Coherent diffractive imaging of
electron dynamics in the attosecond
domain
vorgelegt von
M. Sc.
Björn Senfftleben
ORCID: 0000-0003-1716-5445
an der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Andreas Knorr
Gutachterin: Prof. Dr. Daniela Rupp
Gutachter: Prof. Dr. Stefan Eisebitt
Gutachter: Prof. Dr. Per Eng-Johnsson
Tag der wissenschaftlichen Aussprache: 5. Juli 2022
Berlin 2023
Zusammenfassung
Mit der zunehmenden Verfügbarkeit intensiver Attosekundenpulse und -pulszüge im Röntgen-
und extrem ultraviolettem (XUV) Bereich an Freie-Elektronen-Laser und mit Hilfe von
der Erzeugung hoher Harmonischer wird die schnelle elektronische Dynamik in einzelnen
Nanostrukturen für bildgebende Verfahren zugänglich. Diese Arbeit befasst sich mit dem
Nachweis ultraschneller elektronischer Dynamiken bis in den Attosekundenbereich in einzelnen
Nanopartikeln mittels kohärenter Lichtstreuung. Im speziellen werden Streubilder von
Heliumnanotröpfchen, welche als einfaches Modellsystem dienen, mit XUV Licht erzeugt.
Damit die notwendigen besonders hohen Intensitäten zur Verfügung stehen, wurde in
einer Zusammenarbeit mehrerer Forschungsgruppen am Max-Born-Institut eine intensive
XUV-Lichtquelle auf Basis von der Erzeugung von hohen Harmonischen entwickelt. Neben
der Optimierung des Erzeugungs-Prozesses selbst, bestimmt die Geometrie der zugehörigen
Strahlführung maßgeblich die erreichbaren Intensitäten. Um auch bei der Abbildung von
Heliumtröpfchen von maximalen Intensitäten profitieren zu können, wurde zudem ein neuer
Versuchsaufbau konzipiert. Dieser umfasst unter anderem eine speziell für dieses Experiment
entwickelte Fokussieroptik und mehrere Werkzeuge zur Charakterisierung der mit den
Heliumtröpfchen wechselwirkenden Pulse.
Sowohl die Entwicklung der intensitätsoptimierten XUV-Lichtquelle als auch des Ver-
suchsaufbaus haben es ermöglicht, eine durch einen nahinfraroten (NIR) Laserpuls induzierte
Änderung der Streureaktion von Heliumtröpfchen auf der Femto- bis sogar Attosekunden-
Zeitskala nachzuweisen. Es konnte in Übereinstimmung mit einer früheren Messung nun
routinemäßig ein reduziertes Streusignal bei gleichzeitiger Ausleuchtung der Tröpfchen mit
NIR- und XUV-Pulsen beobachtet werden. Darüber hinaus wurde insbesondere eine Änderung
der Menge an gestreutem Licht während der gleichzeitigen Ausleuchtung auch dann beobachtet,
wenn die Zeitverzögerung der beiden Pulse im Attosekundenbereich variiert wurde. Tatsächlich
wurde eine Oszillation des Streusignals mit einer Periode von etwa einem halben NIR-
Laserpulszyklus gemessen und mehrfach reproduziert.
Durch die Modellierung des Experiments über die Dipolantwort eines einzelnen Elektrons in
einem Heliumatom kann die beobachtete Änderung auf aus der Atomphysik bekannte Phäno-
mene zurückgeführt werden, nämlich die AC-Stark-Verschiebung und lichtinduzierte Zustände.
Insbesondere wird die beobachtete Oszillation pro NIR-Halbzyklus Quantenpfadinterferenzen,
die in lichtinduzierten Zuständen auftreten, zugeschrieben.
Das Experiment dieser Arbeit ist eines der Ersten, das eine Elektronendynamik im
Attosekundenbereich in einzelnen Nanopartikeln durch kohärente Lichtstreuung nachweist.
Damit öffnen sich neue Möglichkeiten zur zeitlich und räumlich aufgelösten Untersuchung
elektronischer Prozesse in kondensierter Materie auf der Nanoskala.
Abstract
With the emerging availability of intense attosecond pulses and pulse trains in the X-ray and
extreme ultraviolet (XUV) regime at free-electron lasers, and high harmonic generation sources,
fast electronic dynamics in single nanostructures are becoming accessible for imaging techniques.
This work addresses the detection of ultrafast electronic dynamics down to the attosecond
domain in single nanoparticles using coherent light scattering. In particular, scattering images
of helium nanodroplets, which serve as a simple model system, are produced using XUV light.
In order to provide the necessary particularly high intensities, an intense XUV light source
based on high harmonic generation has been developed in a collaboration of several research
groups at the Max Born Institute. Besides the optimization of the generation process itself,
the geometry of the associated beamline significantly determines the achievable intensities. To
use the maximum intensities also for imaging helium droplets, a new experimental setup was
designed. It includes focusing optics specially developed for this experiment and several tools
for characterizing the pulses interacting with the helium droplets.
Both the development of the intensity-optimized XUV light source and the experimental
setup have made it possible to detect a change in the scattering response of helium droplets
induced by a near-infrared (NIR) laser pulse on the femto- to even attosecond time scale.
In agreement with a previous measurement, a reduced scattering signal was now routinely
observed during simultaneous illumination of the droplets with NIR and XUV pulses. Moreover,
a change in the amount of scattered light during simultaneous illumination was observed when
the time delay of the two pulses was varied in the attosecond range. In fact, an oscillation of
the scattered signal with a period of about half an NIR laser pulse cycle was measured and
reproduced several times.
By modeling the experiment through the dipole response of a single electron in a helium
atom, the observed change can be attributed to phenomena known from atomic physics, namely
the AC Stark shift and light-induced states. In particular, the observed oscillation per NIR
half cycle is attributed to quantum path interference occurring in light-induced states.
The experiment in this work is one of the first to detect attosecond electron dynamics in
individual nanoparticles by coherent light scattering. This opens up new possibilities for the
temporally and spatially resolved study of electronic processes in condensed matter on the
nanoscale.
Acknowledgements
I want to thank all who made this work possible, contributed to it, and supported me and my
work.
In particular, I am grateful to my supervisor - Prof. Daniela Rupp - for her excellent
guidance, support, and advice. I am very thankful that she enabled me to finish my project at
the Max Born Institute (MBI) in Berlin after she established a new research group at ETH
Zurich in 2019. Her and her group joining and contributing to the experiments in Berlin were
essential for the experiments’ success. Thanks to Mario Sauppe and Katharina Kolatzki for
spending several weeks in Berlin for the experiments. Vice versa, I also want to thank Daniela
Rupp for making several multi-week visits to her group in Zurich possible.
Moreover, I thank Prof. Stefan Eisebitt for his supervision and his advice. I express my
thanks to Prof. Per Eng-Johnsson for reviewing this work.
Special thanks go to Mario Sauppe, who was at first involved in setting up the high
harmonic generation (HHG) beamline at MBI, whose expertise was fundamental to the in this
thesis presented experiments’ design, setup, and realization, and who continuously supported
me while writing the thesis.
I thank Andreas Hoffmann for his advice and his efforts to make the experiments possible,
especially taking care of laser and HHG optimization during the experiments. I am thankful
to Prof. Thomas Möller, Bruno Langbehn, and their students from TU Berlin, as well as
the students and student assistants from MBI, for their contributions to the experiments and
analysis.
Martin Kretschmar’s, Tamás Nagy’s, and Johannes Tümmler’s efforts to improve the laser
system, Bernd Schütte’s expertise, and effort to advance the HHG light source, as well as their
and our combined efforts to characterize the generated pulses and set up the beamline in the
first place made much of the here presented work and experiments possible.
My thanks also go to Prof. Marc Vrakking for his leadership of the project I was part of
and his support for modeling the helium droplets.
I am thankful for the technical support from Roman Peslin, Melanie Krause, Wolfgang
Krüger, and Katrin Herrmann. Moreover, thanks to all the other colleagues at MBI and ETH
Zurich, who supported this work.
Before finishing this thesis, I had already started a new position at the European XFEL.
Therefore, many thanks to my colleagues at the SQS group for enabling me to finish my thesis.
Last but not least, I want to thank my family and friends. Thank you for being supportive,
for being encouraging, and for always being there for me.
Table of Contents
Title page i
Zusammenfassung iii
Abstract v
1 Introduction 1
2 Fundamental concepts 5
2.1 Atoms in intense fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Intensity and duration of a light pulse . . . . . . . . . . . . . . . . . . . 6
2.1.2 High harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3
Quantum mechanical description of an atom placed in an external
classically described field . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4
Reconstruction of attosecond beating by interference of two-photon
transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.5 Laser dressing of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Helium droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Electronic structure of helium droplets . . . . . . . . . . . . . . . . . . . 30
2.2.2 Coherent diffractive imaging of helium droplets . . . . . . . . . . . . . . 32
3
Development & characterization of an HHG beamline for non-linear XUV
optics and coherent diffractive imaging 37
3.1 NIR laser source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 XUV pulse generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 General considerations for optimizing a HHG beamline . . . . . . . . . . 41
3.2.2 Analytical description of the XUV intensity scaling . . . . . . . . . . . . 44
3.2.3 Tailoring a beamline for CDI and non-linear XUV experiments . . . . . 46
3.2.4 Implemented, optimized XUV pulse generation setup . . . . . . . . . . . 52
3.3 XUV pulse characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Techniques for XUV detection . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 XUV pulse diagnostic setup . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Delaying two pulses with sub-femtosecond precision . . . . . . . . . . . 58
3.3.4 Temporal characterization of the XUV pulses . . . . . . . . . . . . . . . 60
ix
TABLE OF CONTENTS
4
Development of an NIR-XUV pump-probe coherent diffractive imaging
experiment with sub-fs resolution 65
4.1 Vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Generation of helium nano-droplets . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Optical setup for pump-probe coherent diffractive imaging . . . . . . . . . . . . 70
4.3.1 Beam conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Focusing and delay unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.3 Coincident ion & diffraction detection scheme . . . . . . . . . . . . . . . 74
4.3.4 XUV spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.5 Straylight & background reduction . . . . . . . . . . . . . . . . . . . . . 76
4.4 Diagnostic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1 Focus tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 Beam diagnostic behind the interaction region . . . . . . . . . . . . . . 79
4.5 Design of the focusing optic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Characterization of the NIR and XUV intensities in the interaction region . . . 87
4.6.1 Characteristics of the NIR in the interaction region . . . . . . . . . . . . 87
4.6.2 Characteristics of the XUV in the interaction region . . . . . . . . . . . 90
4.7 Establishing spatial & temporal overlap . . . . . . . . . . . . . . . . . . . . . . 93
4.7.1 Establishing spatial overlap . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.7.2 Establishing temporal overlap . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Ultrafast dynamics in helium nano-droplets 97
5.1 Overview, calibration & processing of the acquired data . . . . . . . . . . . . . 97
5.1.1 Reference and calibration data . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.2 Pre-processing of diffraction detector images . . . . . . . . . . . . . . . 101
5.1.3 XUV spectrum calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Laser dressing of helium droplets on the few femtosecond scale . . . . . . . . . 109
5.2.1 General observation and representation of the acquired data . . . . . . . 110
5.2.2 Variation of the NIR intensity . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.3 Limitations of the acquired data . . . . . . . . . . . . . . . . . . . . . . 114
5.3 Observation of sub-cycle dynamics in helium nano-droplets . . . . . . . . . . . 116
5.3.1 Setting null hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Testing of sub-cycle oscillations on null hypotheses . . . . . . . . . . . . 117
5.3.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Modeling of the delay-dependent diffraction response . . . . . . . . . . . . . . . 123
5.4.1 Modelling the scattering response of helium droplets . . . . . . . . . . . 123
5.4.2 Few-cycle resolved simulations . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.3 Sub-cycle resolved simulations . . . . . . . . . . . . . . . . . . . . . . . 144
5.4.4
Applicability and limitations of the simulated optical response and
diffraction intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.5 Discussion of experimental and modeling results . . . . . . . . . . . . . . . . . 159
5.5.1 Dynamics on the time scale of few NIR cycles . . . . . . . . . . . . . . . 159
5.5.2 Dynamics in the attosecond domain . . . . . . . . . . . . . . . . . . . . 164
x
TABLE OF CONTENTS
6 Summary & outlook 169
References 173
Appendix A Data processing 189
A.1 Additions to pre-processing of diffraction detector data . . . . . . . . . . . . . . 189
A.1.1 Validation of the diffraction image quality . . . . . . . . . . . . . . . . . 190
A.1.2 Robustness of the background classification algorithm . . . . . . . . . . 190
A.2 Toward spatial analysis of the diffraction patterns . . . . . . . . . . . . . . . . . 190
A.2.1 Identifying patterns among hits . . . . . . . . . . . . . . . . . . . . . . . 191
A.2.2 Routine to find the center of diffraction patterns . . . . . . . . . . . . . 192
Appendix B Additional experimental observations 195
B.1 Influence of the spatial overlap on time-scale of few-femtoseconds . . . . . . . . 195
B.2 Sub-cycle dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Appendix C Non-linear microscopic & macroscopic response 205
C.1 Derivation of the non-linear macroscopic response . . . . . . . . . . . . . . . . . 205
C.2
Limitations of the non-linear dynamic polarizability-approach on the microscopic
level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
C.2.1 Analytical discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
C.2.2 Numerical investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.3 Simulated response to realistic spectra . . . . . . . . . . . . . . . . . . . . . . . 214
xi
1
Introduction
The development of the laser [1] is one of the most important technological advances originating
from fundamental research [2]. This technology not only has numerous applications in everyday
life, but it also has had a formative influence on science and research since [3, 4, 5]. Further
advancement of the technology enabled the production of shorter and shorter pulses pushing
the boundaries of investigatable speeds of dynamics: the shorter the produced pulse, the
shorter the exposure duration of any measurement of interest, which effectively represents
the snapshot of an observed dynamical process at the time of exposure. By reaching a pulse
duration in the femtosecond regime, fundamental ultrafast processes like chemical reactions
became accessible. The field of femtochemistry was born. [6]
Besides the continuous development toward shorter pulses, the control over virtually all
characteristics of the light including spectral chirps led to techniques like, e.g., chirped pulse
amplification [7], which allowed the production of ultra-short pulses with increasing peak
power [5]. Those technologies made studies of highly non-linear processes possible and further
led to the discovery of high harmonic generation (HHG) [8], a process to produce short and
coherent light pulses at much shorter wavelengths than the used laser has. In fact, HHG can
yield light in the extreme ultra-violet (XUV) and the soft x-ray regime. Experiments could
demonstrate that the produced pulses actually consist of a train of pulses each with a duration
in the attosecond domain [9]. Further, the generation of isolated attosecond pulses could be
demonstrated [10] and further developed towards even shorter pulse durations, down to a few
tens of attoseconds [11]. The attosecond domain, i.e., the timescale on which the dynamics of
electrons within atoms take place, has been reached [12, 13].
Parallel to optical laser and laser-based technology (like HHG), synchrotron and free-
electron-laser (FEL) facilities have been developed. Those can provide XUV to hard X-ray
pulses with increasing brilliance and thereby produce significantly more photons than HHG-
based short wavelength light sources [14]. Due to the achievable high intensities with a pulse
duration in the tens to hundreds of femtosecond range at FELs [15], the high temporal resolution
could be combined with high spatial resolution facilitated by a range of imaging techniques,
such as holography [16] or coherent diffractive imaging [17, 18]. These imaging techniques
1
1. Introduction
are widely applied, among others, to investigate biological samples [19] such as viruses [20,
21] and nanocrystals of bio-molecules [22] or to study fundamental physical properties of and
processes in nano-sized structures [23, 24, 25, 26, 27, 28].
Just recently, efforts at FEL sources to further shorten the pulse duration have succeeded
in the demonstration of sub-femtosecond pulses [29, 30]. On the other side, also efforts at
HHG sources towards high intensity pulse trains are made [31]. As a result, some intensity
demanding experiments such as the investigation of multi-XUV-photon ionization [32] and
single-shot coherent diffractive imaging with fixed targets [33] or free-flying single-particles
[34] have been successfully realized at HHG light sources. These exciting developments have
the potential to enable a range of novel imaging experiments with ultimate spatio-temporal
resolution.
This work is concerned with studying near-instantaneous electronic changes in condensed-
matter nanoparticles induced by an ultra-short NIR light pulse. Helium nano-droplets are
used as ideal model systems. They offer a scalable size and reduced complexity and thus are
well accessible with theoretical simulations. A particularly strong change of the electronic
structure in a laser field can be expected near resonances, so XUV light in the 20 to 25 eV
photon energy range, matching the strong helium resonances in this region, is used for probing.
In CDI, a diffraction pattern is formed by the superposition of light coherently scattered by
the sample’s individual atoms (primarily their electrons). Therefore, the technique is sensitive
to the position of the scatterers, i.e., the structure of the sample, and their respective scattering
behavior, i.e., in a bulk perspective, the complex refractive index of the sample. As a result,
CDI can be used here to directly detect rapid changes in the electronic structure, corresponding
to an equally rapid change in the complex refractive index in the helium nano-droplets.
For this experiment, the XUV light is generated by HHG. Due to the attosecond pulse
structure, which is coupled to the phase of the NIR pulse, a stroboscopic detection of the
electronic response can be performed depending on the phase of the NIR pulse. Thus, it
can be investigated whether a different electronic response occurs at the zero-crossing of
the NIR’s electric field in helium droplets than when the probing occurs at the extrema
of the field. Indeed, the most prominent result of the experiments presented in this work
is the demonstration of changes in the diffraction response of a helium nano-droplet on a
sub-laser-cycle time scale and correspondingly small delays between NIR and XUV pulses.
More precisely, we observed an oscillation of the scattering response for every half cycle of the
NIR field.
Furthermore, as a second result of this thesis, a general decrease in the amount of diffracted
light was observed when both pulses are present simultaneously. This behavior was also seen in
a previous work using longer optical and XUV pulses [35], which also inspired the experiments
of this work. Our reproduction confirms the previous finding and the assumed direct relation
of the temporal width of the decreased scattering strength to the convolution of both involved
pulses.
In order to make such a CDI experiment with attosecond accuracy possible, in the framework
of this work, in a collaborative effort of multiple groups at the Max Born institute, an HHG
light source capable of providing particularly intense XUV pulses was developed, constructed,
and characterized. With this beamline, the intense and short XUV pulses could be focused to
2
very high intensities in the 10
14 W/cm2
regime, allowing for the generation of record-setting
highly ionized Argon, up to
Ar5+
[36]. To further achieve maximal intensities in the CDI
experiment, a focusing optic has been designed from scratch. To enable a time delay between
the XUV and NIR pulses with sufficiently high precision, a new delay unit was constructed
and evaluated by measuring sub-laser-cycle oscillations in electron spectra side-bands.
The thesis is structured as follows. First with chapter 2, fundamental concepts for the
understanding of necessary physical principles and applied techniques on the atomic level and
also specifically for helium droplets are introduced. The development and characterization of
the HHG light source are discussed, including the optics used to generate particularly intense
pulses and the design of the delay unit, in chapter 3. Chapter 4 explains the experimental
setup for the CDI experiment, also presenting the focusing optic’s design and the focus
characterization. The experimental observations of the ultra-fast dynamics, their theoretical
modeling, and interpretation are presented in chapter 5. Finally, a summary with outlook
follows in chapter 6.
3
2
Fundamental concepts
In this work, atoms and atomic clusters - helium nano-droplets, to be precise - are exposed to
intense near infrared (NIR) and extreme ultra-violet (XUV) light pulses. The interaction of
intense pulses with matter spans a broad research field and accordingly, the research discussed
in this thesis touches diverse topics. This chapter aims to provide a concise guide, focusing
on the most relevant fundamental concepts. The first section deals with the, for this work,
crucial atomic physics effects with an emphasis on the derivation of the AC Stark effect and
Autler-Townes-splitting, fundamental to the simulations used later in this thesis. In the second
section of this chapter, the helium droplets are introduced. Here, in particular the current
knowledge about the electronic structure of helium droplets is summarized and coherent
diffractive imaging (CDI) as a technique to investigate their properties is explained.
2.1 Atoms in intense fields
For this work, intense, short XUV pulses are generated by high harmonic generation (HHG),
for which an atomic gas is exposed to an intense NIR laser pulse. The XUV pulses
reaching the experiment are characterized employing reconstruction of attosecond beating
by interference of two-photon transitions (RABITT) or frequency-resolved optical gating
for complete reconstruction of attosecond bursts (CRAB). Later on, the experimentally
investigated ultra-fast dynamics in helium nano-droplets will be interpreted using an atom-like
perspective (see chapter 5.5). Therefore this section introduces the atomic effects related to
the laser-dressed-atom-picture. These effects include the AC Stark shift and light-induced
states (LISs).
The microscopic descriptions in this section are based on a semi-classical picture, where
the atomic system is treated quantum mechanically, but the light fields are introduced to it
classically. Such a treatment is justified as the relevant laser fields are of high intensity, i.e.,
consisting of many photons and thus approaching the classical limit.
The section is structured as follows; after an initial definition of a laser pulse and its
characteristic properties in section 2.1.1, HHG will be introduced using a qualitative classical
description on a microscopic scale (section 2.1.2). As the efficient production of XUV light
5
2. Fundamental concepts
10 0 10
time (fs)
1.0
0.5
0.0
0.5
1.0
1.5
electric field (arb. u.)
FWHM
2
(a)
env
10 0 10
time (fs)
0.00
0.25
0.50
0.75
1.00
1.25
power per unit area (arb. u.)
FWHM
(b)
instantaneous
I (cycle averaged)
Figure 2.1: (a) Exemplary electric field according to eq. 2.1 with its envelope. (b) Temporal
instantaneous power per unit area perpendicular to the propagation direction and intensity (i.e.
the cycle average of the former) profile with the associated FWHM
τ
defining the pulse’s duration.
As a consequence the FWHM indicated for (a) is wider than the defined pulse duration by a factor
of √2.
is a crucial precondition for the photon-demanding diffraction experiments presented later,
in addition, the relevant macroscopic effects of HHG are summarized. Then, following an
introduction to the semi-classical mathematical formalism for an atom placed in an external
field (section 2.1.3), the RABITT-technique-enabled bound-continuum interaction with external
fields is introduced in section 2.1.4. Finally, bound-bound transitions are discussed in section
2.1.5 using the picture of a laser-dressed atom, and related effects are explained.
2.1.1 Intensity and duration of a light pulse
Before getting started with the introduction of physical effects, some fundamental definitions
of pulse duration and intensity shall be made. Analog definitions and derivations can be
found e.g. in chapter 1.1 in the textbook "Ultrashort Laser Pulse Phenomena" by J. Diels and
W. Rudolph [37]. The electric field is denoted by
E
. Later on in this work, it will become
important to differentiate between a microscopic electric field
E
acting upon a single atom and
a macroscopic electric field Eacting upon the bulk. Because of that, the atomic description
will use the microscopic field notation
E
by default. In this work, vectors are denoted by bold
symbols.
For a plane wave, the time-dependent, classical electric field of an idealized short light
pulse E(t)at a fixed point in space can be written as
E(t) = E0·Eenv(t)·cos(ωt +ϕCEP ).(2.1)
Thereby, the field amplitude is given by
E0
=
nEE0
, where
nE
is the unit vector indicating
the polarization direction and
E0
the scalar field amplitude. The (normalized) pulse envelope
constraining the pulse’s duration is given by Eenv(t). The electric field’s oscillation is defined
through a central frequency
ω
and a phase
ϕCEP
. Because the otherwise continuous oscillation
6
2.1 Atoms in intense fields
Figure 2.2: Composition of a typical spectrum resulting from HHG; Beyond the perturbation
regime, in which the observed spectral intensity decreases towards higher orders, a plateau is
observed. Characteristic is a sudden decrease of the spectral intensities at high photon energies
behind the plateau, known as cut-off. (reprinted from [39])
defined by the cosine term is super-imposed with an envelope, this phase is also referred to as
the carrier-envelope phase (CEP).
From the Poynting theorem (see, e.g., [37]), the instantaneous power per unit area
perpendicular to the propagation direction of an electromagnetic field
1
can be obtained.
Following a widely used convention, its average over one optical cycle
2
is taken to define the
intensity I[37, 38]. For the pulse defined in eq. 2.1, it is given by:
I(t) = 1
2·ϵ0·c·n·|E0|2·Eenv(t)2([37],eq.1.21) (2.2)
and consequently the peak intensity of a pulse (where Eenv = 1) is described by
Ipeak =1
2·ϵ0·c·n·|E0|2.
In this work, following [37], the duration of a light pulse
τ
that is enforced in eq. 2.1 through
the pulse envelope
Eenv
(
t
)is defined with respect to the pulse’s intensity. In fact, it is defined
for the squared field envelope
E2
env
(
t
). Based on eq. 2.2, for a pulse with Gaussian function as
envelope centered around t= 0, the intensity is given as:
I(t) = 1
2·ϵ0·c·n·|E0|2·e−4·ln 2·t2
τ2.
Fig. 2.1 shows (a) the electric field and (b) the intensity of a Gaussian light pulse. In particular,
it illustrates the different FWHM measurements for the electric field and the intensity of
the very same pulse: while the FWHM duration of the intensity is defined as
τ
, the FWHM
duration of the electric field is given by
√2τ
. Thus, the envelope of a Gaussian pulse’s electric
field may be written in terms of the pulse duration τthe following way:
Eenv(t) = e−2·ln 2·t2
τ2.
7
2. Fundamental concepts
Figure 2.3: Illustration of the three-step model providing a semi-classical explanation for the
HHG process in a single atom. A: a strong linearly polarized electric field affects the potential
of an atom so that an electron may tunnel out of it into the continuum. B: the trajectory of the
electron in the continuum is controlled by the electric field, whose oscillation allows the electron
to gain energy. C: eventually, the electron recombines with the ionic core and emits the gained
energy as an XUV photon. (reprinted from [39])
2.1.2 High harmonic generation
High harmonic generation (HHG) describes a phenomenon occurring when atoms are exposed
to intense laser pulses. There, photons at energies of many multiples of the laser’s photon
energy are generated [8, 40]. As illustrated by fig. 2.2, particular features of the spectra are
1.
the presence of photon energies that are odd harmonics of the intense laser field, [8] and
2.
a plateau of harmonics with similar intensities in between a regime of lower harmonics
which substantially decrease in intensity with harmonic order [8], and a cut-off at a
certain photon energy [40].
While a full description of HHG demands a quantum mechanical treatment, the fundamental
physical process yielding the generation of high harmonics from an atom placed in an intense
driving laser field is well described by a semi-classical model developed by Paul Corkum [41].
It is also known as the simple man, or three-step model [13]. Following [42] and as illustrated
by fig. 2.3, the summarized steps are:
1.
The atom is ionized due to electrons tunneling out of the atom’s potential into the
continuum, where the potential is modified by the driving laser field.
2.
In the continuum, accelerated by the laser field, the electron follows a classical trajectory.
Thus some electrons may remain in the vicinity of the ionic core.
3.
Electrons returning to the ionic core, i.e., ’recolliding’ with it, can recombine with it into
the ground state and thus will emit an XUV photon.
The observation of only odd harmonic orders is explained by the symmetry of the process.
Since the driving laser field consists of multiple optical cycles and, following the three-step model,
radiation is generated every half-cycle, even orders will interfere destructively. The emission of
1One could name this also the ’instantaneous intensity’.
2in time
8
2.1 Atoms in intense fields
Figure 2.4: Illustration of the phase-matching within the gaseous medium that is required to
achieve efficient high harmonic generation on the macroscopic scale. (reprinted from [44] with
modified labels: the illustration was originally labeled for X-rays, mid-NIR light, and a capillary.
As the same principle also applies to XUV with NIR and a gaseous medium, the labels have been
adapted accordingly.)
harmonic radiation every half cycle further leads to the characteristic temporal structure of
HHG pulses, attosecond pulse trains (APTs). Those consist of consecutive attosecond pulses
spaced by half a laser cycle. The spectral cutoff is explained by the maximum energy an
electron may gain quivering in the driving laser’s field in combination with the ionization
energy
Ip
of the atom that is released in addition during recombination. It occurs at an the
harmonic order HOmax, which is given by:
HOmax =Ip+ 3.17 ·Up
ω[41,42].
Thereby,
Up
is the ponderomotive energy, i.e., the average energy of an electron quivering in an
oscillating field [43] and
ω
is the central frequency of the driving laser. A detailed theoretical
treatment of the HHG process reaching beyond the here introduced three-step model can be,
e.g., found in chapter 7 in [13].
To achieve efficient HHG, in practice, in addition to the response of a single atom,
macroscopic effects have to be taken into account. Fig. 2.4 illustrates the practical realization:
an incident NIR laser drives the HHG process in a macroscopic, gaseous medium, which
produces XUV exiting the medium together with the NIR laser light. For efficient HHG
on macroscopic scales, as highlighted in fig. 2.4, it is particularly important to achieve
a good match between the phases of the emitted light from different atoms, the so-called
phase-matching. The following presented summary follows the more detailed overview in [45].
A phase mismatch of newly generated light with the already generated, co-propagating
high harmonics may occur due to [46]:
•
different Gouy phase shifts (see, e.g., [47]) experienced by the driving laser at different
distances from its focus, where the HHG takes place
•
the intensity-dependent phase of the emitted high harmonic (dipole) radiation. Regarding
the three-step model, it can be seen as the phase gained by the electron during acceleration
in the driving laser’s field.
9
2. Fundamental concepts
•
different refractive indices of the medium’s neutral fraction and thus different phase
velocities for the generated high harmonics and the driving laser
•
different refractive indices of the medium’s ionized fraction and thus different phase
velocities for the generated high harmonics and the driving laser
Since different contributions to the phase can positively or negatively impact the phase-
mismatch and depend on different parameters, it is possible to find conditions, and spatial
regions with a low mismatch, i.e. where the counteracting individual contributions cancel out.
