing this technique is the energy

scaling, in order to use efficiently the more energetic CPA laser pulses available

while maintaining good performances at higher pulse energies. The trade off

between power and performance is not trivial and has been investigated in our

laboratory [32].

The first experiments exploiting SPM for spectral broadening of a pulse

were performed in bulk material [85], but it was soon realised that high inten-

sities could not be handled this way, due to the intensity induced damage to

the material, and also the non spatially uniform SPM induced in free propa-

gation. From here the idea of guided propagation in gases [80] was explored.

Solving Maxwell equations for a hollow wave-guide with inner radius a � λ

and infinite thickness (i.e. the outer interface does not contribute to the field)

shows [86] that the modes with the lowest propagation constant1 α for glass

waveguides (n < 2.02) are the hybrid EH1m modes, as shown in figure 3.6.

Such modes are essentially transverse modes (negligible z component of the

1P (z) = P (0) exp(−αz)


fields if 2πa/λ � 1) and polarised. The radial intensity profile within the

hollow core is given by:

I1m = I0 · J02(u1mr/a) (3.1)

In (3.1) J0 is the zero order Bessel function of the first kind, u1m is the mth

zero of J0, and a is the fibre radius. Figure 3.5 shows the first four modes,

with the hollow core represented by the black dashed line.

EH11 EH12

EH13 EH14

Figure 3.5: First four EH1m modes that can be excited by a linearly polarised fieldin a hollow core fibre. The black-dashed line is the fibre entrance, the red continuousline the radial intensity profile of the mode. Note that the excited mode is zero inthe cladding of the fibre (outside the black-dashed line), while here the field hasbeen plotted at all radii for reference.

To efficiently excite the EH11 mode the laser spot-size has to be chosen

appropriately. The goal is to achieve the maximum overlap between the Gaus-

sian profile of the incoming beam Eg, and the mode to excite E1m. Using the

normalised fields and assuming a linearly polarised pulse one can obtain [87]:


Eg =

√2

πw20

exp (−r2/w20) (3.2a)

E1m =J0(u1mr/a)√∫ a

02πrJ2

0 (u1mr/a)dr(3.2b)

∣∣∣∣∫ a

0

Eg · E1m2πrdr

∣∣∣∣2 ==4[∫ a

0rJ0(u1mr/a) exp (−r2/w2

0)dr]2

w20

∫ a

0rJ2

0 (u1mr/a)dr

(3.2c)

From equations (3.2) it is possible to see that there is an optimum w0/a

such that the coupling of the EH11 mode is optimised. Figure 3.6 shows that

this optimum is w0/a = 0.64, where the coupling efficiency is 98%.

L (m)0 0.2 0.4 0.6 0.8 1

Mod

e at

tenu

atio

n (%

)

0

10

20

30

40

50

60

70

80

90

100

EH11

EH12

EH13

EH14

w0/a0 1 2 3

Cou

plin

g ef

ficie

ncy

(%)

0

10

20

30

40

50

60

70

80

90

100

110

98% at w0=0.64a

Figure 3.6: Fibre parameters to consider when optimising the fibre throughput.On the left is the natural attenuation of modes in the fibres, showing that themode EH11 is the one experiencing least attenuation. On the right is the couplingefficiency, i.e. the overlap between each mode and a Gaussian profile at the focus,as a function of w0/a. The optimum coupling is obtained for the mode EH11 atw0/a = 0.64 with a coupling of 98%

When trying to scale the energy throughput of a fibre to higher levels other

parameters need to be considered. In fact non-linear effects can start to play

an increasingly important role, for example self-focusing and plasma-induced


defocusing can alter significantly the focal spot size at the fibre entrance, de-

teriorating the coupling efficiency and even damaging the fibre entrance. The

input peak power P0 has to be lower than then critical power for self-focusing

Pcrit since self focusing would prevent an efficient coupling into the fibre. This

can be expressed [88] in terms of the pressure of the gas p as:

Pcrit =λ202k2p

, (3.3)

where k2 = n2/p with n2 the non linear refractive index responsible for the Kerr

effect as in (2.25). For example for 1 bar of Ne the critical power at 800 nm

is 0.5 TW (1012 W) [87], power levels that can be reached by multi-millijoule

femtosecond systems. Different strategies have been proposed to replace the

initial static filled fibre approach and overcome such limitation. Differential

pumping of the fibre allows one to evacuate the fibre entrance to few mbar or

less, limiting the gas density and therefore the non-linear effects in the coupling

step [32, 81, 83, 89, 90]. Circularly polarised light can be used due to the lower

Kerr-effects (χ(3)) and the lower ionisation rate it induces [91–93]. Pre-chirped

pulses have been used to lower the non-linearity at the fibre entrance as well

[81, 83].

Let us consider the case where a fixed spectral broadening is required, for

example to compress a 28 fs pulse from a CPA system, down to the few cycle

regime. Defining Δω as the final spectral bandwidth, and the initial bandwidth

Δω0, the broadening factor F = Δω/Δω0 can be shown to be [94]:

F =

√1 +

4

3√3ψ2m (3.4)

The variable ψm is the the phase shift accumulated in the fibre and can be

expressed as

ψm =n2ω0P0Leff

cAeff

= γP0Leff (3.5)


In this case P0 is the input peak power, Aeff is the effective area of the

fibre mode (≈ 0.48πa2 [94]) and n2 = k2p is the non-linear refractive index of

the medium. The effective length Leff depends on the experimental conditions

implemented [32, 94]. For a linearly polarised (LP) pulse and static fill (SF) of

the fibre the Leff is expressed as [94]:

LSFLPeff =

∫ L

0

e−αzdz =1− e−αL

α(3.6)

with L being the actual fibre length and α the attenuation constant of the

mode.

For a circularly polarised pulse (CP) the only modification required is re-

lated to the non-linear refractive index in (3.5) and since nCP2 = 2

3nLP2 [95] one

can write:

LSFCPeff =

2

3LSFLPeff =

2

3

(1− e−αL

α

)(3.7)

For a differentially pumped (DP) fibre the pressure is varying along the

fibre length [89]:

p(z) = p(L)

√z

L(3.8)

Including this dependence in equation (3.5) gives another expression for the

effective length [32]:

LDPLPeff =

1√L

∫ L

0

√ze−αzdz ≈ 2

3

(1− e−1.21αL

1.21α

)(3.9)

In the experimental campaign that will be presented in the next section,

these different experimental conditions have been tested and the corresponding

effective lengths are shown in equations (3.10).


LSFLP =1− e−αL

α= 0.94 m (3.10a)

LDPLP ≈2

3

(1− e−1.21αL

1.21α

)= 0.62 m (3.10b)

LSFCP =2

3

(1− e−αL

α

)= 0.63 m (3.10c)

Focusing on the static fill case and linearly polarised pulse, after choosing

the required broadening with the parameter F , one needs to determine the fibre

radius. Numerical simulations [94] showed that requiring the plasma effects

to be small compared to the Kerr effects sets a minimum radius available

for a given pulse duration and energy. The panel (a) of figure 3.7 shows the

minimum radius to be used with different gases in order to prevent detrimental

plasma effects.

Once the fibre radius is chosen, the required gas pressure pF for a given

spectral broadening F can be expressed as in equation (3.11) and it is shown in

panel (b) of figure 3.7 in the case of He gas target (k2 = 2.9 ·10−25m2W−1bar−1

[94]).

pF =cAeff

2k2ω0P0Leff

[3√3(F 2 − 1)

]1/2(3.11)

In order to scale the power to higher levels for a fixed broadening factor in

the SFLP case, one can select the minimum radius amin at each input energy,

and compute the minimum pressure required with that radius, and with it the

critical power limit. The result is that the input power cannot be increased at

will. At some point the peak power P0 will be comparable to Pcrit. The panel

(c) of figure 3.7 shows that this condition happens at about 4 mJ pulse energy

level (28 fs input pulse duration, He gas medium) preventing a straightforward

energy scaling of a hollow core fibre pulse compressor setup beyond this level.


0 1 2 3 4 5

a min

(m

)

0

200

400

600(a)

HeNeArKr

0 1 2 3 4 5

p min

(bar

)

0

1

2

3

4

5(b)

SFLPDPLPSFCP

Energy (mJ)0 1 2 3 4 5

Pow

er (T

W)

0

0.05

0.1

0.15

0.2(c)

P0Pcrit

Figure 3.7: The minimum radius of a SF fibre is shown as a function of pulseenergy for a 28 fs (FWHM) pulse and for different gases in (a). Below such values theplasma effects are comparable to the Kerr effects, degrading the fibre performance.The pressure required to achieve a broadening factor for a given fibre diameter isshown as a function of pulse energy in (b). The intersection between the input powerP0 (black) and the critical power at the minimum radius, shows that straightforwardpower scaling of hollow core fibres is prevented by self focusing in (c).


3.2.1 Experimental setup

An experimental campaign [32] was performed to to investigate the fibre per-

formance under the different experimental condition mentioned in the previous

paragraph. We wanted to test whether differentially pumping the fibre is a vi-

able method to achieve few cycle pulses while maintaining good CEP stability.

Figure 3.8 shows the hollow core fibre set-up in our laboratory.

pump

pum

p

gas

2.5 mJ, 1 kHz, 28 fs

f-2f interferometer

poin

ting

stab

ility

Figure 3.8: Fibre setup. The CPA output is focused with a 1.5m mirror. Thebeam pointing is stabilised with a feedback loop. The entrance and output of thefibre can be pumped independently. Pulse compression is achieved with 10 bounceson multilayer chirped mirrors. The CEP of the fibre output is monitored with a f-2finterferometer.

The output of the CPA is focused with a f = 1.5 m focal length mirror into

the fibre entrance. The fibre has an inner diameter of 260 μm and is one meter

long. For this inner diameter the optimal focal spot size is 169 μm. Figure 3.9

shows the measured spot size, resulting in 162×185 μm. The astigmatism has

to be attributed to the CPA output quality, as no spatial filter is implemented

in the system.

The laser pointing is stabilised by recording the focal spot position on a

CCD camera and adjusting the pointing with a motorised mirror controlled in

a feedback loop. The fibre entrance and exit chambers can be pumped indepen-

dently to perform measurements both with the fibre statically filled or differ-


x ( m)-150 -100 -50 0 50 100 150

y (

m)

-150

-100

-50

0

50

100

150

Figure 3.9: Focal spot size at the fibre entrance. The optimum value is 170 μm

entially pumped. To switch between circular and linear polarisations regimes

a quarter wave plate was positioned before the fibre entrance to make the in-

put beam circularly polarised, while a broadband achromatic wave plate2 was

positioned at the fibre output, before the chirped mirrors, to bring the pulse

polarisation back to linear. To avoid non-linear effects the entrance window is

positioned 85 cm from the fibre entrance, similarly the output window is 45 cm

from the fibre end. The beam is then re-collimated with a f = 87.5 cm mirror.

The pulse is compressed with 10 reflections on double angled chirped mirrors

(UltraFast Innovations GmbH) providing -36 fs2 dispersion per bounce in the

spectral range 500-1000 nm. Figure 3.10 shows the typical spectral properties

of the input and output pulses of the fibre.

The CEP stability of the fibre output was measured with an f-2f interfer-

ometer, similar to the one illustrated in figure 3.4. The main difference is that

now, an octave spanning spectrum is already available, so no WLG generation

is required, and the fibre spectrum is directly focused in the frequency dou-

bling crystal. The delay between the two frequency components is provided

by 1 mm of glass dispersion. The fringe pattern is then recorded as a function

of time and the phase difference between long and short wavelengths can be

2λ = 0.25± 0.007 in the 600-950nm range


lambda (nm)500 600 700 800 900 1000

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1 InputOutput

time (fs)-50 0 50

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1InputOutput

Figure 3.10: Input spectrum (left) and FTL temporal profile (right) at the fibreinput (black line) and at the fibre output (red dotted line).

extracted and stabilised with a feedback loop on the AOM in oscillator pump

laser. Figure 3.11, shows the typical CEP-stability performance of the fibre

output. The CEP values distribution can be fitted with Gaussian profile with

σmean = σ/√Nshots, where σ is the statistical error on each shot measurement.

(a)

0 1 2 3 4

lam

bda

(nm

)

480

490

500

510

520

530

540

lam

bda

(nm

)

480

490

500

510

520

530

540

(b)

time (hours)0 1 2 3 4

CE

P (m

rad)

-1000

0

1000 (c)

CE

P (m

rad)

-1000

0

1000

=194 mrad

(d)

Figure 3.11: Fibre output CEP stability. The interference pattern between longand short wavelengths of the beam are measured as a function of time in (a). theintegrated signal in (b) shows that the coherence is preserved over time. The CEPcan be extracted for each spectrum measured and is shown as a function of time in(c). The resulting distribution has a gaussian profile with σ = 194 mrad as shownin (d).


3.2.2 Results and discussion

The main results from this experimental campaign are a set of measurements

at different input energies, for different fibre operation modes to explore the

energy scaling of hollow fibre pulse compression. In all measurements the

pressure of the fibre was adjusted to achieve a 4 fs FTL spectrum.

The first results presented in figure 3.12 are the output energies achieved

under the different experimental conditions, as a function of input energy.

vacuum

SFLPSFCPDPLPtheory

68%

Figure 3.12: Fibre output energy as a function of input energy, showing the clamp-ing of the transmission at higher energies. Adapted from [32].

Below 0.6 mJ input energy all modes of operation yield results close to the

vacuum transmission of 67%. At higher energies even the vacuum transmission

is not maintained, most probably due to the onset of non-linear effects even at

a pressure level of 10−2 mbar. The interesting aspect is the difference between

the SFLP case with respect to the SFCP and DPLP cases. It can be seen how

for SFLP, increasing the power input above 1.5 mJ does not affect the power

output that remains around 0.6 mJ. On the other hand, for DPLP and SFCP

the power output can be increased up to 0.8 mJ.

The theoretical predictions come from propagation simulations performed

by D. Austin (Imperial College), with a coupled-mode, split-step code which

includes the dispersion and losses of each mode, Kerr effects (self-steepening

included) and ionisation. The ionisation rate was calculated using the ADK

model as introduced in equation (2.38). The simulations were performed for


the DPLP case, with a gaussian spatial profile (1/e2 = 166μm) and 28 fs pulse

duration. The required pressure was set according to equation (3.11), in order

to achieve a 4 fs FTL spectrum. Figure 3.13 shows the theoretical pressure

required and the experimental data points.

SFLPSFCPDPLP

Figure 3.13: Fibre gas pressure required for different experimental condition andenergies. The points are the experimental values, lines are the expected pressureaccording to equation (3.11). Adapted from [32].

From these simulations it is possible to clarify the origin of the losses at

higher energies. We compared simulations where the direct ionisation losses

were taken into account (i.e. the energy required to ionise the gas atoms in

the fibre was subtracted from the system), to ones where ionisation was not

accounted in the energy balance. The results show that direct ionisation losses

are negligible and therefore suggest ionisation defocusing and related coupling

into higher order modes as main mechanism of energy loss.

The next results presented are related to the CEP stability in a hollow

fibre system. The propagation through any dispersive medium shifts the CEP

due to the difference between the group and phase velocity vg and vp. Over a

distance L, such shift can be expressed therefore as:

Δϕ = ωL

(1

vp− 1

vg

)≈ 2πL

dn

dλ=ω2L

c

dn

dω(3.12)

One source of CEP fluctuations arises from SPM. In fact an intensity fluc-

tuation δI0 induces a change in the refractive index and therefore a phase shift


δϕSPM[96]:

δϕSPM =ω20

c

∂n2

∂ω

∣∣∣∣ω0

LδI0 (3.13)

Another instability source is related to free electron dispersion experienced

by the pulse in the plasma produced by ionisation. The plasma refractive index

in fact depends on the pulse intensity and gas pressure via the free electron

density ρe:

nplasma =

√(1− ω2

p

ω2

)≈ 1− ω2

p

2ω2, ωp =

√e2ρe/ε0me (3.14)

where ωp is the plasma frequency. From this expression one can derive the

plasma induced CEP shift due to an intensity fluctuation as

δϕplasma ≈ e2L

meε0ω0c

∂ρe(I0, p)

∂I0δI0 (3.15)

To simulate the CEP stability, different intensities were set as input param-

eters (1% intensity fluctuations) and the distribution of the CEP output was

computed and σCEP extracted. Figure 3.14 shows the results of the simulations

(solid line) and the experimental data for SFLP and DPLP.

SFLPDPLPtheory

Figure 3.14: Fibre CEP rms for different pulse energies and experimental config-urations. Adapted from [32].


It can be seen how the CEP stability degrades while increasing the pulse

energy from 0.5 to 1.25 mJ. The simulations confirmed this degradation is

driven by ionisation-induced instabilities (δϕplasma) as switching SPM off in

the simulations does not change the CEP performance significantly. Above

1.25 mJ, σCEP does not increase any further. The mechanism responsible for

this stabilisation is identified as energy loss. As shown in figure 3.12, above

1.25 mJ more and more energy is coupled into higher order modes, preventing

a higher intensity to be reached and therefore limiting the related instabilities.