A low mismatch corresponds to a long coherence length of the generated harmonics. The Gouy
phase shift and dipole induced phase shift largely depend on the spatial characteristics of
the driving laser, that is, the tightness of the focusing (e.g., controlling the intensity gradient
determining the dipole phase contribution) and the position of the medium relative to the
driving lasers focus. The refractive indices, however, are controlled by the medium’s density.
That is for a gaseous medium, given by the pressure, as used in this work.
Consequently, by optimizing the driving laser’s properties, such as focusing and intensity,
and the medium’s properties, e.g., the gas pressure, regions with large coherence lengths of
the generated high harmonics can be identified. According to those regions, the medium
length shall be chosen. An optimized amount of high harmonics can be generated when their
coherence length
Lcoh
is much larger than their absorption length
Labs
in the medium [48]. As
an orientation, with a coherence length
Lcoh >
5
·Labs
and a medium length
Lmed >
3
·Labs
at least half of the maximal possible amount of light will be generated [48].
After optimal microscopic and macroscopic conditions are found, it is possible to scale the
setup to match the demands of the application or experiment, which utilize the generated
light. One may optimize for a high intensity, pulse energy, or a combination of such in the
application or experimental setup. Additional constraints and parameters, like available length
in the laboratory and where to place the HHG medium become relevant. This optimization
will be discussed in detail concerning the experimental setup presented in this thesis in chapter
3.2. For example it will be shown, that a particular high XUV intensity in the experimental
setup can be achieved by placing the HHG medium far away from the experiment.
2.1.3
Quantum mechanical description of an atom placed in an external
classically described field
So far, mainly qualitative arguments have been used to describe the HHG process. Now, a
basis will be established for an analytical description of other effects, which will be carried out
in subsequent sections 2.1.4 and 2.1.5. Therefore, this subsection is heavily based on [47] and
[13]. The description in these two books uses partially SI and partially atomic units. Here the
description is consistently kept in atomic units.
Most general the time evolution of a quantum mechanical system may be described by the
time-dependent Schrödinger equation (TDSE):
i∂
∂t |Ψ(r, t)⟩=H
ˆ|Ψ(r, t)⟩.(2.3)
10
2.1 Atoms in intense fields
An analytical solution of the TDSE, even for a single atom, ion, or molecule, can only
be obtained for a few specific cases, and even precise numerical results are limited to certain
assumptions. A widely applied one is the single active electron (SAE) approximation. It
neglects the interaction of multiple electrons with each other and external fields: multiple
electrons are in this picture only accounted for by adding a contribution to a static, effective
atomic potential V(r)and only the one "active" electron is considered dynamically. [47]
Applying the SAE approximation, the Hamiltonian
H
ˆ0
of an isolated atom in the length
gauge may be described by kinetic energy and potential:
H
ˆ=H
ˆ0=−1
2∆ + V(r).
Furthermore, the size of atoms, ions, and molecules is typically on the order of one
Ångström, which is small compared to wavelengths of light on the order of 10
nm
and above.
Hence, for light the XUV and longer wavelength regimes, the spatial variation of the electric
field over the extent of the atom and thus its spatial dependency can be neglected [13]. With
this so-called dipole approximation, the Hamiltonian of an atom exposed to an field can be
written as
H
ˆ=−1
2∆ + V(r)
⏞ ⏟⏟ ⏞
H
ˆ0
−r·E(t)
⏞ ⏟⏟ ⏞
H
ˆ1
[13].(2.4)
It can be seen as two parts, the previously introduced Hamiltonian of an isolated atom
H
ˆ0
and the added interaction with the electric field
H
ˆ1
. Together with the general form of the
TDSE given in eq. 2.3, for the interaction of an atom with an external electric field in dipole
and SAE approximation, the TDSE can be written in length gauge as:
id
dt |Ψ(r, t)⟩={︃−1
2∆ + V(r)−r·E(t))}︃|Ψ(r, t)⟩([13],eq.8.1).(2.5)
Depending on what is most convenient for the problem to be solved, alternative to the
length gauge, the TDSE may be used in the so-called velocity gauge. In this work, both are
used: For example, the analytical derivations for the description of the laser dressing of atoms
uses the length gauge (see section 2.1.5). However, a numerical algorithm to solve the TDSE
that is applied later in this work is based on the velocity gauge (see chapter 5.4). The TDSE
in velocity gauge may be written as
id
dt |Ψv(r, t)⟩={︃1
2[−i∇r−A(t)]2+V(r)}︃|Ψv(r, t)⟩([13],eq.8.2).(2.6)
A(
t
)describes the vector potential of the present light in coulomb gauge. It can be obtained
from the electric field or vice versa:
A(t) = −∫︂t
−∞
dt′E(t′) or E(t′) = −d
dtA(t) ([13],eq.8.3/chapter 8.1) (2.7)
One elegant approach to avoid a purely numerical treatment of the TDSE is provided by
the strong field approximation (SFA). In this picture, only one state of the atom - the ground
11
2. Fundamental concepts
state - is accounted for and the electron may only be excited directly into the continuum [47].
Alternatively, in other words, for any excited state, the atomic potential V(r)is neglected.
With the SFA, the TDSE as stated in eq. 2.5 has the following analytical solution (see [47]
for details):
|Ψ(r, t)⟩=e−iIpt[︃a(t)|0⟩+∫︂d3vb(v, t)|v⟩]︃([47],eq.5.15).
Here,
Ip
is the ionization energy from the ground state, and
a
(
t
)is the time-dependent
amplitude of the ground state’s (
|0⟩
) wavefunction. Further,
b
(v
, t
)relates to the population
of the respective state
|v⟩
in the continuum. SFA applies to situations involving strong laser
fields, such as the process of generating high harmonics, but also those involving weaker fields
with higher photon energies, like the pulses resulting from HHG, that can ionize an atom in a
single photon process. [47]
Alternatively, for weak fields, perturbation theory may be applicable. If the field
E
(
t
)is
weak, in eq. 2.4 the part of the Hamiltonian relating to the interaction with the external field
H
ˆ1
can be treated as a perturbation to the time-independent Hamiltonian of the isolated atom
H
ˆ0
. Analog to the Hamiltonian in eq. 2.4 that is given in length gauge, this can be done for
the Hamiltonian velocity gauge H
ˆv, for which the TDSE is given in eq. 2.6:
H
ˆv=−∇2
r
2+V(r)
⏞ ⏟⏟ ⏞
H
ˆv,0
+1
2[A(t)·i∇r+i∇r·A(t)] + 1
2A2(t)
⏞ ⏟⏟ ⏞
H
ˆv,1
([47],eq.7.65 −7.67).
For weak fields one may approximate A
2
(
t
)
≈
0[47]. When further considering A(
t
)in
Coulomb gauge (∇rA(t) = 0), the perturbing Hamiltonian H
ˆv,1can be simplified to
H
ˆv,1=A(t)·i∇r.
For a periodic perturbation, a very important result of first-order perturbation theory
is Fermi’s second golden rule [49] (see standard textbooks like, e.g., [50] for a derivation
3
).
Similar to the SFA, the final state must be in the continuum, and the Coulomb attraction of
the ionic core must be negligible [50]. With Fermi’s golden rule the average transition rate
Ri→f
from an initial state
|Ψi⟩
(e.g., the ground state) into an excited, final state
|Ψf⟩
, e.g.,
due to a periodic external field, can be computed [47, 50].
Ri→f∝2π|⟨Ψf|H
ˆv,1|Ψi⟩
⏞ ⏟⏟ ⏞
Mf,i
|2([47],eq.7.68) (2.8)
Mf,i
is the so-called transition matrix element, providing a scale for the rate of the transition
from the initial state i to the final state [47]. By integration of the transition rate
Ri→f
with
regards to time, the overall transition probability can be obtained.
Fermi’s golden rule, i.e., a first-order perturbation, can only take into account direct
transitions. In some cases, particular transitions are only possible through intermediate
states. Transitions involving one intermediate state, i.e., effectively two transitions, are
3Fermi refers in his lecture to the derivation in [51]
12
2.1 Atoms in intense fields
described by second-order perturbation theory. For transitions involving more intermediate
states, accordingly, higher orders of perturbation have to be taken into account [51]. For
time-dependent perturbations, obtaining transition rates or probabilities is more complicated.
4
According to Dirac ([52]), in second order perturbation (assuming a direct transition is not
possible and its probability thus 0) the transition probability
Pi,f
from initial state i into final
state f through any possible intermediate states m is given by
Pi,f =−i∑︂
m∫︂t
t0⟨Ψf|H
ˆ′
v,1(t′)|Ψm⟩dt′∫︂t′
t0⟨Ψm|H
ˆ′
v,1(t′′)|Ψi⟩dt′′
2
([52],eq.26).(2.9)
Again,
⟨Ψf|H
ˆ′
v,1
(
t′
)
|Ψm⟩
and
⟨Ψm|H
ˆ′
v,1
(
t′′
)
|Ψi⟩
describe a matrix element for the transition
from
m→f
and
i→m
respectively [52]. Note, to get from
H
ˆv,1
to
H
ˆ′
v,1
, the following unitary
transform needs to be applied:
H
ˆ′
v,1=eiHv,0(t−t0)·H
ˆv,1·e−iHv,0(t−t0)([52],eq.18) .
This result from second order perturbation theory shows the reaching of a final state f by an
intermediate state m. However, it also indicates that when a final state can be reached through
different intermediate states, then there is interference between these different transitions
5
[52]. Such interference is known as quantum path interference or "which-way interference"
[53]. As addressed in the discussion of the "dressed atom" picture in section 2.1.5, this type of
interference occurs not only in this perturbative approach to transitions into the continuum. It
also appears in other approximations or in the numerical solution of TDSE, even for transitions
below the ionization threshold (see, e.g., [54]).
2.1.4
Reconstruction of attosecond beating by interference of two-photon
transitions
The fundamental property of light produced from HHG to consist of an attosecond pulse train
(APT) was first proven experimentally using the RABITT technique [9]. The used approach
was suggested theoretically by Véniard et al. [56]. For this technique, a sample, e.g., an
atomic gas, is exposed to a superposition of the generated higher harmonics (weak field) and a
weak copy of the fundamental laser pulse driving the HHG process. The time delay between
fundamental and HHG pulses is then scanned, and the kinetic energy spectrum of electrons
appearing in the continuum, as, e.g., depicted in fig. 2.5 (a), is observed.
Provided a sample with a sufficiently low ionization threshold was chosen due to the HHG
radiation, transitions occur directly into the continuum. Because each suitable harmonic may
yield a transition into the continuum from the same ground state, the electron’s kinetic energy
spectrum (in the continuum) resembles the harmonics spectrum minus the ionization energy
that is necessary to reach the continuum. As illustrated by fig. 2.5 (b), states in the continuum
between the harmonic energies cannot be reached through a direct path.
4
Note, Fermi’s first golden rule that relates to the transition rates for a second-order perturbation applies
only for time-independent perturbations [49, 51].
5
To state it explicitly, the sum over the different transitions involving different intermediate states m implies
the interference.
13
2. Fundamental concepts
Figure 2.5: (a) exemplary RABITT spectogram showing a beating in the sidebands (SBs) between
the photoelectron energies originating from single photon excitation of electrons by the XUV. (b)
aligned the the spectogram a schematic drawing illustrating this single photon excitation and the
pairs of quantum paths leading to the same sideband. (reprinted from [55])
However, indirectly it is possible. As introduced with the second-order perturbation
approach for solving the time-dependent Schrödinger equation (TDSE) assuming weak fields
in the previous section, it may be reached by excitation through an intermediate state. Since
the middle between two neighbored harmonic energies is at a distance of the fundamental’s
frequency (or photon energy), the continuum state reached directly with one harmonic can act
as an intermediate state from which the fundamental laser field can enable the transition into
a final state that lies energetically between the harmonic energies. Such a final state is also
known as a sideband (SB).
In fact, as can be seen from fig. 2.5 (b), when involving one intermediate state (i.e.,
two single-photon transitions), each sideband can be reached via two different quantum
paths. That is through the intermediate state reached by the harmonic below the sideband
in combination with the absorption of one fundamental photon or through the one reached
by the harmonic above the sideband in combination with the stimulated emission of one
fundamental photon. According to the conclusions drawn from eq. 2.9, both quantum paths
will interfere. Accordingly, when scanning the time delay between XUV and fundamental laser
light, a beating in the sidebands, as shown in fig. 2.5 (a), can be observed.
In the initial theoretical proposal of the RABITT technique, Véniard et al. could show
utilizing second-order perturbation theory that this interference encodes the relative phase
between the involved neighbored high harmonics [56]. Based on the results presented in [56], a
very intuitive derivation of the time-delay-dependent
6
amount of electrons that can be observed
reaching the sideband is given by Isinger et al. in [57]: Considering the following complex
amplitudes for the fundamental field E0,1and for the jth harmonic order E0,j:
E0,1=E0,0,1·e−iωtd
E0,j =E0,0,j ·e−iϕj.
6referring to the time delay between the light from HHG and the weak fundamental pulse;
14
2.1 Atoms in intense fields
The complex amplitudes,
E0,1
and
E0,j
, consist of a real amplitude
E0,0,1
and
E0,0,j
respectively in combination with a phase shift. This phase shift is due to a variable time delay
td
relative to the high harmonics for the fundamental field. For the harmonics, it must be
expected that each has an individual phase, which is reflected by
ϕj
for the jth harmonic order.
The transition matrix elements containing the probabilities for the excitation from the
ground state to the sideband via the (q+1)th harmonic and the stimulated emission of a
fundamental photon and via the (q-1)th harmonic and the absorption of a fundamental photon
are denoted by
Memi
and
Mabs
respectively. Then, according to the results in [56] and in the
notation of [57], the amount of signal (arb. scale) at the sideband q Sqis given by:
Sq=|E0,1E0,q−1Mabs
⏞ ⏟⏟ ⏞
Aabs
+E∗
0,1E0,q+1Memi
⏞ ⏟⏟ ⏞
Aemi
|2([57],eq.2.1).
It can be converted to the following form:
Sq=|Aabs|2+|Aemi|2+ 2|Aabs||Aemi|cos(arg(Aabs)−arg(Aemi)) ([57],eq.2.1).(2.10)
Here the cosine term is of interest. Its argument can be split into the contributions by the
fundamental field’s amplitude
E0,1
, the amplitudes of the harmonics
E0,q−1
and
E0,q+1
, and a
contribution from the atomic properties introduced by the transition matrix elements
Mabs
and Memi.
arg(Aabs)−arg(Aemi) = [arg(E1)−arg(E∗
1)]
⏞ ⏟⏟ ⏞
2ωtd
+[arg(Eq−1)−arg(Eq+1)]
⏞ ⏟⏟ ⏞
∆ϕq−1,q+1=ϕq−1−ϕq+1
+[arg(Mabs)−arg(Memi)]
⏞ ⏟⏟ ⏞
∆θq−1,q+1=θq−1−θq+1
The phase difference between the two involved harmonics
q−
1and
q
+ 1 is introduced
as ∆
ϕq−1,q+1
. In addition, the transition matrix elements may introduce a phase (see [56] eq.
5), here denoted by
θj
. The phase difference for the two involved harmonics is denoted by
∆θq−1,q+1.
With this eq. 2.10 can be rewritten into
Sq=|Aabs|2+|Aemi|2+ 2|Aabs||Aemi|cos(2ωtd+ ∆ϕq−1,q+1 + ∆θq−1,q+1) ([57],eq.2.2).
(2.11)
This result shows that when varying the time-delay
td
between fundamental and high
harmonics, the sideband signal will oscillate at twice the fundamental’s frequency. The phase of
the oscillations indicates the phase difference between the harmonics; however, it also contains
a phase contribution from the atomic gas. In first approximation, the phase contribution of
the atomic gas ∆
θq−1,q+1
may be assumed to be the same for all harmonics and thus their
difference can be assumed to be zero [56]. By measuring the oscillations and their phases at
all available sidebands, the relative phases of the harmonics can be determined. Together with
the amplitudes of the harmonics retrieved from a measured spectrum, the attosecond pulses in
the attosecond pulse train can be reconstructed. However, at this point, the envelope of the
attosecond pulse train remains unknown.
15
2. Fundamental concepts
In a real measurement, in addition to the predicted sideband signal
Sq
in eq. 2.11 some
’real world effects’ need to be accounted for. Those may include spatial averaging as the
interaction occurs in a given volume, the so-called interaction region, or temporal jitter. The
influence of such effects on the measurement is simulated under the SFA and discussed in [57].
Figure 2.6: (a) CRAB spectogram for an APT showing the decrease in sideband oscillation
amplitude away from temporal overlap. (b) depicts the original XUV pulse (solid red line) and its
reconstruction (dotted blue line) with a FWHM duration of 12 fs. (reprinted from [58])
So far, in this work, the discussion and derivations were carried out assuming monochromatic
harmonics and fundamental laser field. A monochromatic laser field is inherently of infinite
duration. A laser pulse therefore cannot be strictly monochromatic. In fact, the shorter the
used pulses and pulse trains, the less precise the monochromatic approximation. Likewise, also
the sidebands show no monochromatic signal. Furthermore, as figure 2.6 (a) exemplarily shows,
the envelope of the pulses also plays a role: If the pulses do not overlap in time, then only the
XUV can excite the electrons into the coninuum, but a quantum path into the sideband does
not exist. Consequently, the amplitude of the oscillations in the sidebands depends on the
time delay of both pulses to each other.
By modelling the complete pulse structure, using iterative retrieval algorithms, the complete
pulses (fundamental laser pulse and APT with its envelope) can be reconstructed. For the
spectogram in 2.6 (a), fig. 2.6 (b) shows the constructed APT. This method is known as
frequency-resolved optical gating for complete reconstruction of attosecond bursts (CRAB). A
detailed description is e.g. given in [47].
2.1.5 Laser dressing of atoms
Until now, in this chapter, excitation from the ground state into the continuum was discussed.
For RABITT, particularly the simultaneous presence of two fields, i.e., the light from HHG and
the fundamental laser, was important since those enabled quantum path interference within the
continuum. For this subsection, transitions between bound states of the atom, i.e., below its
ionization threshold, are of interest. Instead of taking all possible states of an atomic potential
V
(r)into account, simplified systems consisting of two or three states will be discussed in this
section. Those are sufficient to explain the relevant physical effects for this thesis. As detailed
below, an important aspect is that a laser field depending on its photon energy
ωL
and its
intensity modifies the position of resonances (AC Stark effect and Autler Townes splitting),
16
2.1 Atoms in intense fields
gives rise to new resonances (light-induced states) and fast time-delay-dependent oscillations
least twice the laser frequency. These findings will be compared to the experimental results of
this work in chapter 5.5.
In this section, the use of the semi-classical approach will be continued and it is also at the
basis of the numerical TDSE simulation presented in chapter 5.4. At the high intensities used
in the experiment, many photons are interacting with the quantum-mechanically described
atomic system. Thus the light can still be well approximated as a classical field. The derivation
presented here follows the description in [59], [54] and chapter 5 in [60]. However, it shall
be noted that many references introduce the AC stark shift and dressed atom picture with
a fully quantum mechanical description (see, e.g., [61, 62, 63]), because it allows for a very
intuitive picture. Such description commonly uses Fock states as a basis for the light field,
which by themselves are incoherent (maximum uncertainty in their phase) [62]. Only by their
superposition coherent states can be described [62]. Because the AC stark shift is independent
of the dressing laser’s phase, it is well described by considering only a few Fock states (and
thus using a very basic system). However, the sub-cycle effects observed and studied in this
work do rely on the coherence properties of the laser field, and therefore a description based
on Fock states would severely complicate the understanding. Instead, the chosen semi-classical
approach is able to provide a more simple and intuitive picture for the here studied effects.
A two-level atom exposed to a laser field
In general, an atomic system placed in an external field is well described by the TDSE in
eq. 2.5 (or alternative eq. 2.6). The Hamiltonian, as stated in eq. 2.4, can be split into a
time-independent operator valid for the unperturbed atom
H
ˆ0
and a time-dependent operator
H
ˆ1describing the interaction with the external field.
For a two level atom, the wavefunction
|Ψ(t)⟩
can be expressed in terms of a basis formed
by the two (time-independent) eigenstates of the unperturbed atom
|1⟩
and
|2⟩
, weighted by
the time-dependent factors C1(t)and C2(t), respectively:
|Ψ(t)⟩=C1(t)|1⟩+C2(t)|2⟩([60],eq.5.2.1).
Please note, for convenience, the explicit spatial dependence is not stated.
By multiplication with unity (
|1⟩⟨1|
+
|2⟩⟨2|
= 1) the matrix form of the Hamiltonian’s
components
H
ˆ0
and
H
ˆ1
can be obtained. Suppose,
|1⟩
and
|2⟩
(the eigenstates of the
unperturbed atom) are associated with the eigenenergies
ω1
and
ω2
, then follows
H
ˆ0|1⟩
=
ω1|1⟩
and H
ˆ0|2⟩=ω2|2⟩, respectively:
H
ˆ0= (|1⟩⟨1|+|2⟩⟨2|)H
ˆ0(|1⟩⟨1|+|2⟩⟨2|) = ω1|1⟩⟨1|+ω2|2⟩⟨2| ⇔ H
ˆ0=(︄ω10
0ω2)︄
For the treatment of
H
ˆ1
, the electric field may be split in its normalized polarization
vector
nE
and its time-dependent scalar amplitude
E
(
t
):
E
(
t
) =
nEE
(
t
). Thus, formerly used
17
2. Fundamental concepts
approach yields:
H
ˆ1=−r·E(t) = −(|1⟩⟨1|+|2⟩⟨2|)rnE(|1⟩⟨1|+|2⟩⟨2|)·E(t)
=−
⟨1|rnE|1⟩
⏞ ⏟⏟ ⏞
0|1⟩⟨1|+⟨2|rnE|2⟩
⏞ ⏟⏟ ⏞
0|2⟩⟨2|+⟨1|rnE|2⟩
⏞ ⏟⏟ ⏞
M1,2
|1⟩⟨2|+⟨2|rnE|1⟩
⏞ ⏟⏟ ⏞
M2,1
|2⟩⟨1|
·E(t).
Analog to eq. 2.8
⟨1|
r
|2⟩
and
⟨2|
r
|1⟩
are transitions matrix elements that will be denoted
by
M1,2
and
M2,1
respectively (obviously
7
,
M1,2
=
M∗
2,1
). Due to the symmetry properties of
the wave functions
|1⟩
and
|2⟩
,
⟨1|
r
|1⟩
=
⟨2|
r
|2⟩
= 0. Consequently, in its matrix form
H
ˆ1
can be written as:
H
ˆ1=(︄0−M1,2E(t)
−M∗
1,2E(t) 0 )︄(2.12)
And thus the Hamiltonian is given by:
H
ˆ=H
ˆ0+H
ˆ1=(︄ω1−M1,2E(t)
−M∗
1,2E(t)ω2)︄.
Already here, from the off-diagonal elements of the matrix, it becomes evident that its
eigenstates are no longer given by
|1⟩
and
|2⟩
. Instead, the eigenstates are modified, and their
deviation from the original eigenstates scale with the amplitude of
E
(
t
)and the strength of
the coupling, i.e.,
M1,2
. One may say, the electric field
E
(
t
)’dresses’ the atom so that instead
of the unperturbed atomic eigenstates, the system exhibits ’dressed’ states.
Thus with the basis {︄(︄|1⟩
0)︄,(︄0
|2⟩)︄}︄
the TDSE may be written as
(︄id
dt C1(t)
id
dt C2(t))︄=(︄ω1−M1,2E(t)
−M∗
1,2E(t)ω2)︄(︄C1(t)
C2(t))︄
This matrix form is equivalent to the following set of linear differential equations:
d
dtC1(t) + iω1C1(t) = iM1,2E(t)C2(t)
d
dtC2(t) + iω2C2(t) = iM∗
1,2E(t)C1(t)
Mathematically it is very convenient to substitute
C1
(
t
)and
C2
(
t
)by
c1
(
t
)
·e−iω1t
and
c2(t)·e−iω2t, respectively and further (ω1−ω2)by ∆ω:
d
dtc1(t) = iM1,2E(t)c2(t)ei∆ωt (2.13)
d
dtc2(t) = iM∗
1,2E(t)c1(t)e−i∆ωt (2.14)
7ris an Hermetian operator
18
2.1 Atoms in intense fields
The solutions for
c1
(
t
)and
c2
(
t
)to this system of linear differential equations will enable
insight into the response of the medium. As will be shown in the following paragraphs, this
includes the optical response to an ultra-short pulse probing the system behavior when exposed
to the field
E
(
t
). Therefore, two analytical approaches will be presented: First, the rotating
wave approximation, which effectively neglects any sub-cycle effects but is not limited in
the accepted intensity of the electric field and thus can reproduce the AC Stark effect, and
second, a solution applying second-order perturbation theory, requiring a sufficiently weak
perturbation, i.e., a weak field
E
(
t
). It will be able to demonstrate sub-cycle oscillations
together with light induced states, but not the AC Stark effect. The analytical descriptions
thus facilitate the understanding of the phenomena that may occur. Unfortunately, no simple
analytical description is known that combines the description of high intensities with sub-cycle
resolution. After all, precisely under these conditions, the data discussed in chapter 5 has
been acquired. Therefore, a numerical solution will be discussed after the analytic derivations.
The numerical solution shows that both AC Stark Shift and light-induced states exhibiting
sub-cycle oscillations occur together, beyond the previously used approximations.
The AC Stark effect
When an atom is placed in an external (dressing) laser field, this external light field modifies
the response of the atom - such as its absorption properties. It ’dresses’ the atom. One
modification is an average shift of the absorption lines. Such shift is associated with the AC
Stark effect. It is due to the coupling of two states by the dressing laser field. In particular,
the AC Stark effect occurs when the energetic distance between these states does not match
the light field’s photon energy or frequency, i.e., if it is off-resonant. In resonance, the effect is
known as Autler-Townes splitting. Instead of a shift, the resonance splits symmetrically into
two.
An analytical derivation will be provided in the following, which is valid for both effects,
level shifting and splitting. For simplicity, the dressing laser field that modifies the response
E
(
t
), is assumed to be a continuous
8
, linear polarized wave (cw) with the amplitude
E00
,
frequency
ωL
and phase
ϕL
. The phase can be used later to model a time delay
td
, i.e.,
ϕL=ωLtd.
E(t) = E00 cos(ωLt+ϕL) = E00
2(︂ei(ωLt+ϕL)+e−i(ωLt+ϕL))︂
This expression can now be inserted into the differential eq. 2.13 and 2.14, describing the
behaviour of the two level-system. Then, the rotating-wave approximation can be applied.
It neglects terms that are oscillating on very short time scales, i.e., those oscillating with
∆ω+ωL.9Thus, the set of linear differential equations becomes:
d
dtc1(t) = iM1,2E00
2c2(t)(︂ei[(∆ω−ωL)t−ϕL]+e−i[(∆ω+ωL)t+ϕL])︂≈iM1,2E00
2c2(t)ei[(∆ω−ωL)t−ϕL]
d
dtc2(t) = iM∗
1,2E00
2c1(t)(︂ei[(∆ω+ωL)t+ϕL]+e−i[(∆ω−ωL]t−ϕL])︂≈iM∗
1,2E00
2c1(t)e−i[(∆ω−ωL)t−ϕL]
8A derivation using a light pulse can be found in [59].
9
This assumes that the energy difference between the involved states (∆
ω
) is on the order of the laser
frequency (ωL).
19
2. Fundamental concepts
(a) (b)
∆ω
|0> (ω0 = 0)
|2> (ω2)
|1> (ω1)
∆ω
|2> (ω2)
|1> (ω1)
Figure 2.7: Schematic drawing of the simple two and three level systems that are discussed in
the test. (a) Two-level system for which the TDSE is solved. (b) Three level system that includes
a third, far-off-resonant state to enable a probing of the states
|1⟩
and
|2⟩
coupled by the dressing
laser.
For a more intuitive solution the (time-independent) phase of the dressing laser field
ϕL
can
be grouped together with the other constants:
d
dtc1(t) = iM1,2e−iϕLE00
2c2(t)ei(∆ω−ωL)t
d
dtc2(t) = iM∗
1,2eiϕLE00
2c1(t)e−i(∆ω−ωL)t
The solution can be obtained analytically in a lengthy but straightforward way. With
M1,2
=
|M1,2|eiϕ1,2
,Ω
R
=
|M1,2|E00
and Ω =
√︂Ω2
R+ (∆ω−ωL)2
, according to [60] (eq.
5.2.21/22)10, it is given by:
c1(t) = [︃c1(0)(︃cos Ωt
2−i(∆ω−ωL)
Ωsin Ωt
2)︃+iΩR
Ωe−i(ϕ1,2−ϕL)c2(0)sin Ωt
2]︃ei(∆ω−ωL)t/2
(2.15)
c2(t) = [︃c2(0)(︃cos Ωt
2+i(∆ω−ωL)
Ωsin Ωt
2)︃+iΩR
Ωei(ϕ1,2−ϕL)c1(0)sin Ωt
2]︃e−i(∆ω−ωL)t/2
(2.16)
So far, the response of a two-level system, as illustrated in fig. 2.7 (a), was discussed.
However, no mechanism to probe the laser-induced change of the response was brought up.
This step is still missing and needed, as it determines the initial conditions c1(0) and c2(0).
The basic idea for probing the changed eigenstates of the system is to add a third eigenstate
|0⟩
to the simplified system that is not affected by
E
(
t
)as a reference, as illustrated in fig 2.7.
This third eigenstate must be far off-resonant so that any coupling or other effects induced by
E
(
t
)can be neglected. Suppose that third eigenstate
|0⟩
is the ground state (with eigenenergy
ω0
= 0). Since due to the coupling of
|1⟩
and
|2⟩
selection rules permit transitions between
both,
|0⟩
can only couple into one of them. Without limiting the generality, here, the transition
from
|0⟩
to
|1⟩
is permitted. Then with a weak, secondary, probing light field - this time a
10
Compared to [60], here also the phase of the laser field
ϕL
is included, with the substitution
M
˜1,2
=
M1,2e−iϕL=|M1,2|ei(ϕ1,2−ϕL)=|M
˜1,2|eiϕ
˜1,2, the equations have again the same form as given in [60].