The performance of the SFLP and DPLP is very similar, proving of the validity

of differentially pumping schemes with respect the static filled ones.

3.3 HHG for isolated VUV and XUV pulses

production

In this section I will present results related to the spectral characterisation and

manipulation of the VUV and XUV attosecond pulses. These measurements

represent the first step in order to fully characterise the attosecond pulses, as

will be shown in chapter 5, and were used to identify optimal experimental

parameters, such as CEP, gas pressure and target position, for the streaking

measurements.

3.3.1 Gating the HHG emission

As explained in [75], to obtain attosecond pulses from the HHG process, phase-

matching is crucial. Moreover, obtaining an isolated attosecond pulse, rather

than an attosecond pulse train, requires a careful experimental design. The

standard generation of isolated attosecond pulses requires temporally gating

the high harmonics emission produced with an intense femtosecond NIR laser

to a single burst. When no gating technique is applied, the harmonics emis-

sion happens every half cycle of the driving NIR laser, resulting in a train of


attosecond pulses, not ideal for pump-probe experiments.

The more straightforward technique is amplitude gating [58]. It consists in

limiting the pulse duration of the driving NIR field to the few cycle regime.

In this way, with the correct choice of CEP, the highest energy portion of the

harmonics spectrum will be emitted only once, corresponding to the peak of

the electric field waveform. It is then possible to select this emission with a

spectral filter covering the appropriate energy range.

In order to achieve isolated attosecond pulses not only in the cutcut-off

energy range, and with longer driving fields, other gating techniques have been

developed. In polarisation gating [97, 98] the HHG dependence on polarisation

[99] is exploited. With circularly polarised light, no electron trajectories are

allowed to return to the proximity of the parent ion, therefore frustrating the

harmonics emission. The driving electric field can therefore be shaped so that

the polarisation changes within the pulse duration, going from circular, to

linear and then back circular again (different sign). It is possible to limit the

linear state to only one half cycle, obtaining a single burst of harmonics at all

energies.

In double optical gating (DOG) [100], the polarisation gating technique has

been further developed to be used with even longer pulses.

In fact depletion of the ground state before reaching the linearly polarised

part of the pulse can be detrimental when trying to implement polarisation

gating with long (> 6 fs) pulses. The problem with using longer pulses for

the production of isolated attosecond pulses is that, not only the emission has

to be frustrated for longer, but in this interval of time the population of the

ground state of the target atom must not be depleted.

In DOG a 2ω0 component is added to the driving field, breaking the sym-

metry of the driving field so that the harmonic emission occurs every cycle

rather than every half cycle. In this way pulses up to 9 fs can be used to

obtain isolated attosecond pulses [100].

In order to use directly the longer output of standard CPA laser systems


(≈20-30 fs) one can implement generalised double optical gating [30] (GDOG),

where DOG is optimised by using elliptically polarised pulses rather than the

circular ones. With GDOG the intensity of the leading edge of the pulse is

lowered significantly with respect to DOG, allowing the generation of isolated

attosecond pulses with driving fields as long as 28 fs [30].

The ionisation technique exploits the effects of depletion of the ground state

during HHG [101]. By applying a few cycle pulse with an intensity greater than

the ionisation saturation intensity it is possible, for certain CEP values, to con-

fine the attosecond pulse emission to a single burst. The CEP in a few cycle

pulse in fact allows a smooth and precise way to modify the instantaneous

intensity delivered on target. Experimentally this implementation is rather el-

egant since the optimum CEP value can be chosen by looking at the generated

harmonics, where a clear transition between separated harmonics and a con-

tinuum spectrum can be seen, both in the cut-off and in the plateau region of

the generating harmonics, confirming the transition from multiple to isolated

attosecond pulse generation [101].

The latest addition to the gating techniques for isolated attosecond pulses is

the so-called lighthouse effect [102, 103]. Here, rather than spectrally selecting

the cut-off part of the HHG spectrum as in amplitude gating, or in frustrating

the harmonics emission either via polarisation (DOG-GODG) or depletion of

the ground state(ionisation gating), the gating exploits the spatial properties

of the pulse. If an appropriate wavefront rotation is imprinted on the driving

field, different half cycles will be instantaneously propagating in different di-

rections. Wavefront rotation can be achieved by misaligning a pair of wedges,

and therefore imprinting an angular chirp on the pulse (angular dispersion).

When such a pulse is focused, mapping angles to positions, the pulse will be

spatially chirped, i.e. with a rotating wavefront. In this way, the harmonics

emission will be into slightly different directions at each half cycle and can be

spatially isolated after some propagation [104].

In our laboratory, amplitude gating is implemented with the few cycle pulse


described in 3.2 for the production of isolated attosecond pulses [79].

In the following sections I will present the experimental apparatus for the

HHG process and the spectral characterisation of HHG radiation, and the

results obtained in the process of optimising the XUV and VUV pulse produc-

tion.


The Attosecond beamline at Imperial College consists of a series of vacuum

chambers accommodating different experimental set-ups as seen in figure 3.15.

1 m

focusing mirror

gas jets Al filterflat fieldgrating

imaging MCP CCDcamera

two partmirror

gas targetfilters

irisirisCCD

camera

Figure 3.15: Beamline layout. The first chamber accommodates the gas jets forHHG. The second chamber contains an Al filter for the calibration of the flat fieldspectrometer in the third chamber. In the last chamber is positioned the electrontime-of-flight spectrometer, the metallic filters for spectral selection of the harmonicsradiation and the two part mirror assembly.

The first chamber is dedicated to the HHG process. The NIR pulse enters

the chamber through a 1-mm thick fused silica window and is then focused

with an f = 75 cm mirror. Within the Rayleigh range (15 mm) are positioned

the gas targets. For the production of synchronous VUV and XUV pulses

I used an in-line solution [105], positioning first the gas target for the VUV

production (Kr), and then the gas target for the XUV production (Ne). These

targets, operating at 1 kHz, are piezo actuated for the release in the vacuum


system of gas plumes (≈ 1017cm−3), as shown in figure 3.16. The focal spot

position was determined from the onset of plasma formation in air (chamber

partially evacuated at ≈ 10−1 mbar). This allowed positioning of the gas jets in

approximately the correct position in the propagation direction (z), as shown

in figure 3.16 where the plasma formation can be seen between the two gas

jets.

gas inlet

piezo disk

1.5 mm

HV

laserbeam

Figure 3.16: Gas jet schematics (left) and experimental implementation of a in-linepair (right) where the plasma formation is visible as a bright spot between the twojets.

The gas jet valve opening is triggered with the laser pulses. The two gas

targets were mounted on fully motorised x-y-z stages for alignment and opti-

misation purposes.

The second chamber serves as a differential pumping stage, and contains

an Al filter for the spectral calibration of the UV radiation. This chamber

houses the double slit used for the ablation plume harmonics spatial coherence

experiment [33] (see section 3.4).

The flat field spectrometer consists of a variable groove density, “flat field”,

gold coated grating (1200 l/mm, 001-0437, Hitachi) [106]. This grating is de-

signed to disperse the harmonics on a flat plane, where a micro channel plate

(MCP) is positioned to record the spectral intensity. The imaging MCP is

mounted on a moving flange so that different photon energies (16-200) eV can

be measured. The imaging MCP has two stacked plates with an active area

of 40 mm diameter. A phosphor screen (P46) allows detecting the UV photo-

electrons with a CCD camera (14 bit Coolview FDI, Photonic Science) in the


visible spectral range (530 nm).

The spectral calibration of the flat field spectrometer was performed with

an Al filter. The Ne harmonics in fact go beyond the Al absorption L-edge at 72

eV. Therefore the measurement of Al filtered harmonics allows the calibration

(pixel/wavelength) of the spectrometer by using the known grating dispersion

equation and the Al absorption edge position, as shown in figure 3.17

Photon energy (eV)40 45 50 55 60 65 70 75 80 85

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1Input spectrumfiltered spectrumAl transmission

Figure 3.17: Full spectrum (solid blue) and Al-filtered spectrum (dashed-red) of Neharmonics. The position of the Al L-absorption edge (dotted black line in overlay)allows the calibration of the flat-field spectrometer.

3.3.3 Spectral results

In figure 3.18, two typical VUV and XUV spectra can be seen. These spectra

were measured as a function of several parameters to find the best compromise

for the production of the two pulses simultaneously. For future reference, when

only the spectral information is reported, a spatial integration was performed

over a small window around the harmonics signal.

The choice of Ne gas for the production of the XUV pulse is a standard

one. It provides a high ionisation potential (IpNe = 21.6 eV) that allows a

higher harmonics cut-off with respect to noble gases with a lower Ip such as Ar

or Kr. On the other hand I did not use He, with an even higher Ip, because I

wanted to implement a loose focusing geometry to optimise the phase-matching


Kr

Energy (eV)16 18 20 22 24 26 28

Div

erge

nce

(mra

d)

-10

-5

0

5

10

Ne

Energy (eV)40 60 80 100

Div

erge

nce

(mra

d)

-10

-5

0

5

10

Figure 3.18: XUV (right) and VUV (left) spatially resolved spectra.

conditions and conversion efficiency of the HHG process. The use of a mirror

with focal length f = 75 cm allows us to reach an intensity at focus of 3 ·1014

W cm−2 (100μm waist), that is suitable for Ne, but too low for He that has

a ionisation saturation intensity almost twice as high compared to the one of

Ne ( 1.4 ·1015 W cm−2 for He compared to 8 ·1014 W cm−2 for Ne).

The choice of HHG gas target for the VUV pulse generation was restricted

to Kr and Xe, due to their low Ip that allowed the production of low harmonics

at 15-20 eV. Figure 3.19 shows that Kr was more efficient in the low harmonics

range in our experimental conditions.

Energy (eV)16 18 20 22 24 26 28

Inte

nsity

(arb

. u.

0

0.2

0.4

0.6

0.8

1KrXe

Figure 3.19: Kr (black) and Xe (dotted red) low harmonics comparison.


I performed measurement of the XUV and VUV HHG spectra as function

of each gas target position along the propagation axis, in order to optimise the

phase-matching conditions for the XUV and VUV pulses. Figure 3.20, shows

the results obtained.

Target position (mm)0 2 4 6 8 10 12 14 16

Ene

rgy

(eV

)

1618202224

Target position (mm)0 2 4 6 8 10 12 14 16

Ene

rgy

(eV

)

40

60

80

100

120

Figure 3.20: XUV (top) and VUV (bottom) spectra as a function of gas jet positionwith respect the focal plane (z = 0).

It can be seen that the XUV emission is optimised, (as described in section

2.3), after the focus (z = 0). On the other hand the VUV emission shows a

much weaker z-dependence, allowing the positioning of the two gas targets in

series, with the Kr gas jet first, 3 mm downstream from the focal plane, for

the production of the VUV pulse, and the Ne gas jet 7 mm downstream from

the focal plane for the production of the XUV pulse.

The NIR pulse compression and CEP optimisation was performed with a

pair of motorised wedges in the NIR beam (see figure 3.8) to fine tune the

material dispersion, while observing the XUV harmonics. In fact the cut-off

region is the most sensitive to the NIR pulse duration and CEP, and the NIR

intensity could be adjusted with an iris to set the cut-off to the desired spectral

region (90 eV).

Figure 3.21, shows the results obtained by measuring the XUV and VUV


spectra as a function of wedges position (i.e. glass insertion).

Glass insertion ( m)-100 -50 0 50 100

Ene

rgy

(eV

)

70

80

90

100

110

Glass insertion ( m)-100 -50 0 50 100

Ene

rgy

(eV

)

1618202224

Figure 3.21: XUV (top) and VUV (bottom) spectra as a function of glass insertionin the NIR beam.

It can be seen that for the XUV harmonics the choice of wedge position is

crucial in order to obtain an isolated XUV pulse. For the VUV spectra, no

CEP dependence is visible in this dataset. For this reason the NIR pulse CEP

was always optimised looking only at the XUV harmonics.

I then measured a series of spectra with the two pulsed valves either on or

off, so that the effect of each gas medium on the pulse generated in the other

gas jet could be verified.

The spectra in figure 3.22 show both the VUV range (left side) and the

XUV range (right). The color scale is the same for all spectra (note the x2

factor applied to some panels). For the XUV spectral region the cut-off of the

Kr harmonics (panel (d)) does not reach 90 eV, and therefore no interaction

is expected between the Ne and the Kr spectra at these energies. The signal

obtained with both gas targets on (panel (f)) is lower than when only the Ne

gas target is on (panel (b)), with an attenuation factor of about 2. However,

the signal for the VUV spectral range is much brighter and therefore the gas

pressure in the Kr gas target can be set to a lower value, depending on the


experimental conditions required.

(b)

x2

-5

0

5(d)

Div

erge

nce

(mra

d)-5

0

5(f)

x2

Energy (eV)50 100

-5

0

5

(a)

x2

-5

0

5(c)

Div

erge

nce

(mra

d) -5

0

5(e)

Energy (eV)15 20 25

-5

0

5

Figure 3.22: Interaction effects between the two gas targets. XUV HHG spectra onthe right, VUV on the left. In the top row (a)-(b), shows the spectra obtained withonly the Ne gas jet activated. Similarly the middle row (c)-(d) shows the spectraobtained with only the Kr gas jet on. the bottom row (e)-(f) shows the spectrameasured with the two gas jets operated simultaneously.


In the VUV range the production of sub threshold (below Ip) harmonics in

Ne is not efficient at all (panel (a)) and no significant interaction between the

Ne and Kr spectra is expected in this range. Running both gas targets at the

same time has a similar effect in the VUV range as it had in the XUV one, with

an attenuation of the Kr signal when the Ne gas jet is on. Comparing the Kr

only condition (panel (c)) and the condition with both gas targets activated

(panel (e)) shows an attenuation factor of about 2.

These results allowed me to develop a procedure for the optimisation of the

XUV and VUV pulse production. Since the XUV pulse production sets much

stricter generation conditions, both in terms of CEP and phase-matching, I

always optimised the XUV harmonics first (position and dispersion/CEP) and

then set the position of the Kr target to be few mm before the Ne target.

The effect of the Kr gas jet was tested also by comparing a CEP scan on

the XUV harmonics, with the Kr jet on and off. In figure 3.23 we can see the

results of a CEP scan similar to the one presented in figure 3.21, performed

with the Kr jet on (top) and off (mid). This time the CEP was modulated by

acting on the slow loop feedback signal, rather than moving the wedges. In

this way the dispersion experienced by the beam was not altered in any way

and the CEP effect be more easily analysed. Note how the cut-off maximum

position is not altered in the full 4π scan, in contrast with the data shown in

figure 3.21. By plotting the integrated signal around 90 eV (within the black

dashed lines) an average shift of 0.23±0.06π between the two measurements

(3 central maxima considered) can be extracted, that corresponds to the CEP

shift of the NIR pulse in the Ne target due to the presence of the Kr gas.


VUV on

0 0.5 1 1.5 2 2.5 3 3.5 4

Ene

rgy

(eV

)

80

100

VUV off

0 0.5 1 1.5 2 2.5 3 3.5 4

Ene

rgy

(eV

)

80

100

CEP ( )0 0.5 1 1.5 2 2.5 3 3.5 4

Cou

nts

(nor

mal

ised

)

0.5

1

CEP offset = 0.23 0.06

VUV onVUV off

Figure 3.23: HHG spectra as a function of CEP set on the slow loop feedbacksignal. The top panel corresponds to the case with both gas targets operating (VUVon). The middle panel corresponds to the case with only the Ne target operating(VUV off). The bottom panel shows the integrated signal around 90 eV, comparingthe VUV on case (solid blue) with the VUV off case (dashed red).


3.4 Plasma HHG

In this section I will present some results obtained in the experimental cam-

paign lead by PhD student Z. Abdelrahman (Imperial College) and in col-

laboration with Dr. R. Ganeev (Voronezh State University). The aim of the

campaign was to develop a high harmonics source based on ablation plumes

target rather than the more standard noble gas media.

3.4.1 Introduction

High Harmonics generation is a very promising method to extend the exper-

imental capabilities involving short wavelength coherent light. However the

low efficiency of the process can still be a limiting factor and the presence of

multiple harmonics is not ideal for some experiments where a specific spectral

region is of interest. For this reason, efforts are being made to optimise the

high harmonics photon yield and shape its spectral features. In this context,

high harmonics generation in ablation plumes has been developed as an alter-

native to the more standard noble gas media, extending the choice of targets

and showing some interesting spectral properties such as resonance-induced

enhancement of spectral portions of the HHG spectrum [107]. These charac-

teristics are of interest when looking for methods to develop ultrafast XUV

pulses, as they may provide naturally isolated spectral features, removing the

necessity of spectrally selecting a portions of the HHG emission either with

metallic filters or monochromators. Isolated spectral emission from ablation

plumes have been shown to occur not only at high energies (50 eV for Mn)

[108], but also at lower energies (20 eV for In) [33] and in some cases should

be able to support sub-fs pulse durations [24].

While the standard HHG spectrum is described by the 3-step model intro-

duced in 2.3, the resonance-induced enhancement in ablation plumes harmonics

can be described by the 4-step model [109]. While the first two steps (tunnel

ionisation and propagation in the continuum) remain the same, recombination


now happens in two steps. The electron interacts first with an autoionising

state (step 3) and then recombines with the parent atom (step 4) as shown in

figure 3.24

Figure 3.24: Four step model to explain the resonance-induced enhancement inablation plumes high harmonic generation. Reprinted figure with permission from[109]. Copyright 2010 by the American Physical Society.

It can be seen how, instead of directly recombining with the parent atom

(3* path) the electron wave-packet first recombines with the parent ion set-

tling in an auto-ionising state and from here relaxing to the ground state via

photoemission. As explained in [109], the 4-step model has comparable, or

even higher, probability with respect to the 3-step dynamics, due to the high

transition cross sections , which can be viewed as a resonant enhancement of

the process.

3.4.2 Experimental results

The experiments were performed at Imperial College in the beamline showed in

figure 3.15. In the HHG chamber a solid rotating target was positioned instead

of the gas target. Target rotation is necessary to have a new surface at each

laser shot, resulting in a more stable and stronger harmonics signal [110]. A

portion of the uncompressed beam from the CPA system is used as the ablation


pulse, with an energy of 120 μJ and duration of 23 ps. It is then focused with

a f = 40 cm lens to achieve an intensity of Iablation ≈ 8 · 109 Wcm−2. The

ablation pulse arrives first on the surface to produce the ablation plume that

is the target for HHG of the driving 3.5 fs NIR pulse 35 ns later.

In order to examine for the first time the spatial coherence of the high

harmonic emission from ablation plume targets, we inserted a double slit (50

μm spacing, 6 μm width) in the beam path (second chamber of figure 3.15),

and recorded the fringe pattern with the flat field spectrometer.

The degree of spatial coherence of the harmonic emission has been evalu-

ated with the visibility parameter:

V =Imax − Imin

Imax + Imin

(3.16)

With perfect coherence V = 1, while lower values of this parameter indicates

a lower level of coherence. In figure 3.25 typical results are shown.

For Zn and In (panels (a) and (c) of figure 3.25) the harmonics emission

shows the resonance feature introduced in the previous paragraph, with an

isolated bright emission at 18 eV in zinc and at 20 eV in indium. The en-

hancement at 18 eV is due to the overlap of the laser 11th harmonic (≈ 18

eV) with the 18.3 eV Zn III transition (3d10 → 3d9(2D)4p) [33, 111, 112].

The enhancement at 20 eV is due to the overlap of the 13th harmonic of the

laser (≈ 20 eV) with the 19.9 eV In+ ground to autoionising state transition

(4d105s2 1S0 → 4d95s2(2D)1P1) [33, 113]. Table 3.1 summarises the visibility

values obtained with the different targets.

Target VisibilityZn 0.74± 0.04In 0.66± 0.06

C (H13) 0.63± 0.03Ar (H15) 0.47± 0.08

Table 3.1: Visibility of harmonics generated in ablation plumes.

The reason for the presented values to be significantly lower than unity can


(a)Zn In

(c)

C(b) (d)

Ar

V=0.47V=0.63

V=0.66V=0.74

Figure 3.25: HHG spectra generated in ablation plumes of zinc (a), carbon (b)andindium (c). For comparison the harmonics produced in argon are shown in (d).The interference pattern is obtained with a pair of slits inserted in the beam path.Each panel shows the integrated spectrum (below), and a line-out of the interferencepattern on the right. Adapted from [33].

be attributed to the input coherence of the laser beam. The visibility of the

interference pattern of the q-th harmonic can in fact be computed to be [114]:

Vq = 1− q(1− V1), (3.17)

where V1 is the fringe visibility of the fundamental beam. A driving beam

with V1 = 0.98 is enough to lower the visibility of the harmonics to the levels

measured. The mechanism lowering the driving beam visibility can be due to

non-linear propagation effects in the path to the interaction region (i.e. in the

air path or at chamber entrance window).

The target with the highest visibility is Zn. The degradation of visibility for


the other targets can be attributed to an increase in the free electron density

in the interaction region. The presence of free electrons can lower the spatial

coherence of the beam due to a rapidly varying refractive index across the beam

profile [115]. ADK simulations confirmed that the free electron population in

Ar can be up to three times higher than in Zn [33].

In conclusion we showed that the spatial coherence of harmonics generated

in plasma plumes of C, Zn, and In, is comparable to that of standard har-

monics generated in Ar. This conclusion implies that harmonics generated in

plasma plumes are suitable for experiments requiring high spatial coherence,

for example diffraction imaging [116]. This result confirm the interesting fea-

tures of harmonics radiation produced in plasma plumes, and the possiblity of

using this type of source as a valid alternative to more standard HHG sources.

99

Chapter 4

New methods for attosecond

pump-probe experiments

The first two experiments presented in this chapter are focused on the charac-

terisation of short pulses in the few femtosecond range. As already introduced

in section 2.2, the characterisation of ultrafast pulses is a challenging task [15].

More and more sophisticated pulses can nowadays be produced [16] and they

are implemented in experiments heavily dependent on the precise waveform of

the electric field [3, 117]. For this reason, the development of new characterisa-

tion techniques is driven mostly by two factors. On one hand, the simplicity of

the implementation is essential to have easy access to the most important char-

acteristics of an ultrafast pulse. On the other hand, some experiments require

high accuracy in the pulse characterisation, due to the fact that the precise

waveform of the pulse applied to the experiment is critical for the dynamics

under investigation.

I will present some new developments of the dispersion scan “d-scan” tech-

nique [17, 118], an example of a simple experimental setup able to characterise

few cycle pulses.

Then I will present the Attosecond Resolved Interferometric Electric field

Sampling (ARIES) technique, an example of a powerful new technique for the

precise characterisation of the full carrier waveform of a pulse.

100 CHAPTER 4. NEW METHODS FOR ATTOSECOND PUMP-PROBE EXPERIMENTS

I will then describe the design of an electron velocity map imaging spec-

trometer and ion time-of-flight side-by side diagnostic. This device has re-

placed the electron time-of-flight spectrometer used to perform the streaking

experiments of the VUV and XUV pulses.

4.1 Dispersion-Scan

4.1.1 Standard d-scan

The basic idea behind the d-scan technique is to use the dependence of a

non-linear process, for example SHG, on both the spectral amplitude and

spectral phase of the pulse under investigation [17]. As the name indicates,

it is necessary to scan different amount of dispersion (i.e. input phase) for a

given pulse. The simplest way to implement this scan is with glass wedges and

a motorised translation stage. Depending on the wedges position the pulse

travels through a different amount of material, and therefore experiences a

different amount of dispersion as shown in equation (2.10). In this case the

only adjustable parameter available is L, the length of dispersive medium the

pulse has to travel through, and the dispersion applied is determined by the

type of material chosen. A more sophisticated way to introduce an arbitrary

amount of dispersion on a pulse is via an acusto-optic modulator [119], where

the spectral phase shape can be defined in a more general way (for example a

purely quadratic phase can be imposed to the pulse).

Regardless of the method chosen to perform a d-scan, it is necessary to

measure the non-linear signal spectrum multiple times, once for each change

in dispersion, and then feed the spectra obtained to an iterative algorithm for

the extrapolation of the spectral phase and amplitude corresponding to the

measured data. Performing this measurement can be straightforward. It is

possible to use SHG and a simple 1-D spectrometer to obtain the non-linear

signal spectra as a function of insertion of a pair of motorised wedges. All

CHAPTER 4. NEW METHODS FOR ATTOSECOND PUMP-PROBE EXPERIMENTS 101

these elements are usually already present in an ultrafast laser beamline, so

no particular beam manipulation is required for the pulse measurement. The

non-trivial part of the pulse characterisation is the iterative algorithm required

to find the spectral amplitude and phase that correspond to the measured

trace. In figure 4.1 it can be seen how the matching between measured (a)

and retrieved (b) traces is the first indication of the robustness of the retrieval

process, as in the FROG technique already introduced. Moreover the algorithm

is being further developed and studied to determine its capabilities and limits.

One useful feature of the algorithm developed so far is that it is capable of

dealing with partial traces [118] and an unknown spectral response curve in

the measured signal, due either to phase-matching conditions or experimental

constraints (i.e. spectral response of the spectrometer).

Figure 4.1: Example of pulse characterisation with the d-scan technique. Measuredtrace (a), retrieved trace (b), spectral (c) and temporal (d) properties of the pulse.Adapted from [17].

4.1.2 Single shot implementation

In the process of developing the d-scan technique, I designed an experimental

setup capable of measuring the required spectra for the d-scan technique in


a single shot basis in collaboration with Helder Crespo, Fransico Silva and

Warein Holgado from the Department of Physics & Astronomy and IFIMUP-

IN (University of Porto). The experimental results were obtained in a joint

experimental campaign in Porto. The laser system is similar to the one avail-

able at Imperial College and delivers a few cycle pulse after hollow core fibre

spectral broadening and chirped mirror compression of a Ti:Sapphire CPA sys-

tem [120]. The single shot d-scan implementation is of particular interest to

make the technique more general, e.g. for use with low repetition rate laser

systems, where a shot-to-shot analysis of the experimental data is sometimes

necessary.

As in most single shot measurements, we increased the dimensionality of

the measured data to map the repeated measurement onto an axis experimen-

tally measurable in single shot mode. For example in the single shot FROG

implementation, the delay axis is mapped onto the spatial axis of an imaging

spectrometer [121]. To implement this idea in the d-scan technique we decided

to apply different amount of dispersion to different spatial portions of the beam

and use an imaging spectrometer to measure the dispersion dependent spec-

tra as a function of spatial position. We used a BK7 Littrow prism to insert

different amount of glass at different spatial positions. Figure 4.2 shows the

optical characteristics of this element on the beam.

The first plot 4.2(a) shows the ray-tracing calculations I performed to ob-

tain the data presented in the following plots In 4.2(b) it can be seen how the

introduced dispersion is linear with respect the spatial position on the prism

entrance face. The angular dispersion due to refraction is not detrimental,

as a spectrum in the 400-1100 nm range has a total divergence of only 1.1◦

(19 mrad) as shown in 4.2(c). Since the d-scan is mapped on the spatial pro-

file of the beam it is now necessary to use an optical system to image the

beam onto the SHG crystal and then onto the imaging spectrometer. The

experimental setup, shown in figure 4.3, needs to be able to handle the spa-

tial properties of the beam very precisely in order to preserve the spatially


posi

tioin

(mm

)

0

20

40

60

80(a)

(nm

)

400

500

600

700

800

900

1000

1100

position (mm)10 20 30 40 50

GD

D (f

s2 )

500

1000

1500 (b)

(nm)400 600 800 1000

OU

T (deg

)

0

0.2

0.4

0.6

0.8

1(c)

Figure 4.2: BK7 Littrow prism used for the single shot d-scan. In (a) it is possibleto see how ray at different positions travel through different amount of glass. At theoutput surface there is no spatial dispersion as the beam enters the prism first faceperpendicularly. The range of GDD introduced as a function of position shown in(b) is enough to record a full d-scan trace. The angular dispersion is shown in (c)where θOUT is referenced to the angle of diffraction of λ = 400 nm.

varying dispersion from the prism to the imaging spectrometer. For this rea-

son the first manipulation to the beam is an astigmatism-free [122] reflective

beam expander (Telescope M1/M2 in figure). In this way I ensured spatial ho-

mogeneity of the beam, thus preventing spatio-temporal coupling that would

result in ambiguous traces (different pulses at different spatial positions). An

iris was used to select an 8 mm diameter section of the central part of the

expanded beam. The output surface of the Littrow prism was then imaged

onto the SHG generation crystal (10 μm thick BBO crystal) with a f=45cm

focusing mirror (imaging M1). This first imaging system, with a magnifica-


tion factor M=-0.33, not only mapped the dispersion information correctly to

the SHG crystal, but also allowed us to reach enough intensity in the image

plane to obtain a detectable second harmonic signal. A second imaging system

(Imaging M2, f=50cm M=-0.64) delivered the second harmonic signal to the

astigmatism-free [123] imaging spectrometer. From figure 4.3 it can be seen

how, with a flip mirror, it was possible to have a parallel measurement of the

pulse duration with a standard d-scan setup.

1D spectrometer

Imaging spectrometer

Standard D-scan

Flip M

Telescope M1

Telescope M2

Littrow Prism

Imaging M1

Imaging M2

BBO

BBO

1mJ 30fs 1kHzFemtolaser

Laser set-up

Figure 4.3: Experimental setup for the single shot implementation of the d-scantechnique. See text for details. Note the flip mirror (Flip M) allowing an easycomparison between the single shot implementation, and the standard d-scan.

The calibration for the dispersion axis of the standard d-scan setup consists

simply in mapping the phase introduced by the material used, to the wedges

position, and can be therefore considered completed once the geometry of the

wedges is known together with the material properties. For the calibration of

the single shot implementation, we measured a number of traces at different

wedges position, thus shifting the overall trace in the dispersion direction and

obtaining a pixel-to-dispersion calibration as shown in figure 4.4.


(a)

Lambda (nm)350 400 450

Pix

#100

200

300

(b)

Lambda (nm)350 400 450

Pix

#

100

200

300

Glass insertion (stage steps)80 90 100 110 120 130 140 150 160

Pix

#

100

150

200

250(c)

Peak positionLinear fit

Figure 4.4: Calibration of the dispersion axis for the single shot implementation ofd-scan. Traces are recorded as a function of glass insertion. (a) and (b) are the twoextrema of the calibration scan, showing how the trace is moving downwards. (c)shows the linear fit between pixel and glass insertion that calibrates the dispersionaxis. the error bars correspond to ±3σ to make them more visible.

4.1.3 Results

The measured traces are presented in figure 4.5. In the left column is plot-

ted the measured data with the scanning technique (top) and the single shot

implementation (bottom). On the right hand side the corresponding retrieved

traces are presented.

Comparing the top two traces of figure 4.5 it can be seen how the algorithm

reproduces accurately the measured scanning trace. In the single shot case

(bottom row), the camera sensitivity below 320 nm was very poor, resulting in

lack of signal with respect to the scanning technique, where the spectrometer

was able to measure signal down to 300 nm. In any case, the retrieval algorithm

can cope with partial traces, and managed to retrieve a trace similar to the

scanning measurement.

In figure 4.6 I show the results of the retrieval algorithm analysis performed

by F. Silva comparing the temporal profile and spectral phase obtained from

the single shot and the scanning data.


Scanning M.

BK

7 in

serti

on (m

m)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Scanning R.

BK

7 in

serti

on (m

m)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Single Shot M.

Wavelength (nm)300 350 400 450

BK

7 In

serti

on (m

m)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Single Shot R.

Wavelength (nm)300 350 400 450

BK

7 In

serti

on (m

m)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 4.5: Comparison between the scanning (top) and single shot(bottom) mea-sured (left) and retrieved (right) traces. Analysis by F. Silva

The main result from the comparison between the two temporal profiles

obtained is that they are compatible, within the retrieval error, with a pulse

duration of 3.2±0.1 fs for the scanning technique, and 3.3±0.3 fs for the single

shot measurement. The shaded area around the single shot data represents

the standard deviation between multiple retrievals run with different initial

parameters (64 retrievals in total, with different optimisation variables). The

main discrepancy between the two techniques is the modulation of the spectral

phase. The retrieved phase obtained from the single shot dataset (solid blue

line) is the average of the 64 retrievals performed. Some single shot retrievals

reproduce such oscillations, but the average over all of them washes out such

oscillations. Note the shaded area has an amplitude comparable to the am-

plitude of the oscillations measured with the standard scanning method. A


Pha

se (r

ad)

-3

-2

-1

0

1

2

3(a)

Single ShotScanning

Wavelength (nm)500 600 700 800 900 1000

Inte

nsity

(arb

. u.)

0

0.5

1(b)

Time (fs)-20 -10 0 10 20

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1

3.3 0.3 fs(Single Shot)

3.2 0.1 fs (Scanning)

(c)

Figure 4.6: Spectral phase (a) and corresponding temporal profile (c), obtainedfrom the single shot d-scan (solid blue lines) and the standard scanning d-scantechnique (dashed-red lines). the input spectrum is shown in (b). the shaded areais the standard deviation of the single shot measurement, obtained from differentretrievals settings.

rigorous method to choose one particular set of retrieval parameters, rather

than average over them, is yet to be determined and is currently under inves-

tigation by F. Silva and co-workers.

In conclusion, the experimental results confirmed the validity of the single

shot implementation. The quantitative agreement between the single shot im-

plementation and the standard technique is satisfactory with a pulse duration

of 3.3±0.3 fs retrieved in the single shot measurement, and 3.2±0.1 fs in the

scanning measurement. More work in the retrieval procedure is necessary to

define a robust way to select a set of optimisation variables representing the

phase information without ambiguities.

While the results presented can be considered a proof-of-principle of the


single shot implementation of the d-scan technique, they also pointed to the

aspects that should be improved in the future. To begin with, a more compact

design of the experiment would be desirable, as with the current focusing optics

the beam path in the single shot implementation, from the Littrow prism to the

imaging spectrometer was ≈ 4m. On one had this made the alignment easier,

as the image plane could be found manually without the need of translation

stages. On the other hand the use of shorter focal length optics would allow

the design of a more compact setup with an easier handling of the dispersed

beam after the prism (in our setup the beam filled almost fully the imaging

mirror M1). Moreover an imaging system with a shorter focal length would

relax the intensity constraint at the BBO crystal, since a smaller spot-size

could be achieved.