20
2.1 Atoms in intense fields
pulse - the
|1⟩
-state will be excited. More specifically, the pulse envelope is assumed to be a
delta function centered around
t
= 0. Consequently, the excitation occurs instantaneously. For
the coupled two levels, the initial conditions are:
c1(0) = C1(0) = C0
1and c2(0) = C2(0) = 0
Hence the analytically obtained solution in eq. 2.15 and 2.16 simplifies to:
c1(t) = C0
1(︃cos Ωt
2−i(∆ω−ωL)
Ωsin Ωt
2)︃ei(∆ω−ωL)t/2
c2(t) = iΩR
Ωei(ϕ1,2−ϕL)C0
1sin Ωt
2e−i(∆ω−ωL)t/2
The optical response of the medium with respect to that secondary pulse can now be
obtained through the induced time-dependent dipole moment p(t)that is given by [60, 59]:
p(t) = ⟨Ψ|r|Ψ⟩=C0(t)∗C1(t)M0,1+ c.c.=c0(t)∗c1(t)M0,1e−iω1t+ c.c.
≈c1(t)M0,1e−iω1t+ c.c.(2.17)
C0
(
t
),
M0,1
, and
c0
(
t
)are the analog parameters for the third off-resonant reference state
to
C1
(
t
),
M1,2
, and
c1
(
t
)that where introduced before. In eq. 2.17, it is assumed that the
secondary light field responsible for the transition is weak and depletion of the ground state
can be neglected, i.e.
c0
(
t
)
≈
1. Inserting
c1
(
t
)as given by eq. 2.1.5 one can write for the
dipole moment:
p(t) = M0,1C0
1
2[︃Ω−∆ω+ωL
Ωe−i(ω1−Ω+∆ω−ωL
2)t+Ω+∆ω−ωL
Ωe−i(ω1+Ω−∆ω+ωL
2)t]︃+ c.c.
(2.18)
Ultimately, the measurable response of the atom, i.e. absorption or emission of light, can
be described in the frequency domain using the response function S(ω)11 [64] given by:
S(ω) = −2 Im [p(ω)Eprobe(ω)] (2.19)
Since the probing pulse
Eprobe
(
t
)is approximated with a delta function with an integral
value of Eprobe,0, its Fourier transform yields Eprobe(ω)a constant:
Eprobe(t) = Eprobe,0δ(t)⇔ Eprobe(ω) = Eprobe,0
√2π
Obtaining the dipole moment’s Fourier transform is lengthy but due to the time-dependence
confined to the exponential functions straight forward from eq. 2.18:
p(ω) = √2πM0,1C0
1
2
·{︃Ω−∆ω+ωL
Ω[︃δ(︃ω−(ω1−Ω+∆ω−ωL
2))︃+δ(︃ω+ (ω1−Ω+∆ω−ωL
2))︃]︃
+Ω+∆ω−ωL
Ω[︃δ(︃ω−(ω1+Ω−∆ω+ωL
2))︃+δ(︃ω+ (ω1+Ω−∆ω+ωL
2))︃]︃}︃
11positive values of S(ω)relate to absorption of light, negative value to the emission of light.
21
2. Fundamental concepts
1012 1014
dressing laser
intensity (W/cm2)
19
20
21
22
23
24
absorption photon energy (eV)
1 (2p-like)
2 (2s-like)
2+L
(LIS2s + 1)
(a)
0 1 2
dressing laser
photon energy (eV)
20.5
21.0
21.5
22.0
22.5
absorption photon energy (eV)
1 (2p-like)
2 (2s-like)
2+L
(LIS2s + 1)
Autler-Townes
splitting
(b)
0.0
0.5
1.0
1.5
2.0
absorption strength (arb. u.)
Figure 2.8: Absorption lines in a helium-like three-level system in the presence of a cw dressing
laser field depending on the dressing laser’s intensity and photon energy. The color scale indicates
the absorption strength in arb. units.
ω1
and
ω2
are set to 21
.
22
eV
and 20
.
55
eV
to mimic the
2p and 2s states in atomic helium, respectively. This is why they are dubbed 2p-like and 2s-like.
Accordingly,
M1,2
is set to 2
.
75
a.u.
[54]. (a) AC Stark effect for a dressing laser photon energy
of 1
.
55
eV
(800
nm
): with increasing dressing laser intensity, the stronger the
ω1
-like (2p-like)
resonance shifts to lower photon energies. Its absorption strength decreases. In parallel, another
absorption line builds up strength at ca. one photon energy above the
ω2
state. This resonance,
also known as light-induced state (LIS), shifts to higher photon energies for increasing dressing
laser intensity. (b) The position and strength of the absorption lines depend on the dressing laser
photon energy. In resonance (see arrow), Autler-Townes splitting occurs. The
ω1
-like state is
evenly split in terms of shift and absorption strength. Away from that, the shift is called the AC
Stark effect, and absorption lines can be described to be
ω1
-like and (
ω2
+
ωL
)-like. The green line
indicates the fixed value of the respective parameter used with the other plot.
Under the assumption of a weak probing pulse,
C0
1
is imaginary [54]. Consequently, the
response function S(ω)is given by:
S(ω) = Eprobe,0Im(C0
1)
2Re(M0,1)
·{︃Ω−∆ω+ωL
Ω[︃δ(︃ω−(ω1−Ω+∆ω−ωL
2))︃+δ(︃ω+ (ω1−Ω+∆ω−ωL
2))︃]︃
+Ω+∆ω−ωL
Ω[︃δ(︃ω−(ω1+Ω−∆ω+ωL
2))︃+δ(︃ω+ (ω1+Ω−∆ω+ωL
2))︃]︃}︃
As expected in rotating wave approximation - the response function is independent of
the dressing laser field’s phase
ϕL
and thus also independent of the time delay between laser
dressing and probing field. Note, a delay dependency arises for limited pulse duration due
to the slowly varying pulse envelope (see also the later following discussion of numerical
results). The scaling of the absorption strength and resonance positions depending dressing
laser properties can be observed in fig. 2.8. The parameters used to plot the result in fig. 2.8
were chosen to mimic the coupling of 2s and 2p states in a helium atom.
The dependence on the dressing laser’s intensity in fig. 2.8 (a) reveals that in a helium-like
system, with a 800
nm
dressing laser, the 2p-like (
ω1
-like) state experiences an AC Stark shift
22
2.1 Atoms in intense fields
to lower photon energies. As expected, the higher the intensity, the stronger the AC Stark
shift, but the weaker the resonance’s response, i.e. absorption. It further shows that together
with the AC Stark shift, a second resonance arises one photon energy above the 2s-like (
ω2
-like)
state. Such a state is known as a light-induced state (LIS). It experiences an AC Stark shift in
the positive direction.
The scaling of the system’s response with the dressing laser’s photon energy in fig. 2.8
(b) reveals, that the direction of the AC stark shift can switch. Between both directions of
the AC Stark shift, in resonant coupling, i.e. dressing laser photon energy equals spacing of
2s- and 2p-states, a distinction between LIS-like ((
ω2
+
ωL
)-like) and 2p-like (
ω1
-like) is not
possible. The 2p-state is split equally. This specific case is known as Autler-Townes splitting.
Response on sub-cycle timescales
Up to this point in the derivation of the optical response of a laser dressed atom, any sub-cycle
dynamics were neglected because of the used rotating wave approximation. As a result, the
AC Stark effect, Autler-Townes Splitting, and a LIS were found. In this paragraph, by instead
applying a weak dressing laser field (i.e., low intensity), sub-cycle effects can be investigated
analytically. At the same time, limiting the dressing field to low intensities leads effectively to
a neglection of the AC Stark effect.
Already in the continuum, sub-cycle effects could be derived from second-order perturbation
theory in the context of side-bands and the RABITT technique (see section 2.1.4). The same
approach can be transferred to bound states.
H
ˆ1
as given in matrix form in eq. 2.12 is treated
as a perturbation to the time-independent Hamiltonian
H
ˆ0
(see eq. 2.1.5). Analog to the
derivation of the AC stark effect, for this analytical description, a cw dressing electric field
with amplitude E00, frequency ωLand phase ϕLis assumed:
E(t) = E00 cos(ωLt+ϕL) = E00
2(︂ei(ωLt+ϕL)+e−i(ωLt+ϕL))︂.(2.20)
Equations for zeroth to second order time-dependent perturbation theory in a two level
system are e.g. given in [65] (eq. 9.15 - 9.18). The initial conditions
c1
(0) = 1 and
c2
(0) = 0
are assumed. For the presented case the second order is given as:
c1(t) = 1 −∫︂t
0
(−M1,2E(t′))ei∆ωt′(︄∫︂t′
0
(−M∗
1,2E(t′′))e−i∆ωt′′ dt′′)︄dt′
c2(t) = −i∫︂t
0
(−M∗
1,2E(t′))e−i∆ωt′dt′
Except for the above mentioned approximations, the derivation in this section follows the
same structure as for the AC Stark effect. Consequently, also here the analytical solution
for
c1
(
t
)is of particular interest. Together with the dressing laser field in eq. 2.20 it can be
written as:
c1(t) = 1 −|M1,2|2E2
00
4
·∫︁t
0(︂ei(ωLt′+ϕL)+e−i(ωLt′+ϕL))︂ei∆ωt′
c1,inner(t′)
⏟ ⏞⏞ ⏟
[︄∫︂t′
0(︂ei(ωLt′′ +ϕL)+e−i(ωLt′′ +ϕL))︂e−i∆ωt′′ dt′′]︄dt′.
23
2. Fundamental concepts
First the inner integral
c1,inner
(
t′
)is to be solved. Therefore the following notation is
introduced: ω+=ωL+ ∆ωand ω−=ωL−∆ω:
c1,inner(t′) = ∫︂t′
0[︂ei(ω−t′′ +ϕL)+e−i(ω+t′′ +ϕL)]︂dt′′
=ei(ω−t′+ϕL)
iω−
+e−i(ω+t′+ϕL)
−iω+−eiϕL
iω−−e−iϕL
−iω+
.
Now, the outer integral can be solved:
c1(t) = 1 −|M1,2|2E2
00
4
{︄it −(ω+)−1eiω+t−(2ωL)−1e−i(2ωLt+2ϕL)+ (ω−)−1e−i(ω−t+2ϕL)
ω+
+it + (ω−)−1e−iω−t+ (2ωL)−1ei(2ωLt+2ϕL)−(ω+)−1ei(ω+t+2ϕL)
−ω−
−(ω+)−1+ ((ω−)−1−(2ωL)−1)e−i(2ϕL)
ω+
−−(ω−)−1+ ((2ωL)−1−(ω+)−1)ei(2ϕL)
−ω−}︄
According to eq. 2.17, the dipole moment is then given by12:
p(t) = C0
1M0,1e−iω1t−C0
1M0,1|M1,2|2E2
00
4
·
(︄−(ω+)−1
ω+
+−(ω+)−1ei(2ϕL)
−ω−)︄
⏞ ⏟⏟ ⏞
Cω1−ω+
e−i(ω1−ω+)t+(︄(ω−)−1e−i(2ϕL)
ω+
+(ω−)−1
−ω−)︄
⏞ ⏟⏟ ⏞
Cω1+ω−
e−i(ω1+ω−)t
−(2ωL)−1e−i(2ϕL)
ω+
⏞ ⏟⏟ ⏞
Cω1+2ωL
e−i(ω1+2ωL)t+(2ωL)−1ei(2ϕL)
−ω−
⏞ ⏟⏟ ⏞
Cω1−2ωL
e−i(ω1−2ωL)t+(︃1
ω+
+1
−ω−)︃
⏞ ⏟⏟ ⏞
Cω1,0
ite−iω1t
−(︄(ω+)−1+ ((ω−)−1−(2ωL)−1)e−i(2ϕL)
ω+
+−(ω−)−1+ ((2ωL)−1−(ω+)−1)ei(2ϕL)
−ω−)︄
⏞ ⏟⏟ ⏞
Cω1,1
e−iω1t
+ c.c.
(2.21)
12
Note, as here
c1
(0) was assumed to be 1instead of
C0
1
, this scaling must be countered when computing the
dipole moment
24
2.1 Atoms in intense fields
3 2 1 0 1 2 3
delay (fs)
18
19
20
21
22
23
24
absorption photon energy (eV)
1
2
12L
1+ 2 L
1+ = 2+L
1 + =2L
2p-like
2s-like
LIS2p 2
LIS2p + 2
LIS2s + 1
LIS2s 1
2.08
2.10
2.12
2.14
2.16
2.18
absorption (arb. u.)
0.15
0.10
0.05
0.00
0.05
0.10
0.15
Figure 2.9: Discrete absorption lines in a helium-like three level system in the presence of a cw
dressing laser field depending on the dressing laser’s delay with respect to the probing pulse. The
oscillation at sub-cycle timescales in the color-scale indicates the varying absorption strength in
arb. units by delay, where negative value represent the emission of light. For visualization purposes
two different color maps are used.
ω1
and
ω2
are set to 21
.
22
eV
and 20
.
55
eV
to mimic the 2p and
2s states in atomic helium, respectively. Accordingly,
M1,2
is set to 2
.
75
a.u.
[54]. The dressing
laser’s photon energy and intensity was set to 1.55 eV (800 nm) and 1012 W/cm2, respectively.
As before with the AC Stark effect, the dipole moment must be Fourier transformed to
obtain the absorption lines via optical response S(ω).
p(ω) = C0
1M0,1δ(ω+ω1) + C0∗
1M∗
0,1δ(ω−ω1)−C0
1M0,1|M1,2|2E2
00
4
·[︁Cω1−ω+δ(ω+ω1−ω+) + Cω1+ω−δ(ω+ω1+ω−) + Cω1+2ωLδ(ω+ω1+ 2ωL)
+Cω1−2ωLδ(ω+ω1−2ωL) + Cω1,0δ′(ω−ω1) + Cω1,1δ(ω+ω1)]︁
−C0∗
1M∗
0,1|M1,2|2E2
00
4
·[︂C∗
ω1−ω+δ(ω−ω1+ω+) + C∗
ω1+ω−δ(ω−ω1−ω−) + C∗
ω1+2ωLδ(ω−ω1−2ωL)
+C∗
ω1−2ωLδ(ω−ω1+ 2ωL) + C∗
ω1,0δ′(ω+ω1) + C∗
ω1,1δ(ω−ω1)]︂
(2.22)
In principle, the Fourier transformed dipole moment can then be again inserted into eq.
2.19 and as before, the probe pulse is assumed to be weak, and thus
C0
1
is imaginary. As
before with the rotating wave approximation, inserting the dipole moment to the response
function
S
(
ω
)will change eq. 2.22 only in the form of constant factors and the restriction to
the imaginary part of the equation. Thus, one can understand from eq. 2.22, which frequency
components occur and how their amplitudes scale. In addition, the solution for the response
function
S
(
ω
)is visualized in fig. 2.9, in which a color scale indicates the optical response
depending on (x) the time delay between dressing laser field and probe pulse and (y) the pump
pulse’s frequency (photon energy). Again parameters were chosen to model the excitation into
and coupling of the 2s and 2p states in atomic helium.
25
2. Fundamental concepts
As can be seen from fig. 2.9, but also from eq. 2.22, absorption at five discrete frequencies,
i.e. photon energies, may occur. First, there is the resonance that is also present without a
dressing laser field,
ω1
, i.e., the 2p-like resonance at 21
.
22
eV
. In addition, two absorption lines
spaced each by two times the dressing laser’s frequency from the 2p-like resonance are present
(at 24
.
32
eV
and 18
.
12
eV
), as well as two absorption lines (at 22
.
1
eV
and 19
eV
) spaced each
by the laser the dressing laser’s frequency from the
ω2
or 2s-like state into which selection
rules forbid direct transitions. These four additional absorption lines are known under the
term light-induced states (LISs). The LIS located at one photon energy above the 2s-like state
is particularly strong, which is why fig. 2.9 uses an secondary color scale for its visualization.
The time delay (
td
) dependence between dressing field and probe pulse is also visualized
in fig. 2.9 and becomes apparent for eq. 2.21 and 2.22 when substituting
ϕL
=
ωLtd
. As
evident from the pre-factors containing the
e2ϕL
terms (see also substitution in eq. 2.21), the
absorption observed with the probing pulse oscillates at twice the dressing laser’s frequency.
In fact, fig. 2.9 also shows: all resonance, i.e. the 2p-like and the LIS, exhibit these sub-cycle
oscillations. Thereby, the oscillations of the 2p-like and the 2-laser-photon-LIS are of the same
phase, while in comparison, the 1-laser-photon-LIS oscillate at a phase shifted by half a period.
Moreover, over one cycle in some LIS also negative absorption values occur, which imply the
emission of light.
Opposed to the result under rotating wave approximation, here, where a low dressing laser
field strength is assumed, the positions of the absorption lines do not depend on this field
strength. However, as the pre-factor
E2
00
4
that applies to all oscillation amplitudes in eq. 2.22
indicates, their intensity and the strength of the delay-dependent oscillation do: the stronger
the field, the higher the oscillation’s amplitude.
Numerical solution of the TDSE in comparison to the presented analytical
approaches
So far, an analytical, semi-classical investigation of the response of a three-level system placed
in a cw dressing laser field probed by a weak ultra-short broadband pulse has been carried out.
Under the rotating wave approximation, where effects on timescales below one optical cycle
of the dressing laser are effectively neglected, the AC stark effect / Autler-Townes splitting
was derived. Moreover, one LIS was found. It becomes particularly relevant in high dressing
laser intensities. Opposed to that, when assuming low dressing laser intensities, second-order
perturbation theory is applicable. It includes effects on sub-cycle timescales, providing further
insights beyond the rotating wave approximation. In result, multiple LISs - in fact 4 of them -
are predicted. Each of them and also the 2p-like state show a delay-dependent oscillation in
the optical response to the probing pulse.
Now, two open points are to be resolved. First, how does the system behave in a realistic
scenario, that involves dressing laser pulse instead of cw field. A duration of 15
fs
(FWHM)
is chosen as such has been published in [54]. Second, how do the discussed approximations
compare to the numerical solution of the TDSE, which may resolve effects on sub-cycle
timescales at high dressing laser intensities.
The delay-dependent numerical solution in the rotating wave approximation is depicted
in fig. 2.10 (a). At the central time delays, the analytically predicted LIS and AC Stark
26
2.1 Atoms in intense fields
Figure 2.10: Time delay resolved response (
S
(
ω
)) of a three-level system for a dressing laser
pulse duration of 15
fs
. (a) result based on the rotating wave approximation used above to derive
the AC Stark effect. The approximation effectively neglects any sub-cycle effects. (b) numerical
result obtained from solving the full TDSE that in addition to the result from (a) shows sub-cycle
oscillations similar to such obtained analytically from second order perturbation theory. (reprinted
from [54], the order of the plots was switched and the x-axis label was moved up from (c) and (d),
which are not shown)
shifted resonance are present: on the top is a single LIS and at the bottom a slightly shifted
2p-like resonance. Towards more positive and more negative time-delays, the LIS is shifted to
slightly lower frequencies before it vanishes. The 2p-like state mirrors the behavior by shifting
back to slightly larger frequencies outside the center part of the plot in fig. 2.10 (a). The
delay-dependence arises from the dressing laser’s pulse envelope, which was assumed constant
in the presented analytical treatment. More specifically, the bending of the absorption lines is
due to the strength of the AC Stark shift being dictated by the dressing laser field envelope’s
intensity when the system is probed. For a time delay other than zero, this intensity decreases
and as shown in fig 2.8 (a), the shift of the LIS and 2p-like absorption line is lower. Besides
the LIS and AC Stark shift also negative absorption (emission) at negative time delays around
the 2p-like absorption line can be observed. It is due to the dressing laser acting upon the
dipole moment already excited by the intended probe pulse. In that context, the underlying
effect is known as perturbed free induction decay (PFID) [54].
Now, looking at the complete numerical solution of the TDSE, as can be seen in fig.
2.10 (b), the main features yielded by the solution with the rotating wave approximation
are preserved. However, in addition, the LISs and sub-cycle oscillations as predicted by
second-order perturbation theory appear as well. In fact, the relative strength of the LIS and
their oscillations are comparable to the second-order perturbation theory result (compare to fig.
2.9). Moreover, comparing the phases of the sub-cycle oscillations related to the 2p- and 2s-like
associated LIS (
ω1±
2
ωL
and
ω2±ωL
), one sees that like for the second-order perturbation
theory solution the phases are shifted by half a period to each other. Interestingly, a closer
look at fig. 2.10 (b) around a time delay of zero reveals that weak sub-cycle oscillations also
occur between the LISs.
While in this section, two and three level systems were discussed to provide a profound
understanding of the emergence of light-induced states (LISs), the AC Stark shift and perturbed
27
2.2 Helium droplets
Figure 2.11: Phase diagram for bulk helium; Liquid helium is characterized by two different
phases. First, there is the normal liquid phase known as He I and second, the superfluid phase also
known as He II. (reprinted from [69])
2.2 Helium droplets
Helium appears to be an ideal model system to study fundamental physical processes. Its two
electrons populate a single closed shell. Thus, it has a straightforward electronic structure.
This simple structure, among others, aids the theoretical understanding by enabling precise
numerical modeling, e.g., by solving the TDSE, for many scenarios. Its properties in the gas
phase have been extensively studied experimentally and theoretically. Amongst others, optical
effects, such as the already discussed AC Stark shift and light-induced states (see section 2.1.5)
have been investigated at time scales down to the attosecond domain (e.g., [66, 67, 68, 54]).
Other studies relate to the investigation of helium’s properties in its liquid phase, which is
characterized by "weak interaction between [...] very loosely packed [atoms]" [70]. Although
helium is a very simple atom, it exhibits very peculiarly unique properties in bulk. As depicted
in the phase diagram in fig. 2.11, in ambient pressures helium liquefies at only 4
.
2K [69].
Moreover, it never reaches a solid phase even at 0 K [69]. Instead, only under high pressures,
solid helium is possible [69]. At temperatures below 2
.
17K (at ambient pressures), it even
reaches a special phase, namely, superfluidity, which is also referred to as He II [69, 71]. Analog
to superconductivity for electrons, in superfluidity, atoms experience no friction. In other
words, the viscosity of the liquid is zero [72], leading to a number of fascinating phenomena
such as the formation of quantized vortices [73].
Optically, superfluid helium is difficult to access in experimental investigations: it must be
excited with light at photon energies around 20
eV
[70, 74] that requires a vacuum environment
for propagation, but it must also be kept at extremely low temperatures in some form of
cryostat. A lack of suitable window materials (i.e., materials transparent around 20eV) with
sufficient mechanical stability [75] prevents easy access. Instead, superfluid helium droplets can
be used. Those can be produced free-flying in a vacuum, where they reach quickly temperatures
29
2. Fundamental concepts
down to 0
.
38K by evaporative cooling [76]
13
. Consequently, no additional cryostat and no
windows are required.
14
Moreover, due to their low temperature, superfluidity, and their
transparency in the spectral range from NIR to the vacuum ultra-violet (VUV), they are widely
used as a matrix to embed molecules, and other samples, e.g., for spectroscopy purposes [77].
Helium droplets can be produced at various sizes ranging from just a few nanometers [78]
over the micrometer scale [79] into the millimeter regime [80, 81]. Droplets have a thin surface
region of ca. 6
−
8Å, where their density transitions from 10 to 90 % of bulk liquid helium
[82]. For large droplets, i.e., with diameters larger than 10
nm
, a sharp surface boundary and
a uniform density equal to bulk liquid helium may be assumed [83]. Furthermore, assuming a
spherical droplet, the relation between droplet radius
rdroplet
and the number of atoms in a
droplet NHe is given by:
rdroplet = 2.22N
1
3
He (in Å) [84,77].
The droplets that are relevant to this work have a radius on the order of 100
nm
and therefore
will consist of around 108atoms. Hence they will be referred to as helium nano-droplets.
While many investigations use helium droplets as an environment to embed the actual
samples of interest [77], despite the atom’s comparably simple structure, already pure droplets
show complex behavior with many remaining open questions [74, 83].
2.2.1 Electronic structure of helium droplets
The electronic structure of bulk helium and helium nano-droplets was investigated experi-
mentally by a number of studies (e.g., [70], [75], [78] or [85]). Since close agreement with the
absorption of liquid helium is expected for sufficiently large droplets (
>∼
10
4
atoms) [75], the
electronic structure of the droplet and liquid are discussed together.
Early on, it was found for liquid helium that the electronic structure is very similar to the
helium atom. At an energy of ca. 21
.
4eV a transition similar to the one from the 1s state to the
2p state in atomic helium could be observed [70]. Compared to the atomic transition, however,
this one is strongly broadened and shifted to higher energies [70]. Fig. 2.12 shows a compilation
of different spectroscopic measurements in helium droplets (wit approximately 10
4
atoms).
Particularly, the figure points out transitions in the droplets that could be associated to an
atomic counter part (indicated by the post-fix "-like"). More specifically, observed resonances
could be associated in fluorescence measurements to the atomic 2s and 2p transitions [75] and
with VIS/NIR luminescence yields also to the 3s, 3p, and 3d transitions originating from the
1s ground state [85]. Those kind of measurements consistently reproduce shifted resonances
(see also [75], [85] or [78]). In fact, the observed shift is different for each feature that could be
associated to an atomic-like transition.
Compared to atomic helium, the droplets’ transition energies from the ground state are
shifted to larger energies (i.e., blue-shifted). Besides intrinsic properties (e.g., the molecular
interaction potential), the strength of the shift is found to depend on the density [85]. In
particular, the shift of 3p-like resonance approximately scales with the cube root of the density
13
0
.
38K are achieved in droplets made up of the isotope
4He
. Droplets made up of the much rarer isotope
3He reach even temperatures down to 0.15K [76]
14A comparison between bath cryostats and helium droplets can also be found in [76].
30
2.2 Helium droplets
20 21 22 23 24 25
photon energy (eV)
0.0
0.5
1.0
1.5
Intensity (arb. u.)
2s 3s
3d
2p 3p
4p 5p
2s-like
2p-like
3p-like
3d-like
3s-like
fluorescence yield
photoabsorption
luminescence
Figure 2.12: Compilation of three different spectroscopic measurements on helium nano-droplets
with ca. 10
4
atoms in the XUV regime indicating possible transitions from the 1s ground state.
Specifically, the fluorescence yield (blue), the XUV photoabsorption (orange) and luminescence in
the NIR/VIS spectral regime (green) are plotted. Each individual trace is scaled arbitrarily along
the y-axis. The x-axis indicates the incident XUV photon energy. Some atomic transitions from
the ground state are indicated by vertical dotted lines for dipole-allowed and dashed-dotted lines
for atomic dipole-forbidden transitions. The identified analog transitions in the droplet, indicated
by arrows and the post-fix "-like", are shifted towards higher photon energies. The identification
for the principal quantum number
n
= 2 is shown according to [75] and for
n
= 3 according to the
identification in [85]. (Data extracted from [75] for fluorescence and [85] for XUV photoabsorption
& NIR/VIS luminescence)
[85]. According to von Haeften et al. [85], this is consistent with the behavior observed in
dense helium gas. The shift is attributed to the excited Helium atom being located amongst
neighboring atoms in the ground state to whom it experiences repulsion [85].
In fact, earlier work concerning the theoretical interpretation of experimental data regarding
bulk helium from Surko et al. [70] and helium bubbles in aluminum from Rife et al. [86]
considered two mechanisms that may cause a shift of the resonances [87]. First, there is a
red-shift occurring due to the influence of the bulk polarization of material on the dipole
induced by the excitation in a single atom [87]. It means that the field locally acting on a
single atom is different from the external field the material is exposed to (see also chapter
5.4 where this is included with the theoretical model applied in this work). Such effect is
generally known for dielectric materials and in linear materials can be accounted for with the
Clausius-Mossotti relation (see, e.g., [88]). Second, so-called Pauli-repulsion leading to a blue
shift may play a role [86, 87]. Opposed to the dipole interaction that occurs at mesoscopic
distances, Pauli-repulsion takes only effect on a range of Ångströms to a few nanometers [87],
thus the effect applies to orbitals that usually would overlap with those from their neighboring
atoms. Hence, the ground state remains mostly unaffected [87]. A very peculiar example for
Pauli repulsion in liquid helium is the phenomenon of cavity formation around free electrons
[87, 89]. This is because liquid helium poses for electrons a repulsive potential barrier of
approximately 1
eV
[90]. Similarly, excited helium atoms within liquid helium also form a
bubble around themselves [91, 92, 93]. A concept to model the Pauli repulsion in liquid helium
using a perturbative approach can be found in [87]. Following the idea of Pauli repulsion,
31
2. Fundamental concepts
furthermore, the position of the shifted resonances could be approximately reproduced using a
statistical approach in combination with the TDSE for a single atom, where a 1
eV
potential
step was applied to the otherwise atomic potential at a statistically varied radius from the
atomic core [94].
Because the features in the fluorescence, VUV photoabsorption, and NIR/VIS luminescence
spectra can be associated with a counterpart seen in atomic helium (i.e., unique, well-defined
quantum numbers), following the argumentation by von Haeften et al., the electronic excitations
may be described as perturbed atomic-like [85]. Moreover, such a description is further
motivated by the perturbative approaches taken by Lucas et al. [87] and Kornilov et al. [94]
to model the Pauli-repulsion leading to the energetic shift of the resonances compared to the
atomic case.