Another aspect to consider for the further development of the single shot

implementation of the d-scan technique is related to the dispersion range re-

quired to characterise a pulse. In fact, characterising a few-cycle pulse requires

a smaller dispersion range compared to a longer pulse. To make the single shot

implementation possible for longer pulses it may be necessary to adopt a more

dispersive material for the Littrow prism.

In general, the single shot implementation of d-scan maintains all the ad-

vantages of the scanning technique, since no splitting of the pulse and no

temporal overlap is required for the measurement to be taken. The alignment

of the imaging optics can be performed in a straightforward manner since it

only requires finding the image plane of the dispersive element (in our case the

Littrow prism) and of the BBO crystal. The iterative algorithm necessary to

retrieve the pulse duration is becoming more and more robust, with the pos-

sibility of coping with partial traces [118], making the d-scan technique and

its single shot implementation a viable alternative characterisation technique

with respect the more standard ones based on FROG or SPIDER.


4.2 Attosecond Resolved Interferometric

Electric-field Sampling

With the development of more and more complex pulses [16], one of the main

challenges in ultrafast pulse characterisation is the retrieval of the complete

field waveform. The characterisation techniques presented so far can only

characterise the spectral phase from the second order onwards (see equation

(2.9)). For this reason, different methods have been developed to characterise

ultrafast pulses more precisely, for example retrieving the CEP of a pulse or

even the full carrier waveform information [18, 60, 124]. Apart from a few

recent developments [125], such measurements usually rely on HHG [18] or

photoelectron spectroscopy [124]. In the case of HHG, it is possible to modify

the CEP value looking at the continuum cut-off and attribute CEP=0 to the

position corresponding to the smoothest and highest cut-off [126], as shown in

figure 4.7.

Glass insertion ( m)-50 0 50

Ene

rgy

(eV

)

70

75

80

85

90

95

100

105

110

0 0.5 1

CEP=0CEP= /2

Figure 4.7: CEP scan showing how it is possible to distinguish pulses with CEP=0with highest and smoothest cut-off (dashed black line) from pulses with CEP=π/2with lower and modulated cut-off (red dotted line).

While this is a reliable method for optimising the generation of isolated


attosecond pulses, it does not provide a full pulse characterisation and, until

recent developments [18], the only way to obtain the full waveform of an elec-

tric field was with a photoelectron spectroscopy technique known as attosecond

streaking [127]. This technique is a powerful method for ultrafast pulse charac-

terisation and will be presented in detail in chapter 5. However, the detection

of photoelectrons is technically demanding as it usually requires a dedicated

experimental vacuum chamber and appropriate electron detectors. For this

reason the development of other techniques for the full characterisation of the

electric field waveform not relying on photoelectron spectroscopy are of par-

ticular interest, especially if it can be based on equipment already necessary

for the characterisation of the HHG radiation, such as an XUV spectrometer.

I will present some results obtained in our laboratory at Imperial Col-

lege obtained in collaboration with Adam Wyatt, Andrea Schiavi and Ian

Walmsley from the Clarendon Laboratory (University of Oxford). We de-

veloped the Attosecond Resolved Interferometric Electric-field Sampling tech-

nique (ARIES), an all optical ultrafast pulse characterisation technique.

4.2.1 Concept

ARIES uses the modulation of the HHG cut-off to track the instantaneous

waveform amplitude of the field. As shown in figure 4.7, the HHG cut-off

is strongly dependent on the exact shape of the generating waveform. It is

possible to exploit this dependence to extrapolate the full electric field. Let

us consider two pulses, linearly polarised along the same axis, one driving

the HHG process, that I will call the probe-pulse, and one playing the role

of the pulse to characterise, defined as the test-pulse. These two pulses are

combined coherently in the interaction region of the HHG process. A time

delay can be introduced between the two, and HHG spectra can be recorded

as a function of this delay. If the probe-pulse driving the HHG process is

a few-cycle pulse, the resulting HHG cut-off is determined in the sub-cycle

time interval occurring in the more intense half-cycle of the laser pulse. The


interference between the test and probe pulse occurring at the most intense

half-cycle of the probe-pulse will therefore modulate the cut-off position with

femtosecond to attosecond time resolution, corresponding to the time window

over which the cut-off is generated and emitted.

From a semi-classical point of view, ARIES exploits the fact that the cut-

off emission happens for a very precise electron trajectory. The interference

between the test and probe pulses modifies the instantaneous intensity of the

waveform, leading to a shift of the cut-off position. When the interference is

constructive (destructive), the cut-off is shifted to higher (lower) energies as

shown in figure 4.8.

Times (fs)-8 -6 -4 -2 0 2 4 6 8

Am

plitu

de (a

rb. u

.)

-1

-0.5

0

0.5

1 (a) Probe pulseTest pulse 1Test pulse 2

Harmonic Order45 50 55 60 65 70 75 80 85 90

Am

plitu

de (a

rb. u

.)

0.4

0.5

0.6

0.7

0.8(b)

Figure 4.8: Aries concept. The probe pulse, solid black line in (a), driving the HHGprocess, interferes with the test pulse. For constructive interference (test pulse 1,blue-dashed line) in the high amplitude region of the probe-pulse (grey shaded area),the HHG cut-off is shifted to higher energies as shown in the corresponding spectrum(blue-dashed line and shaded area) presented in (b). For destructive interference(red-dotted line) the opposite happens with a shift of the cut-off to lower energies.Data courtesy of A. Wyatt.

This is, of course, a simple model for the treatment of HHG, but both


SFA (strong filed approximation) simulations (performed by A. Wyatt) and

semi-classical simulations (performed using a code I developed), show similar

results, proving that the main dynamics involved in the HHG cut-off shifting

are mainly driven by the interference process as described above. Moreover,

the correct retrieval of the test-pulse obtained from the cut-off position proves

that the mapping between these two quantities (cut-off shift and waveform

amplitude) can be considered linear. Some simulations results are shown in

figure 4.9, where two test pulses (white overlay) are correctly mapped by the

cut-off position both by an SFA code in (a), and by a classical trajectory code

in (b).

(a)

Delay (fs)-5 0 5

Ene

rgy

(eV

)

100

110

120

130

140

150

160(b)

Delay (fs)-5 0 5

Ene

rgy

(eV

)

140

145

150

155

Figure 4.9: Aries traces obtained with an SFA code (a), and a classical trajectorycode (b). SFA data courtesy of A. Wyatt. The white line in overlay is the inputtest-pulse, correctly mapped in both cases by the cut-off position.


The experimental implementation of the ARIES technique requires combining

the test-pulse and the probe-pulse in the interaction region (gas target) for

HHG with attosecond resolution. This was achieved with a Mach–Zehnder

interferometer positioned after the hollow fibre setup of our laboratory (see fig


3.8). The few cycle pulse after hollow fibre pulse compression was used as the

probe-pulse. As shown in figure 4.10, a beam-splitter was used to split 10%

of the beam for the test-pulse which passed through a pair of wedges before

being recombined with the probe-pulse with another 10% beam-splitter (i.e.

the test pulse was 1% of the probe-pulse).

HHGFlat field

spectrometer

test-pulse

probe-pulse

delay

BS BSW1

W2

3.5 fs 0.5 mJ

Figure 4.10: Aries setup. The fibre output was delivered to a Mach-Zehnderinterferometer. The test-pulse is obtained from the reflection of two 10% beam-splitters. The probe pulse is delivered to a double stage (mechanical plus piezo)to obtain the necessary high precision, long delay line. The two pulses were thenrecombined in the second beam splitter and focused in the HHG chamber.

The probe-pulse compression and CEP control was performed with a pair

of fused silica wedges before the interferometer and after the chirped mirrors

by looking at the HHG emission produced with the probe-pulse only. The

test-pulse dispersion and CEP was controlled with the second pair of wedges

(W2) positioned in the test-pulse arm of the interferometer. This allowed us to

tailor different test-pulses to be retrieved with the ARIES technique. Precise

temporal delay between the two pulses was achieved with a piezo stage (Physik

Instrumente P-753 Lisa 50 pm resolution over 28 μm travel range) positioned

in the probe-pulse arm of the interferometer. This piezo stage was mounted on

top of a coarser stage (Thorlabs PT1-Z8, 200nm resolution over 25 mm travel

range) to obtain at the same time high stability and long delay range.

To test the ARIES technique capabilities we measured different test-pulses


obtained with different amount of dispersion. The wedge dispersion in the

test-pulse arm was calibrated via spectral interferometry and therefore we

can compare the expected effects of dispersion (CEP shift and chirp on the

waveform) with the measured one.

The first results I present are the measurement of test-pulses tailored so

that they had the same duration but different CEP. These test-pulses were

obtained with very small wedge displacement. In fact to achieve a CEP shift

of 2π it is necessary to insert ≈ 56 μm of fused silica, changing the pulse

duration of a pulse at 790 nm from 3.5 fs to 3.9 fs. The results obtained are

shown in figure 4.11.

On the left-hand side the raw, spatially integrated, spectra are shown,

showing modulations both in amplitude and in the cut-off position. The am-

plitude modulations are attenuated on the right column by normalising the

spectrum at each delay to unity. This allows an easier determination of the

cut-off position that is marked by the black line. It can be seen how the CEP

of the pulse changes from top to bottom performing a full 2π cycle. Each step

correspond to π/2 and it is possible to recognise the different sin-like and cos-

like shapes of the waveform. The sensitivity of the technique to the electric

field waveform is clearly evident.


Delay (fs)-8 -6 -4 -2 0 2 4 6 8

110115120125

110115120125P

hoto

n en

ergy

(eV

)

110115120125

110115120125

Delay (fs)-8 -6 -4 -2 0 2 4 6 8

110115120125

Delay (fs)-8 -6 -4 -2 0 2 4 6 8

110115120125

110115120125 P

hoto

n en

ergy

(eV

)

110115120125

110115120125

Delay (fs)-8 -6 -4 -2 0 2 4 6 8

110115120125

Figure 4.11: Aries CEP measurement. Test pulses with different CEP are suc-cessfully characterised. Each row corresponds to a π/2 shift in CEP. On the leftis the raw, spatially integrated, spectrum, showing the cut-off energy and intensityfluctuations. On the right the same data is scaled to unity per each delay, enhancingthe energy shift visibility of the cut-off. Analysis of A. Wyatt


We then proceeded to introduce an even higher amount of dispersion in

order to significantly modify the test-pulse duration. In figure 4.12 the results

of such measurements are shown.

Time (fs)-50 0 50

Am

plitu

de (A

rb. u

.)

(a)

Wavelength (nm)500 600 700 800 900 1000

Pha

se (r

ad)

-30

-25

-20

-15

-10

-5

0

5

10

15(b)

Figure 4.12: Aries dispersion measurement. Test pulses with different dispersionhave been characterised. (a) is the collection of different test pulses retrieved. (b) isplotted the corresponding phase measured (solid line) and the expected one (dashed)according to the amount of glass inserted. The confidence interval plotted correspondto ±σ. In overlay (solid blue line) is the retrieved spectrum. The order of the pulsesin (a) is the same as the phases shown in (b). The third pulse from the top has beenpicked as reference for the phase calculations. Analysis of A. Wyatt


In figure 4.12(a) are plotted the retrieved waveforms of a series of test

pulses. The analysis of the waveforms allowed us to extract the spectral phase

of each of them and compare it with the expected phases provided by the glass

inserted with the wedges pair. This comparison is shown in 4.12(b), where the

retrieved spectral phase is plotted in solid lines, and the calculated phase from

the glass insertion is plotted in dashed lines. The confidence interval plotted for

each phase in red lines corresponds to the RMS around the expected dispersion

(solid black line), weighted on the retrieved spectrum (shown in overlay solid

blue line).

The agreement between the two data series is good showing the accuracy

of the ARIES technique.

In conclusion ARIES successfully characterised the complete electric field

of pulses with different CEP and different chirps and is a promising method

for the characterisation of ultrafast pulses waveforms without the need of an

attosecond streaking setup.

A valuable aspect of the ARIES technique is the fact that it characterises

the pulse in the interaction region dedicated to the HHG process. For other

techniques the difference in dispersion between the beam path to the charac-

terisation set-up, and to the experimental setup has to be taken into account,

and therefore the pulse used in the actual experiment must be calculated from

the pulse characterised by applying the expected amount of dispersion expe-

rienced on the way to the experimental chamber. With the ARIES technique,

on the other hand, the pulse is already characterised at the interaction region,

giving direct access to the waveform that will be used for HHG.

The data acquisition speed is another interesting aspect of the ARIES

technique. In principle it is possible to record HHG spectra on a single shot

basis. To improve the signal to noise level the HHG signal can be integrated

for longer. To perform a delay scan of 30 fs, with 0.1 fs step size, integrating

for 1 second the image at each delay step (1000 laser shots at 1 kHz repetition

rate), the total integration time required would be of just 5 minutes.


Some aspects of the technique are to be further investigated. For example

it would be interesting to verify the range of test-pulse intensities for which

ARIES is still applicable. In the experiment presented, a test-pulse of 5 μJ

has been characterised with a probe-pulse of 500 μJ. In principle the ARIES

technique can be scaled to lower energies using a tighter focus for the HHG.

Regarding the limitations on the test-pulse bandwidth and duration, the results

presented have shown that the ARIES technique can characterise long pulses,

as the most chirped pulse presented in figure 4.12 has a duration of ≈ 18 fs and

the carrier modulation were measured in a 40 fs range. Therefore, from this

point of view, it seems that the maximum duration measurable is limited to

the precision of the delay line, since attosecond resolution is required at each

time step.

Regarding the measurable bandwidth with the ARIES techniques, the re-

sults presented indicate that it is possible to measure a bandwidth comparable

to the one of the probe-pulse. However the technique seems sensitive to higher

frequencies. Classical simulations, shown in figure 4.13, predicts the possibility

of extracting a field with spectrum centred at ω0test = 2ω0probe .

In the simulation a 3.5 fs pulse centred at 800 nm was used as probe-pulse.

A pulse centred at 400 nm and with 30fs3 chirp was used as test-pulse. The

computed trace shown in figure 4.13(a) shows the expected oscillations at 2ω0.

However a delay can be noticed between the carrier of the test-pulse (plotted

in overlay) and the edge of the signal. This difference is more evident in figure

4.13(b) where the temporal profiles of the probe-pulse (dashed blue), of the

test-pulse (solid red) and of the retrieved pulse (dotted black) are shown. This

delay may be due to the fact that the timing of the trajectory responsible for

the highest photon energy does not correspond to the peak of the envelope

of the probe-pulse. Nevertheless the spectral properties of the test-pulse are

retrieved. This is of particular interest as it would open the possibility of a

direct characterisation of the pulses used in two color (ω + 2ω) schemes [128,

129], or with extreme bandwidths [16].


test-p

(a)

Ene

rgy

(eV

)

140

142

144

146

148

150

152

Delay (fs)-10 -5 0 5 10

Am

plitu

de (b) probe-ptest-pretrieved

0 0.5 1 1.5 2 2.5 3

Inte

nsity

00.5

11.5

(c) probe-ptest-pretrieved

Figure 4.13: Semi-classical simulation of an ARIES measurement with a test-pulsewith a central frequency corresponding to the second harmonic of the probe-pulse.(a) Shows the computed trace, with in overlay the test-pulse. In (b) the temporalprofiles of the pulses are shown. A delay between the input test-pulse and theretrieved pulse can be noticed. In (c) are plotted the spectral properties of thepulses.


4.3 Electron Velocity Map Imaging and ion

time-of-flight

In this section I describe the development and design of a dual [130] electron

velocity map imaging (VMI) spectrometer and ion time-of-flight spectrometer

(see figure 4.14) that was installed in our beamline, replacing the electron time-

of-flight spectrometer used to record the streaking traces presented in chapter

5.

e-

i+

ion TOF MCP

electron VMI imaging MCP

repellerextractorcorrector

a) b)

Figure 4.14: Schematics (a), and a picture (b), of the installed dual electron VMIand ion TOF spectrometers. The top part is the VMI side, with the imaging MCPphosphor screen read by a camera mounted on the lid of the chamber. At the bottom(not visible) is the MCP dedicated to the detection of the ions hits.

This device allows simultaneous VMI and TOF measurements, thus allow-

ing, for example, covariance analysis of the recorded signals. The covariance

mapping technique [131] has been demonstrated useful in deconvolving com-

plex dynamics, by decoupling signals originating from different channels of the

dynamics under investigation [132].

4.3.1 VMI background theory

With the existing electron time-of-flight spectrometer, only the photoelectrons

with momentum pointing towards the TOF aperture are recorded. This allows


for a high angular resolution at the price of a low collection efficiency (accep-

tance angle 7.85 · 10−3 sterad, 0.25% of the full 4π angle). To overcome this

limitation an electron VMI was implemented. It also allows the measurement

of the full 3-D momentum distribution from a 2-D projection [133]. This pro-

jection is performed with a set of electrodes, and the electrons are detected on

an imaging MCP. The projection process is described in figure 4.15. Assum-

ing an initial electron distribution �p(px, py, pz), the transformation providing

the 2-D distribution on the detector P (xd, zd) is known as an Abel transform

[134]. While the projection action does not require particular assumptions,

the inverse procedure requires a degree of symmetry to recover the full 3-D

information. Assuming cylindrical symmetry around an axis in the xz plane

allows an Abel inversion to be performed, the transformation providing �p from

P (xd, zd).

x

y

z

Abel transform

Abel inversion

3D 2D

3D 2D (plus symmetry)

L

Figure 4.15: Concept of Abel transform and inversion, applied to retrieve the3-D momentum distribution p from the measured 2-D projection P (xd, yd) on thedetector plane.

In the case of a constant �E field applied to the electrons, the position on

the detector plane is related to the initial kinetic energy by

KE =p2x2me

+p2z2me

=eE

4L(x2d + y2d) (4.1)

This relation allows the distribution, measured in pixels, to be converted


into the momentum distribution using an Abel inversion algorithm. I used a

PBASEX [135, 136] algorithm implementation, developed by former Imperial

College PhD student M. Siano. It is an efficient method as it describes the 3-D

momenta angular distribution in the natural Legendre polynomial basis (and

spherical coordinates). The projected image is fitted with a set of functions

defined as [135]:

f(k,l)(R, θ) = e(R−Rk)2

2σ Pl(cos θ). (4.2)

This basis is convenient since common processes such as single photon ionisa-

tion of a randomly aligned target (atom or molecule) have an angular distri-

bution given by [137, 138]:

P (θ) ∝ 1 + β2P2(cos θ) (4.3)

where θ is the angle between the polarisation vector and the direction of the

ionised electron. For this reason a limited number of Legendre orders (0 and

2) need to be included in expansion (4.2). Note that to ensure a non-negative

cross-section at every θ, the value β2 is limited in the range [−1, 2], where

β2 = −1 means electron ejection peaked along the plane orthogonal to the

laser polarisation, β2 = 0 means uniform emission, and β2 > 0 corresponds

to emission peaked along the polarisation axis. Figure 4.16 shows simulated

images for different β2 values.

4.3.2 Design

The design of the VMI spectrometer was based on an existing VMI instrument

at Artemis, Central Laser Facility, Rutherford Appleton Laboratory, UK, and

was optimised for resolving electrons in the 5-90 eV kinetic energy range. This

range is chosen based on the photon energies provided by the available VUV

and XUV pulses (see 5), and assuming the ionisation of small molecules with

an Ip of few eV (see chapter 6).


0 5 10 15 20 25 30 35

Figure 4.16: Simulated VMI images for different β values. Each simulation con-sists of 1E6 particles with angular distribution set by the β parameter and spatialdistribution given by a 3D Gaussian with (σx, σy, σz) = (1000, 5, 5)μm

The design was optimised by simulating different photoelectron distribu-

tions with Simion R©, a program dedicated to the simulation of electric and

magnetic fields once the geometry and voltage of the electrodes is given. The

expected projections of the electrons distribution were therefore obtained tak-

ing into account the geometry of the electrodes and the consequent electric

field modulations. The position, size and voltages of the electrodes can be

then optimised to achieve a resolution limited to the pixel size of the camera

used to record the spatial distribution of the electrons. In fact the simple model

presented in the previous paragraph does not take into account that the parti-

cles do not originate from a point-like source. This means that particles with

different initial kinetic energies may arrive in the same detector positions and

equation (4.1) would not hold. For this reason a focusing electrode is added

in order to minimise the spreading due to the initial spatial distribution.

Once the VMI geometry was chosen, minimal modifications were required


to include the ion time-of-flight spectrometer. I will only present the final

design, where we reached a reasonable trade-off between VMI performance,

ion TOF resolution and the voltages required. The resolution of the VMI

spectrometer is in fact lowered by the presence of the aperture for the ion TOF

spectrometer. The presence of this aperture does not completely screen the

electrons from the voltages applied on the ion TOF side. Applying a mesh to

lower the penetration of the field between the two spectrometers would result

in a lower collection efficiency for the ion TOF. Applying the technique of

covariance mapping to measurements biased by different collection efficiency

is detrimental, and therefore the option of using a mesh was discarded. In

figure 4.17 the set of electrodes simulated in Simion is shown, together with a

summary of the device performance. D. Walke in collaboration with Dr. T.

Barillot found the necessary technical solutions and design for the construction

of this diagnostic.

On the left of figure 4.17 a 3D rendering of the simulated electrodes can be

seen. A section of these electrodes is presented on the top plot 4.17(a). The

voltages have been chosen to obtain a grounded interaction region (at 98.5

mm in the plot). This will simplify the design of any target delivery system

to the interaction region. In fact, to minimise field distortions at the interac-

tion region, any external component should be introduced at the potential for

the spatial region it occupies. Grounding the interaction region means that

components placed in the interaction region (e.g. gas needles) will be close

to ground as well. A consequence of this choice is that the imaging MCP is

floated at 6 kV, while the MCP for the ion TOF is floated at -3kV. The two

electrodes around the interaction region are called the repeller and the extrac-

tor (from an electron point of view) and are held at ±333V. The repeller on

the ion TOF side has a smaller aperture than the other electrodes in order

to better shield the VMI side from the penetrating voltage of the ion TOF

electrodes. A third electrode, named the “corrector”, has been optimised to

further compensate the presence of the ion TOF with a voltage of 4.8 kV.


(a)

(b)(c)

(d)

1 cm

Figure 4.17: A summary of the expected performance of the electron VMI andion TOF diagnostic. On the left is a 3-D rendering of the electrodes simulated inSimion. The top plot provides a section of the electrodes and their voltages. Inthe middle raw is the simulated calibration of the pixel-to-energy curve. Note theerror bars in the simulated points showing the expected resolution. The plot at thebottom shows the resolution of the ion TOF for fragments with different masses.

In figure 4.17(c) I present the simulated calibration curve corresponding

to equation (4.1) together with the image 4.17(b) used to obtain the peaks

positions. The blue dots represent the peak position obtained from the inverted

images. Each distribution consists of 5·105 mono-energetic particles with a

3-D Gaussian spatial distribution ([σx, σy, σz] = [1000, 5, 5]μm to simulate an

elongated interaction region. The angular distribution in these simulations was

uniform (β2 = 0). The error bars in the plot represent the expected resolution

at the given voltages, optimised for high energy photoelectrons. At 85 eV the

resolution is 0.5% (Δpix/pix) and the pixel size limit is almost reached with a


FWHM of the radial distribution of just 2.2 pixels. This means that a mono-

energetic distributions of electrons will be mapped to a radial distribution with

a peak width comparable to the pixel size of the camera recording the image,

and therefore no further optimisation of the VMI electrodes can improve the

energy resolution of the device. At low energies the resolution is lower, but

still satisfactory with a resolution at 5 eV of 6.2% and a FWHM of the radial

distribution of 6.2 pixels. These are promising simulations as the voltages can

be rescaled to optimise the resolution in the energy range required for the

specific experiment.

In figure 4.17(d) I present the performance of the ion time-of-flight. It can

be seen that the simulated fragments peaks start to merge when considering a

difference of 1 atomic mass out of 80. This is again satisfactory as it allows us

to discriminate heavy fragments of molecular targets with different hydrogen

amounts.

The electron VMI and ion TOF setup has been successfully installed in our

beamline and figure 4.18 shows some benchmark results. In figure 4.18(a) the

ATI rings generated in Xe can be seen in the recorded VMI picture, while in

4.18(b) the different isotopes of Xe have been distinguished in the recorded ion

TOF spectra. Data courtesy of D. Walke and Dr. T. Barillot.

(a)

0

5

10

15

20

Mass (a.u.)128 130 132 134 136

coun

ts

101

102

103

104

129Xe 131Xe132Xe 134Xe 136Xe(b)

Figure 4.18: Benchmark measurements for the electron VMI (a) and for the ionTOF (b). In (a) the ATI rings generated in Xe are visible. In (b) the differentisotopes of Xe have been distinguished.

127

Chapter 5

Attosecond streaking of XUV

and VUV pulses

This chapter is dedicated to the main result of my PhD, the streaking experi-

ments performed to characterise the pulses corresponding to the spectra shown

in figure 3.22 generated with the in-series gas target setup. In the case of the

XUV pulse, the main result is that the effect of the second gas jet on the XUV

pulse is negligible and we confirmed a pulse duration similar to the one already

measured in previous experimental campaigns [84].

Streaking experiments in the VUV range have been rare [21], and never with

pulses obtained with the amplitude gating technique. The results I am going

to present show that attosecond streaking can be successfully implemented in

this spectral range and, similarly to the XUV case, the second gas target does

not affect the pulse duration of the VUV pulse significantly.

The characterisation experiments were performed in the three configura-

tions already presented in 3.22: XUV only, VUV only and XUV and VUV

generated at the same time. In this way we characterised the effect of each

gas target on the temporal profile of each pulse. First I will provide the the-

oretical background of attosecond streaking and then I will present a series of

measurements obtained with an electron time-of-flight spectrometer, together

with their analysis with the least squares generic projection algorithm (LS-

128 CHAPTER 5. ATTOSECOND STREAKING OF XUV AND VUV PULSES

GPA) [139] applied the FROG-CRAB measurements performed [124].

5.1 Theoretical background

Attosecond streaking is, so far, the only technique that allows the full char-

acterisation of an attosecond pulse. The first step of this technique is pho-

toionisation, where a photoelectron wavepacket is produced by the attosecond

pulse. Assuming only one electron participates to the ionisation event, the

transition amplitude a(v) between the initial ground state to the final state

with momentum |v〉, can be written as [140]:

a(v) = −i∫ ∞

−∞dvEUV (t)e

i(W+Ip)tdt, (5.1)

where W is the energy of the final state, EUV (t) is the attosecond electric

field, Ip is the ionisation potential of the target atom, and dv is the dipole

transition matrix element between the ground state and the final state. It can

be seen that, if dv does not include energy dependent factors, either in phase

or amplitude, then a(v) corresponds to the Fourier transform of the incident

pulse EUV (t).

If now an intense NIR field is added to the ionisation process, the transition

amplitude can be rewritten as a function of the delay τ between the two pulses

[124]:

a(v, τ) = −i∫ ∞

−∞eiφ(t)dp(t)EUV (t− τ)ei(W+Ip)tdt (5.2)

The new term with respect to equation (5.1) is the phase term eiφ(t) with:

φ(t) = −∫ +∞

t

v ·A(t′) +A2(t′)

2dt′, (5.3)

where A(t) is the vector potential of the applied NIR field. Note that the

presence of the scalar product v · A(t) implies that the streaking effects are

different for different angles between the momentum of the emitted photoelec-

CHAPTER 5. ATTOSECOND STREAKING OF XUV AND VUV PULSES 129

tron and the streaking electric field. Moreover, the momentum p(t) takes into

account the streaking action with p(t) = A(t) + v.

In figure 5.1 I show the two most relevant streaking regimes for the exper-

iments that will be presented in the next sections. In 5.1(a) I show the case

where the attosecond pulse is much shorter than the gate modulation. This is

the case when streaking the photoelectron spectrum from our XUV attosecond

pulse with a NIR pulse. The gate can be approximated by the portion of the

NIR vector potential overlapping with the short XUV pulse, and therefore can

be considered at any time either as linear or as quadratic with respect time.

The effect of a linear phase is a shift in the Fourier conjugate domain (the

same as GD and the corresponding delay shift in the time domain). Similarly,

the effect of a quadratic phase is broadening or compression of the signal dis-

tribution in the Fourier conjugate domain. These are the main features of an

XUV-NIR streaking trace.

In the case of a phase gate with modulations across the duration of the

UV pulse, sidebands arise, as can be seen in 5.1(b). This is the regime of our

VUV-NIR streaking measurements. Note that measuring the photoelectron

spectrum corresponds to measuring the intensity spectrogram of the photo-

electron wavepacket since I(�ω, τ) = |a(v, τ)|2.

Recalling what was introduced in section 2.2, it can be seen that equa-

tion (5.2), describing a streaking measurement, has the same form as equation

(2.30), describing a FROG measurement, interpreting the effect of the added

NIR field a phase gate on the photoelectron temporal profile. From this obser-

vation a streaking trace can be interpreted as a FROG trace and the pulse and

gate information can be retrieved [124]. In the next section I will discuss the

two main retrieval algorithms that can be used to extract from the measured

I(�ω, τ) the corresponding phase and amplitude of the attosecond pulse and

of the gate.


(a)

-5 0 5

Ene

rgy

(eV

)

85

90

95

Delay (fs)-5 0 5

Am

plitu

de (a

rb. u

.)

-1

-0.5

0

0.5

1(c)

(b)

-5 0 5

Ene

rgy

(eV

)

85

90

95

Delay (fs)-5 0 5

Am

plitu

de (a

rb. u

.)

-1

-0.5

0

0.5

1(d)

Figure 5.1: Theoretical streaking traces (a) and (b) with their respective pulses(c) and (d) showing the case where the UV pulse (dashed blue line) is much shorterthan the wavelength of the streaking pulse (red solid line) in the left column. In theright column the case where the two durations are comparable.

5.1.1 Retrieval algorithms for streaking experiments

The concept behind using an iterative algorithm to retrieve the pulses produc-

ing a given streaking trace is illustrated in figure 2.18. The idea is to apply two

independent constraints, in our case the measured I(�ω, τ) in the frequency

domain and equation (5.2) in the time domain, to predict a pulse (P) and

gate (G) pair that minimises, iteration after iteration, the estimated error of

the retrieval. Applying these two constraints in different domains is defined

as generalised projection and will be the approach shared between the two

presented algorithms. One is the Principal Component Generalised Projection

Algorithm (PCGPA) [139] the second one is the Least-Squares Generalised

Projection Algorithm (LSGPA) [55]. The algorithms are described schemati-

cally in figure 5.2, adapted from [139].

The two algorithms differ in the way they predict the next iteration P and

G. Despite this difference the flow of actions is the same in both cases. First,


Pj, GiOj,i=Pj Gi

O=>S

S'=>O'

Sj,i=PjGj+L(i-1)

S=FFT(S)S'=IFFT(S')

S'= exp[i arg(S)]

time domain constraint

frequency domain constraint

powe

r met

hod

least-squaresm

ethod

LSGPAPCGPA

Figure 5.2: Retrieval algorithms, in blue are the LSGPA steps, in green the PCGPAones. Adapted from [139], with kind permission of Springer Science+Buisness Media.

from the current best guess of P and G the signal matrix S(t, τ) in the time

domain is produced according to (5.2). The trace obtained is then converted

via Fourier transform to the signal trace in the frequency domain S(ω, τ). This

allows the phase information of S(ω, τ), that is not present in the measurement

of I(ω, τ), to be obtained. At this point it is possible to apply the intensity

constraint. The phase obtained from the calculation in the time domain is used

together with the amplitude information measured in the frequency domain

resulting in a signal matrix S ′(ω, τ) =√I(ω, τ) exp

{i · arg

[S(ω, τ)

]}. From

this matrix it is possible then to extract a better estimate of P and G for

the next iteration. For more details about the implementation of the two

algorithms see appendix A.

I will now present an example of performance of the two algorithms. The

numerical implementation of the PCGPA was carried out by Dr. T. Witting,

while I implemented the LSGPA. To test the algorithms I opted for a chal-


lenging streaking trace, of two pulses, at slightly different photon energy (81

eV and 90 eV), and delayed by 3 fs. The results of the comparison for the two

algorithms is shown in figure 5.3.

It can be seen how, in presence of double pulses and isolated spectral fea-

tures the PCGPA fails while the LSGPA is still able to retrieve the correct

pulse structure as expected [139]. This is a simple example to show why the

LSGPA has been chosen over the PCGPA for the retrieval of the VUV pulses.


input(a)

Delay (fs)-10 0 10

Ene

rgy

(eV

)

70

80

90

100

110

LSGPA(c)

Delay (fs)-10 0 10

Ene

rgy

(eV

)

70

80

90

100

110

PCGPA(e)

Delay (fs)-10 0 10

Ene

rgy

(eV

)

70

80

90

100

110

iteration200 400

FRO

G e

rror

0.05

0.1

0.15

0.2(b) LSGPA

PCGPA

Time (fs)-2 0 2

Inte

nsity

(arb

. u.)

0

0.2

0.4

0.6

0.8

1(d) LSGPA

input

Time (fs)-2 0 2

Inte

nsity

(arb

. u.)

0

0.5

1(f) PCGPA

input

Figure 5.3: Retrieval algorithms comparison, in blue are the LSGPA results, inred dashed the PCGPA ones. In (a) the input trace is represented. The reitrevedones are shown in (c) for the LSGPA, and in (e) for the PCGPA. The FROG errorof the two algorithms is shown in (b). The retrieved Temporal profiles are shown in(d) and (f).


5.2 XUV and VUV streaking experiment


Figure 5.4 shows the TOF chamber, dedicated to the streaking experiments.