2.2.2 Coherent diffractive imaging of helium droplets
With the upcoming availability of intense, coherent, short-wavelength light sources in the past
two decades [95, 96, 97, 98, 45], the investigation of individual freestanding nano-scale objects,
such as single viruses, aerosols, atomic clusters, or superfluid helium nanodroplets through
coherent diffractive imaging (CDI) became feasible [21, 99, 100, 101]. CDI is a lensless imaging
technique that, in combination with intense light sources in the XUV and x-ray regime, among
others, can provide insight into the structural and optical properties of single particles at high
resolution. The fragility of Helium nanodroplets restricted studies of their structure to indirect
measurements until CDI allowed to gain unique insights into the distorted shapes, and even
quantum vortices within the short-lived objects in the pioneering work by the Vilesov group
[101]. Since then, a growing community has studied the structure and dynamics of helium
nanodroplets and immersed nanostructures [102, 103, 34, 104, 105, 106].
In the following, after a short general introduction to the framework of CDI, Mie scattering
- the analytical solution for light scattered from spheres - will be described. Also, the issue of
coherent but not monochromatic light sources will be brought up.
Basic principles
When light, i.e., electromagnetic radiation, interacts with matter, such as a helium droplet, the
electromagnetic field acts upon charges in the matter. The electrons will be displaced relative
to their ionic cores by the field and forced into an oscillatory motion. A rapidly varying dipole
moment is induced. Besides other dissipative processes
15
, the accelerating charges re-radiate a
fraction of the incident light omnidirectional. The emitted light is referred to as scattered light
[107]. In conventional optical imaging, a lens is used to collect the scattered light to produce a
projection of the original object from it.
For coherent incident light, the coherence is preserved in the scattered light
16
[107]. The
scattering process affects the relative phase between different pathways the light may take. On
a plane behind the object, by interference, an intensity distribution - the diffraction pattern - is
formed [107], which is recorded for CDI. The specific pattern encodes the structural properties
15The absorption of a photon can be seen as a dissipative process in this oscillator picture.
16
Note: depending on the material, shortly after the excitation, dephasing mechanisms take effect due to
which the coherence of the oscillating dipoles gets lost [37].
32
2.2 Helium droplets
Figure 2.13: Illustration of the dependence of the projection plane (blue bar) encoded in the
diffraction image (red bar) (a) at small scattering angles (small
q
) the projection plane may be
approximated to be parallel to the imaging plane. (b) when also wide diffraction angles are
recorded, the former approximation cannot be made. Now many different projection planes are
encoded with the diffraction pattern, i.e. three dimensional information can be retrieved from a
single image. (reprinted from [100])
of the sample (e.g., size, shape, orientation), but also its specific dipole response to the incident
light, i.e., its optical properties [107]. With recording the intensity distribution, the phase
information of the light in the diffraction pattern’s plane is lost [107]. Computational methods
are then used to retrieve this phase information (see, e.g., [108, 109]).
In order to image single individual nano-sized objects in a single shot, low diffraction
efficiencies demand light at high intensities [110]. In fact, the required intensities are so high
that they destroy the particle. Facilitated by ultrashort pulses, the imaging process can be
completed faster than the destruction process, which is referred to as the "diffraction before
destruction" principle [111, 96]. It could be first experimentally demonstrated by Chapman et
al. at the FLASH-facility [110].
The scattering angle
θscat
describes the angle between the incident and scattered light.
Depending on the size and distance of the detector from the sample, different maximum
scattering angles are captured. Commonly, it is differentiated between techniques capturing
wide and small diffraction angles [100]. Generally, the wider the angle, the larger the fraction
of scattered light that is captured. However, for small diffraction angles, mathematically aided
by the small-angle approximation, effectively, only a single 2d projection is encoded in the
pattern [100]. As illustrated by fig. 2.13 (a) for small angles the projection plane’s scattering
angle-dependent rotation can be neglected. Established algorithms can be used to reconstruct
the phase distribution of the captured intensity distribution to finally obtain a two-dimensional
image of the captured sample [108]. On the other hand, wide-angle diffraction patterns contain
information beyond a single projection plane, i.e., three-dimensional information of the sample
[100]. As shown in fig. 2.13 (b) the angle between the plane normal to the incident light and
the scattering angle-dependent projection plane cannot be neglected. As demonstrated by
Barke et al., it allows the identification of three-dimensional particle shapes in a single image
[100].
33
2. Fundamental concepts
Mie theory
Previous experiments with helium nano-droplets found that they can take various shapes [101,
105]. There are spherical, elliptical, prolate (pill-shaped), and dumbbell-like structures [101,
105, 34]. However, those experiments have also shown that, at all expansion conditions tested,
the majority of diffraction patterns (
≈
93% in [105],
≈
76%
17
in [34] and
≈
60%
18
in [101])
stems from spherical droplets. Since in this work the focus was on the light-matter interaction
and the use of helium nano-droplets as ideal model systems, known expansion conditions with
mostly spherical droplets were chosen here and the analysis refers to those.
The diffraction of light from spherical droplets is described by Mie theory, which solves
Maxwell’s equations analytically for propagation through a sphere [107]. While Mie theory
assumes spherical symmetry, it also provides a first-order description for effects in non-spherical
particles [107]. The original Mie theory makes the following assumptions:
1.
The scattering occurs by a particle that is spherical with sharp boundaries to the
surrounding medium. The particle must be homogeneous, isotropic, have a linear optical
response, and be non-magnetic [112]. Furthermore, it is assumed to be static during the
scattering process [107].
2.
The particle must be placed in a homogeneous, dielectric or vacuum environment [112].
3.
The incident light is a plane monochromatic electromagnetic wave with a constant
intensity across the particle [112].
Various extensions to the original theory have been developed to accommodate cases that
only partially fulfill the stated assumptions, such as non-plane incident waves (e.g., incident
Gaussian beams), a sphere with non-sharp boundaries, slightly distorted spheres, coated
spheres, or non-monochromatic incident light (see, e.g., ch. 2 in [112]). In this work, the
original Mie theory is applied with the only extension to polychromatic light by superimposing
the result for different contributing wavelengths.
With the original Mie theory, in a vacuum environment, from the radius of the the sphere
rsphere
relative to the incident’s light wavelength
λinc
and the complex refractive index of the
sphere
nsphere
, dependent on the direction of the scattered light rthe scattered fraction of the
incident light
S(︂rsphere
λinc , nsphere,r)︂
can be computed. The observed intensity on a spherical
surface at radius robserve from the spherical particle is then given by:
Iscat =λ2
inc
(2π)2r2
observe S(︃rsphere
λinc
, nsphere,r)︃
2
Iinc [113]
The intensity of the observed pattern
Iscat
scales linearly with the incident intensity
Iinc
.
Since the total amount of light on the observation surface needs to be preserved, and the
surface’s area increases with
r2
observe
, the local intensity is corrected by the inverse. The
17
When including elliptical droplets, in [34] both droplet shapes together make up approximately 97 % of the
recorded data set. A fraction of the elliptical droplets may be sufficiently well approximated by a spherical
shape.
18
When including elliptical shapes, in [101] the combination of both droplet shape types makes up
approximately 99 % of the data set. A fraction of the elliptical droplets may be sufficiently well approximated
by a spherical shape.
34
2.2 Helium droplets
analytical expression for
S(︂rsphere
λinc , nsphere,r)︂
is lengthy, but straightforward. An instructive
derivation can, e.g., be found in [113].
To illustrate the results computed from Mie theory, fig. 2.14 (a.1) and (a.2) show the
diffraction patterns produced from a helium nano-droplet with a radius of 300
nm
for the
wavelengths 61
.
5
nm
and 53
.
3
nm
which correspond to the 13
th
and 15
th
harmonic of a 800
nm
NIR laser. In both cases, the intensity of incident light is equal. However, the 13
th
harmonic
order is scattered at much higher efficiency, and thus the pattern is much brighter.
Typically, for a more straightforward analysis of the diffraction patterns, CDI experiments
rely on monochromatic light sources. However, HHG produces radiation consisting of multiple
harmonic orders, i.e., multiple colors. Since monochromatizing the light would alter the
temporal structure of the incident light, and drastically reduce the pulse energy, it is impractical
for this work. The incident light must be polychromatic. Polychromatic light can be integrated
with the Mie theory by computing for each wavelength an individual pattern (like, e.g., those
shown in fig. 2.14 (a.1) and (a.2) or (b.2) and (b.3)) and superimposing those incoherently,
weighted by spectral intensity, on the detector [114]. This superposition approach has been
successfully implemented in the past to interpret and simulate multi-color diffraction patterns
[34, 35].
Fig. 2.14 (a.3) shows the superposition of fig. 2.14 (a.1) and (a.2). In the illustrated case
(a) the ring pattern is generally dominated by the 13
th
harmonic due to the high scattering
efficiency compared to harmonic order 15. Still, the radial profiles in fig. 2.14 (a.4) indicate
a decrease in the contrast of the rings in the superimposed harmonic orders. In other cases,
where no dominance of one pattern is obvious, e.g., case (b) in fig. 2.14 that assumes a
fundamental wavelength of 826
nm
, a stronger impact is observed. Both the radial profiles in
fig. 2.14 (b.1) and the superimposed patterns in fig. 2.14 (b.4) show clearly a reduced contrast
and deviate strongly from each monochromatic pattern in fig. 2.14 (b.2) and (b.3).
With the development of light sources that provide isolated pulses with a duration in the
attosecond domain, not only multiple discrete colors need to be supported, but broadband
spectra. Such pulses would yield a significant blurring of the produced diffraction patterns [115].
In that scope in [114], algorithms to monochromatize diffraction patterns before successful
reconstruction could be demonstrated.
35
2. Fundamental concepts
45 0 45
scattering angle (deg)
45
0
45
scattering angle (deg)
(a.1) HO13
45 0 45
scattering angle (deg)
45
0
45
(a.2) HO15
45 0 45
scattering angle (deg)
45
0
45
(a.3) HO13 + HO15
0 20 40
scattering angle (deg)
10 3
10 1
101
103
intensity (arb. u.)
(a.4) radial profiles
HO13 + HO15
HO13
HO15
0 20 40
scattering angle (deg)
10 2
100
102
(b.1) radial profiles
HO13 + HO15
HO13
HO15
45 0 45
scattering angle (deg)
45
0
45
scattering angle (deg)
(b.2) HO13
45 0 45
scattering angle (deg)
45
0
45
(b.3) HO15
45 0 45
scattering angle (deg)
45
0
45
(b.4) HO13 + HO15
Figure 2.14: Diffraction patterns for spherical helium droplets (
rdroplet
= 300
nm
) illustrate
the influence of two overlapping colors. Specifically the two overlapping harmonic orders 13 and
15 are simulated for two cases (a) and (b) that assume fundamental wavelengths: (a, light blue
background) 800
nm
and (b, light orange background) 826
nm
. For each case, patterns are shown
for the isolated harmonic orders in (a.1)/(a.2) & (b.2)/(b.3) and for the superimposed harmonic
orders in (a.3) & (b.4). The patterns are depicted with a false-color map on a logarithmic scale. In
addition, for a direct comparison between the isolated and superimposed harmonic orders, their
radial profiles are also shown in (a.4) and (b.1). (For the diffraction patterns and radial profiles, a
radial symmetry was assumed, which is valid for scattering angles away from 90
◦
[113]. Here the
Mie theory computations were made for scattering angles ranging from 0◦to 45◦)
36
3
Development & characterization of an
HHG beamline for non-linear XUV
optics and coherent diffractive imaging
Performing single shot, isolated nanoparticle CDI experiments requires an intense light source
in the approached short wavelength regime. The generation of intense XUV light is challenging.
The highest pulse energies are achieved on large-scale facilities, specifically XUV-FELs [116]
such as FERMI [117, 97] or FLASH [118, 95]. Despite generating intense pulses in daily
operation, pushing the duration of the pulses to the attosecond regime and achieving high
temporal control of the pulses at FELs was demonstrated only very recently [30]. HHG based
XUV light sources intrinsically generate attosecond pulse trains (APTs) and exhibit exquisite
temporal control by being phase-locked to the HHG driving laser pulse. Nevertheless, achieving
high intensities based on HHG is difficult; only few light sources are capable of generating
APTs or isolated attosecond pulses (IAPs) with high pulse energies [119, 120, 32, 121].
This chapter guides through the development of the HHG setup which has been optimized
for performing the CDI experiments presented in this thesis and non-linear experiments
1
in
the XUV regime. Moreover, tools for characterization and in-situ diagnostics of the generated
pulses are introduced. This includes an experiment employing the RABITT-technique to
characterize the temporal structure of the generated pulses.
An overview over the components of the beamline setup is shown in fig. 3.1. The NIR
pulse is generated by an in-house developed laser system (fig. 3.1, yellow) and after final
compression in vacuum handed to the HHG setup (fig. 3.1, blue). The HHG setup includes a
straight beamline of 18 m length: from focusing the NIR over generation of HHG to diagnostic
(fig. 3.1, purple) and experimental (fig. 3.1, pink) setups.
The structure of this chapter follows the structure of the beamline setup: Starting with an
introduction to the NIR laser source in section 3.1, and followed by a description of the XUV
1
The HHG beamline was developed and setup in a collaborative effort with Bernd Schütte and colleagues,
who are pursuing research on non-linear XUV effects at this HHG setup.
37
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
OPA pump
pulse generation
front-end seed signal
preparation
& OPA
pulse
com-
pres-
sion
NIR
focusing
HHG HeNe &
pointing
beamline
diagnostics
experi-
ment
Figure 3.1: Schematic overview indicating the essential components of the beamline setup; For
orientation, the gray boxes represent the optical tables on which the setup components are located.
Yellow components relate to the HHG-driving NIR laser, blue are sections directly related to the
generation of high harmonics, purple the permanently installed diagnostic section, and pink the
space allocated for experiments that can be attached.
700 800 900 1000
wavelength (nm)
0.0
0.2
0.4
0.6
0.8
1.0
norm. intensity (arb. u.)
FWHM
126 nm
(a)
40 20 0 20 40
time (fs)
0.0
0.2
0.4
0.6
0.8
1.0
norm. intensity (arb. u.)
= 8.3 fs
(b)
Figure 3.2: Laser characteristics: (a) broadband spectrum of the compressed pulses produced by
the beamline’s OPCPA setup (b) temporal pulse shape retrieved using the SHG-FROG technique.
The pulse is well approximated by a Gaussian function. It has a FWHM duration of 8.3fs. Both
(a) and (b) use the data from fig. 6 in [122].
generation, in particular discussing layouts tailored to the anticipated experiments, in section
3.2. Finally the characterization of the generated pulses utilizing the installed diagnostics or
for temporal characterization an attached experiment is presented in section 3.3.
3.1 NIR laser source
The HHG beamline is driven by an MBI developed terawatt laser system based on optical
parametric chirped-pulse amplification (OPCPA). This system has recently been described in
detail [122]. In the following, primarily based on that publication, the basic working principle
is introduced, and an overview focusing on the most relevant features for HHG is given.
The system produces pulses with a duration below 9
fs
FWHM with pulse energies of up to
42
mJ
and consequently a peak power of 4
.
4
TW
at a repetition rate of 100
Hz
. The ultra-short
pulses are supported by the spectrum shown in fig. 3.2 (a) with a FWHM spectral width of
38
3.1 NIR laser source
Figure 3.3: Operation principle of an OPCPA-scheme: a seed pulse is stretched by inducing a
chirp (exemplary a negative chirp is depicted), then by amplifying the pulse by optical parametric
amplification (OPA) in a non-linear crystal and finally, by reversing the chirp to compress the
stretched, high energy pulse into an ultra-short, high-power pulse. Image reprinted from [45, fig. 1
(b)].
126
nm
and a central wavelength of 810
nm2
. Active carrier-envelope phase (CEP) stabilization
is possible. Thereby an r.m.s. fluctuation of 350
mrad
is observed. The shot-to-shot measured
pulse energy acquired over the duration of approximately one hour varies by 0
.
8% (r.m.s.).
[122]
In an OPCPA-based scheme, illustrated in fig. 3.3, an ultra-short seed pulse is amplified
by combining OPA with chirped-pulse amplification (CPA) [7]. Nowadays, CPA is widely
used with high peak-power laser systems producing ultra-short pulses [123]. In this scheme,
an initial ultra-short but low energy pulse is stretched by introducing a temporal chirp, i.e.,
exploiting the spectral width of the pulse and gradually delay either its shorter (negative
chirp) or longer (positive chirp) wavelengths. Due to the longer pulse duration, the optical
elements used for amplification are exposed to much lower intensities. After the amplification,
an inverted chirp is applied to the pulse, and thereby the pulse is compressed, resulting in an
ultra-short high peak-power laser pulse [124]. One method to amplify the stretched pulse is
OPA, where a strong pump pulse together with a weak seeding signal pulse is supplied into a
non-linear medium. Due to second-order polarization, the pump pulse transfers energy into
the signal pulse, which is thereby amplified. Following the principle of difference frequency
generation, a third pulse, the so-called idler, is generated. The sum of generated signal and idler
photon energy equals the pump photon energy, i.e., energy is conserved. In addition to energy
conservation, also momentum conservation needs to be satisfied. Accordingly, the incident
beam needs to be aligned at a specific angle relative to the optical axis of the crystal. Moreover,
the support of broadband spectra is significantly improved by a non-collinear incidence of
the pump and signal beam. Introducing an angle between those beams allows reaching a
phase-matching (momentum conservation) condition that is satisfied independent of the signal
and idler wavelengths. [125]
For the beamline’s OPCPA-scheme that is illustrated in fig. 3.4, a Yb:KGW-based oscillator
provides a seed pulse to the pump and signal pulse generation. Using a combination of regional
amplifier, white light generation, and OPA, a seeding signal pulse with a duration of (250
±
50)
fs
and pulse energy of 70
µJ
is generated. Its bandwidth reaches from 700
nm
to 1050
nm
. Using
a GRISM setup (a combination of a grating and a prism) combined with an acousto-optical
2
The central wavelength of 810
nm
refers to the number stated in [122]. However, the spectrum shown in fig.
3.2 (a) that is also published in [122] suggests a central wavelength of 797 nm instead.
39
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
Figure 3.4: Strongly simplified layout of the OPCPA that drives the HHG beamline (for more
details see [122]): The front end provides seeds to the pump and signal pulse generation for OPA.
The seeding signal pulse for OPA is stretched with a GRISM and amplified in three OPA stages
to pulse energies exceeding 55
mJ
. Compression is achieved by transmission through glass and
in-vacuum placed chirp mirrors. (image based on [122, fig. 1], but simplified. Among others for
simplification stabilization and beam conditioning systems that are not discussed in this thesis are
omitted.)
programmable dispersive filter (AOPDF), the pulses are chirped negatively. 2
µJ
pulses with a
duration of ca. 4
ps
are provided to the OPA. Using three
β
-barium borate (BBO) crystals,
that are pumped by 35
mJ
(stage 1), 65
mJ
(stage 2), and 100
mJ
(stage 3) 515
nm
pulses,
before re-compression, signal pulses with a pulse energy of over 55
mJ
are produced. The
uncompressed beam is guided over a deformable mirror to correct the phase-front, of particular
importance for optimizing the HHG process.
3
The pulses are then compressed in a first step
by propagation through a 25
mm
thick N-SF6 glass block followed by a 15
mm
thick fused
silica window into vacuum. In a second step, compression down to below 9
fs
is realized with
eight chirped mirrors. One of the chirped mirror’s leakage is used to provide a signal to
the CEP measurement and stabilization system. The final output pulse has a diameter of
approximately 25
mm
. With a second-harmonic generation (SHG) frequency-resolved optical
gating measurement (FROG), a Gaussian-like pulse shape that is shown in fig. 3.2 (b) with a
duration of 8.3 fs (FWHM) was reconstructed. [122]
3.2 XUV pulse generation
The fundamental concept of HHG has been introduced in chapter 2.1.2. This section discusses
their optimized practical implementation and the realized setup. Especially, an approach
for tailoring the beamline setup for a maximized signal yield in anticipated experiments is
presented. Instead of a standard optimization of the layout for the highest number of XUV
photons, the achievable intensity is of interest. The work presented here has already been
published in parts in [36].
For such optimization, an existing initial setup is used as a starting point. Within this
initial setup, settings like, e.g., pulse energy of the laser or gas density of the HHG medium,
3
The deformable mirror was added after [122] has been published. It was part of the experimental setup
only for data presented in chapter 4. It is not depicted in fig. 3.4
40
3.2 XUV pulse generation
Figure 3.5: Schematic drawing of the essential components relevant for the layout of an HHG
beamline: coming from the right, the incident NIR beam with diameter
D0
is guided over a
focusing mirror with focal length
fNIR
. Afterwards it is steered by a plane mirror - the "last"
NIR mirror - into the beamline. At a distance
dNIR
from this "last" NIR mirror, the focus and
HHG medium is located. Approximately in the focus of the NIR light, XUV light is produced by
HHG. From the HHG medium on, the NIR and the generated XUV propagate together toward
the experimental setup. The NIR is stripped from the XUV light using a metal foil filter. At a
distance
dXUV
behind the NIR focus and XUV source point, a mirror that is part of the attached
experiment refocuses the XUV at a focal distance
fXUV
. In sum, the lengths
dNIR
and
dXUV
make up the total length of the beamline.
are supposed to be tuned to conditions producing a maximal XUV pulse energy. Thus, for
this initial setup, it can be assumed that the conversion efficiency from NIR to XUV light is
maximized. The geometry of the initial setup is then modified while preserving this maximized
conversion efficiency aiming for experiment-specific ideal conditions (e.g., maximized pulse
energy or maximized intensity). Particularly, the conditions for a focused XUV pulse located
in the considered specific experiment is considered.
3.2.1 General considerations for optimizing a HHG beamline
When designing a beamline geometry, one of the fundamental questions is where to place the
HHG medium, such as a gas jet or gas cell. To be more specific, the distance of that medium
from the beginning of the beamline
dNIR
, i.e., the last mirror guiding the NIR into the linear
beamline, needs to be defined. Accordingly, this also defines the distance
dXUV
of the medium
to the end of the beamline. As sketched in fig. 3.5, on a fundamental level, an HHG beamline
geometry is constrained by its total length. It is based on an incident, unfocused NIR pulse
with diameter
D0
and a pulse energy
Epulse,NIR
as delivered from a laser system that is guided
onto focusing optics. For simplicity, a single focusing mirror is considered, from which the
beam is guided onto a last plane mirror directing the beam into the beamline.
4
Thus, the
focal length of that mirror
fNIR
must be at least the distance of the HHG medium to the
last plane mirror:
fNIR >
=
dNIR
. Otherwise,
fNIR
is an independent parameter. After the
HHG medium (in the presented specific case, a cell) XUV and NIR propagate toward the
diagnostics/experiment that typically includes some focusing of the XUV.
For any given experiment that comes with a defined focal length
fXUV
, only the following
free geometrical parameters determine the beamline geometry:
•dNIR, the distance of the gas cell from the last plane mirror
4
In practice, potentially more complex focusing optics are used and/or the NIR might be guided over multiple
mirrors before entering the beamline.
41
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
•fNIR, the NIR’s focal length
In addition, the NIR pulse energy
Epulse,NIR
may be adjusted.
5
As a first step, given an initial
NIR focal length
fNIR
and pulse energy
Epulse,NIR
, the propagation lengths before (
dNIR
)
and after (
dXUV
) the HHG medium are to be minimized. For each distance, two questions
arise:
1.
What determines the minimal distance achievable for given conditions (e.g., beam
dimensions, pulse energy, focal length)?
2.
When tailoring the conditions to the experiment, why and for what outcome may a
longer distance be beneficial?
Answering these two questions, first, the minimal lengths on each end of the beamline are
defined depending on other free parameters. Second, by identifying the effect of extending
each length, for the anticipated experiment, both can be weighted to find the optimal working
point. Therefore, in next two paragraphs, both questions are discussed first for the distance
before the HHG medium
dNIR
and then for the distance from HHG medium to the experiment
dXUV .
The minimal distance of the beginning of the beamline to the HHG medium
is limited by the last mirror’s laser-induced damage threshold (LIDT), i.e., the maximum
laser flux per pulse the mirror can sustain. The minimal distance is achieved by operating the
mirror close to its LIDT. Therefore, it is placed after the focusing mirror as close as possible
to the focus. Under this condition, the distance to the HHG medium
dNIR
scales with NIR’s
focal length fNIR and the NIR’s pulse energy Epulse,NIR6:
dNIR =dNIR(fNIR , Epulse,NIR).
A longer distance
dNIR
is beneficial as it is associated with longer focal length of the NIR
fNIR
and/or higher pulse energies
Epulse,NIR
. The larger the focal length of the NIR
fNIR
,
the larger the produced NIR focal spot
7
. When other parameters, such as the NIR pulse
energy, are scaled accordingly, the conversion efficiency is preserved, and with a higher incident
NIR pulse energy more XUV photons will be generated [45].
The minimal distance behind the HHG medium to the end of the beamline is
also determined by the LIDT of used optical components. In particular, ultra-thin metal foils,
which are used to block the NIR light but transmit the generated XUV light [127], pose a
concern [128].
8
Too much NIR light may induce thermal effects leading to the foil’s destruction
[130, 128].9
5D0
is assumed to be a fixed parameter as a variation of
D0
can be accounted for by appropriately scaling
dNIR and fNIR.
6not necessarily linearly
7
It has been shown that the product of divergence and minimal spot size are invariant for any given beam
[126], a longer focal length implies a smaller divergence and hence a larger minimal spot size.
8
Wang et al. measured the LIDT of a 50
nm
Al foil to be 0
.
045
J/cm2
(for a 50
fs
pulse at 800
nm
) [128].
For comparison, commercially available optimized silver mirrors (as may be used before the HHG medium) can
achieve LIDTs of 0.38 J/cm2(for a 50 fs pulse at 800 nm) [129].
9
The otherwise minimal distance can be further reduced by using reflective optics that absorb a majority of
the NIR light [131].
42
3.2 XUV pulse generation
Nevertheless, a longer propagation distance after the HHG medium increases the XUV
beam size. For the focused XUV pulse in the experiment, it yields a higher intensity in the
interaction region.
10
Opposed to NIR light, XUV mirrors have comparably low reflectivity.
Thus, extending the distance to the focusing mirror using an additional plane mirror behind
the HHG medium is very costly in terms of pulse energy [132, p. 155], and, consequently, not
beneficial for many scenarios.
The above discussion on the distances
dNIR
and
dXUV
, i.e., the placement of the HHG
medium along the fixed-length beamline, associate the scaling of each distance with a different
aim:
•high XUV pulse energy favoring experiments prefer a large dNIR.
•high XUV intensity favoring experiments prefer a large dXUV .
For both cases however, the minimal distance of the respective other parameter is dictated by
damage thresholds.
Influence of the XUV pulse energy scaling on the XUV intensity
The scaling of the XUV pulse energy under preservation of conversion efficiency is described
by Heyl et al. in [45]. It implies, among other factors, that a constant conversion efficiency
may be achieved by preserving the NIR intensity where the HHG takes place11.
Any given focusing optics images the source point into the focal spot. The sizes of both are
linked by the demagnification
12 γ
that is determined to a large extent by the focusing optic
and layout. Given that the beam waists are related by factor
γ
(straight-forward relationship:
w02
=
w01/γ
), then the intensities
I01
in the XUV generation plane and
I02
in the focusing
plane are related by
I02
=
γ2·I01
. As
I01
remains approximately constant, when preserving
the conversion efficiency, the achieved XUV intensity in the experiment does not depend on
the output XUV pulse energy but rather on the demagnification of that output intensity
achieved by the focusing optics implemented with the experiment. In simple approximation,
the demagnification factor γis described by geometrical optics as
γ=dXUV
fXUV
[133, p. 279]. (3.1)
In conclusion, this confirms: for a given experiment (with given
fXUV
), only the free
parameter
dXUV
determines the achievable intensity. In fact, scaling the pulse energy under
preservation of the conversion efficiency has no impact on this intensity.
10
It has been shown that the product of divergence and minimal spot size are invariant for any given beam
[126], a larger incident beam diameter with a constant focal length implies a stronger divergence and hence a
smaller minimal spot size.
11
Elaboration: here, the location, where the HHG takes place, is approximated by the NIR’s focal plane to
be located at the end of the HHG cell. When scaling the NIR’s focal length by factor
η2
, accounting for an
increased/decreased incident beam diameter required when providing higher/lower pulse energies to stay close
to the LIDT of the last mirror, the NIR focus size increases by factor
η
. Due to the requirement of an according
scaling of the NIR pulse energy by
η2
and the subsequent scaling of the generated XUV pulse energy by
η2
[45], the NIR intensity and thus also the XUV intensity in the plane of generation
I01
remains approximately
constant (exactly constant for Gaussian beams).
12
Demagnification is the inverse of magnification
M
(
γ
= 1
/M
) that is described in general physics textbooks,
see e.g. [133].
43
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
Figure 3.6: Overview of the optical layout and associated parameters used to describe the XUV
focus’s intensity scaling to the beamline to the attached experiment. The NIR beam enters the
beamline from the right-hand side with a diameter of
wentrance,NIR
and is focused at a distance
dNIR
. In the focus, i.e. at its beam waist it has a radius of
w01,NIR
. There also the XUV light is
generated. The XUV source plane and beam waist with a radius
w01
is to be assumed at the NIR
beam’s focal plane. Both beams co-propagate until a metal foil filter - here a filter made of Al,
where the NIR is blocked. The XUV light reaches a focusing mirror at a distance
dXUV
after its
generation and is refocused at a distance
fXUV
into a beam waist with a radius
w02
. There, the
XUV light has an intensity of I02.
3.2.2 Analytical description of the XUV intensity scaling
While the conversion efficiency preserving scaling in [45] directly yields a scaling relation with
regards to the pulse energy, now an analytical description of the scaling of the XUV intensity
in an attached experiment will be presented. Like in the previous section, an initial setup
with maximized conversion efficiency provides the basis. Relying on Gaussian optics applied
to Gaussian beams [134] and the conversion efficiency preserving scaling an expression for
this XUV intensity depending on beamline parameters will be derived. An overview over the
relevant geometrical variables, also those relating to the Gaussian beams, is shown in fig. 3.6.
The propagation of Gaussian beams in the paraxial approximation is a common approach
to analyze an optical system. Using ABCD-matrices, one can calculate how an initial beam
waist
w01
translates into a second beam waist
w02
after propagating a distance
d
and then
passing a focusing element with focal length
f
. In good approximation, for
d >> f
, the
secondary beam waist is located at a distance
f
from the focusing mirror. Furthermore, the
focus size (
w02
) depends on the wavelength
λ
and the beam quality
M2
.
13
Hence for the here
described geometry, after the HHG medium the refocused XUV’s beam waist radius
w02
is
given by:
w02 =⌜
⎷f2
XUV w2
01
π2w4
01
M4λ2+ (dXUV −fXUV )2(3.2)
The conversion-efficiency-preserving scaling relations in [45] are given for two scaling
schemes. Here, the "global scaling" approach is of relevance. However the scaling relations do
not include an additional plane mirror before the HHG medium.
13for an introduction into the propagation of Gaussian beams / Gaussian optics see [134, 135]
44
3.2 XUV pulse generation
scaling with
η
scaling with
κ
Epulse,NIR η2κ
fNIR η2κ
wentrance,NIR η√κ
w01,NIR η√κ
Lη2κ
ρ1/η21/κ
Table 3.1: Scaling of parameters relating to the HHG process following the global scaling presented
in [45]. The scaling with
η
is directly taken from [45] (except
w01,NIR
, which is given here assuming
Gaussian beams). The formalism introduced in this chapter uses scaling with κwith κ=η2.
As already indicated previously, when operated at a constant NIR intensity (i.e., near
the LIDT) there is a fixed relation to the focusing mirror. In fact, for a minimized
dNIR
,
the beam diameter on the mirror and distance from the HHG medium
dNIR
depend only on
the free parameters focal length
fNIR
and incident pulse energy
Epulse,NIR
- see also fig. 3.5.
Consequently, as schematically shown in fig. 3.6, at the position of the plane mirror, one can
define an effective single focusing mirror with focal length
dNIR
and variable incident beam
diameter D1. Then, later on, one may determine fNIR from dNIR and D1.
Now, using Gaussian optics on the basis of eq. 3.2 together with the simplified NIR focusing,
XUV focus properties in the attached experiment are to be described quantitatively depending
on the scaling of the beamline under preservation of the conversion efficiency. Starting from
an initial configuration, for the beamline a total beamline length
lBL
is considered with an
initial propagation distance
dXUV,0
after the HHG medium and an effective NIR focal length
dNIR,0≈lBL −dXUV,0
. Further, the XUV beam waist radius in the XUV generation plane is
given with
w01,0
for the initial configuration. It is assumed to scale linearly with the NIR’s
beam waist radius
w01,NIR
where the HHG is generated. According to the scaling shown in
table 3.1, the following relations are to be considered:
κ=dNIR
dNIR,0
=lBL −dXUV
lBL −dXUV,0⇔dXUV =lBL −κ·(lBL −dXUV,0)(3.3)
w01 =√κ·w01,0Epulse,XUV =κ·Epulse,XUV,0(3.4)
Combining those relations due to the scaling behavior with eq. 3.2, describing the beam
waist radius of the refocused XUV, its behavior in terms of the scaling factor
κ
can be expressed:
w02 =⌜
⎷f2
XUV ·κ·w2
01,0
π2
M4·λ2·κ2·w4
01,0+ (lBL −κ·(lBL −dXUV,0)−fXUV )2(3.5)
This directly leads to an expression for the demagnification factor given by the size relation
of the XUV source and focus beam waist w01 and w02 respectively:
γ=w01
w02
=⌜
⎷
π2
M4·λ2·κ2·w4
01,0+ (lBL −κ·(lBL −dXUV,0)−fXUV )2
f2
XUV
(3.6)
45
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
Finally, using the expression for the beam waist size in eq. 3.5 in combination with the
pulse energy scaling from eq. 3.4 the peak intensity of the refocused XUV pulse can be
described:
I02,XUV =2·PXUV
π·w2
02
I02,XUV =2·PXUV
π
π2
M4·λ2·κ2·w4
01,0+ (lBL −κ·(lBL −dXUV,0)−fXUV )2
f2
XUV ·κ·w2
01,0
(3.7)
The peak power PXUV is given by:
PXUV =2√ln 2
√π
Epulse,XUV
τ=2√ln 2
√π
κ·Epulse,XUV,0
τ.
By substituting the peak power
PXUV
into the intensity expression in eq. 3.7, the intensity
is given only in terms of the fixed beamline and experiment parameters, the initial setup’s
geometry and the scaling parameter κ:
I02,XUV =4√ln 2
π√π
Epulse,XUV,0
τ
(a)
⏟ ⏞⏞ ⏟
π2
M4·λ2·κ2·w4
01,0+
(b)
⏟ ⏞⏞ ⏟
(lBL −κ·(lBL −dXUV,0)−fXUV )2
f2
XUV ·w2
01,0
.
(3.8)
The resulting expression for the scaled XUV intensity, contains two important terms that
are indicated by (a) and (b). Within those, the scaling factor
κ
is contained for which both
terms exhibit opposite behavior: Increasing
κ
, corresponding as can be seen from eq. 3.3 to
an increased dNIR, yields an increased term (a), but decreased term (b) and vice versa.
So far, it was expected that a small
dNIR
will result in higher intensities. This behaviour
is represented by term (b). However, term (a) indicates that also particularly large
dNIR
could be beneficial. For a long beamline, term (a) becomes only a relevant contribution if
κw2
01,0> M2λ
. In such case a large XUV source beam waist
w01,0
corresponding to a large
dNIR
would be required. It is not clear whether, the LIDT-limited minimum distance
dXUV
allows for such a case.
3.2.3 Tailoring a beamline for CDI and non-linear XUV experiments
CDI and non-linear XUV experiments benefit from a high XUV intensity. Based on the
qualitative description in section 3.2.1, it is expected that an asymmetric setup concerning the
HHG cell’s distances from the last NIR mirror and to the end of the beamline is desired. The
realization of such asymmetric setup at the described HHG beamline was initially proposed by
Bernd Schütte, discussed, and carried out by the project team, including myself. To apply the
previously introduced quantitative scaling, as a starting point, the initial HHG beamline setup
described in the following is considered.
Initial setup
The parameters of the initial setup are given in table 3.2. The initial beam waist size
w01,0
could not be measured directly. It was estimated based on a beam diameter of 4
.
2
mm
(1
/e2
)
46
3.2 XUV pulse generation
Parameter Value Parameter Value
dNIR,05m M21
dXUV,013 m τ4fs (FWHM)
w01,0240µm(1/e2, diameter) λ60nm
Epulse,XUV,070 nJ fXUV 75mm
Table 3.2: Parameters of the presented HHG beamline, which were used to apply the intensity
scaling relations that were previously described; (see eq. 3.1 to 3.8) Note: the XUV pulse energy
Epulse,XUV,0
is given in the experiment, considering a total transmission of 10 % into the focus due
to metal foil filter and focusing optics. The initial beam waist size
w01,0
was estimated from a
given beam diameter at a known distance from the focus. For this and all scaling calculations, for
simplicity, monochromatic radiation (λ= 60 nm) with a beam quality M2= 1 are assumed.
[136] measured at a distance of 13 m from the source point. Assuming a central wavelength
of 60
nm
(
∼
13th harmonic) and a beam quality factor
M2
of 1(ideal Gaussian beam), the
source size is estimated with 240
µm
(1
/e2
, diameter, 44% of the NIR focus size). An initial
XUV pulse energy of 56
nJ
could be measured using an AXUV photodiode (see also section
3.3) and assuming an upper limit of the conversion efficiency of 0
.
27
A/W
[137]. In addition
to an Al filter in the beamline, a secondary Al filter was placed in front of the photodiode.
The transmission of both was estimated by directly measuring the transmission of one of the
filters to be 28 %. Consequently, the initial pulse energy is effectively given before any optical
components by 700
nJ
. It is determined by dividing the measured value behind the filters by
the transmission of both filters.
Already with the initial setup, comparable high intensities were achieved at the beamline.
This was demonstrated with an experiment mainly carried out by Bernd Schütte and Martin
Kretschmar, in which multiple ionization in argon up to
Ar5+
was measured using a velocity
map imaging setup (VMI) that implemented a focal length
fXUV
of 7
.
5
cm
(results shown in
[36]). The achieved intensity is estimated from the pulse energy discussed above in combination
with a filter transmission of 40 %
14
and an average focusing mirror reflectivity of 25%. The
total transmission of 10 % corresponds to a pulse energy of 70
nJ
in the experiment’s XUV
focus. Furthermore, a beam waist radius of 1
.
3
µm
could be retrieved via the Rayleigh range
obtained from VMI measurements imaging the spatial extent of the ionization region. Based
on published results from other HHG experiments [138, 139], the pulse duration was estimated
with 4
fs
. Combining the reasonable estimations and measurements, the achieved peak intensity
is estimated with 7
·
10
14 W/cm2
.
15
Based on theoretical predictions on temporal pulse structure
and focus size [140] the intensity of 9·1015 W/cm2can be reached in the focus. [36]
Applying the intensity scaling relation
Potential improvement of the achievable intensities will now be discussed by applying the
previously derived scaling scheme to the initial setup. The qualitative discussion in the
beginning suggested that an intensity optimized scheme will have an HHG medium placed
asymmetrically along the beamline. This is to be confirmed now quantitatively. Therefore all
14A filter with less oxide contamination was used for the measurement [36].
15
Here the expected attosecond pulse train temporal structure of the XUV pulse is neglected. Taking an
attosecond pulse duration in the train of 600
as
into account increases the peak intensity approximately by a
factor of 2.
47
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
51015
dXUV (m)
5 10 15
dNIR (m)
2
4
XUV beam
waist radius ( m)
100
200
demagnification
51015
dXUV (m)
5 10 15
dNIR (m)
1
2
XUV pulse energy
(arb. u.)
0
10
20
30
Peak IXUV (arb. u.)
468101214
dXUV (m)
4 6 8 10 12 14
dNIR (m)
50
100
150
200
demagnification
1.00
0.75
0.50
0.25
dXUV/fXUV
(a) (b)
(c)
Figure 3.7: Scaling of (a) XUV focal size & demagnification and (b) pulse energy & XUV intensity
in the interaction region by applying the beamline describing parameters shown in table 3.2 to
equation 3.1 to 3.8; The vertical green line indicates the HHG gas cell position in the initial setup.
(c) compares the influence of various XUV source sizes given as fraction from the NIR focus size to
the demagnification scaling approximated by eq. 3.1.
48
3.2 XUV pulse generation
parameters are shown depending on the HHG medium position directly, instead of the scaling
parameter κ. The optimization is visualized in three steps:
1.
The scaling of purely optical beam parameters, i.e., the final XUV focus size and
demagnification.
2. The XUV pulse energy scaling from [45].
3.
The achievable intensity, which combines the scaling of optical parameters and the XUV
pulse energy scaling.
In addition, the influence on the assumed XUV source size with respect to the NIR focus size
will be presented.
The behaviour of the XUV pulse parameters depending on the HHG medium’s position
are visualized in fig. 3.7:
1.
In 3.7 (a) the scaling of the optical beam parameters is shown. Like qualitatively
motivated in the previous section, a shorter
dNIR
(bottom x-axis) and hence a larger XUV
propagation distance
dXUV
(top x-axis) yields a smaller XUV focus. Furthermore, it is
apparent that in the vicinity of the initial configuration (green line), the complex analytical
expression for the demagnification in eq. 3.6 is well approximated by geometrical optics
as given in eq. 3.1.
2.
Including the pulse energy scaling, fig. 3.7 (b, blue curve) shows the increase of the XUV
pulse energy with increasing dNIR directly resulting from the scaling discussed in [45].
3.
However, though when disregarding the previous discussion it may seem counter intuitive,
fig. 3.7 (b, orange curve) clearly shows that higher intensities in the experiment are
achieved with shorter NIR focal lengths
dNIR
and thus with opposite optimization than
that achieving high pulse energies.
The biggest uncertainty is given by the assumed XUV source size or, in other terms,
the assumed fraction of the NIR focus size acting as an XUV source. So far a ratio of 0.44
was assumed. Fig. 3.7 (c) shows that indeed different assumptions on the source size affect
the scaling of the demagnification. In fact, the larger the ratio, the more prominent is the
contribution from the term (a) from eq. 3.8, which yields higher intensities for larger
dNIR
.
Nevertheless, the qualitative arguments for an effective focal length
dNIR
are certainly valid
in the discussed region below 8m. For smaller fractions the result is getting closer to the
approximation of the demagnification (eq. 3.1). In all cases, a shorter NIR focal length yields
a stronger demagnification associated with a higher intensity.
Indeed, the results strongly support the approach to realize a highly asymmetric HHG
beamline setup. They further indicate that even higher intensities may be achieved by reducing
dNIR
with respect to the initial setup even further. However, higher intensities are achieved at
the cost of smaller XUV foci in the experiment. Different experimental schemes may require a
compromise between focus size and achieved intensity. As the key applications of this beamline
are the investigation of non-linear XUV effects and CDI, a compromise for each is discussed in
the following.
49
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
51015
dXUV (m)
5 10 15
dNIR (m)
5
10
XUV beam
waist radius ( m)
25
50
75
demagnification
51015
dXUV (m)
5 10 15
dNIR (m)
1
2
XUV pulse energy
(arb. u.)
0
2
4
Peak IXUV (arb. u.)
468101214
dXUV (m)
4 6 8 10 12 14
dNIR (m)
20
40
60
80
demagnification
1.00
0.75
0.50
0.25
dXUV/fXUV
(a) (b)
(c)
Figure 3.8: Scaling of the intensity as shown in fig. 3.7, but for a focal length of 178
mm
, which
is used with the carried out CDI experiment described in chapter 4.
Tailoring the beamline geometry for CDI experiments
For CDI, the target dimension poses a relevant constraint. For example, Helium nano-droplets
produced with the source that is used within the CDI experiment presented in chapter 4.2,
have an average radius of 273
nm
ranging from 80
nm
to 700
nm16
[141]. To image only the
properties of the sample and not the intensity gradient of the beam profile, nearly homogeneous
illumination of the droplets is desired. Assuming a variation of the intensity of less than 5 %
(peak to peak) over a droplet with radius 300
nm
in focus (Gaussian beam profile), a beam
waist radius of at least 1.3µmis required 17.
To learn the optimal position of the HHG medium for this experiment, the data previously
presented in fig. 3.7 needs to be recomputed for the XUV focal length of 178
mm
used with
the CDI experiment (see chapter 4.5). The results are shown in fig. 3.8, which for comparison
resembles fig. 3.7. The observed trends remain the same. The results in fig. 3.8 indicate that
no shorter distance dNIR than 3.7 m should be implemented.
16The range of measured sizes is limited by the resolution of the used detector. [141]
17
But assuming the droplet is hit by the laser pulse when placed 1
µm
offset from the peak intensity in the
focal plane (due to the varying droplet position in the focus), the same requirement yields a minimal beam
waist radius of 4.8µm.
50
3.2 XUV pulse generation
2 4 6 8 10 12 14 16
dNIR (m)
0.0
0.2
0.4
0.6
0.8
1.0
normalized expected
signal strength (arb. u.)
order of
non-linearity
1
2
4
6
8
10
Figure 3.9: Normalized scaling of the expected signal (e.g., photo-electrons) that can be acquired
from a non-linear process in the VMI setup; The to each curve associated order of non-linearity is
given in the legend.
Tailoring the beamline geometry for nonlinear XUV experiments
When measuring non-linear effects in gaseous targets, the amount of signal generated (e.g.,
photo-electron signal) and measured needs to be maximized. It does depend not only on the
present intensity but also on the focal volume, which provides a scale for the amount of target
gas contributing to the signal. The focal volume
Vfoc
depends on the beam cross-section in
the XUV focus (proportional to the beam waist radius of the focused XUV
w02
:
∝w2
02
) and
the XUV’s Rayleigh range (
∝w2
02
) along the beam axis. Consequently, the focal volume scales
with
Vfoc ∝w4
02
. Assuming that signal is detected from the full focal volume, the acquired
signal for an
nth
order non-linear process is directly proportional to the focal volume
Vfoc
and
available intensity Ito the power of n:
signal ∝Vfoc ·In∝w4
02 ·In(3.9)
Because the XUV beam waist
w02
and the achieved intensity scale opposite with the HHG
medium position (see fig. 3.7 (a) blue curve in comparison to (b) orange curve), according
to eq. 3.9, depending on the order of non-linearity, different optimal working points can be
expected. This can be verified by combining the result for
w02
and
IXUV
shown in fig. 3.7
with the relation in eq. 3.9. Fig. 3.9 presents the result for different orders of non-linearity.
Indeed, it shows a different optimal working point, i.e. a different position for maximum signal,
depending on the targeted non-linearity.
It turns out that the initial experimental setup was already operated close to the optimum
for the carried out non-linear ionization experiment of argon. At a HHG medium distance
dNIR
of 5m, the optimal for a non-linearity of 8(corresponding to the generation of
Ar4+
) is nearly
matched. Furthermore, the working point is very close to the optimum for a non-linearity of
10 which is the minimum non-linearity expected for the generation of Ar5+ [36].
51
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
3.2.4 Implemented, optimized XUV pulse generation setup
For the experiments presented within this thesis, the optimized HHG beamline set up reduces
the distance of the HHG medium from the beginning of the beamline
dNIR
from 5m to 4 m
compared to the initial setup. Consequently, up to 14m propagation distance follow after the
HHG cell. Compared to the presented scaling that assumes ideal focusing optics, in the real
setup one needs to account for optical aberrations. During this work, two different focusing
schemes for the NIR were used:
•
two spherical mirrors, a focusing and a defocusing one, forming a telescope; With these
two spherical mirrors, the astigmatism arising from not perfect, but near normal incidence
can be corrected [36].
•
A single spherical focusing mirror (
fNIR
= 5m) that uses a deformable mirror to correct
wavefront errors and astigmatism caused by the near normal incidence of the NIR pulses
onto the focusing mirror.
So far, a strongly simplified beamline setup (see fig. 3.5) was assumed. In practice
additional components are used for easy in-situ tuning of the HHG process and to stabilize
the laser beam:
•
the efficiency of the HHG process and phase-matching strongly depends on the generated
NIR focus. NIR properties in the focus such as pulse energy and Rayleigh length can be
influenced by cropping the unfocused NIR beam using a motorized iris18.
•
to ensure a stable pointing of the NIR beam and control the pointing, a stabilization
system consisting of two motorized mirrors and two camera sensors for feedback
(commercial product by TEM) is implemented with the setup.
Beampath
Fig. 3.10 shows the optical in-vacuum setup preparing the NIR pulse for HHG. The compressed
NIR pulse generated with the OPCPA laser system (see section 3.1) enters the setup on the
bottom left. It propagates via a motorized mirror that is part of the pointing stabilization
system (a second one is placed along the beam path before this setup) through a motorized
iris to control the HHG process. It further propagates through a thin glass slide, whose back
reflex is used for the pointing stabilization feedback system (fig. 3.10, orange/yellow beam
path). The beam is then folded over five mirrors in near-normal incidence before entering
the beamline. One of the mirrors is a spherical focusing mirror. Alternatively, the layout
is also compatible with the telescopic focusing optics. The last mirror guiding the beam
into the beamline is also motorized, allowing to steer the beam independent of the pointing
stabilization. To measure the NIR pulse energy entering the HHG cell behind the motorized
iris (see fig. 3.10), a mirror can be moved into the beam path coupling the beam out of the
vacuum onto a powermeter. Simultaneously, another mirror is moved on top of the standard
NIR beam path. It couples a continuous-wave (cw) Helium-Neon laser (543
nm
) into the
18
Reducing the diameter of a beam to be focused, increases the beam waist in the focus and also increases
the Rayleigh length.
52
3.2 XUV pulse generation
powermeter
alignment HeNe laser
NIR light from OPCPA
pointing
feedback 1
pointing
feedback 2
pointing
motorized mirror
motorized iris
NIR outcoupling /
alignment laser incoupling
motorized linear stage
glass slide
“last” NIR mirror,
motorized
focusing mirror
Figure 3.10: Optical setup preparing the laser-provided NIR pulse to produce HHG; The red
lines indicated the main NIR beam path: entering on the bottom from the OPCPA compression it
is guided over a motorized mirror associated with the pointing stabilization through a motorized
iris to control the HHG process. After being reflected by multiple plane mirrors, it reaches the
focusing mirror. Finally, the "last" NIR mirror that is also motorized for manual steering guides
the NIR beam into the beamline. A motorized stage (dark blue) may drive two additional mirrors
into the beam path. In that case, the main NIR is guided along the dashed pink line towards
a power meter measuring the laser pulse energy. Moreover, instead of the NIR, a HeNe laser,
represented by the green lines, is coupled into the beampath for alignment purposes (dotted green
line). Originating from reflexes by thin glass slides in the beam, orange and yellow beam paths are
related to the pointing stabilization feedback provided to sensors outside of the vacuum.
53
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
otherwise NIR beam path. This so-called "alignment laser" enables safer alignment operations
in the experiments and provides a weak but spectrally sharp coherent reference if needed. For
the pointing stabilization, the used back reflex is split up to be provided to two sensors located
out of the vacuum. One sensor is placed directly next to the chamber and another far away
next to the XUV diagnostics toward the end of the beamline.
The high harmonic generation takes place in a 30
cm
long cell filled with Xe or Kr. While
HHG takes place only in the very last part of the cell, a long propagation of the NIR through the
gas target enables a reshaping process that flattens the NIR’s intensity profile and introduces
a blue shift of its spectrum. The theoretical modeling of the reshaping process suggests that
this enables the efficient usage of more NIR light since it effectively increases the NIR focus
size, normally achieved by implementing longer focal lengths. [140]
Summary
Already the initial setup was based on an asymmetric setup allowing for a longer propagation
distance of the XUV behind the HHG medium. With it a highly non-linear process, namely the
generation of
Ar5+
could be demonstrated. A discussion of this setup’s optimization showed
in particular that for the anticipated CDI experiment the HHG medium should not be placed
less than 3.7 m from the beginning of the beamline.
Finally, a 4 m distance was implemented with the setup. In practice, the beamline setup
before the HHG cell is extended to stabilize and characterize the NIR beam and to optimize the
XUV beam. Also the generated XUV pulses need to be characterized. This will be discussed
in the next section.
3.3 XUV pulse characterization
Characterizing the generated XUV pulses relies on various experimental techniques. While
some characterizations can be performed routinely and are available during other experiments
performed at the beamline, others require the attachment of their own complex experimental
setups.
In this section, first the technologies to detect XUV radiation are introduced to lay the
foundation for the diagnostic characterization tools in the beamline, which are explained
with the subsequent subsection. The temporal profile of the XUV pulses cannot be measured
directly by any (opto-)electronic devices. Only indirect measurements are possible. Those may
be based on pump-probe configurations precisely scanning the HHG driving NIR pulse against
the XUV APT. A tool to produce such precise delays between two pulses is described in
section 3.3.3 . The concept for one particular method, namely RABITT, has been introduced
in chapter 2.1.4. Finally in section 3.3.4, it is applied to characterize the XUV pulses generated
by the presented beamline.
3.3.1 Techniques for XUV detection
XUV light can be detected by converting it to visible light, or by creating electronic or ionic
signal. Depending on the specific quantities to be characterized, multichannel plate based
54
3.3 XUV pulse characterization
Figure 3.11: (a) Operation principle of a single channel in the MCP: primary incident radiation
ejects secondary electrons from the channel wall. Due to high voltage along the channel, the
electrons get accelerated and by the produced electrons hitting the wall over and over again an
avalanche of electrons is produced. (b) Side view of the operation principle of a Chevron-type
MCP stack with a metal anode: the MCP stack consists of two MCPs with many channels tilted
at a bias angle. Primary radiation produces an avalanche of electrons in the first channel on the
first MCP. From there, the avalanche of electrons enters multiple channels on the second MCP, in
which the electron signal is further amplified. The so produced electron pulse is then measured as
a current utilizing a metal anode. ((a) and (b) reprinted from [145, fig. 2 & 5])
systems, XUV photo diodes or ion/electron time-of-flight measurements can be applied. In
this section, the methodologies and techniques are briefly described.
XUV photo diodes
The pulse energy of the XUV APT can be determined by using XUV sensitive photo diodes.
Each incident XUV pulse train generates a current proportional to the number of photons
it contains. Spatial resolution can be achieved using an array of those photo-diodes (low
resolution) [137].
Phosphor screens
Phosphors are materials that exhibit strong luminescence properties. Either by fluorescence
or phosphorescence, incident light or also electrons are converted into visible light. An XUV
detector with spatial resolution is created by coating a screen with phosphor materials. Those
screens are a very cheap solution with the potential to support high repetition rates. However,
their response may be non-linear [142] and the quantum efficiency only around one [143] (i.e.,
there is none to low signal amplification).
Standard materials and their central emission wavelengths are, for example, P1
(
Zn2SiO4:Mn
,525
nm
), P43 (
Gd2O2S:Tb
,544
nm
), P46 (
Y3Al5O12:Ce
,530
nm
), or P47
(
Y2SiO5:Ce
,400
nm
). Besides their peak emission wavelength, the main differences are given
by decay time, and wavelength-dependent quantum efficiency. [144]
Micro-channel plates
Micro-channel plates (MCPs) consist of many few micrometer diameter channels in a glass
plate. Every single channel acts as a photo-multiplier amplifying an initially weak photon
or particle (e.g. electron) signal. Electrons entering the channel, so-called primary electrons,
55
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
are accelerated by a high voltage (on the order of few kV) applied across the plate. When
hitting a wall of the channel, the kinetic energy of the primary electrons causes the ejection of
more electrons, so-called secondary electrons and the signal gets amplified. Alternatively, as
illustrated in fig. 3.11 (a), also light can cause the ejection of secondary electrons due to the
photo-electric effect. The secondary electrons are further accelerated by the applied voltage
and the process repeats, initiating an avalanche of electrons. To increase the probability for
the incoming photons or particles to hit a wall, thus to create a secondary electron and initiate
an avalanche, the channels are usually tilted against the plate’s surface normal by the so-called
bias angle. [145]
A single plate amplifies the electronic signal by up to 4 orders of magnitude [146]. To
increase the amplification even further, MCPs can be stacked. The MCP stacks used in the
scope of this thesis use the Chevron geometry [147], shown in fig. 3.11 (b), where the tilted
channels are oriented in opposite directions. With a Chevron geometry an amplification of up
to 7 orders of magnitude is achieved. [145]
Generally, as already indicated, MCPs are sensitive to particles, such as electrons or ions,
and photons in the UV to X-ray regime. However, a significant increase in sensitivity for light
is achieved by coating the MCP with a photo-cathode material such as
MgF2
increasing the
photo-electron yield [148]. The electron pulse exiting the MCP can be captured using a metal
anode (see fig. 3.11 (b)). It can be measured with a temporal resolution of 1
ns
and below.
Accordingly, MCPs commonly serve as a fast detector for charged particles.
Furthermore, spatial resolution can be achieved by replacing the anode shown in fig. 3.11
(b) by a phosphor screen converting the electron avalanches into visible light. Typically for
devices with 12 µmchannel diameter, resolutions of 40 −45 µmand 80 −90 µmare achieved
for a single and two stacked MCPs, respectively [149]. [145]
Detecting XUV light through ionization
Furthermore, XUV radiation can be detected and characterized by ionizing gaseous targets
and detecting the generated ions or photo-electrons (see e.g. [150]) [47]. For example, kinetic
energy resolved photo-electron spectra provide information on the photon energy spectrum
of the XUV. Those can be measured using time-of-flight spectrometers. A simple time-of-
flight spectrometer resolves the velocity (kinetic energy), mass and charge of ions or electrons.
Therefore it disperses the particles by their velocities: Introducing a free flight distance between
the particles origin and detection, the different velocities cause the electrons or ions to reach
the detector at different times. A temporal spread in the nano- to microseconds is expected.
In an ion time-of-flight spectrometer, the ions can be accelerated to imprint their mass and
charge onto their velocity before the free flight. Finally, electrons or ions are detected using an
MCP with an anode readout. [151, 152]
3.3.2 XUV pulse diagnostic setup
Toward the end of the HHG beamline, the co-propagating XUV and NIR pulses arrive in the
diagnostics’s chamber. Its layout is shown in figure 3.12. To reduce the amount of undesired
NIR radiation, including straylight, a motorized iris crops outer parts of the incident light
containing mostly NIR radiation, as the NIR is more divergent than the XUV. With the
56
3.3 XUV pulse characterization
Figure 3.12: Layout of the diagnostics’s chamber; co-propagating NIR and XUV light reach
the diagnostics’s chamber from the right-hand side. The NIR is more divergent and partially
blocked by a motorized iris. The following aluminum (metal foil) filter blocks the remaining NIR.
Alternatively, the filter can be removed from the beam. To measure the XUV spectrum a slit and
grating can be moved into the beam. The spectrum is then projected onto an MCP stack, whose
phosphor screen can be recorded with a camera. The grating can also be rotated to make use of
the reflected zeroth order. With no grating in the beam, either the pulse energy can be measured
with a photo diode or the XUV beam propagates towards an attached experiment.
following aluminum filter, the remaining NIR can be heavily suppressed by many orders of
magnitude [127, fig. 20], while a significant fraction of the XUV passes through (for a 100
nm
Al filter 30 −40 %19).
XUV spectrometer
For measurements on a day-to-day basis, the diagnostics’s chamber implements basic XUV
characterizing instruments. The beam profile and spectral components of the XUV beam can
be characterized utilizing a simple spectrometer consisting of an (optional) slit, a gold-coated
grating in grazing incidence, and an MCP-phosphor-stack. It is intended to be used in three
configurations:
1.
The XUV spectrum can be recorded by inserting the slit and rotating the grating so
that the first diffraction order is visible on the MCP-phosphor-stack.
2.