TOF

2-part mirror

filters

TOF

2-part mirror

filters

Figure 5.4: TOF chamber, picture on the left, schematics on the right. Theelectron time-of-flight spectrometer is visible on the right portion of the chamber,with the nozzle encasing the electrostatic lens. The 2-part mirror assembly focusesthe incoming beam in front of the nozzle aperture. A set of filters (green box)spectrally select a portion of the harmonic emission. The NIR beam is directedoutside the chamber through a lens for the re-imaging the focus on a CCD cameraoutside the chamber.

Not visible in figure 5.4 is the effusive needle used to deliver the gas target

in the interaction region in front of the TOF nozzle. It was mounted on a x-y-z

manipulator on the lid of the chamber so that the optimum alignment could

be chosen looking at the signal count rate. The filters (shown in the green

box in figure 5.4)are used to spectrally filter the HHG emission to obtain an

isolated attosecond pulse.

The 2-part mirror assembly consists of a multilayer focusing mirror (f =

10.5 cm) with the inner part (4 mm diameter) cut out and mounted on a

piezo-stage allowing a spatial offset between the two parts of the mirror to be

introduced that corresponds to a delay between pulses reflected off the two

different portions of the multilayer mirror. The tip-tilt screws of the mirror


mount were equipped with motorised actuators for spatial alignment of the

outer part of the mirror with respect to the inner, as shown in figure 5.5.

inner part

outer part

tip-tilt adjustment5mm

Figure 5.5: Two part mirror assembly, with the detail for the inner part cut outfrom the outer one. The inner part is mounted on a piezo stage providing a delayrange of 80 fs with a resolution of 0.3 as.

The electron time-of-flight spectrometer consists of a nozzle, that houses

an electrostatic lens, and a field-free drift-tube at the end of which the photo-

electrons are detected with an MCP.

In the following paragraphs I will describe more thoroughly the steps per-

formed to record and calibrate the streaking traces for the characterisation of

the XUV and VUV pulses.

Spectral filtering

The beam (harmonics +NIR) goes through an iris that allowed us to aperture

the NIR to reach the desired intensity (in the range of 1012 Wcm−2) at the in-

teraction region in front of the TOF aperture. So far the harmonic emission has

not undergone any manipulation, therefore, to obtain an isolated attosecond


pulse, as explained in 3.3, a portion of the HHG spectrum has to be selected.

The beam can go through a set of filters (green box in figure 5.4, shown out

of the beam path). The filters were mounted side by side on holes with 3 mm

aperture and 1 mm separation, as shown in figure 5.7. The foils used were

Kapton (7.5 μm thick) for the selection of the NIR pulse only, zirconium (200

nm thick) for the selection of the cut-off region of the harmonics, and indium

or tin (200 nm thick) for the selection of the VUV harmonics. The selected

pulses were then directed to the 2-part mirror assembly, where the inner part

of a focusing multilayer mirror (Mo:Si, selecting ≈8 eV bandwidth at 90 eV)

was translated on a piezo stage (P-753, PI GmbH, 80 fs delay range, 0.3 as

resolution) in order to provide a delay manipulation between the beam on the

inner part of the mirror, with respect the beam on the outer part.

Energy (eV)10 20 30 40 50 60 70 80 90 100

Tran

smis

sion

(%)

0

10

20

30

40

50

60

H9

H11

H13

H15

MoSiZrSnIn

Figure 5.6: Filters for the spectral selection of different portions of the HHGspectrum. The laser central wavelength is λ0 = 765 nm.


The spectral properties of the metallic filters and of the multilayer mirror

are shown in figure 5.6. It can be that the Mo:Si and Zr filter combined provide

a spectral bandpass of 8 eV centred at 90 eV. The Sn filter bandpass contains

2 harmonics (11th and 13th) while the In filter bandpass contains only one

harmonic (9th). While it is difficult to precisely define the central wavelength

of a few cycle pulse, we use λ0 = 765 nm, according to the position of the

main peak of the fibre spectrum shown in figure 3.10. Assuming a central

wavelength closer to 800 nm would imply the presence of three harmonics

under the transmission window of Sn. The photoelectron spectra measured

detected only two of peaks, confirming a central wavelength between 760-780

nm.

Spatial and temporal overlap

Figure 5.7 shows how different pulse combinations can be delivered to the inter-

action region. The first two combinations (XUV-NIR, VUV-NIR) were used to

perform the characterisation of the attosecond pulses. The last combination

(XUV-VUV) will enable us to perform future attosecond-pump attosecond-

probe experiments.

XUV-NIR VUV-NIR XUV-VUV

Figure 5.7: Schemes to combine different pulse pairs in the TOF chamber. On theleft a picture of an actual filter assembly.

Once the main geometrical alignment is obtained according to figure 5.7,

it is necessary to verify the spatial and temporal overlap of the two pulses at

the interaction region. To do so, an imaging system was designed to re-image


the focus inside the chamber onto a CCD camera (see the mirror and the

lens after the focus in figure 5.4). Spatial overlap was achieved with motorised

actuators on a tip-tilt mount holding the outer part of the 2-part mirror. Once

the spatial overlap is satisfactory, the focal image is recorded as a function of

delay between the inner and outer part of the multilayer mirror. In this way

it is possible to record the spatial interference pattern occurring between two

delayed replicas of the NIR pulse (no filters used in this case). Figure 5.8 ,

shows the results of such a measurement.

Delay (fs)45 50 55 60 65 70 75 80

Inte

nsity

(arb

. u.)

104

1

2

3

4

5(a)

(b) (c)

Figure 5.8: Measurement to temporally overlap the inner and outer components ofthe 2-part mirror. The interference pattern is measured to find the temporal overlapand the spatial overlap is optimise to maximise the fringes contrast.

Note that the zero delay t0, found with this method is not the t0 for the

pulses going through the filter assembly, since in that case there will be delays

due to propagation in the filters relative to vacuum (1-2 fs through the metallic

filters, 50 fs through Kapton). However the piezo stage has enough delay range

(80 fs) to compensate for these.


Electron time-of-flight spectrometer calibration

The design of the electron time-of-flight spectrometer (now replaced by the

electron VMI and ion TOF) used to record the streaking traces, is similar

to the one presented in [141]. The photoelectrons are detected by an MCP

producing a hit signal with an amplitude of ≈10 mV and a time duration of ≈3

ns. The signal from the MCP is amplified with a x10 fast amplifier (TA1800B,

Fast Comtec GmbH, 1.8 GHz bandwidth) and acquired on a computer by

a time to digital converter (P7889, FAST Comtec GmbH, 0.1 ns time bin).

The TOF nozzle, equipped with an electrostatic lens (set of 4 electrodes), is

positioned 2 cm from the interaction region and has an aperture of 4 mm. This

allows for a high angular resolution at the expense of the collection efficiency

(0.05 % assuming uniform 4π emission and including inner TOF geometry).

The low collection efficiency was not a problem for the XUV measurements,

but the electrostatic lens was required for the VUV measurements to boost

the collection efficiency and therefore reduce the data acquisition time. The

lower count rate in the VUV case is due mainly to the poor reflectivity of

the Mo:Si mirror in this spectral region, estimated to be as low as 0.2% (see

paragraph describing the VUV photon flux measurement). Figure 5.9 shows

the schematics of the electrodes.

For the lens-free case, the calibration is straightforward, as the TOF length

was already carefully measured [79]. The measurement of t0 for the time-of-

flight axis was performed by looking at the photon peak obtained with the

NIR pulse (scattered photoelectron directly hitting the MCP), and the time

to energy conversion followed the simple formula:

E =1

2me

(L

TOF

)2

, (5.4)

where me is the electron mass, L is the spectrometer length, and TOF is the

time-of-flight measured. The presence of an electrostatic lens makes equation

(5.4) not valid, since the path from the interaction region to the detector is


V2=0.90 VsetV3=0.96 Vset

V1=0.8 Vset

V4=Vset

V5=Ground

20 mm

Figure 5.9: TOF lens electrodes schematics. A potential divider applied differentvoltages to different electrodes.

not field free. For the calibration of the electrostatic lens I used three ap-

proaches. First the calibration was performed experimentally, looking at the

shift of ATI peaks in Ar and of the main peak of the selected VUV harmonics.

Then I solved the equation of motions of electrons with different initial veloc-

ities, assuming the lens behaves as a capacitor, with an effective voltage Veff

applied, accelerating the electrons. And finally I performed simulations with

Simion, including the full geometry of the nozzle. The results of the calibration

experiments and simulations are shown in figure 5.10.

The experimental points were taken looking at the shift of ATI peaks op-

erating the TOF with the lens on and off. I also included the shift of the

strongest harmonic from the VUV spectra, recorded in the same way. The

collection enhancement factor is obtained with Simion simulations, from a

uniform 4π emission distribution. The effective voltage Veff to use in the sim-

ple model to fit the experimental and simulated points is Veff = 10.46 V. The

analytical function, not requiring any interpolation, was used to calibrate the

recorded streaking traces.


distance (mm)320 340 360 380 400 420di

stan

ce (m

m)

20

40Simion Simulation

(a)

Veff

Simple Model(b)

energy measured WITH lens (eV)0 10 20 30 40 50ac

tual

ene

rgy

(no

lens

) (eV

)

0

10

20

30

40(c)

Energy (eV)0 5 10 15 20 25 30 35C

olle

tion

enha

ncm

ent f

acto

r

1

2

3

4

5(d)Simulated

Smoothed

Figure 5.10: TOF calibration. The electrodes geometry is showed as a section in(a). The schematics of the simple model for the analytical effect of the electrostaticlens is shown in (b), with the electrostatic lens simplified to an ideal capacitor.The experimental and simulated results are presented in (c) with the relationshipbetween the energy measured with the lens on, and off. Finally the ratio betweenthe number of electrons collected with the lens on and with the lens off is shown in(d). An enhancement factor up to 3 is expected.


5.3 Results

5.3.1 XUV Results

With a few cycle pulse, a main attosecond pulse is produced, corresponding

to the main peak of the carrier. The lower following peaks of the carrier

generate satellite pulses. When the XUV wavepackets are streaked by the

same generating NIR pulse, they are shifted in opposite directions, as they are

produced half a cycle apart. The main and the secondary attosecond pulses

are therefore streaked to the highest energy values, in turn, every π shift of

the CEP. To optimise the XUV pulse contrast, I recorded some photoelectron

spectra without any filter and with the iris almost closed, to allow only the

inner part of the mirror to be illuminated. In this way the XUV pulse and the

generating NIR pulse reach the interaction region with the best spatial and

temporal overlap possible. It is then possible to record photoelectron spectra

as a function of the CEP. From the recorded spectra, the best CEP value for

the generation of a isolated attosecond pulse can be selected [79, 142]. The

results of these measurements, obtained in Ne, are shown in 5.11.

The first peak at high energies (105 eV) corresponds to the streaked wavepacket

of the main attosecond pulse. The same peak is then streaked to low energies

(75 eV) for a CEP shift of π. Following the analysis suggested in [79] a max-

imum contrast of Coff = 6.0 ± 0.9 has been measured in the VUV off case,

and a maximum contrast of Con = 6.9 ± 0.7 in the case of the VUV on case,

as shown in figure 5.11(d).

From this dataset it is also possible to measure the CEP offset between the

case with the VUV on and off, as shown in figure 5.11(c). The measured offset

is 0.15±0.07π corresponding to the average of the two shifts obtained from the

two measured peaks. This value is consistent, within the experimental errors,

with the 0.23±0.06π value extracted from the data obtained from the flat field

spectrometer in figure 3.23 confirming that the second gas jet has little effect

on the XUV pulse generation.


VUV on

(a)0 0.5 1 1.5 2 2.5 3 3.5 4E

nerg

y (e

V)

80

100

120

VUV off

(b)0 0.5 1 1.5 2 2.5 3 3.5 4E

nerg

y (e

V)

80

100

120

CEP ( )0 0.5 1 1.5 2 2.5 3 3.5 4C

ount

s (n

orm

.)

00.5

1

CEP offset=0.15 0.07(c)

VUV onVUV off

CEP ( )0 0.5 1 1.5 2 2.5 3 3.5 4

Con

trast

0

5(d)

VUV onVUV off

Figure 5.11: XUV photoelectron spectra recorded as a function of CEP to optimisethe NIR pulse CEP to obtain an isolated XUV pulse with the best contrast possible.(a) corresponds to the case where both the Kr and Ne jet were on. (b) correspondsto the XUV only case. In (c) the integrated signal (within the black dashed lines)in the two above cases is shown for direct comparison. In (d) the contrast of theattosecond pulse is shown as a function of CEP.

To fully quantify this effect, complete XUV-NIR streaking traces were

recorded with the VUV on and off. Photoelectrons ionised from the 2p level

of the Ne gas target provide the photoelectron wavepacket replica of the XUV

pulse that is then streaked by the few cycle NIR pulse. The NIR intensity is

adjusted with the iris in the TOF chamber to obtain a streaking amplitude of

a few eV, corresponding to an intensity of INIR ≈ 1012 Wcm−2. An integration

time at each delay step of 10 s was used (150 delay steps, 0.1 fs step size) with

an integrated peak height of 80 counts at 90 eV. The recorded raw traces, and

the smoothed ones are shown in figure 5.12.


(a)

Delay (fs)-5 0 5P

hoto

n en

ergy

(eV

)

70

80

90

100

110(b)

Delay (fs)-5 0 5 P

hoto

n en

ergy

(eV

)

70

80

90

100

110

(c)

Delay (fs)-5 0 5P

hoto

n en

ergy

(eV

)

70

80

90

100

110(d)

Delay (fs)-5 0 5 P

hoto

n en

ergy

(eV

)

70

80

90

100

110

Figure 5.12: XUV-NIR raw (left) and smoothed (right) streaking traces. (a)-(b)correspond to the case of the Kr jet off, while (c)-(d) correspond to the case of theKr jet on.

The smoothing procedure consists in isolating the signal coming from the

Ne 2p level. First, the signal below 70 eV, corresponding to ATI photoelectrons

and photoelectrons ionised from other levels is removed. After applying a

threshold corresponding to 2% of the maximum signal, the center of mass

and the second moment of the spectrum at each delay is computed and used

to select the streaking trace with a spectral Gaussian window with position

set by the first moment and a width proportional to the second moment of

the spectrum recorded at each delay (minimum filter width of 12.4 eV in the

non-streaked region of the trace is larger than the wavepacket bandwidth to

ensure no clipping). The first moment E(τ), or Center Of Mass (COM), of the

recorded spectrum as a function of delay provides a measurement of the vector

potential A(τ) [127] through the relationship:

A(τ) =

√me

2E0ΔE(τ)e cos θ

. (5.5)

Here E0 is the COM of the recorded spectrum in absence of the NIR vector

potential (i.e. at early/late delays) that is taken as a reference to compute

the shift ΔE(τ) as a function of delay. The photoelectron collection angle θ

is small in this setup (≈ 5◦) and therefore cos θ ≈ 1. The CEP shift of the


NIR pulse between the two experimental configuration can be measured from

A(τ). In figure 5.13 are shown the two NIR pulses retrieved from the smoothed

traces.

(a)

VUV OFFPho

ton

ener

gy (e

V)

70

80

90

100

110RawFiltered

(b)

VUV ONPho

ton

ener

gy (e

V)

70

80

90

100

110

Time (fs)-4 -2 0 2 4 6 8

Am

plitu

de (a

rb. u

.)

-1

-0.5

0

0.5

1

CEP shift =0.28 0.07

(c) VUV OFFVUV ON

Figure 5.13: NIR pulse extraction from streaking traces. (a) and (b) are again thestreaking traces in the two cases of the Kr jet off and on. In (c) are plotted the twovector potentials extracted from the COM of the streaking traces.

A Fourier filter has been applied to the raw COM obtained. Only the

portion within the 450-1100 nm spectral range has been selected. The pulse

durations obtained are 3.5±0.1 fs in both cases. The uncertainty is due to the

small temporal widow over which the NIR pulses are sampled. The raw data

provides only ten points in the spectral window of interest, making it difficult

to select an appropriate central wavelength for the expression of the CEP

shift. For this reason, smoothing and oversampling are required. Moreover

the CEP fluctuations of the system impacts directly on the sharpness of the

trace. While fluctuations should not change the overall NIR pulse duration,

they blur the trace. For example, a 200 mrad RMS fluctuation on the CEP


corresponds to a jitter of 0.08 fs of the carrier. Selecting the maximum of the

oversampled spectrum as the central wavelength, the measured CEP shift is

0.28±0.07π. Given that there is no statistical option to compute the error from

multiple measurements, I attributed to this measurement the error obtained

from CEP scan recorded in the TOF chamber shown in figure 5.11. In fact

both measurements rely on the same parts of the experimental apparatus,

and on the same physical process. However, I expect the error to be bigger

in the streaking trace method due to the smoothing procedures, and related

uncertainties, required to evaluate the CEP. For this reason, even if the CEP

shift obtained in this case is bigger than the two previous estimates, I consider

it still in reasonable agreement with the previous values and once again small

enough not to disrupt the production of the XUV pulse.