The XUV beam profile is imaged by the MCP-phosphor-stack, by removing the slit and
rotating the grating to project the zeroth diffraction order onto it. At the same time,
although the MCP potentially provides a non-linear response, the signal brightness gives
a scale for the XUV pulse energy. Making this configuration a suitable tool for the
optimization of the HHG process.
3.
As a combination of first and second mode, the first one can be used without a slit.
Then, due to the expected spectral distinct harmonics in an attosecond pulse train, the
approximate beam profile at each harmonic can be resolved.
Grating and slit can be moved out of the beam, which then can pass through.
Pulse energy characterization
Behind the grating, instead of passing the beam to the attached experiment, an XUV photodiode
(Optodiode AXUV100G [153]) can be moved into the beam path, enabling the XUV pulse
19
Experimentally, this is consistent with data shown in [127, fig. 21], depending on the oxide layer thickness.
57
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
Figure 3.13: CAD rendering showing the technical realization of the delay unit. In the presented
rendering, plane gold mirrors are mounted between two PEEK spacer pieces. Two beams colored
in red and purple are guided over the mirrors, representing NIR and XUV light, respectively. The
different colored parts indicate their different purpose as follows; blue: motorized tip/tilt mounts
for each mirror; green: high resolution piezo-driven linear stage for fine delays; red: low resolution
linear stage enabling a long range of delays; purple: linear stage to adjust the gap between the
mirrors; yellow: two linear stages to move the complete unit right/left and forward/backwards.
orange: removable slabs to mount additional mirrors for an active delay stabilization in the future;
energy measurement. Being also sensitive to the NIR, the photodiode requires heavy shielding
against NIR radiation. Although, on the direct beam path the NIR light is suppressed by
several orders of magnitude by the Aluminum filter, a little fraction remains in the beam.
Moreover, a significant fraction of the NIR light blocked by metal foil filters is diffusely reflected
by it causing significant NIR straylight background in the diagnostics’s vacuum chamber.
Therefore it is enclosed in a light-tight housing with an additional aluminum filter at its
entrance.
3.3.3 Delaying two pulses with sub-femtosecond precision
Split-mirror setup
Resolving temporal dynamics of materials or light requires introducing a highly controlled
time delay between a pulse initiating and a second pulse probing a dynamic process. Such
experimental access to a dynamic process is crucial for the temporal characterization of the
XUV pulses. The volume, in which the investigated dynamic process takes place, is called
interaction region. In the presented experiments, a delay between two pulses is achieved by a
split mirror, where one half is translated along its surface normal. Thereby the optical path
length of the beam hitting this mirror is changed. For near-normal incidence, the introduced
58
3.3 XUV pulse characterization
delay corresponds approximately to the duration the light needs to travel twice the translated
distance.
A delay unit has been developed for near-normal incidence geometries, since a near-normal
focusing geometry is also chosen in the CDI experiment (see chapter 4.5). The delay unit is
shown in fig. 3.13. It is designed to hold two mirrors side-by-side with surface dimensions of
up to 25
×
13
mm
. An overall compact unit is desired to avoid the amplification of vibrations
through long components. Nevertheless, the compactness is limited by the size of the mirrors
and the motorized stages used.
Motorized axes
As both beams are reflected by the mirrors on the delay unit side-by-side, they reach the
interaction region in slight non-colinearity. When delaying the two pulses by moving one
mirror, the beams will also move horizontally relative to each other. Such movement could
potentially shift the region where the beams overlap out of the interaction region. To preserve
the overlap during delay scans and also to establish it initially, each mirror is mounted to
a motorized two-axis kinematic mirror mount, which are shown in blue in fig. 3.13. The
tilt axis of one of the mirrors is equipped with a closed-loop motor to enable a reproducible
reestablishment of the overlap in the desired plane, also during a delay scan. This way, the
delay stage can address a total delay range of ca. 77ps.
This picosecond-spanning delay range is enabled by a highly stable DC-motor driven
closed-loop stage that is colored red in fig. 3.13 with a travel range of 11
.
5
mm
with the
mounted mirrors. It can achieve a step size of 50
nm
, which corresponds to a delay step size
of ca. 333
as
. The temporal resolution of the delay unit is enhanced to theoretically 2
.
7
as
(0
.
4
nm
) and practically 43
as20
(standard deviation) by a piezo-driven closed-loop stage (green
in fig. 3.13), over a range of 100µm(i.e., ca. 660fs).
In addition to moving the mirrors relative to each other, the complete delay unit can be
moved along the mirror’s surface normal. This is particularly useful when it is equipped with
focusing mirrors to move the focus relative to the designated interaction region. Moreover,
the complete delay unit can be moved sideways (parallel to the mirror’s surface) to align the
incident beams relative to the gap between the mirrors. These two stages are highlighted in
yellow in fig. 3.13. Furthermore, one of the mirrors can be moved right-left separately to
adjust the gap in between the mirrors. The associated stage is depicted in purple in fig. 3.13.
Extension for active delay stabilization
As an additional feature, the stage is ready for the implementation of an active delay
stabilization. Such an active stabilization monitors the actual delay of two additional beams
propagating over the delay stage, using a camera sensor to look at the interference. The
computed change in delay (shot to shot) can then be used as closed-loop feedback. Short focal
lengths and other spatial restrictions make it challenging to implement such a system using
beam propagating over the mounted mirrors. Instead, mirrors can be attached to extensions
mounted on top of the mirror mounting.
20
measured in a vented setup in the interaction region with the alignment HeNe laser; During an experiment
under vacuum conditions additional vibrations (e.g. due to vacuum pumps) will worsen the achievable resolution.
59
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
For example Huppert et al., who implemented an active delay stabilization, measured
without their stabilization a peak-to-peak drift of 300
as
((41
±
24)
as
r.m.s.) over the duration
of two hours. With an active stabilization, long term drifts are eliminated and the delay varies
only by (26 ±10) as r.m.s. on a shot-to-shot basis. [154]
However, the experiments discussed in this thesis do not implement an active delay
stabilization.
3.3.4 Temporal characterization of the XUV pulses
Characterizing the temporal structure of the generated XUV pulse may yield either single
characteristics or a complete reconstruction of the XUV pulse. Single characteristics can be
derived e.g. from autocorrelation measurements, which measure the pulse against itself. Here
the r.m.s. width of the pulse but not the complete pulse shape is retrievable unambiguously.
Only with assumptions on the pulse shape (e.g. Gaussian or
sech2
shape), other measures like
FWHM can be determined. [155, ch. 4]
Pulse characterization methods
For a complete understanding of a light pulse’s temporal shape, amplitude and phase need
to be obtained. In frequency domain, this corresponds to retrieving the spectral amplitudes
and associated spectral phases
21
(phase for each spectral component) [156]. While obtaining
spectral amplitudes is straight-forward using spectrometers, it is not for the spectral phases.
For optical lasers, the frequency-resolved optical gating (FROG) technique [157] enables the
unambiguous retrieval of the full pulse. The temporal characterization of the attosecond
pulses in XUV APTs is achieved with the reconstruction of attosecond beating by interference
of two-photon transitions (RABITT) [9] and a full retrieval of XUV APTs is possible by
frequency-resolved optical gating for complete reconstruction of attosecond bursts (CRAB)
[58].
At this beamline, two different characterization methods were deployed. Martin Kretschmar
and Bernd Schütte performed an autocorrelation measurement. Andreas Hoffmann and Björn
Senfftleben carried out RABITT/CRAB measurements. The fundamental concepts of RABITT
and CRAB were already introduced in chapter 2.1.4. In the following, the setup and analysis
of this experiment are discussed.
Experimental setup
As schematically shown in fig. 3.14 (a), the RABITT experiment realizes a pump-probe scheme
by filtering out the co-propagating NIR on the XUV pump pulse and deriving an NIR probe
pulse from the co-propagating NIR that is not overlapping with the XUV
22
. Both beams are
guided over a delay unit and via a focusing mirror into the interaction region. The RABITT
technique relies only on linear effects and thus requires only low XUV intensities. Too high
XUV intensities can even result in too many generated photo-electrons, whose repulsion distorts
their acquired kinetic energy spectrum (space charge effects) [160, 161]. Consequently, despite
21assuming linear polarization and neglecting spatial variations
22Residual XUV light is removed by propagating through a 100 mum thick glass slide.
60
3.3 XUV pulse characterization
Figure 3.14: (a) Optical setup of the RABITT experiment: in the diagnostics’s chamber a split
Al filter in combination with an special aperture for the NIR is installed to form two distinct beams
out of the incident co-propagating NIR and XUV pulses. Both pulses can be temporally shifted
to each other using the delay unit that is equipped with gold-coated plane mirrors. From there
the beams are guided over a focusing mirror into the interaction region, where the target gas Ar
is provided from the top and an electron time-of-flight spectrometer (eTOF) captures produced
electrons. The beams can be monitored behind the interaction region or in the interaction region
on a screen. When sending NIR light also over the XUV’s beam path a mirror inserted after the
focusing mirror may couples the beams out of vacuum onto a camera sensor. (b) XUV reflectivity
of the involved mirrors: (blue) reflectivity of a gold coated mirror in near normal incidence (data
according to [158]), (orange) reflectivity of a
B4C
coated mirror in near normal incidence (data
according to [159]), and (green) combined reflectivity of both mirrors;
the mirror’s low reflectivity, the experimental setup can afford to implement two near-normal
incidence reflections (ca. 3
.
7
◦
). In fact, as can be see from fig. 3.14 (b), the transmission of
the incident XUV pulse is reduced to below 5 %.
The previously introduced delay unit (see section 3.3.3) is equipped with two bare-gold
coated mirrors. A spherical mirror with a focal length of 45
cm
and a
B4C
coating facilitates
the focusing of both - NIR and XUV - pulses into the interaction region. There, a 120
µm
inner diameter gas capillary is placed less than 2
mm
above the focus in the interaction region.
It introduces the Argon target gas. Argon is chosen for its low ionization energy of 15
.
76
eV
that limits the minimal XUV photon energy to produce electrons and exhibits still comparable
low spin-orbit splitting, which would impair the measurements (see [47, ch. 7.2] for details).
During the experiment, in short, the XUV ionizes the Argon atoms, and the NIR causes
a quantum path interference by acting upon the excited electrons. Then, the kinetic energy
spectrum of these free electrons is to be measured depending on the delay. Here, a commercial
electron time-of-flight spectrometer (Stefan Kaesdorf, ETF11) resolves the kinetic energy of
those electrons that were ejected sideways.
A phosphor-coated screen behind the interaction region sensitive to NIR and XUV radiation
is used for alignment and monitoring the beam pointing. Similar to the overlap alignment in
the CDI experiment (see chapter 4.7 for a detailed description
23
), the spatial overlap between
XUV and NIR is established visually on a phosphor-coated screen that can be placed in the
interaction region. However, beforehand, in addition to the scheme, the spatial and temporal
overlap between both beam paths when transporting only the NIR is established out of vacuum
23here instead of a microscope, a camera with a 50mm lens is used
61
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
0 5 10 15 20 25
delay (fs)
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
photo-electron energy (eV)
SB12
SB14
SB16
SB18
SB20
107
108
photo-electron signal
Figure 3.15: Acquired RABITT trace for HHG in Xenon. The sidebands are indicated by white
boxes and indexed according to the multiple fundamental photon energy they are associated with.
after neutral density filters in focus on a camera sensor. Therefore, a plane mirror can be
moved into the beam directly after the focusing mirror. It then guides the focused laser beams
through a window outside the vacuum environment.
Measurement
The experiment was performed for high harmonics generated either from Xenon or Krypton.
The results for HHG generated from Xenon are most relevant for the scope of this thesis and thus
only those are discussed. Fig. 3.15 shows an energy-calibrated RABITT spectrogram obtained
by performing a delay scan and acquiring a photo-electron signal with the electron time-of-
flight spectrometer. Spatio-temporal overlap is indicated by the presence of sidebands
24
. The
expected beating due to the quantum path interference (see fundamental concept introduced
in ch. 2.1.4) during the presence of the sidebands is apparent.
Retrieval of the pulse structure
As the shown trace contains the beating signal of the sidebands and their appearance and
disappearance by delay, it appears to be suitable for retrieval of properties using either the
RABITT or CRAB concept. However, the attempts to reconstruct the pulse using a CRAB
algorithm developed to retrieve the APT’s shape, when fluctuations, for example, caused by
an unstabilized CEP occur [162], failed. A possible cause could be too weak contrast in the
beating, likely caused by delay instabilities.
Instead, a simplified retrieval approach is taken:
•
The spectral phases, shown in fig. 3.16 top-left, are retrieved only for the central photon
energies of the harmonics, by directly using the sideband oscillation’s phase. The spectral
24
i.e., electron energies between those associated to the XUV spectrum of the (odd) high harmonics (i.e.,
between the main bands)
62
3.3 XUV pulse characterization
20 25 30
photon energy (eV)
10
0
spectral phase relative to
harmonic order 11 (rad)
spectral phases
inter- & extrapolation
20 25 30
photon energy (eV)
0.0
0.5
1.0
amplitude (arb. u.)
spectral amplitudes
corrected for Au mirror
5 0 5
time (fs)
10
0
10
electric field
(arb. u.)
retrieved XUV pulse:
electric field
5 0 5
time (fs)
0
100
200
intensity (arb. u.)
retrieved XUV pulse:
intensity
Figure 3.16: Retrieval of the XUV APT using the spectral phases shown in the top left in
combination with the for one near-normal incidence reflection on a gold-coated mirror corrected
spectral amplitudes in the top-right subfigure. The gray dashed line indicates the original spectral
amplitudes. The resulting pulse is given in terms of its electric field (bottom left) and intensity
(bottom right). The gray dashed line indicates the Gaussian envelope of the retrieved intensity
profile.
63
3. Development & characterization of an HHG beamline for non-linear XUV optics and
coherent diffractive imaging
phases in between the data points are interpolated with linear segments. The spectral
phases below the lowest harmonic order visible in the electron spectrum are extrapolated.
•
The spectral amplitudes are obtained from the mean electron spectrum acquired at a
relative delay of 0
fs
(most left trace in spectrogram in fig. 3.15). Since still, minor
sideband contributions are visible, the harmonic peaks are fitted using Gaussian functions.
The resulting fitted spectral amplitudes are shown in fig. 3.16 top-right. In addition, the
spectral amplitudes are corrected for the reflectivity of the gold-coated mirrors on the
delay stage25.
Combining both, spectral phases and spectral amplitudes, the APT shown in terms of the
electric field in fig. 3.16 bottom-left and in terms of intensity in fig. 3.16 bottom-right are
retrieved. It is to be noted that similar to the single pulse CRAB retrieval performed by
Osolodkov et al. [162], the retrieved pulse with an FWHM of 3
.
1
fs
of the pulse envelope
is extremely close to its Fourier transform limit. Taking the conclusions from Osolodkov et
al. [162] into account, the retrieved APT likely underestimates the real pulse duration. The
retrieved attosecond pulses within the envelope have a FWHM duration of 420as.
Summary
In summary, with the temporal characterization experiment, the presence of an attosecond
pulse train (APT) could be confirmed. The results show the duration of the individual
attosecond pulses to be 420
as
and the duration of the APT’s envelope to be 3
.
1
fs
in FWHM.
As can be seen from fig. 3.16 (d), this corresponds to 5 attosecond pulses making a notable
contribution. Due to a simplified retrieval method, the pulse trains envelope duration is likely
underestimated. Nevertheless, it can be used as an approximation for the analysis of CDI
experiment in chapter 5.
Concerning the CDI experiment, which will be introduced in the next chapter, the presented
characterization experiment confirms that in practice the delay unit is capable of resolving
dynamics in the attosecond domain. In fact, it has been shown that the time resolution is
sufficient to resolve a beating at at least twice the laser frequency in a long term measurement.
25
Further experiments relevant to this thesis were performed using a single near-normal incidence reflection
on a
B4C
-coated mirror. To retrieve a pulse that can be expected in these experiments, only the gold mirror’s
reflectivity was corrected for, but not the
B4C
mirror’s. Note: the correction has only minor influence on the
retrieved pulse.
64
4
Development of an NIR-XUV
pump-probe coherent diffractive
imaging experiment with sub-fs
resolution
The presented thesis aims to investigate dynamics in helium nano-droplets induced by a
near-infrared light pulse through coherent diffractive imaging. The HHG beamline discussed
in the previous chapter can provide the NIR and the XUV light for inducing the dynamics and
probing them by CDI, respectively. A former experiment of such kind on helium nano-droplets
revealed dynamics on timescales of a few tens of femtoseconds [35]. However, this experiment
ultimately aims to resolve NIR-induced changes in the droplets, e.g., induced transparency
in the XUV regime on the sub-femtosecond scale - similar to a recent observation in solids
[163]. Enabling pump-probe CDI experiments of isolated nano-droplets at unprecedented
time-scales, the setup differs significantly from previous NIR-XUV CDI setups [35, 164]. A
high-resolution delay stage for the generation of sub-fs delays between the NIR and XUV pulse
in combination with a normal incidence focusing geometry has been developed (see chapter
3.3.3). In addition to the delay unit, in particular, the NIR pump-pulse delivery, the tight
focusing and extensive options for characterization of the NIR and XUV pulses that enable an
improved interpretation of the data distinguish this setup from previous ones.
Fig. 4.1 provides a schematic overview of the main components of the setup. The XUV
beam, originating from the NIR-driven HHG process, arrives with the copropagting NIR at
the beamline diagnostic chamber’s (see chapter 3.3.2) entrance iris. A combination of filters
and apertures prepare the incident light into two isolated beams, separating the NIR from the
XUV light. After passing through a tube, the beams arrive at the delay stage and focusing
mirrors from which they propagate through several stray light apertures into the interaction
region. There the light intersects with a beam of nano-droplets. An ion mass spectrometer
combined with an MCP-Phosphor stack allows for the coincident detection of generated ions
65
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.1: (a) Schematic drawing of the experimental setup in the context of the beamline; (b) Layout of the experimental setup; The relevant units in (b) are
indicated with dashed boxes and referred to in subsequent paragraphs. Components not used for CDI are shown in light gray tones; The beams enter from the
bottom right, are separated and focused into the interaction region by mirrors in near-normal incidence on a delay stage. In the interaction region the target is
introduced sideways, the wide-angle diffraction image is collected by an MCP detector and ions are collected from the top by a time of flight mass spectrometer.
The undiffracted beam is guided into a spectrometer setup.
66
4.1 Vacuum apparatus
and diffracted light. The undiffracted beam passes through a hole in the diffraction detecting
MCP-Phosphor stack into an XUV spectrometer providing additional coincident spectral
information on a shot-to-shot basis.
The overall experimental assembly can be divided into the vacuum apparatus (see section
4.1) that encapsulates helium nano-droplet source described in section 4.2 and the in-vacuum
setup described in 4.3 together with out of vacuum components necessary for CDI. This
chapter puts special emphasis on the setup’s capabilities regarding XUV & NIR alignment
and characterization, which discussed separately in section 4.4. Section 4.5 highlights the
design aspects of and section 4.6 presents the characterization performed for the focusing optic.
Finally, the procedures to establish spatial and temporal overlap of both pulses are introduced
in section 4.7.
4.1 Vacuum apparatus
Due to strong absorption in air, working with XUV radiation requires working in a vacuum
environment: At an exemplary photon energy of 20
eV
and a pressure of 1 mbar air on a
pathlength of 10
cm
more than 99
.
9% of the light are absorbed. Reducing the pressure to
10
−4mbar
more than 99
.
9% are transmitted [165]. Moreover, the high-voltage operated MCP
detectors used with the ion and XUV spectrometer as well as the scattering detector require an
environment with pressures below 10
−5mbar
for safe operation. Achieving the desired pressure
is complicated by the high gas load introduced by the helium nano-droplet source into the
system. Contamination of the vacuum environment can affect the performance of the helium
nano-droplet source from less effective cooling up to freezing the valve during operation and
degrade the focusing mirrors. Carbon contamination is a common problem for XUV mirrors
operated under vacuum conditions [166]. Due to the dissociation of hydrocarbons indirectly
induced by XUV photons on the mirror surface, carbon layers grow on it, scaling with the
fluence of XUV light to which the mirror surface is exposed [166]. To reduce the effects of
contamination, without additional gas load in the chamber, pressures below 10
−7mbar
are
targeted.
Achieving high vacuum (10
−3
to 10
−7mbar
) to ultra-high vacuum (10
−8
to 10
−11 mbar
)
conditions is facilitated by high pumping speeds that e.g. can be achieved by turbo-molecular
pumps [167]. Their working principle requires molecular flow, i.e., a mean free path length of
the gas molecules larger than distances between flow guiding components within the device
[167]. This condition can be met using an additional pumping stage that first generates a
pre-vacuum on the order of 10
−1
to 10
−3mbar
at the outlet of the turbo-molecular pump. The
experiment’s vacuum setup is based on such a scheme.
Fig. 4.2 depicts the layout of the vacuum chamber setup: the main experimental chamber
that houses the optical setup (section 4.3) and diagnostics (section 4.4) is evacuated using a
turbo-molecular pump that achieves pumping speeds of up to 1300
l/s
(
He
) [168]. The helium
nano-droplet source (section 4.2) that introduces the nano-droplet beam (in fig. 4.2 from the
top) causes a significant gas load. Therefore, the region where the droplet beam is introduced
to the vacuum is separated by a differential pressure stage. It consists of a plate on which a
cone with a hole at the tip, a so-called skimmer, is mounted to ’skim off’ not desired parts of
67
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.2: Schematic drawing of a horizontal cut through the vacuum apparatus that is attached
to the HHG beamline: the main experimental (light green) and the skimmer (light blue) vacuum
chambers are connected through a fixed conical skimmer combined with an adjustable skimmer slit;
Their vacuum generation (grey) consisting of turbo-molecular and pre-vacuum pumps is separated;
In the skimmer chamber the helium nano-droplet source, surrounded by a heat shield (brown),
introduces a beam of nano-droplets (blue), which is reduced by the skimmers to a stream of isolated
droplets reaching the interaction region; For orientation, parts of the optical setup from fig. 4.1
are shown.
the droplet beam. This part of the apparatus, where the nano-droplet source is attached, is
called the skimmer chamber. It features two additional turbo-molecular pumps - each achieving
pumping speeds up to 390
l/s
for helium [169]. Their pre-vacuum is also generated separately
from the main chamber’s turbo-molecular pump and designed to handle the high gas load.
The specific apparatus used for this experiment is based on the same set of chambers
within which the temporal characterization of the generated XUV pulses has been realized (see
chapter 3.3). The apparatus, shown in fig. 4.3, is built upon a spherical stainless steel chamber.
Its DN 250 CF flanges facilitate an extension of the chamber with additional components as
needed, interface with the turbo-molecular pump, and with the skimmer chamber extending
to the He nano-droplet source that can be aligned along all three axes. The top flange is used
to access the interaction region with the ion time of flight mass spectrometer mounted on a
three axes manipulator and at the same time to capture the diffraction patterns with a camera.
Furthermore, 4 DN 40/63 CF ports provide a direct diagonal view of / access to the interaction
region aligned with the chamber’s center. Two of them are actually used for their viewing
angles. The other two accommodate a leak valve, vacuum gauge, venting valve, and electric
feed-throughs. The experiment is enormously profiting from many strategic placed viewports
that not only allow to monitor detectors and screens but also critical movable components.
68
4.2 Generation of helium nano-droplets
Figure 4.3: 3D CAD rendering of the vacuum apparatus; green and blue pointers indicate parts
relating to the main chamber and skimmer chamber, respectively; orange arrows highlight movable
axes enabled by out of vacuum manipulators or a shiftable flange: aside from the nano-droplet
source, also the ion spectrometer and XUV spectrometer’s MCP are mounted to such devices.
4.2 Generation of helium nano-droplets
For the presented experiment, large helium nano-droplets are produced through supersonic
expansion of pressurized cooled helium gas through a pulsed valve and attached nozzle into
a vacuum. By cooling the valve to 5
.
2
−
8K and supplying helium with a pressure of min.
20bar, during the isentropic expansion of the helium (supercritical fluid) into the vacuum, it
expands and cools down below the critical point of 2
.
27
bar
/5
.
2K (
4He
) [170] into the liquid
phase. Reaching the vapor pressure, as illustrated in fig. 4.4 (a), gas bubbles form within the
liquid helium causing it to break up into droplets. While propagating towards the interaction
region, they further cool down by evaporation of helium atoms. Thereby, temperatures in the
droplet down to 0
.
37K (
4He
) [77] are reached quickly. Droplets with more than 10
5
atoms are
formed. [83, 141]
In essence, the used helium nano-droplet source, which is schematically shown in fig. 4.4
(b), consists of a pulsed valve (Even-Lavie valve) attached to a cooling head. A temperature of
40K is reached by a first cooling stage connected to a shield protecting the second stage and
the mounted valve from thermal radiation. The second cooling stage achieves temperatures
down to 4 K (at the cooling head). The temperature at the nozzle is mainly limited through the
power of the cooling head and the generated heat due to the motion of the pulsed valve. The
valve temperature is controlled through a heater element and the feedback from a temperature
sensor. In addition, also the shield’s temperature can be measured. The source used with this
experiment was built by Bruno Langbehn (TU Berlin) in the course of his Master’s thesis. It
69
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.4: (a) sketch of the generation of helium nano-droplets by liquid fragmentation: the
break up of liquid helium due to formation of gas bubbles into liquid droplets (reprinted section
from [77, fig. 6]); (b) CAD of the helium nano-droplet source (modified from [141]) with a schematic
drawing of a cut of the heat shield and outgoing droplet beam.
is characterized to produce droplets with a mean radius of 273
nm
at a nozzle temperature
of 5
.
2K and a backing (stagnation) pressure of 80
bar
when operated at a repetition rate of
10Hz. [141]
4.3
Optical setup for pump-probe coherent diffractive imaging
The optical setup, especially the mirrors delaying the NIR and XUV pulses, demand high
mechanical stability with minimized vibration to achieve a temporal resolution in the sub-
femtosecond regime. Already displacements on the nanometer-scale may affect the pump-probe
experiment. Most optical components are mounted on solid Aluminum breadboards (in-
vacuum) or out-of-vacuum on optical tables. In the main experimental chamber, which houses
the delay stage, external vibrations are dampened by 2mm thick FKM rings between the
breadboard and the chamber. In addition, the experimental chamber is located on a part of
the building’s foundation which is isolated from the part that supports the strongly vibrating
fore-vacuum pumps.
The optical setup of the HHG beamline has already been described in chapter 3. In
this section, the optical setup specific to this experiment will be explained. An overview is
depicted in fig. 4.1. There, grey boxes highlight the setup’s "building blocks" by which a
more detailed description in this section is organized: "beam conditioning", "focusing & delay
unit", "coincident ion & diffraction detection scheme", "XUV spectrometer", and "straylight &
background reduction".
4.3.1 Beam conditioning
Like for the experimental setup used for temporal characterization of the XUV pulses (see
chapter 3.3.4), the design of the pump-probe experiment relies on the delivery of separated, well-
defined pump and probe beams. Therefore the incident co-propagating NIR and XUV pulses
need to be spatially isolated and shaped into suitable beams. Based on the beam conditioning
required for the temporal characterization experiment, special pump-probe enabling filters,
a filter wheel, and (aperture) masks are added to the spectrometer chamber setup of the
beamline that was previously described in chapter 3.3.2.
70
4.3 Optical setup for pump-probe coherent diffractive imaging
Figure 4.5: (a) Schematic of the beam conditioning setup: XUV and co-traveling NIR reach
the motorized entrance aperture from the right (see colored arrows). After two insertable XUV
pump-probe filters (i.e., filters that have a gap to let some NIR through) and a variety of single-
color filters provided in a filter wheel, the NIR arm is defined by one of two aperture masks in
combination with NIR passing filters. When the XUV diode is inserted, the XUV arm is blocked.
Orange arrows indicate movable axes. In (b), a three-dimensional CAD rendering illustrates the
splitting of the incident light into an NIR pump and an XUV probe arm.a
a
3D model for the "entrance iris" is used under the license CC BY-NC-SA 3.0 (
www.creativecommons.
org/licenses/by-nc-sa/3.0/
) by rkagerer, published on October 26, 2017 at
www.thingiverse.com/thing:
2607315
Fig. 4.5 depicts the separation scheme: The central part of the incident co-propagating
NIR and XUV beam contains the majority of the XUV and is chosen to produce an isolated
XUV beam. Due to the more substantial divergence of the NIR, its beam diameter is much
larger than the XUV’s. Thus an off-axis region of the incident beam containing only residual
XUV
1
is used to isolate an NIR beam. On one part, the NIR is removed by filtering the
incident light using a 100
nm
thin Al filter foil; alternatively, a 150
nm
Sn filter foil may be
used to select a narrower frequency band of the XUV light. Spaced by approximately one
millimeter, the designated NIR beam passes through a hole next to the filter. In a second
stage, vice versa, any residual XUV is removed from this designated NIR beam by passing it
through a (100 ±15) µmthin glass slide2.
Finally, the NIR half of the beam is cropped by a circular aperture on which the glass
slide is mounted. Multiple such apertures are available to control the amount of NIR light
entering the experiment roughly. Moreover, fine adjustments can be made using the motorized
iris at the entrance of the spectrometer chamber. The XUV photodiode in the beamline’s
spectrometer chamber can block the XUV beam to deliver only the isolated NIR beam into
the experiment. Vise versa to block the NIR beam either an available pump-probe filter mount
or the motorized iris can be used.
For using only the XUV or the only the NIR beam during the experiment, a filter wheel
holds up to 6 filters. Two are neutral density filters used for NIR alignment operations. The
others are Zr (150
nm
), Sn (150
nm
), and Al (70 & 1000
nm
) filters for transmission of 22
−
27
eV
,
15 −24eV, or 15 eV and above, respectively.3
1Residual XUV light that overlaps with the NIR light is not depicted in fig. 4.5.
2
The glass slide is made from Schott D 263 M glass. See also product website:
https://www.thorlabs.de/
newgrouppage9.cfm?objectgroup_id=9704.