To mitigate the short delay range of the experimental data, a few fem-

toseconds of unstreaked spectra have been added to the trace at early and late

delays, in order to provide the algorithm with more points to converge to a

stable spectral solution and the spectrum integral at each delay has been nor-

malised to unity. In figure 5.14 the results of the LSGPA algorithm analysis

on both traces, with the VUV on and off, are shown.

The input matrix for the algorithm has 512×401 elements (energy × de-

lays). The algorithm converged to a stable solution in 500 iterations, reaching

a FROG error of 5% in the case with the VUV off, and a FROG error of 6%

with the VUV on. The two retrieved pulse durations are of 271±25 as and

257±21 as. The error has been computed as the standard deviation of the

distribution of the XUV pulses obtained from the final iteration signal matrix

according to equation (A.12). The pulse durations in the two experimental

configurations are therefore comparable, confirming that the presence of the

Kr gas jet for the production of the VUV pulse does not perturb the generation

of the XUV attosecond pulse.


VUV OFF(a)

Ene

rgy

(eV

)

80

90

100

110VUV ON

(b) Energy (eV

)

80

90

100

110

(c)

Ene

rgy

(eV

)

80

90

100

110 (d) Energy (eV

)

80

90

100

110

Time (fs)-5 0 5

Inte

nsity

(arb

. u.)

0

0.5

1

271 25as

(e)

Time (fs)-5 0 5

Intensity (arb. u.)

0

0.5

1

257 21 as

(f)

Energy (eV)80 90 100 110

Inte

nsity

(arb

. u.)

0

0.5

1

Pha

se (

)

-1

0

1(g)

Energy (eV)80 90 100 110

Intensity (arb. u.)

0

0.5

1(h)

-1

0

1

Figure 5.14: XUV pulse retrievals in the case of having the VUV pulse off (leftcolumn) and on (right column). The top row shows the input traces for the LSGPAanalysis, the second row shows the retrieved trace. The third row shows the retrievedpulses in the time domain, and the last row in the spectral domain.


5.3.2 VUV Results

For attosecond streaking in the VUV energy range, the target gas choice is

limited by the low photon energy. For example, when filtering the VUV pulse

with indium the available photon energy is peaked at 15 eV, close to the Ip

of Xe (12.1 eV from level 5p3/2, 13.4 eV from level 5p1/2) [143] and Kr (14.1

eV from level 4p). The proximity to the Ip of these atoms, and the presence

of features in the electronic structure (such as the spin orbit splitting of Xe)

may also affect the streaking measurement, since the dipole transition matrix

elements dv at these energies are not well known, and they may introduce

some dispersion effects. This calls into question the assumption that the pho-

toelectron wavepacket perfectly replicates the incoming photon wavepacket.

The role of the ionisation step has been of interest in attosecond physics, for

example in relation to the time delay between the ionisation of different en-

ergy levels [7, 144]. Such distortion effects, if limited to a time delay, do not

prevent the temporal characterisation of a pulse via attosecond streaking. The

open question is whether there is a further effect, introducing a significant

reshaping of the photoelectron wavepacket with respect to the incident pulse.

From an experimental point of view, it is difficult to address this question, as

it would require the measurement of the same pulse with different gas targets.

This task is particularly challenging in the low energy range where no mea-

surements have been performed. Palatchi et al. [145] presented experimental

measurements of these delays as a function of harmonic order for He Ne Ar

and Kr. The low energy range 1-3 eV (above Ip), which would be of specific

interest for this work, is not covered experimentally nor theoretically.

While all these aspects should be taken into account, it is important to

remember that the spectral range of interest for the VUV pulses measured

in this work is limited to one harmonic (when filtering the harmonics with

indium) or two (when using a tin filter). This implies that the dispersion

effects should be characterised in a relatively narrow range. More comments

on these aspects will be given when discussing each dataset.


Photon Flux Measurement

In collaboration with PhD student Paloma Matia-Hernando we calibrated the

photon flux of the VUV pulses. We used a scintillator based on sodium salicy-

late (common aspirin) that has the property of absorbing light in the UV-VUV

(30-300 nm) and converting the absorbed photons to the 400 nm range with

a nearly constant quantum efficiency [146–149]. The calibration procedure in-

volves producing a 266 nm signal with a 3ω setup (Eksma optics kit) from

the output of the CPA laser system. At this wavelength we used a NIST

calibrated and PTB traceable photodiode powermeter (Thorlabs S120VC) to

determine the pulse energy. With this known source we could then calibrate

a sodium salicylate coated window. The coating was produced by applying

few drops of a 10:1 mass solution of methanol and aspirin on a fused silica

window. To obtain a coating as uniform as possible the fused silica window

was on an even surface and each drop was deposited after the previous one

had dried almost completely. No additional heating was applied to speed up

the drying process. The 400 nm fluorescence signal from the coating was de-

tected with a photon multiplier tube (PMT). Illuminating the window with a

variable amount of 266 nm radiation, that could be measured independently

with the power meter head, allowed us to obtain a voltage to photon-number

calibration. The constant quantum efficiency allows this same calibration to

be used in the VUV range, and thereby convert the PMT voltage measured

when illuminating the coated window with the VUV pulses to a photon flux.

In figure 5.15 the calibration and measurement steps are shown (left) together

with the relevant spectral signals (right)

Particular care was taken in the calibration procedure in order to remove

any stray light. The 266 nm signal was selected with a prism, and the remain-

ing 400nm and 800nm components were rejected with beam blocks well before

the position of the powermeter head. A blue filter selected only the 400nm

component generated by the coating. The sodium salicylate coated window


lambda (nm)200 300 400 500 600 700 800 900

Inte

nsity

(arb

. uni

ts)

0

5

10

15coating signalTHGCPA spectrumPMT response

coating filter

PMT266 nm V

PMTVUV V

mea

sure

men

tca

libra

tion

Figure 5.15: With a known source at 266 nm it is possible to calibrate the sodiumsalicylate coating and then perform the photon flux measurement on the actualVUV signal. On the right the signals obtained from the coating and the 3ω setupare shown.

was mounted on the TOF chamber, and the VUV radiation delivered to it

with an UV enhanced mirror (Thorlabs PF10-03-F01). The reflectivity of the

2-part mirror was estimated by performing two measurements, one with the

2-part mirror in the beam line, and one with only the UV-enhanced mirror.

An estimate of the photon flux was also derived from the count rate mea-

sured in the TOF chamber. Table 5.1, shows the relevant quantities for this

estimate.

Quantity value

Count Rate 5 counts/shot

TOF coll. eff. 0.05%

Events=count rate/TOF coll. eff. 104events/shot

Gas density 5 · 1023 m−3

Volume 315 μm3

Cross section 38 Mb [150]

VUV photons at interaction 1.4 · 104 photons/pulsePulse energy at interaction 0.045 pJ

Table 5.1: Photon flux estimate from count rate

Table 5.2 summarises the results obtained with this method, together with

the photon flux estimated with the sodium salicylate method. The VUV radi-

ation was filtered with tin. No measurement is available in the case of indium

due to the lower signal level.


Quantity PMT estimate Count rate estimate

Phot. flux delivered (photons/pulse) 2 · 104 1.4 · 104Energy delivered (pJ) 0.064 0.045Phot. flux generated (photons/pulse) 1.3 · 108 6.5 · 107Energy generated (pJ) 427 299

Table 5.2: Results of the photon flux measurement comparing the sodiumsalicylate and PMT method with the estimate from the count-rate from theTOF spectrometer.

The reflectivity of the 2-part mirror in the VUV was found to be 0.2% and

together with the small aperture of the TOF spectrometer are the main reasons

for the low level signal obtained in the streaking traces that will be presented.

This is understandable as our Mo:Si multilayer mirror is optimised for the

XUV range (≈ 90 eV). Changing the 2-part mirror coating is a straightforward

way to improve the number of photons delivered to the interaction region, for

example a gold coated mirror has a reflectivity up to 15% in the VUV range

[151].

Indium filtered pulse

Selecting the 9th harmonic with a photon energy of ≈ 15 eV, the only options

for rare gas targets for streaking are Xe (12.1 eV from level 5p3/2, 13.4 eV

from level 5p1/2)[143] and Kr (14.1 eV). I opted to use Xe to make sure all the

available bandwidth of the generated pulse was able to ionise the target atom.

From a theoretical point of view, the generation of a short pulse using In as

a spectral filter had already been confirmed by the theoretical work of our

collaborator J. Henkel et al. [31]. The two most interesting results obtained

from the TDSE simulations performed can be summarised as follow. First,

the pulse duration can be sub-femtosecond, but the temporal structure can be

quite modulated, depending on the CEP of the generating field. For this reason

we will refer to the pulse duration both using the standard FWHM definition,

and also with the second moment of the temporal intensity distribution that


takes into account the contributions present in the low pedestal of the pulse.

The second interesting point of this theoretical work was the study of the

polarisation state of the pulse. While a pulse in the cut-off region of the

generated high harmonics is inherently limited to a single emission, a pulse

selected in the low energy range can have contributions from the different

half-cycles of the generating pulse. As already introduced in section 3.3, it

is possible to implement polarisation gating to isolate the harmonic emission

even for a many cycle pulse. With this idea in mind, the simulations performed

considered the effect of polarisation gating in the generation of the VUV pulse.

The main advantage of this scheme is the production of a more stable and

less structured VUV pulse, in the sense that sub-femtosecond durations is

achieved for a wider CEP range and with a less modulated structure. However,

the results showed that the NIR pulse time-dependent polarisation state is

partially imprinted in the generated VUV pulse, while the pulse duration is

not significantly shortened.

For these reasons, polarisation gating has not been adopted to shorten the

pulse, as it would have a minimal effect on the overall pulse duration, at the

price of complicating the experimental setup. Our beamline, in fact, is opti-

mised for the collinear propagation of the streaking field and of the attosecond

pulse. Implementing the polarisation technique would require the delivery of a

linearly polarised pulse to the experimental chamber from a different path with

respect the generating NIR field. Moreover, a VUV pulse with a polarisation

state not well defined is not the ideal candidate for pump-probe experiments

due to the difficulties in modelling such pulse while interacting with the system

under investigation.

In figure 5.16 are presented the raw and smoothed traces obtained by ion-

ising a Xe gas target with the In filtered harmonics generated in Kr. The

spectrum at each delay (86 delay steps, 0.2 fs step size) was integrated for 45

seconds to reach a peak height of 15 counts for the unstreaked portion of the

spectrum.


(a)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

11

12

13

14

15

16

17

18

19(b)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

11

12

13

14

15

16

17

18

19

Figure 5.16: Raw(left) and smoothed (right) streaking traces obtained in Xe fil-tering the VUV radiation with indium.

The smoothing procedure was similar to the one introduced for the XUV

traces. In this case the threshold level was increased to 12% of the maximum

signal, given the lower statistics. One technical difficulty was the application

of the Jacobian in the conversion from the TOF axis to energy axis due to

the proximity of the signal to zero (see the Ip level of Xe in figure 5.16 that

corresponds to the zero energy of the measured photoelectrons). The Jacobian

can be computed from equation (5.4) to be:

d(TOF )

dE=

√meL2

8E3, (5.6)

which goes to infinity when the energy goes to zero. This means that this factor

would amplify all signal around zero (saturated region in the raw data of figure

5.16), where most of the noise is condensed. All electrons with a time-of-flight

longer than 2 μs correspond to an energy below 0.1 eV. Considering that there

is a start signal (laser shot) every 1 ms, it is clear that slow electrons are

heavily affected by any random noise collected for high time-of-flight. For this

reason, the collection of signal was interrupted after 2 μs (notice how the raw

signal does not reach the Ip level in figure 5.16), and a super Gaussian filter of


order 16 was used to cancel the signal in the very close proximity to the origin

(< 0.5 eV), without altering the structure of the streaking trace. Applying

the 12% threshold cancels the small signal of a secondary sideband appearing

at 17.5 eV. This modification was found not to disrupt the LSGPA analysis.

In fact, the presence of secondary sidebands is related to the higher intensity

of the streaking NIR field and they mostly provide an information about the

intensity of the NIR field rather than about the VUV pulse structure. The

results of the LSGPA analysis are shown in figure 5.17

12 14 16 18

Pha

se (r

ad)

0

0.5

1

1.5

Delay (fs)-5 0 5P

hoto

n en

ergy

(eV

)

12

14

16

18

Delay (fs)-5 0 5 P

hoto

n en

ergy

(eV

)

12

14

16

18

Time (fs)-5 0 5

Inte

nsity

(arb

. u.)

0

0.5

11.7 0.1 fs

Photon energy (eV)12 14 16 18

Inte

nsity

(arb

. u.)

0

0.5

1

1.5

Figure 5.17: LSGPA analysis of the streaking traces obtained in Xe filtering theVUV radiation with indium. Plots as in 5.14

The input matrix for the algorithm had 1024×160 elements (energy × de-


a FROG error of 2%. The pulse duration is 1.7±0.1 fs, and despite being longer

than expected, it is close to the FTL which is 1.66 fs. The discrepancy with

the expected attosecond duration obtained by the TDSE calculations [31] can

be attributed to a number of factors. First of all, the TDSE simulations were

performed in a Ne gas target. In the experiment, I opted for Kr as the gen-

erating medium due to the higher signal levels. However, the lower Ip of Kr

allows the less intense half-cycles of the generating NIR to give more contri-


bution to the HHG yield. For this reason, the pulse temporal profile may be

more modulated than estimated for a Ne target. Moreover, it is possible that

the CEP value was not fully optimised for the shortest pulse VUV duration.

In fact the CEP was always optimised according to the signal generated in

the Ne target in the XUV region, to make sure I could preserve the XUV

attosecond pulse. Finally, the proximity of the photon energy to the Xe Ip and

the presence of the spin-orbit split might introduce some distortion effects in

the photoionisation step of the streaking measurement.

Tin filtered pulse

With a tin filter the 11th and 13th harmonics of the driving laser are selected.

With this photon energy it is possible to ionise Ar and therefore I decided

to use Ar as the streaking gas target to avoid the spin orbit splitting of Xe.

In this case however, to boost the signal, I used the electrostatic lens in the

TOF nozzle. As shown in figure 5.10, I performed both experimental and

simulated calibrations for the performance of the TOF with the electrostatic

lens, from which I could use a simple analytical formula to estimate the shift in

the measured time-of-flight with respect to the field free case. The parameter

found to fit the experimental calibration is Veff = 10.46 V. However, once I

tried to apply this parameter to the actual measured trace, the two harmonics

peaks were not shifted to the “expected” positions, as shown in figure 5.18.

The “expected” positions are given by the presence of only two harmonics

under the transmission window of the tin filter. Applying Veff = 10.46 V

moves the measured harmonics to a spacing corresponding to a laser central

of λ0 = 820 nm. If this was the case, some signal should be present for the

15th harmonic (top solid thin black line), as it would fall within the spectral

transmission window (dashed red line red). Moreover it can be seen that the

signal of the secondary sideband, that should correspond to the 15th harmonic,

does not match the expected position very well. In the left plot of figure 5.18 I

show the same trace calibrated with Veff = 9 V. The agreement is better in this


Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

14

16

18

20

22

24

26

28

30

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

14

16

18

20

22

24

26

28

30

Figure 5.18: Calibration of the nozzle effect while recording streaking traces ob-tained in Ar filtering the VUV radiation with tin. On the left the trace is calibratedwith Veff = 9 V, on the right with Veff = 10.46 V. The thin black lines are the har-monics positions according to λ0. The red-dashed line in overlay is the transmissionof tin.

case, with λ0 = 780 nm and the 15th harmonic positioned just at the edge of the

spectral window. The reason for the discrepancy between the calibrated Veff

and the value used to position the trace correctly has not been determined. It

is possible that the grounding conditions of the electrodes between experiment

and calibration (performed a few days apart) may have changed, for example

due to a loose cable. It is important to note, however, that this disagreement

is not detrimental for the LSGPA analysis. In fact the uncertainty on the Veff

parameter corresponds simply to an uncertainty on the central wavelength of

the VUV pulse.

Note how the signal is close to the Ar Ip level. Using the electrostatic

lens improved the signal to noise ratio in the low energy range of the signal.

The corresponding time of flight for a photoelectron with an energy of 0.1

eV is 0.35 μs with the electrostatic lens on, and 2 μs with the lens off. This

means that more random noise can be rejected as the maximum time of flight

can be reduced significantly when the electrostatic lens is on. In fact, the

data presented in figure 5.19 did not require any smoothing procedure at low


energies, as was the case in 5.16.

Two streaking traces for the Sn-filtered VUV pulses are presented, one with

the XUV on (Ne jet on) and one with the XUV off (Ne jet off), to quantify

the effect of the generation of the XUV pulse on the VUV one. In figure 5.19 I

show the measured raw traces and the smoothed traces corresponding to these

two experimental configurations.

(a)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

15

20

25

30(b)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

15

20

25

30

(c)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

15

20

25

30(d)

Delay (fs)-5 0 5

Pho

ton

ener

gy (e

V)

15

20

25

30

Figure 5.19: Raw (left) and smoothed (right) streaking traces obtained in Arfiltering the VUV radiation with tin. (a) and (b) correspond to the case of XUV off,(c) and (d) to the case of XUV on.