3
The photon energy range is specified for 100
nm
thick filters (not concerning the used thicknesses or oxide
layers), where the transmission overcomes 10% according to [165].
71
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
In the experiment, not only an NIR-XUV but also NIR-NIR pump-probe schemes are
realized. To avoid confusion between the designated NIR beam path (with regards to the
NIR-XUV scheme) and the actual NIR beam, the designated NIR beam path will be called
the NIR arm. Correspondingly the designated XUV beam path will be called the XUV arm.
Hence, in an NIR-NIR pump-probe situation, one NIR beam is delivered via the NIR arm and
the other via the XUV arm.
4.3.2 Focusing and delay unit
After conditioning, the beams enter the experimental chamber and reach the focusing optics
combined with the delay unit (see chapter 3.3.3) under an incident angle of approximately 12
◦
to the surface normal. Each beam is hitting one of the focusing mirrors, as shown in fig. 4.6
(b), which are mounted directly to the delay unit. With a focal length of only 178
mm
, those
can achieve XUV focus sizes on the order of few micrometers. Depending on the specific HHG
parameters and surface quality of the mirrors, high XUV intensities in the interaction region
ranging from 10
13
to 10
14 W/cm2
are possible. With an aperture mask diameter of 4
mm
in
the beam conditioning, NIR focus diameters of several tens of micrometer and intensities on
the order of 10
14 W/cm2
can be achieved. The focusing optic design and focus characterization
are described in sections 4.5 and 4.6 respectively. In the following, effects arising from such a
focusing layout with the delay unit are discussed.
The focusing optic’s design is associated with a Rayleigh length of the XUV of at least
250
µm
(see section 4.5). With anticipated delay ranges of
±
70
fs
, the XUV arm mirror, which
is translated along its surface normal to introduce the delay, is moved by
±
10
µm
. Thereby
the XUV focus is shifted along its propagation axis by approximately the same amount. The
influence on the interaction region is negligible because the shift of the XUV’s focal plane (i.e.,
peak intensity) is small compared to the Rayleigh length.
Due to the 12
◦
incidence with respect to the surface normal, setting a delay by translation
of the XUV arm mirror along the surface normal introduces a lateral shift. It limits the delay
range, in which spatial overlap between pump and probe arm is maintained, to approximately
±
100
fs
. For long delays, the overlap can be re-established using a closed-loop controlled tilt
axis of the NIR arm.
A major concern for this experiment is the temporal resolution between pump and probe
pulse that can be achieved in the interaction region. The short focal length causes a noticeable
non-colinearity of the NIR and XUV beams spaced by ca. 5
mm
on the mirror and overlapping
in focus. The angle between the beams of approximately 1
.
7
◦
translates to the wavefronts.
Confirmed by a wavefront propagation computation shown in fig. 4.6 (c), this introduces
a delay error across the XUV beam (assuming a waist diameter of ca. 4
µm
(1
/e2
)) of ca.
±
200
as
. In addition, the non-colinearity causes a phase drift between the NIR and XUV
along the XUV beam axis: the NIR field’s oscillation cycle measured along the XUV beam
axis is larger than measured along the NIR beam axis. The resulting variation of the delay
over the XUV focal region that is a combination of both effects is plotted in fig. 4.6 (d). The
additionally shown overlay of the contours of the simulated XUV beam’s intensity profile
suggests that the error in delay can be reduced when being sensitive to only the highest XUV
intensities. Another way to reduce the drift of the delay across the interaction region is to
72
4.3 Optical setup for pump-probe coherent diffractive imaging
Figure 4.6: (a) top view of the relevant beam path section; (b) front view photo of the focusing
mirrors on the delay stage: the beams on the XUV arm (left) and NIR arm (right) reach the
focusing mirrors via an aperture mask with a diameter of 4
mm
for both arms during the mirror’s
alignment;
a
(c) wavefronts of the overlapping NIR (red/black) / XUV (violet/white) light in
focus spaced by the NIR wavelength showing their relative tilt (ca. 1
.
7
◦
) caused by the non-
colinear pump-probe geometry that introduces uncertainty in delay across the beam (spanning
approximately 400 as over the XUV focus diameter (1
/e2
)). (d) map of the spatial dependent
delay due to non-colinear pump-probe geometry geometry in combination with contours (black) of
the XUV intensity profile based on wavefront propagation simulation.b
a
For this photo two images are overlayed, one photo was taken with an optics cleaning tissue in front to
showing the beam profiles and the other without to capture the mirrors. The tissue causes the structure on the
beam cross-sections. The distance between the mirrors was reduced after the photos were taken.
b
Wavefronts have been calculated using the WavePropaGator framework [171]. Mirror has been simulated
with a thin lens. Assumption of ideal Gaussian beams and focusing optics may yield too small beam diameters
in focus.
73
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.7: Schematic cut through the coincident detection scheme revealing the internal layout
of the ion time of flight mass spectrometer and the geometry used to acquire diffraction patterns.
MCPs
effective area diameter 77 mm
center dead area diameter 5 mm
center hole diameter 3 mm
single channel diameter 25 um
bias angle 8 deg
open area ratio 60 %
Phosphor screen
effective area diameter 75 mm
phosphor type P43
decay time (10%) 1 ms
peak emission wavelength 545 nm
Table 4.1: Technical specification of the chevron-type MCP-phosphor stack used to acquire the
diffraction pattern [172].
confine the interaction region along the beam axis like it can be done with the adjustable
skimmer slit in front of the nano-droplet source.
4.3.3 Coincident ion & diffraction detection scheme
The interaction region is defined by the diameter of the sideways introduced nano-droplet
beam and the size of the XUV’s focus. When the XUV pulse is intersecting a droplet, light is
diffracted immediately. Then, the droplet disintegrates into electrons and ionic fragments. In
the experiment, those ionic fragments are measured coincidentally with light that is diffracted
from the very same droplet.
The diffracted XUV light yields a pattern amplified and converted to visible light using a
chevron-type MCP-phosphor stack (see specification in tab. 4.1 & working principle in chapter
3.3.1) with an effective diameter of 75
mm
. As depicted in fig. 4.7, it is placed at a distance of
38
.
4
mm
behind the interaction region. Thus it collects light from diffraction angles, defined
as half angles, up to approximately 44
◦
. The image visible on the phosphor screen is then
74
4.3 Optical setup for pump-probe coherent diffractive imaging
captured by a camera from the top through a protective NIR-blocking filter (Schott BG39)
and via a mirror under 45
◦
. Since the MCP-phosphor stack cannot be exposed to the direct,
intense XUV beam, it has a central hole through which the undiffracted light passes. It is
motorized along its vertical and horizontal axes to enable a proper alignment relative to this
undiffracted XUV beam.
The ionic fragments are acquired using an ion time of flight mass spectrometer. Following
the already introduced principle of time-of-flight spectrometers (see chapter 3.3.1), temporal
dispersion of the charged particles occurs due to field-free drift with different velocities. Their
signal is detected using an MCP detector, which generates a current signal recorded as a
voltage via a 50 Ω resistance by a digitizer. However, opposite to an electron time of flight
spectrometer, where intrinsically the electrons have different kinetic energy by which they
are resolved, in an ion time of flight spectrometer, the ion fragments are in addition strongly
accelerated to velocities depending on their charge and mass. Therefore they are exposed to a
static electric field generated by applying a voltage between two plates. Assuming the ions are
generated in the center between those plates spaced by a distance
d
, they exit the acceleration
phase after the time ti,acc with velocity v.
ti,acc(m, q) = √︄d2m
qU v(m, q) = √︄2qU
m.
Their velocity depends on the voltage applied across the plates
U
as well as the ion fragments
mass
m
and charge
q
. The drift time
tdrift
depends on the drift velocity (
v
) and the drift
length
L
. In addition, for the total time of flight
tflight
, the time needed for post-acceleration
behind the drift tube
tp,acc
is considered. Though, typically, the duration of acceleration is
small compared to the drift time and can be neglected:
tflight(m, q) = tdrift +ti,acc +tp,acc ≈tdrift =L
√2U√︃m
q.(4.1)
The specific ion time of flight mass spectrometer used for this experiment has been developed in
the scope of the Master’s thesis of Andrea Heilrath [173]. As shown by fig. 4.7, it is introduced
into the interaction region from the top and consists of a drift tube with a length of 11 cm.
In front of the drift tube, two plates - the so-called extractor (above the interaction region)
and repeller (below interaction region) are spaced by 12 mm and provide a static electric field
to accelerate the ions towards the drift tube and MCP detector. Orthogonal to the XUV
beam propagation axis, the extractor has a 1
×
18 mm slit that serves as an entrance aperture
to confine the origin of the measured ions along the beam propagation axis to roughly the
focal volume. The ion mass spectrometer can be aligned to the interaction region through
translation along all three dimensions. In addition, the time of flight spectrometer can be
completely pulled out of the interaction region vertically.
4.3.4 XUV spectrometer
The spectrometer extends the coincident detection scheme by acquiring the single XUV pulse
spectrum associated with a potential diffraction pattern and ion mass spectrum. Since the
expected nano-droplet diameter of several hundred nanometers (see section 4.2) is small
75
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.8: (a) Schematic drawing of the XUV spectrometer: the incident XUV propagates
through a slit onto a rotatable grating; the spectrum is detected with an MCP-phosphor stack and
acquired with a camera; (b) wavelength-dependent diffraction efficiency of the grating in the in (a)
shown configuration measured by the PTB.
compared to the XUV focus size in the micrometer regime, most XUV light does not
interact with the nano-droplets. Accordingly, the obtained spectrum approximates the spectral
composition of the light that interacted with the droplet.
The spectrometer, shown in fig. 4.8 (a), consists of an entrance slit of 250
µm
width and a
1
/
2
′′
bare gold-coated blazed grating (4
.
3
◦
blaze angle) with 600 lines/mm where the incident
XUV light is reflected under a grazing angle of ca. 13
◦
. The slit and grating are mounted on
a combined platform, allowing the slit and grating position adjustment relative to the XUV
beam and pulling the slit and grating out of the beam. The grating’s rotation can be adjusted
as well. The resulting spectrum is then detected by an MCP-phosphor stack combined with a
camera with a protective NIR filter (Schott BG39). Using a manipulator, its position can be
horizontally aligned to the spectrum’s position.
The spectrometer’s primary purpose is to resolve the relative amount of light contained in
each harmonic. To approximate the MCP-detector’s response to be linear, it is operated with
low amplification. At the same time, images acquired from the MCP’s phosphor screen are
taken with a maximum gain of the camera electronics. Furthermore, the intensity distribution is
corrected for the diffraction efficiency of the grating characterized by the PTB. The diffraction
efficiency characterization is presented in fig 4.8 (b).
4.3.5 Straylight & background reduction
The CDI experiment aims to capture light that is diffracted from the sample solely. However,
an MCP detector is also sensitive to ions and diffuse or direct scattered light from other origins,
so-called straylight.
Mirrors and any edge cutting into the XUV beam are a significant source of diffuse straylight.
It can be reduced by blocking paths other than the incident beam’s using several apertures.
Two adjustable and one fixed aperture(s) are positioned after the focusing mirrors, offering a
variety of aperture sizes. The first wraps more tightly around the beams and may even be
a source of low amounts of additional straylight. The second blocks potentially generated
76
4.4 Diagnostic tools
Figure 4.9: 3D CAD of the focus tool assembly; movable axes are indicated by orange arrows.
straylight from the first aperture. To block diffuse light that passes the aperture panels
sideways or through another aperture close to the selected one, the straylight aperture setup
is completed by a large area blocking, fixed-size aperture just before the interaction region.
Any interaction of the XUV light with matter, like, for example, the mirrors or gas, yields
ions. Partially, the shielding for straylight also shields the detector from ions. Those ions
generated far from the diffraction detector reach it on timescales of several hundred nano- to
microseconds. Their detection can be prevented by operating the detector in a gated scheme
by switching the gain-controlling voltage between the MCP plates and the phosphor screen
fast between a low- and high-gain value. The high gain mode can be enabled for only some
hundred nanoseconds.
4.4 Diagnostic tools
Performing the experiment requires, among others, an optimized XUV focal spot, i.e., precisely
aligned focusing optics, and overlap between XUV & NIR pulses in the focal plane. Furthermore,
to ensure proper conditions in the interaction region during the experiment and a proper
interpretation of the results, XUV and NIR pulses in focus must be well characterized.
Therefore, the experimental setup for performing pump-probe CDI is complemented by
a set of diagnostic instruments: a "focus tool" (section 4.4.1) and general "beam diagnostic
behind the interaction region" (section 4.4.2).
4.4.1 Focus tool
The focus tool assembly is shown in fig. 4.9. The target, i.e., phosphor screen or fluence
mapping grating (see below), can be translated along all three axes using compact open-loop
controlled linear stages. The tool is moved into the interaction region using a linear stage with
85 mm range. The long-range stage enables support for up to three targets with dimensions of
up to 12
×
12
mm
. The integration of the tool with the previously introduced optical setup
(section 4.3) can be seen with fig. 4.1: The targets are retracted from the interaction region
toward the helium nano-droplet source during CDI measurements.
77
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Phosphor type P1
Composition Zn2SiO4: Mn [174]
Activator concentration 5−7% [175]
Quantum efficiency 0.9to 1.6(estimate) [143]
Central wavelength 525 nm [174]
Decay time (10%) ca. 12 ms [175]
Manufacturer Phosphor Technology Ltd
Table 4.2: Properties of the phosphor used on the focus screen; Decay time refers to the duration
in which the signal decays to 10% of its original value.
Finding the focal plane, aligning the focusing mirror’s pitch & tilt axes, and overlapping the
NIR & XUV beams in the focal plane relies only on qualitative features, like a minimal spot
size, but not the actual spot size. Hence, a phosphor-coated screen with optimized quantum
efficiencies for XUV, which is also suitable for NIR, is sufficient for these tasks.
This experiment employs a P1-type phosphor (
Zn2SiO4: Mn
). An overview of its properties
is presented in table 4.2. Experimentally, in comparison with P46 and
BaO ·6Al2O3: Mn
, the
P1 phosphor exhibited the highest quantum efficiency (QE). According to literature, for the
phosphors activator concentration of 5
−
7%, a QE of 0
.
9to 1
.
6is expected in a spectral range
of 15 to 25 eV [143, 175]. Its central emission wavelength of 525 nm passes the NIR blocking
filters (Schott BG39) used to protect cameras during this experiment from NIR radiation.
The phosphor-coated screen in the focal plane can be observed outside of the vacuum
through one of the diagonal flanges on the main chamber with a direct view of the interaction
region (see section 4.1). There a long-range microscope with a working distance of 341 mm and
a specified resolution limit of 13
.
34
µm
[176] is used in combination with a camera-based on
an AMS CMV2000 sensor (pixel size 5
.
5
µm
) [177] to resolve the beam profiles and positions
on the phosphor. The diagonal view (not orthogonal to the screen’s surface) reduces the
effective resolution by approximately 29% (vertically) and 7% (horizontally). In practice, using
a calibration target, up to 50 line-pairs per mm, i.e., features between 10 and 20
µm
in the
screen’s plane along the horizontal axis, were resolved.
Nevertheless, although the phosphor screen can be used for qualitative feedback, it may
not produce an accurate spatial image of the focus due to not known non-linear behaviour
and saturation effects. The NIR beam profile can be measured directly in a vented setup (see
also section 4.6). However, quantifying the XUV beam diameters, potentially below 10
µm
,
requires a technique applicable in the vacuum, providing a high resolution.
Utilizing the diffraction detector implemented with the optical setup (see section 4.3),
fluence mapping gratings are used to project the XUV fluence distribution close to the focal
plane, i.e., the XUV’s beam profile, onto the diffraction detector [178]. Those gratings are
essentially application-specific designed off-axis segments of a Fresnel zone plate [179]. For
the diffraction orders 1and
−
1, such grating operate effectively as a pair of lenses with focal
lengths f+1,−1depending on the designed detector distance zand magnification m[179]:
f+1,−1=±z
2m.(4.2)
78
4.4 Diagnostic tools
Figure 4.10: (a) Schematic drawing of the beam diagnostic layout with the XUV spectrometer
behind the interaction region following the diffraction detector; the orange arrows indicate movable
axes. (b-d) show the configuration for beam profile monitoring, NIR characterization and XUV
pulse energy measurement respectively. Bold arrows indicate the moved axes compared to (a).
Acting like co-existing convex and concave lenses with the same focal length, the grating yields
two images on opposite sides of the diffraction detector. One image (
f >
0) depicts the fluence
at an object distance of approximately
f
before the grating. The other mapped fluence is
an image from the virtual object plane at an object distance of approximately
f
behind the
grating. In fact, it represents the fluence distribution that would be in that virtual plane if
there would not be the grating. The normally undesired strong chromatic aberration of zone
plates [180], originating from diffraction, causes the separation of the fluence maps for each
contributing harmonic on the imaging detector. It is possible to design those fluence mapping
gratings so that well-separated fluence profiles are produced.
Practically, multiple 30 nm thick
Si3N4
membranes are provided in sub-millimeter-sized
windows of a 5
×
5
×
1mm silicon chip. Those are coated with 10 nm
Ta
. A focused ion beam
(FIB) is used to mill one grating per window into that coating. For the experiment, two chips
with several gratings of different sizes and magnifications are available. The size of the larger
gratings is about 30
×
30
µm
, and gratings as small as 10
×
10
µm
for this experiment are
available. Moving the grating along the beam to image profiles at several planes around the
focus, already small deviation of the axis of movement from the beam axis cause the beam to
quickly walk off the grating. The fluence mapping gratings for this experiment were designed
and provided by Michael Schneider (Max Born Institute).
4.4.2 Beam diagnostic behind the interaction region
Complementing the in- and near-focus diagnostic provided with the "focus tool", XUV and
NIR properties can be characterized or monitored behind the interaction region. There, the
setup can be set to the four different configurations depicted in fig. 4.10 (a) - (d). By default,
the already introduced XUV spectrometer setup (fig. 4.10 (a)) is used.
79
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
As shown in fig. 4.10 (b), removing the grating with slit from the beam path and instead
inserting a plane gold mirror behind, a phosphor-coated screen next to the XUV spectrometers
MCP-phosphor stack can visualize the XUV and NIR beam profiles. Observing the beam
profiles from both arms enables alignment operations based shadows in the beam profile. Those
include the right-left positioning of the split-mirrors in the beam (the mirror edge creates
a shadow and diffraction features in the beam profile), alignment of apertures in both the
main experimental and HHG beamline’s spectrometer chamber, as well as horizontal and
vertical positioning of the diffraction detector so that the full XUV beam is passed through
the detectors and mirror’s center hole.
By aligning the MCP detector and planar mirror to pass the full NIR beam and further
moving the phosphor screen aside, like shown in fig. 4.10 (c), the beam can be coupled out of
the vacuum. Outside the vacuum, the NIR pulses are then characterized by either a power
meter, to measure the pulse energy, or a spectrometer.
Fig. 4.10 (d) illustrates how with neither the grating with slit nor the plane mirror in the
beam path, and an alignment to pass the full XUV beam, the XUV pulse energy is supposed to
be measured. Therefore, an XUV photodiode with an aluminum filter in front in a light-tight
enclosure is placed in the beam path. However, during the experiment, because the diode’s
position could not be adjusted in the vacuum and the beam alignment was optimized for
the best focusing, it was not correctly exposed. As a consequence, no pulse energy could be
measured with it. Alternatively, the pulse energy in the interaction region can be calculated
from the known spectrum, focusing mirror’s reflectivity, and pulse energy measured in the
HHG beamline’s spectrometer chamber.
4.5 Design of the focusing optic
The diffraction signal scales with the number of photons that interact with the droplet.
Consequently, a high photon flux is required for the acquisition of high-quality diffraction
patterns. Therefore the HHG beamline layout is optimized to enable maximal intensities in
the interaction region. In order to fully benefit from the optimized layout, a tailored design of
the focusing optics is necessary.
This experiment is carried out using a near-normal incidence geometry for the focusing
optics (see overview in fig. 4.1). Having in mind that grazing incidence focusing optics are
commonly used with XUV and X-ray light because of their high reflectivity, this choice may
seem unconventional to use when aiming for high intensities. This section will show that
it is actually beneficial for the spectral XUV regime of this experiment. To make a direct
comparison towards the end of this section, considerations regarding the propagation layout,
the mirror coating, and mirror shape will be discussed for both types.
Propagation layout
A straightforward design approach to achieve high intensities in focus is to minimize the focal
length and consequently to reduce the beam’s waist as implied by Gaussian optics [181, chapter
3]. This layout follows that approach and further takes geometrical constraints into account.
80
4.5 Design of the focusing optic
Figure 4.11: Geometrical considerations for the focusing layout; (a) Contributing factors yielding
a minimal focal distance for near-normal incidence (top) and grazing incidence (bottom), see text
for numbers; (b) Minimum focal length required depending on the angle of incidence (AOI) and
other geometrical parameters for near-normal incidence setups.
The minimal focal length is determined by three limiting factors illustrated in fig. 4.11 (a)
for the near-normal (top) and grazing (bottom) incidence case.
1.
For near-normal incidence mirrors and grazing incidence mirrors, the distance of the
first component behind the mirror’s center is limited to around 50
mm
by either the
proximity of the incident beam or the dimension of the mirror, respectively.
2.
At least two straylight apertures should be used between mirror and focus. The aperture’s
distance to each other should be at least 50mm to reduce the stray light efficiently.
3.
The minimum distance of components to the interaction region is around 30
mm
due to
the dimension of the ion time-of-flight spectrometer and the required safety distance due
to the high voltage applied.
In total, a minimal distance of 130 mm has to be taken into account.
The disadvantages of grazing incidence optics at short focal lengths will be discussed later
on. At this point, the reason for choosing a slightly longer focal length for the near-normal
incidence focusing optics chosen for the experiment will be explained.
For near-normal incidence setups, the unfocused beam is delivered to the focusing mirror,
passing the diffraction detector next to it. This scheme is depicted in fig. 4.11 (b): the detector
radius
rdet
(including the holder) defines the distance of unfocused and focused XUV beam at
the detector’s position (
rdet
+
dbeam/cos
(
α
)). The minimum focal length
fXUV
depending the
angle of incidence (AOI) is then given by:
fXUV =rdet +dbeam/cos 2α
tan 2α−ddet (4.3)
Thus, the shorter the focal length
fXUV
, the larger the AOI
α
on the focusing mirror, the
more challenging it is to fit the single-shot spectrometer and diagnostic behind the detector
81
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.12: Reflectivity of
B4C
depending on wavelength and angle of incidence for (a) s-
polarization and (b) p-polarization of the incident beam (reprinted from [159]).
within the constraints of the available vacuum chamber. Taking these spatial constraints into
account, an AOI of 8to 14
◦
and hence, according to eq. 4.3
4
, a minimum focal length of 17
cm
seems to be feasible for this experiment.
Mirror coating
The mirror design consists of a substrate material with surface specification (shape and others)
and a coating. Beside the substrate’s surface roughness, mainly the coating determines the
reflectivity of the focusing optics. It can be realized using either single-layer or multi-layer
designs. As shown in fig. 4.12, with a single layer coating of
B4C5
, one achieves an even,
constantly decreasing response over the spectral range of 50 to 80 nm (ca. 15 to 25 eV) for
near-normal incidence with reflectivities in the range of 15 -35 % [159]. In contrast, for grazing
incidence, meaning close to the angle of total reflection, 80 % and above are possible. In
near-normal incidence, the reflectivity in the XUV regime depends strongly on the chosen
material. Typically, the reflectivity achievable in near-normal incidence is clearly below a
grazing incidence geometry [165]. Commonly, to increase the low reflectivity in near-normal
incidence, multi-layer coatings exploiting an interferometric approach can be applied to the
design (see also [132]). For this spectral regime, only few multi-layer coatings known to the
author that support the required bandwidth (50 -80 nm) potentially yield notable improvement
[182, 183]. Because of the added complexity to the manufacturing and occurring phase-shifts
affecting the attosecond pulse train structure, the focusing mirror of this experiment uses a
single layer coating of B4C.
4
Result is computed with
rdet
= 65
mm
(i.e., detector radius including the mounting),
dbeam
= 11
mm
and
ddet = 40 mm. In addition, a ±2◦tolerance for manufacturing is included and thus αis set to 10◦.
5for alternative materials see e.g. [132, fig. 8.5 (a)].
82
4.5 Design of the focusing optic
Figure 4.13: Primary aberrations occurring when focusing with spherical and toroidal mirrors:
(a) spherical aberration due to large beam diameter on a spherical mirror; Light from outer mirror
regions focuses at a different distance than from inner regions. (b) Astigmatism due to not-normal
incidence on a spherical mirror; The focus in sagital and tangential plane occur at different distance
from the mirror. (c) Coma due to large beam diameter on a toroidal mirror; Light from outer
mirror regions, partially, does not overlap with the focus produced from inner mirror regions.
Figure 4.14: Geometry of a toroidal mirror; the mirror’s shape is defined by two radii one in the
sagital plane (blue) and the other in the tangential plane (red). The incoming beam’s propagation
axis lies within the tangential plane and reaches the mirror under the angle
α
. (modified from
[184] with permission from Standa Ltd.)
83
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Mirror shape
The shape of the mirror surface is a compromise between optimized focusing, manufacturability,
and availability. While an ideal point-to-point focusing can be realized with an ellipsoidal shape,
its precise manufacturing is highly challenging and most expensive. For many applications,
cheaper and easier producible spherical or toroidal shapes provide sufficient focusing quality.
The achievable focus quality is reduced for spherical mirrors by spherical aberration or not
normal incidence or large beam diameters yielding astigmatism, illustrated in fig. 4.13 (a) and
(b), respectively. A toroidal mirror can correct for astigmatism and thus is suitable for grazing
incidence (large AOI) designs. While a spherical mirror, resembling a surface section of a
sphere, has one radius of curvature (ROC), a toroidal mirror, resembling a surface section of a
toroid, has two ROCs defined in separate planes. Those are shown in fig. 4.14: the tangential
plane is defined as the plane of the AOI. The sagittal plane is oriented orthogonal to the plane
of the AOI. Similar to spherical mirrors, large incident beam diameters relative to the sagittal
radius yield a strong aberration, the so-called coma. There, as sketched in fig 4.13 (c), light
from the mirror’s outer areas is primarily not focused on the focal spot but a region beside it,
resulting in a V-shaped beam profile in the focus. Thus effectively, the light transmission into
the focal spot is reduced, and hence the achievable intensity is decreased.
The ROC rof a spherical mirror depends only on the anticipated focal length fXUV :
r= 2fXUV .
For toroidal mirrors the design depends in addition on the desired angle of incidence relative
to the surface normal (AOI) α[185]:
rsag = 2fXUV cos(α)rtan =2fXUV
cos(α).(4.4)
As can be seen from the toroidal mirror design equations, the sagittal radius
rsag
is always
smaller or equal to the tangential radius
rtan
. Towards larger AOI, it shows a significant
decrease. Hence, for a constant incident beam diameter, a near-normal incidence design (small
AOI, large
rsag
) will exhibit less coma than a corresponding grazing incidence (large AOI,
small rsag) setup for a given focal length.
To illustrate this effect, fig. 4.15 shows the results of two ray-tracing simulations, both for
a focal length of 18 cm, one with 10
◦
AOI and the other one with 80
◦
AOI (equals 10
◦
grazing
incidence) with an incident beam diameter (1/e2) of 4 mm.
Near normal incidence compared to grazing incidence
As shown in fig. 4.12, in near-normal incidence reflectivities in the XUV regime significantly
depend on the specific wavelength. In that scope, the in fig. 4.15 shown ray-tracing simulation
can be taken one step further by accounting for reflectivity. A comprehensive investigation of
the achievable peak intensities for a spectral range of 40 -60 nm (ca. 21 -31 eV) is shown
in fig. 4.16. Down to approximately 44
nm
(28
eV
), a near-normal incidence geometry is
beneficial over the grazing incidence approach. For those wavelengths, at smaller AOIs, the
larger sagittal radius (with respect to the incident beam diameter) causes a less pronounced
84
4.5 Design of the focusing optic
5 0
horizontal ( m)
5
0
5
vertical ( m)
5 0
horizontal ( m)
5
0
5
vertical ( m)
01 1062 1063 1064 106
Intensity in P0/cm2
(b)
AOI 80°
5 0
horizontal ( m)
5
0
5
vertical ( m)
5 0
horizontal ( m)
5
0
5
vertical ( m)
02 1074 1076 107
Intensity in P0/cm2
(a)
AOI 10°
Figure 4.15: Comparison of the optimal achievable foci from (a) 10
◦
(near-)normal and (b) 80
◦
normal (= 10
◦
grazing) incidence toroidal mirrors without considering the mirror’s reflectivity nor
finite dimension. The color scale is given in units of intensity with
P0/cm2
where
P0
represents
the incident XUV peak power.
20 40 60 80
AOI (deg)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
Intensity (P0/cm2)
1e7
40nm
42nm
44nm
46nm
50nm
60nm
Figure 4.16: Comprehensive comparison of the achievable peak intensities for single toroidal
mirror optics coated with
B4C
and a focal length of 18 cm by incident angle (relative to the surface
normal) and wavelength. (Note: due to lack for more specific data, for each wavelength the same
divergence and source size was assumed)
85
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
86420
horizontal ( m)
4
2
0
2
4
vertical ( m)
86420
horizontal ( m)
4
2
0
2
4
vertical ( m)
0
1 107
2 107
3 107
4 107
5 107
6 107
Intensity in P0/cm2
Figure 4.17: Beam profile in the focal plane computed from ray tracing based on the final
focusing mirror design parameters.
coma aberration. This geometrical benefit overcomes the lower reflectivities. Towards smaller
wavelengths, the near-normal incidence reflectivity decreases too rapidly. Consequently, this
effect is not exceeded by the geometric benefit that yields less coma anymore. It is worth
noting that a typical grazing incidence angle of 10
◦
(AOI of 80
◦
) is also not beneficial. The
optimal compromise for the investigated wavelengths below 44
nm
is instead around an AOI of
60◦.