Despite using the electrostatic lens the signal level was low. The spectrum

at each delay was integrated for 3 minutes obtaining a signal with a peak

height of 30 counts when the Ne jet was on, and 60 when the Ne jet was off,

confirming the factor 2 drop in signal when operating the two gas targets at the

same time as already seen in figure 3.22. The drop in signal can be explained

as absorption of VUV photons in the Ne target.

The smoothing procedure is similar to the one performed for the previous

traces. A threshold level of 4% (8%) was applied in the case of the XUV off

(on), together with a 2D Fourier filtering, to preserve as much as possible all

the interference structures present between the two harmonics. Non-streaked

spectra were added at the trace edges and the integral of each spectrum was

normalised to unity. In figure 5.20 I present the results of the LSGPA analysis


on the smoothed streaking traces.

XUV OFF(a)

Ene

rgy

(eV

)

15

20

25XUV ON

(b) Energy (eV

)

15

20

25

(c)

Ene

rgy

(eV

)

15

20

25(d) E

nergy (eV)

15

20

25

Time (fs)-10 -5 0 5 10

Inte

nsity

(arb

. u.)

0

0.5

1

585 31as

(e)

Time (fs)-10 -5 0 5 10

Intensity (arb. u.)

0

0.5

1

576 16 as

(f)

Energy (eV)15 20 25

Inte

nsity

(arb

. u.)

0

0.5

1

Pha

se (

)

-1

0

1(g)

Energy (eV)15 20 25

Intensity (arb. u.)

0

0.5

1(h)

-1

0

1

Figure 5.20: LSGPA analysis of the streaking traces obtained in Ar filtering theVUV radiation with tin. The left column corresponds to the case of the XUV off(Ne jet off), while the right, column corresponds to the case of the XUV on (Nejet on). The top row shows the measured traces while the second row shows theretrieved traces. In (e) and (f) the temporal profile of the retrieved pulses is shown(solid blue) together with the theoretical results from TDSE simulations (dashedblack). In (g) and (h) the retrieved spectral amplitude (solid blue) and the spectralphase (solid red) are plotted.


The input matrix for the algorithm has 1024×370 elements (energy × de-


a FROG error of 2% in the case with the Ne jet off, and a FROG error of

5% with the Ne jet on. The two retrieved pulse durations are of 585±31 as

FWHM (second moment 950±30 as) and 576±16 as FWHM (second moment

860 ± 45 as). From this analysis it is verified that the pulses with the XUV

jet on and off, preserve a comparable pulse duration within the experimental

error. The contrast with respect to the two satellites pulses has been measured

to be 41± 14% with the Ne jet off, and 38± 13% with the Ne jet on.

Note the slight asymmetry of the pulse that goes through the Ne jet, most

probably due to the dispersion experienced in the Ne gas target. The temporal

profile is in close agreement with the theoretical one computed by Jost Henkel

solving the TDSE for a NIR pulse with λ0 = 760 nm and Kr generating

medium. A precise theoretical estimate of the contrast of the satellites pulses

is particularly difficult, as it relies on the precise measurement not only of the

spectral amplitude, but also of the absolute phase difference between the two

harmonics. It is well known that the relative phase between isolated spectral

features is one of the most challenging tasks in ultrafast pulse characterisation

[57].

In conclusion, in this section I presented the characterisation of three

attosecond pulses generated by selecting different portions of the HHG spec-

trum obtained with a 3.5 fs pulse centred at 790 nm. The characteristics of

these pulses, together with the generating NIR pulse, are summarised in table

5.3.

Spectral range Wavelength Photon energy Duration

NIR 790 nm 1.57 eV 3.5± 0.1 fsXUV 14 nm 90 eV 270± 25 asVUV 62 nm 20 eV 585± 31 asVUV 83 nm 15 eV 1.7± 0.1 fs

Table 5.3: Summary of the pulses presented in this work.

161

Chapter 6

Conclusions and future work

In this thesis I have presented new methods for the characterisation of ultrafast

pulses and the development of new attosecond sources and their characterisa-

tion.

In chapter 3 I presented the CEP performance of our hollow core fibre pulse

compression setup. These results showed that when scaling the energy input

to higher levels, ionisation-induced instabilities degrade the CEP stability, and

that ionisation defocusing couples energy to higher order modes, degrading the

transmission performance of the fibre setup. To overcome these limitations,

fibres with a bigger inner diameter are being tested (400 μm compared to the

260 μm used for the measurements presented). Moreover, a better vacuum

system is under development to lower the pressure at the fibre entrance and to

house the focusing optic coupling the NIR pulse into the fibre. This will allow

us to remove any non-linear effect due to the focused NIR pulse going through

the entrance window of the vacuum system. With these improvements we aim

to couple the full 2.5 mJ available from our CPA laser system into our hollow

fibre, giving a 3.5 fs laser pulse with up to 1.2-1.3 mJ of pulse energy.

I described in chapter 3 high harmonics generated in ablation plumes. This

type of source is an interesting alternative for the generation of XUV radiation

via HHG in gas targets. In particular we showed that the spatial coherence

properties of the radiation generated in ablation plumes are comparable to

162 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

the more standard high harmonics generated in rare gases. The next step

is their temporal characterisation. Preliminary measurements preformed by

Z. Abdelrahman show that the photon yield of the harmonics generated in

ablation plumes is comparable to rare gases. For this reason an attosecond

streaking experiment for the temporal characterisation of the isolated spectral

features seen in Zn (see figure 3.25) seems now possible.

I presented two characterisation techniques for few-cycle pulses in chapter

4. The first technique presented was a single-shot implementation of the d-

scan technique. The dispersion scan is mapped on the spatial profile of the

pulse by using a Littrow prism. An imaging spectrometer was used to record

the second harmonic signal as a function of position/dispersion, thus allowing

us to record a d-scan trace on a single shot basis. The retrieved pulse duration

of 3.3 ± 0.3 fs is compatible with the 3.2 ± 0.1 fs duration obtained by an

independent d-scan measurement of the same pulse (see figure 4.6). Further

developments of this single-shot implementation require the design of a more

compact setup, and a study of the maximum dispersion range achievable.

The second characterisation technique presented was the ARIES technique.

Exploiting the dependence of the HHG cut-off position on the precise wave-

form of the generating few-cycle pulse, this technique was able to successfully

characterise test-pulses of few-cycle duration and different CEP values, and

also test-pulses significantly different from their respective FTL with pulse du-

rations as long as 18 fs (see figures 4.11 and 4.12). A very interesting aspect

of this technique is the possibility of measuring pulses with components at

twice the frequency of the pulse used for HHG. This means that, in principle,

the full waveform of a ω + 2ω pulse, for example, can be retrieved with this

method. The measurement of such a waveform has been performed only with

attosecond streaking so far [16]. Future work is planned to synthesise a ω+2ω

field that may improve the contrast of our attosecond VUV pulses. The ARIES

technique could be used to characterise the ω + 2ω field.

I presented in chapter 4 the design of a dual electron VMI and ion TOF

CHAPTER 6. CONCLUSIONS AND FUTURE WORK 163

spectrometer designed to resolve photoelectrons up to 90 eV and with a ion

mass resolution of ions up to 1% at 80 a.u. This device is now installed and

in operation. Future work using this apparatus will proceed in a number

of directions. We will perform streaking measurements of the XUV pulse

to compare the performance of the VMI to the electron TOF for attosecond

streaking. Moreover the higher collection efficiency and the flexibility on the

optimisation of the energy resolution may allow a easier/quicker measurement

of the XUV and VUV pulses. Once the setup is optimised the first experiments

using covariance mapping between the electron VMI and ion TOF signal can

be performed.

The main result of this thesis has been the production and characterisation

of two pulses in the VUV range synchronised with an isolated XUV pulse

generated at the same time in a double gas jet setup. A pulse duration of

1.7±0.1 fs has been achieved at a photon energy of 15 eV (83 nm) and a pulse

duration of 585 ± 31 as has been measured at 20 eV (62 nm). These pulses

have been obtained by filtering the spectrum of high harmonics generated in

Kr with a few cycle NIR pulse (3.5 fs) using In and Sn foils respectively. I

verified experimentally that these pulses could be generated simultaneously

with an XUV pulse (270± 25 as at 90 eV), opening up new opportunities for

attosecond-pump attosecond-probe experiments.

In order to understand better the generation dynamics of these pulses,

especially the VUV pulses, their characterisation could be performed under

different experimental conditions. In particular the installation of the electron

VMI spectrometer should allow a faster measurement of the streaking traces.

For this reason one could explore the dependence of the temporal profile of

the generated pulses for different CEP values and intensity of the generating

NIR pulse, in order to verify if the experimental parameters used in this thesis

are indeed optimal. Moreover the in-line geometry of the gas targets we used

has been shown to produce an enhancement of the XUV yield [152]. This

enhancement has been seen in [152] for high gas pressures in the first gas jet,

164 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

when the low low harmonics (3-7 eV), generated in the first jet, were efficiently

produced, while the efficiency for the higher order harmonics (> 20 eV) was

reduced. In the results I presented for the VUV pulses, the experimental

conditions were rather different. The pressure of the first jet was kept low

in order to minimise the effect on the XUV pulse generation in the second

jet. However it may be possible for us to reproduce the enhancement feature

presented by Brizuela et al. by increasing to higher levels the pressure in the

first jet.

The unique feature of the attosecond source I have developed is the capa-

bility of delivering any pair of the pulses to an experiment, with attosecond

resolution. The in-line geometry of the gas targets does not require any auxil-

iary stabilisation procedure to recombine the two pulses with attosecond preci-

sion. This capability opens up the possibility of performing attosecond-pump

attosecond-probe experiments with different photon energies for the pump and

for the probe step. The VUV pulses are well suited to many molecular tar-

gets which typically have high photoionisation cross sections in this spectral

region. In the work of B. Cooper and V. Averbukh [153] they suggest that a

VUV pulse, similar to the one I characterised, could probe the hole migration in

ionised glycine. The technique proposed is called single-photon laser-enabled

Auger decay (spLEAD). In this case the initial ionisation step does not di-

rectly allow a two-electron Auger-like transition [154] since the ionised state

does not have enough energy. The missing energy can be provided by the

probe-pulse in the so called Laser Enabled Auger Decay (LEAD) [155, 156]. In

sp-LEAD, the energy is provided by a single VUV photon and the simulations

performed showed that the timing of the VUV photon is sensitive to the hole

dynamics and therefore the hole migration can be extracted from the sp-LEAD

photoelectron yield.

This is an example of how the new tools developed in this thesis are a valu-

able resource for the development of new attosecond pump-probe experiments.

165

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doi: 10.1038/nphoton.2013.156.

[162] Felix Frank. “Generation and Application of Ultrashort Laser Pulses in

Attosecond Science”. PhD thesis. Imperial College London, 2011.

187

Appendix A

Retrieval algorithms

A.1 PCGPA

The PCGPA [55] is based on the mathematical properties of the outer product

of two vectors. Let us define the pulse and gate as

P =Pi = P (ti)

G =Gi = G(ti)

i =1 · · ·N

(A.1)

In other words the pulse and gate are represented by an array of N sample

points. The outer product is defined as

O = P⊗G =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

P1G1 P1G2 · · · P1GN−1 P1GN

P2G1 P2G2 · · · P2GN−1 P2GN

......

. . ....

...

PN−1G1 PN−1G2 · · · PN−1GN−1 PN−1GN

PNG1 PNG2 · · · PNGN−1 PNGN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A.2)

The signal matrix from equation (5.2), representation of P · G(t + τ) can

be written as:

188 APPENDIX A. RETRIEVAL ALGORITHMS

S =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

P1G1 P1G2 · · · P1GN−1 P1GN

P2G2 P2G3 · · · P2GN P2G1

......

. . ....

...

PN−1GN−1 PN−1GN · · · PN−1GN−3 PN−1GN−2

PNGN PNG1 · · · PNGN−2 PNGN−1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A.3)

Note how it is possible to obtain O from S and vice versa. S is the direct

matrix representation of the streaking process in the time domain, and has

therefore a straightforward interpretation. O has the key property of having

only one eigenvalue different from zero with only one left and one right eigen-

vector corresponding to P and G. After applying the intensity constraint in

the frequency domain the new matrix O′ is not an outer product matrix, but

the left and right eigenvectors with the highest eigenvalue provide a good ap-

proximation for P and G. A good guess of P and G for the next iteration is

given simply by one iteration of the power method [55]:

P next = (O′O′T )P

Gnext = (O′TO′)G(A.4)

There are two main assumptions or constraints when using the PCGPA

algorithm. First the relation between O and S has physical meaning only if

the gate has periodic boundary conditions. In other words one has to be sure

that shifting the entries of G to obtain a delayed replica of the gate is actually

a good representation of the physical system. This is verified in the case of

streaking with a few cycle pulse that goes to zero at early and late delays.

The second and more stringent assumption for PCGPA is that the axes of

the signal matrix are Fourier conjugate and therefore one has to manipulate

the measured trace so that the maximum energy range measured ΔΩ and the

APPENDIX A. RETRIEVAL ALGORITHMS 189

delay step size δτ are related as

ΔΩ · δτ = 2π (A.5)

This is not usually the case and in most cases interpolation of the mea-

sured trace is necessary before the analysis with PCGPA is possible. Such

manipulation can lead to ambiguities in the retrieved pulses [139].

A.2 LSGPA

The LSGPA [139] differs from the PCGPA in the approach for the prediction

of the next iteration P and G. As the name explains the LSGPA is based on

a least-squares minimisation of a figure of merit in order to find the best P

and G pair. Moreover it decouples the fourier requrements for the two axis of

the signal matrix in the following way:

ΔΩ · δτ = 2π · L (A.6)

This allows the delay step to be L times larger than the one required by

PCGPA. With this new parameter the signal matrix can be written as Sj,i =

PjGj+L(i−1) with j = 1 · · ·N the energy points and i = 1 · · ·Nτ the delay

points. In matrix form S is given by:

S =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

P1G1 P1G1+L · · · P1G1+(Nτ−2)L P1G1+(Nτ−1)L

P2G2 P2G2+L · · · P2G2+(Nτ−2)L P2G2+(Nτ−1)L

......

. . ....

...

PN−1GN−1 PN−1GN−1+L · · · PN−1GN−1+(Nτ−2)L PN−1GN−1+(Nτ−2)L

PNGN PNGN+L · · · PNGN+(Nτ−2)L PNGN+(Nτ−2)L

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A.7)

The figure of merit M used to describe the accuracy of the retrieval is

defined as follow:

190 APPENDIX A. RETRIEVAL ALGORITHMS

M =Nτ∑i=1

N∑j=1

∣∣PjGj+L(i−1) − S ′(j, i)∣∣2 (A.8)

This parameter can be described as the error, in the (t, τ) domain, between

the signal matrix obtained with the measured intensity, and the signal matrix

arising from the pure application of (5.2). When this comparison is performed

in the (ω, τ) domain it is expressed in the form of the FROG error [53]:

εFROG =

√1

Nmatrix

∑∣∣∣∣[S ′(ω, τ)]2

−[S(ω, τ)

]2∣∣∣∣2 (A.9)

This expression corresponds to the rms of the difference between the mea-

sured and retrieved trace intensity. Nmatrix corresponds to N2 in the PCGPA

case where the matrix is square, while it corresponds to NτN in the LSGPA

one.

The least-squares method yields an expression for the pulse and gate of the

next iteration:

Pj =

∑m Sj,mG

∗j+L(m−1)∑

m

∣∣Gj+L(m−1)

∣∣2Gk =

∑n Sk−L(n−1),nP

∗k−L(n−1)∑

n

∣∣Pk−L(n−1)

∣∣2(A.10)

with indexes running according to:

j = 1 · · ·Nk = 1 · · ·N + L(Nτ − 1)

m = max

{1,

⌈R− j + 1

L

⌉+ 1

}· · ·min

{Nτ ,

⌈R− j

L

⌉+Nτ

}n = max

{1,

⌈k −N

L

⌉+ 1

}· · ·min

{Nτ ,

⌈k

L

⌉}(A.11)

The parameter R is required to avoid points at the extrema of the gate axis

to be taken into account and is set to N/2.

APPENDIX A. RETRIEVAL ALGORITHMS 191

It is also possible to estimate the robustness of the retrieval from S ′. Having

the combined information of the temporal and frequency constraints, we can

extrapolate how well they match (or differ) with each other. As proposed in

[82], it is possible to remove the gate action from the pulse information in S ′.

In the case of LSGPA such action is performed by just inverting the definition

of S to obtain P :

Pj,i =Sj,i

Gj+L(i−1)

(A.12)

The result is then a matrix with a collection of pulses, calculated at each

delay. If the temporal and frequency constraints were perfectly matched this

action would be a mathematical inversion, yielding a series of identical pulses.

The discrepancies between the pulses obtained can there fore be considered as

an indication of the error that can be attributed to the retrieved pulse.

193

Appendix B

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375 78

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Physical Society�

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7 131 Figure [139]c©The Author(s) 2008,

Springer�

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Ultrafast light sources and methods for attosecond pump-probe experiments