This plot shows conclusively that for photon energies below 28 eV (above 44
nm
), the
near-normal incidence geometry is beneficial instead of single mirror grazing incidence optics.
Note: For this design, only single mirror optics have been taken into account. While for
near-normal incidence optics, this seems an obvious choice in this spectral regime, for grazing
incidence optics, it is not an uncommon approach to use multiple mirrors [186]. It was
neglected for this design as it introduces an additional amount of complexity in alignment and
controllable degrees of freedom.
Final mirror specification
The final mirror design is based on a focal length of 17
.
7
cm
and an AOI of 11
.
4
◦
. According to
eq. 4.4 this yields radii of 35
.
8
cm
(
rtan
) and 34
.
4
cm
(
rsag
). This specific angle of incidence was
chosen because the manufacturer Pilz-Optics specified the tolerances with 1% concerning the
radii. Accordingly, radii within that tolerance support AOIs ranging from 8
◦
to 14
◦
and focal
lengths between 17
.
5
cm
and 18
cm
. Those align well with the geometrical design requirements
derived in the section "propagation layout". The mirror is coated with a single layer of
B4C
.
86
4.6 Characterization of the NIR and XUV intensities in the interaction region
40 50 60 70 80 90 100
wavelength (nm)
0.1
0.2
0.3
Reflectivity
PTB toroid measurement
Gottwald et al. (2017)
Figure 4.18: Reflectivity measurement by PTB for a manufactured toroidal mirror coated with
B4C for unpolarized light at an angle of incidence of 11
.
4
◦
(blue); Considering a contribution from
s- and p-polarization of 50 % each in the unpolarized light, for reference the mean of
B4C
s- and
p-polarization data from [159] is depicted in fig. 4.12 is shown as a grey line. The gray shaded
area represents the range between the s- and p-polarization data by wavelength.
To determine the expected focus properties, further ray-tracing calculations have been
carried out. The resulting XUV beam profile in focus (beam waist), shown in fig. 4.17,
measures 2.6µm×2µm(1/e2) or concerning FWHM 1.3µm×1.1µm.
The produced mirror’s radii were measured by the manufacturer to be 35
.
836
cm ±
0
.
1%
and 34
.
296
cm ±
0
.
2%, which results in an ideal AOI of 12
.
0
◦±
0
.
4
◦
and focal length of
17
.
75
cm ±
0
.
03
cm
. The reflectivity of one of the manufactured mirrors was characterized by
PTB for unpolarized light at an AOI of 11
.
4
◦
and is shown in fig. 4.18. The comparison with
data from fig. 4.12, which was used for the preceding discussion shows that it is a suitable
reference, but for this particular mirror potentially causes a slight underestimation of the
reflectivity.
4.6
Characterization of the NIR and XUV intensities in the
interaction region
Simulations carried out for a similar NIR-XUV pump-probe experiment on helium nano-
droplets indicate a strong sensitivity of the generated diffraction signal to the present NIR
intensity [35]. Furthermore, analysis of multi-color XUV CDI data can enormously benefit
from a well-characterized XUV pulse in focus [34]. Therefore, a crucial component of this
experiment is a thorough characterization of the XUV and NIR pulse properties, specifically in
focus. In this context, a comprehensive dataset consists of the spatial dimension of the focus,
the pulse energy and duration, and the spectral composition. Hence the flux and intensity in
focus can be determined.
4.6.1 Characteristics of the NIR in the interaction region
The spectral composition and pulse energy of the NIR can be measured behind the interaction
region outside of the vacuum (see section 4.4.2). Therefore, a diffusor combined with a compact
commercial spectrometer and a power meter is used. Similar to previous measurements at this
HHG beamline [140], the NIR spectra in fig. 4.19 (a) show a blue-shift when high harmonics
are generated (i.e., the gas cell is filled with Xe). Both spectra are red-shifted compared
87
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
700 800 900
wavelength (nm)
0.00
0.25
0.50
0.75
1.00
normalized intensity
(arb. u.)
(a)
w/o gas
with gas
10 0 10
delay in fs
0.00
0.25
0.50
0.75
1.00
normalized intensity
(arb. u.)
(b)
Figure 4.19: (a) NIR spectrum measured behind the interaction region with and without (w/o)
gas in the HHG cell shows a clear spectral shift between both settings. The dotted lines indicate
the signal-weighted mean wavelength. Due to the wide, uneven spectrum, there is no obvious
central wavelength; (b) confirmation of a short NIR pulse duration reaching the interaction region
by auto-correlation on helium 1+ ion signal without HHG (i.e., without Xe in the HHG cell).
Helium was delivered by the droplet source with a nozzle heated to 295 K.
to those measured directly after the OPCPA system (see section 3.1) due to the spectral
dependence of reflectivity of the B4Cmirror coating [187]. A dominating central wavelength
is not apparent.
The NIR pulse duration has been measured at the exit of the OPCPA system using
FROG to be 8
.
3fs [122]. The reconstructed pulse envelope can be approximated well with a
Gaussian function. A short pulse duration without gas in the HHG cell can be confirmed in
the presented experiment by an NIR-NIR autocorrelation on the single ionization of gaseous
He in the interaction region. Such a measurement is depicted in fig. 4.19 (b). As the order of
non-linearity for this process is not known, a quantitative determination of the pulse duration
from this measurement was not possible.
The beam profile of the NIR focus can be directly measured
6
on a CMOS chip placed in
the interaction region with several neutral density filters in front of it. Fig. 4.20 shows the
recorded NIR beam profile. The sensor was placed within the Rayleigh range of the focused
NIR (focal plane
±
2mm). The profile measures 30
×
41
µm
(1
/e2
) behind the used 4
mm
aperture mask configuration. The color scale reflecting the local intensity in fig. 4.20 results
from a conversion factor Γapplied to the integral normalized beam profiles:
Γ = P0
d2
px ≈0.94 ·Epulse
τfwhm
d2
px
Using measurements of the pulse energy
Epulse
carried out behind the interaction region with
the pulse duration
τfwhm
= 8
.
3
fs
to determine the peak power
P0
and the pixel size
dpx
of the
sensor, Γand consequently the average intensity per pixel is calculated.
88
4.6 Characterization of the NIR and XUV intensities in the interaction region
0 100 200
m
0
100
200
m
1012
1013
1014
intensity (W/cm2)
Figure 4.20: NIR beam profiles (single shot) close to the focal plane (within the Rayleigh range)
measured in the vented setup on a camera CMOS sensor with several neutral density filters in
front. The colormap indicates intensity calculated with a pulse duration of 8
.
3
fs
(FWHM) and a
pulse energy of 24 µJthat was measured behind the interaction region.
20 30 40
photon energy (eV)
0
2
4
6
intensity (arb. u.)
1e6
(a)
11 13 15 17 19 21
harmonic order
0.0
0.1
0.2
0.3
fraction
(b)
Figure 4.21: (a) Typical XUV spectrum measured behind the interaction region, corrected for
the diffraction efficiency of the grating (see fig. 4.8 (b)). (b) Contribution of each harmonic order
to the spectrum by integration of the harmonic peaks shown in (a).
89
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
(a)
-1000
m
(b)
-800
m
(c)
-600
m
(d)
-400
m
(e)
-200
m
(f)
0
m
(g)
200
m
(h)
400
m
(i)
600
m
(j)
800
m
(k)
1000
m
Figure 4.22: Perspective corrected XUV beam profiles on the phosphor screen for different offsets
between focal plane and screen that were introduced by moving the focusing mirror. The zero is
defined at the profile closest to the focus.
4.6.2 Characteristics of the XUV in the interaction region
The spectral composition of the XUV is measured in-situ on the spectrometer behind the
interaction region on a shot-to-shot basis. A typical spectrum is shown in fig. 4.21.
The pulse energy was characterized before the beam enters the main experimental chamber
within the HHG beamline’s diagnostics chamber (see chapter 3.3.2). In combination with the
XUV spectrum, the reflectivity characterization of the focusing mirror (by the PTB), and an
estimation of the loss caused by the straylight apertures of 10 %, the pulse energy within the
interaction region can be deduced. Accordingly, the pulse energy in the interaction region is
determined to be typically around 4nJ.
The XUV pulse duration has been characterized within the scope of the beamline itself
(see chapter 3.3.4): The individual pulses in the pulse train have a FWHM of approximately
420 as and the envelope of the attosecond pulse train measures 3
.
1fs FWHM. Due to working
under similar conditions at which that measurement was taken, the same temporal profile of
the XUV pulse is assumed for this experiment.
In contrast to the NIR focus’ size measurement, the spatial characterization of the XUV
focus has to take place in vacuum using the focus tool with phosphor screen and fluence
mapping gratings as described in section 4.4.1.
Phosphor Screen
Using the phosphor screen to visualize the XUV beam profile in the interaction region while
shifting the focus position along the beam axis yields a clear qualitative position of the focal
plane, as shown in fig. 4.22. Since the depicted images are taken from a diagonal point of view
(see also section 4.4.1), the perspective had to be corrected for using an affine transformation.
Due to expected non-linear behavior, saturation effects of the phosphor, and the limited
spatial resolution of the microscope, a direct spatial interpretation of the obtained beam
profiles is not feasible. The presence of non-linear response and saturation is confirmed by
signal integration. Here, a decrease around the focus occurs (see fig. 4.23 (a)) when a constant
integral would be observed for a linear, not-saturated detector. For the measured data, the
amount of fluorescence generated on the screen relates inversely to the intensity, i.e., directly
to the area illuminated by the XUV pulse.
Assuming a radially symmetric XUV beam, the illuminated area (i.e., amount of light
emitted by the phosphor) relates to the beam radius squared. Further assuming the beam
behaves like a Gaussian beam, a Rayleigh length of 760
µm
can be fitted to the data, as
6within the vented experimental chamber.
90
4.6 Characterization of the NIR and XUV intensities in the interaction region
1000 0 1000
offset to focal plane ( m)
4000
6000
integrated signal
(arb. u.)
(a)
1000 0 1000
offset to focal plane ( m)
4
6
1/e2 beam radius
( m)
(b)
Figure 4.23: (a) Integrated signal from the phosphor screen at different positions relative to the
XUV’s focal plane. Rayleigh length was fitted according for a Gaussian beam by using the assumed
dependency of the signal on
w
(
z
)
2
. The red data points are excluded due to clear deviation from
w
(
z
)
2
behaviour. This is expected for certain saturation and non-linear response of the phosphor.
(b) Expected XUV beam radius around the focus for a wavelength of 60 nm and the in (a) fitted
Rayleigh length of 760 µm.
0.2 0.4 0.6 0.8 1.0
relative saturation level (arb. u)
10 2
10 1
100
order of non-linearity
761
800
900
1000
fitted rayleigh length ( m)
Figure 4.24: Numerical simulation of the Rayleigh length retrieval from a screen for different
orders of non-linearities and saturation threshold. The simulations were carried out for an original
Rayleigh length of 760 µmand a wavelength of 60nm.
shown in fig. 4.23. For the dominating 13
th
harmonic (wavelength ca. 60 nm, see fig. 4.21),
this would relate to a beam waist radius (1
/e2
) of 3
.
9
µm
. The red data points are excluded
as they clearly deviate from the assumed behavior and thus otherwise would distort the fit.
This deviation could be due to a change in non-linearity for lower intensities or leaving the
saturation regime. The symmetry of those data points around the focus support this argument,
pointing away from the non-even surface of the phosphor coating, which also could cause such
behavior.
Fig. 4.24 depicts the results of a straightforward numerical simulation. The simulation
emulates the phosphor screen for an ideal Gaussian beam by translating beam profiles for a grid
of non-linearities
κ
implemented as
signalκ
and hard saturation levels in a range of
±
750
µm
around the focus. Those are integrated for each tuple of saturation level and non-linearity,
and a Rayleigh length is fitted to the data. The Gaussian beam assumes an original Rayleigh
length of 760
µm
together with a wavelength of 60 nm. The color scale of fig. 4.24 shows that
no fitted Rayleigh length undercuts the original Rayleigh length. Therefore the experimentally
implemented method provides an upper estimation of the Rayleigh length and hence focal size.
91
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
(b)
(a)
25 m
(b)
0.0
0.2
0.4
0.6
0.8
1.0
intensity (arb. u.)
Figure 4.25: (a) mean image produced by a fluence mapping grating on the diffraction detector;
It shows the XUV beam profile spectrally resolved in planes shortly before and after the plane of
the fluence mapping grating as acquired during the experiment without further post-processing.
(b) shows a particular frame from (a) containing the XUV beam profiles closest to the focus.
Thereby the beam profiles are arranged top left to bottom right from high to low harmonic orders.
The shown scale is computed from the known detector geometry and magnification of the fluence
mapping grating (given for a wavelength of 62
nm
, neglecting chromatic aberration). As the
non-linear inhomogeneous response of the detector is not corrected, particular measurements like
FWHM or beam radius are not taken.
In conclusion, characterizing the XUV focus with a phosphor screen yields an upper limit
of the XUVs beam waist radius of 4
µm
under the approximation of a radially symmetric
Gaussian beam. With the pulse energy range and attosecond pulse train temporal profile
stated before, even in the upper limit of the determined focus size, peak intensities up to
1013 W/cm2are possible.
Fluence mapping gratings
The quantitative analysis of the acquired beam profiles is not straightforward due to the
MCP detector’s spatially non-homogeneous, non-linear response. In addition, analysis reveals
a ringing toward at least one edge in many of the acquired beam profiles. It occurs when
the XUV beam hits the grating’s edge along the walk-off direction. For the axis along the
direction of the ringing, the data cannot be analyzed well. As a result, many profiles would be
analyzable along the vertical and fewer along the horizontal axis.
As further detailed analysis of the MCP’s non-linearities and spatial inhomogenieties would
be required, in this thesis no quantitative analysis is carried out. An example image that
shows no artefacts of the focii acquired with the fluence mapping gratings is shown in fig. 4.25.
Although in general an assessment of the spot size is difficult, the observed size seems to be
on the order of the former estimate with the phosphor screen. Furthermore, the focus size
appears to scale as expected from Gaussian optics: higher XUV photon energies, which are
imaged closer to the center of the detector, produce a smaller focal spot. This qualitative
result points out, that the spectral composition of the light interacting with a nano-droplet
depends on its position relative to the focus. Thus the spectrum acquired with the single shot
spectrometer in the experiment, may deviate from the spectral composition experienced by
92
4.7 Establishing spatial & temporal overlap
the droplet. Nevertheless, in a first step the analysis may assume the measured spectrum for
the XUV interaction with the droplet and has to consider possible deviations as an error.
4.7 Establishing spatial & temporal overlap
Spatial and temporal overlap is the basis for a successful pump-probe experiment. The term
"spatial overlap" refers to the intersection of the pump and probe pulse’s beam paths within
the interaction region. The simultaneous presence, precisely the coincident peak intensity, of
both pulses in the interaction region is called temporal overlap. In perfect "temporal overlap",
the relative delay between pump and probe pulse is zero. This setting is also referred to as
time zero.
The presented experiment realizes an NIR-XUV pump-probe scheme. However, for
alignment and characterization, NIR induced and NIR probed dynamics are exploited as
well (see also section 4.6). Both schemes are considered for the following explanation of the
employed procedures to establish spatial (section 4.7.1) and temporal (section 4.7.2) overlap.
4.7.1 Establishing spatial overlap
One straightforward approach to establish spatial overlap is to insert a screen in the focal plane
in the interaction region where the overlap is needed. For feasibility, three major requirements
must be fulfilled: the availability of a screen material luminescent for NIR & XUV that can
sustain the intensities in the focal plane, the ability to resolve the beam positions indicated by
the screen, and knowledge of the focal plane to place the screen there.
As explained in section 4.4.1, the phosphor material used with the focus tool shows a high
quantum efficiency for the XUV light but is also suitable for NIR. Thus, for this experiment,
the phosphor screen provided by the focus tool is suitable for forming spatial overlap between
NIR and XUV or NIR and NIR. However, the phosphor cannot sustain the high NIR intensities.
Hence, neutral density filters are used to reduce the amount of light that reaches the screen.
Consequently, NIR and XUV cannot be present at the same time in the interaction region.
They have to be aligned one after another. Thereby with the XUV arm, the target position
of the NIR arm is defined. The resolution of the long-range microscope used with the focus
tool cannot resolve the sub-10
µm
focal spot itself. Nonetheless, this resolution limit does not
apply to determining the center of a single spot. For a sufficient number of pixels and signal
to noise ratio, a resolution of the position well below the resolution given by the pixels can be
achieved [188]. Since the phosphor screen on the focus tool is used for focus optimization and
characterization (see section 4.6), intrinsically, the screen can be placed in the focal plane.
A general concern for the experiment poses residual NIR light propagating on the XUV
arm. It is due to the mostly neglected NIR transmission of the metal filter foils, because the
foils reduce the NIR signal by several orders of magnitude. During the experiment, a small
transmitted fraction of NIR was observed near the interaction region, but it did not overlap
with the XUV in focus on the phosphor screen. The XUV and co-propagating NIR driving the
HHG process have slightly different pointing when exiting the HHG cell. Accordingly, when
switching between NIR-NIR and NIR-XUV settings during the experiment, adjustment of the
93
4. Development of an NIR-XUV pump-probe coherent diffractive imaging experiment
with sub-fs resolution
Figure 4.26: Observed interference pattern using the focus tool in the interaction region over
a range of approximately 27
fs
in 6
.
7
fs
steps: (a) and (e) show weak interference showing a low
contrast pattern. The patterns in (b)-(d) exhibit a high contrast and thus are close to ideal
temporal overlap. The blue cross marks a reference position close to the center of the overlapping
beams.
spatial overlap setting is required. Hence, it is ensured that the probe pulse in spatial overlap
with the NIR pump pulse contains only XUV.
4.7.2 Establishing temporal overlap
Due to the intrinsic synchronicity between the NIR and HHG XUV pulses, the temporal
overlap between NIR pulses on both arms implies temporal overlap between NIR and XUV.
Accordingly, a method for establishing temporal overlap of two NIR pulses will be presented.
It requires a previously established spatial overlap between NIR pulses on both arms.
In temporal overlap, two coherent pulses of the same color interfere. It can be visualized
using a screen in the interaction region. While in a colinear setup, the overlapping region
would show an oscillation between constructive (bright) and destructive (dark) interference
depending on the delay between the pulses, in this experiment, interference fringes can also be
observed across the beam profile, as shown in fig. 4.26. Judging by contrast, time zero, i.e.,
the perfect temporal overlap between the NIR pulses can be determined.
Comparable to the method for establishing spatial overlap, the focus tool’s phosphor screen
combined with the microscope is used to observe the interference. Again, a neutral density
filter, which is passed by both beams, is used to reduce the intensity hitting the screen, which
would be damaged otherwise. Using the phosphor screen to find temporal overlap bears
the advantage that simultaneously, the spatial overlap can be monitored and, if necessary,
re-established
Spatio-temporal overlap (coexistence of spatial and temporal overlap) between two NIR
pulses can be confirmed by performing a cross-correlation measurement on gaseous helium
introduced by the droplet source. By observing the amount of singly charged helium (
He1+
),
where the ionization by NIR-radiation is a highly non-linear process, an auto-correlation-like
signal is observed. An exemplary measurement is plotted in fig. 4.19 (b).
So far, it was claimed that NIR-NIR temporal overlap implies NIR-XUV overlap because
of the intrinsic behavior of the HHG process. The spatiotemporal overlap for XUV-NIR can
be confirmed by performing a delay scan with gaseous helium in the interaction region. Most
of the available XUV photons are energetically below the ionization threshold of helium of
24
.
59
eV
[190] (see e.g. spectrum in fig. 4.21). Hence without the NIR pulse, the XUV excites
the helium mostly into bound states. With the NIR present, as illustrated in fig. 4.27 (a), one
to three of its photons can yield ionization of the helium atom. This process was investigated
previously in similar setups, either also observing ion yields [67] or using electron spectroscopy
[191, 189]. The onset of ionization is occurring over a range of several tens of femtoseconds [67].
94
4.7 Establishing spatial & temporal overlap
1s
4p
3p
2p
... vacuum
XUV
NIR
probepump
(a)
100 50 0 50
delay (fs)
0.4
0.6
0.8
1.0
integrated ion yield (arb. u.)
(b) He1 + ion yield
low
resolution
high
resolution
Figure 4.27: (a) energetic scheme for NIR assisted photoionization of helium (based on the
concept of [189, fig. 1]); The helium atom is initially excited by the XUV from its 1s ground state.
However, ionization may only occur when NIR photons are present after the XUV pre-excited
the atom, not the other way around. Direct ionization by higher harmonics into the vacuum is
omitted. Only Pathways involving ’np’ states as inter-mediate states are illustrated. (b) a low
(blue) and high (orange) resolution time delay scan between NIR and XUV pulses in a gaseous He
target (droplet source heated to 100 K) showing the
He1+
ion yield to confirm spatio-temporal
overlap by means of NIR assisted photoionization of helium. Negative time delays correspond to
the XUV pulse preceeding the NIR pulse.
This previously observed behavior agrees well with the
He1+
ion signal measured in a delay
scan to confirm the overlap, shown in fig. 4.27 (b). However, particularly, the data presented
in [67] demonstrates that the position of the rising edge relative to time zero strongly depends
on the NIR and XUV spectral composition.
95
5
Ultrafast dynamics in helium
nano-droplets
This thesis ultimately aims to investigate the direct influence of an NIR pulse on the scattering
behavior of helium nanodroplets in the attosecond domain and reveal the droplets’ underlying
dynamics. Such dynamics of the electronic structure of the droplets may occur as a direct
response to the NIR laser field and, therefore, may take place on time scales shorter than a
single near infrared (NIR) laser cycle or short in the sub-cycle regime. In this chapter, the
femtosecond and sub-femtosecond response of single helium nanodroplets to an NIR field is
studied with coherent diffractive imaging and ion spectroscopy, and the results are discussed
based on modeling.
First, an overview of the acquired data together with its processing is given, then
experimental observations are presented, separated into few-femtosecond and sub-cycle
dynamics. To explain the observations made, a modeling scheme is introduced, and simulations
of the measurement are carried out. Finally, in a discussion, observations, simulation results,
and published results by others are combined to form a physical picture of the effects responsible
for the experimental observations.
5.1 Overview, calibration & processing of the acquired data
In the scope of the NIR-XUV pump-probe experiment, a total amount of 20
.
5
TB
of raw
single-shot-resolved data was acquired. A comprehensive overview is given in table 5.1. For a
single shot, data is available from up to three detectors, namely the time-of-flight spectrometer,
the diffraction detector, and the XUV spectrometer. The data is structured in runs that
typically consist of a dataset taken while controllable parameters of the experiment are held
constant (e.g., a constant time delay between pump and probe pulse). Each run is given an
identification number (’run id’) and consists of a given number of shots. A single shot is
associated with a specific laser pulse / XUV APT. The related detector data is labeled with a
97
5. Ultrafast dynamics in helium nano-droplets
all few-fs
dynamics
sub-cycle
dynamics
XUV only droplet
source
timing
other
104
105
106
number of shots
total
hits
patterns
Figure 5.1: Composition of the recorded images from the diffraction detector; The majority
of data was acquired to investigate either few-femtosecond or sub-cycle dynamics in the Helium
nano-droplets. Static XUV-only data and scans of the timing of the pulsed valve in the droplet
source (with only XUV light) make up also significant fractions. The proportion of diffraction
images of Helium droplets (’hits’) and those ’hits’ suitable for structural analysis (’patterns’) are
indicated by blue and orange, respectively.
category raw data amount number of
runs
number of
shots
all data 20.5TB 5092 6.1·106
data with diffraction detector images 19.7 TB 2099 3.6·106
few-fs dynamics 8.5TB 860 1.6·106
sub-cycle dynamics 5.9TB 379 1.1·106
XUV only 1.6TB 109 3.0·105
droplet source timing 1.1TB 344 2.1·105
other 2.6TB 407 5.0·105
Table 5.1: Overview of the acquired amount of data by component of the experiment. A shot
refers to a data set associated with a single laser pulse. A run consists a group shots acquired
in succession while typically parameters of the experiment are static. The diffraction images are
responsible for most of the data amount.
98
5.1 Overview, calibration & processing of the acquired data
’shot id’
1
. Consequently, the tuple of ’run id’ and ’shot id’ uniquely identifies a pulse whose
single-shot data was acquired during the experiment.
The acquired raw data consists of 5092 runs. Images of the diffraction detector are contained
in 2099 runs. In fact, there are more than 3
.
6million of such images. The contributions of
different components of the experiment and the fractions containing data from Helium droplets
are shown in fig. 5.1 and table 5.1. When the detector shows a signal exceeding the background
level, it is declared a ’hit’ (see section 5.1.2 for details). Those hits containing sufficient signal
to show a structure are categorized as a ’pattern’ (see appendix A.2 for details). More than
840 thousand hits in the full dataset, and among them 380 thousand patterns were identified.
5.1.1 Reference and calibration data
To provide an overview of the acquired reference and calibration data, this subsection lists
the experimental procedures and associated data that ensure the data quality and enable
calibration of the available instruments.
On a typical measurement day
2
of the beamtime data was taken according to the following
procedure:
•
Reference data (XUV only) with the droplet source operated at 10
Hz3
, to validate
droplet and HHG source performance (see fig. 5.2 (a-c))
•
Reference data (XUV only) with the droplet source operated at 16
.
7
Hz4
(see fig. 5.2
(d-f))
•
Delay scan of the droplet source’s valve timing compared to the XUV pulses (XUV only)
and setting of the optimized timing (see fig. 5.3)
•Conducting the NIR-XUV pump probe measurements
•Comprehensive set of background data for reference
•
Calibration data for the XUV spectrometer located behind the interaction region (see
section 5.1.3)
•(Sometimes) Characterization of the NIR pulses
The performance of the cluster source was evaluated daily concerning its nozzle temperature
and produced droplet sizes. Diffraction patterns were acquired using only the XUV pulses.
Initial reference data was taken with a repetition rate of 10
Hz5
. The typical repetition rate of
the droplet source during the experiment was 16
.
7
Hz
, yielding a good compromise between
achievable droplet size, gas load in the chamber and amount of data per time. Also there, a
reference data set was taken, ensuring the data quality. Due to the increased repetition rate,
the pulsed valve within the droplet source generates more heat than during 10
Hz
operation,
producing smaller clusters, as shown with experimental data in fig. 5.2.
1starting from zero in each run and increasing in increments of one;
2Refering to a day on which the experiment was working properly
3droplets provided to every 10th laser pulse
4droplets provided to every 6th laser pulse
5The Helium droplet source is well characterized at a repetition rate of 10 Hz [141, 164].
99
5. Ultrafast dynamics in helium nano-droplets
(a) (b) (c)
(d) (e) (f)
104
2 × 104
3 × 104
4 × 104
detector signal (arb. u.)
Figure 5.2: Each row shows the three brightest images of a reference run. (a-c) (id: 3706)
droplet source operating at a repetition rate of 10Hz (d-f) (id: 3707) droplet source operating at a
repetition rate of 16
.
7
Hz
. The larger ring spacing corresponding to on average smaller droplets
was caused by the increased heat input from the more frequent valve openings within the droplet
source. (logarithmic color scale, for visualization purposes the color scale clips a detector signal
below 10000(arb.u.))
1.1 1.0 0.9 0.8
relative trigger delay (ms)
0
2
fraction of
bright images (%)
main pulse
post
pulse
Figure 5.3: Fraction of bright images (arb. threshold) within a run by different delays between
the trigger for the droplet source’s valve and the xuv pulse. A main cluster pulse is followed by a
shorter post pulse.
100
5.1 Overview, calibration & processing of the acquired data
background mean
background
hit
integral
analysis
masking image
classification
Figure 5.4: Pre-processing procedure for coherent diffractive imaging data; The algorithms and
steps described in this section are represented by rectangles. Circles are the resulting classes/labels
from the classification that are treated separately. Finally, the octagon represents the type of
analysis to be performed.
A representative scan of the timing of the pulsed droplet source relative to the XUV
pulses is shown in fig. 5.3. There, the fraction of images of the diffraction detector, whose
integral exceeds a given threshold, is plotted versus the relative delay between XUV pulse
and nano-droplet source. A shorter post pulse follows the main droplet pulse, which has been
assigned to a valve rebounce in previous work [192]. An optimized timing corresponds to the
XUV pulse intersecting with the center of the main pulse.
5.1.2 Pre-processing of diffraction detector images
To deal with the vast amount of available data, in pre-processing, following the scheme depicted
in fig. 5.4, the images from the diffraction detector are masked, then classified and labeled.
Depending on the label, further pre-processing is done until the data is forwarded to the
analysis, based on integrals of defined areas (’integral analysis’, see sections 5.2 and 5.3). In this
work, the spatial structure of the patterns is not analyzed. However, work done toward analysis
of the spatial structure can be found in appendix A.2. In this subsection, the description of the
algorithms and steps indicated by rectangles in fig. 5.4 is structured by respective subheadings.
In order to analyze only signal diffracted from helium droplets, the background signal, e.g.,
originating from diffusely scattered light at the mirror surface or from the edges of apertures,
needs to be removed. A mask excludes regions showing a strong background signal. Classifying
the acquired images into the groups ’background’ and ’hit’ enables their separate treatment. A
mean background image can be calculated from images in the’ background’- group to facilitate
the subtraction of weaker background signals on-demand within the analysis. ’hits’ are then
used within the integral-based analysis routines, which do not rely on structural information.
Masking
During the experiment, the camera obtains images of the diffraction detector’s phosphor screen.
Although the detector’s active area is round, a squared image is recorded. For plotting and
processing, regions not belonging to the detector’s active area need to be masked out. This
basic mask is shown in fig. 5.5 (a)6.
Suppressing straylight is crucial for diffraction image quality. Strong straylight occurring
localized likely drives the detector in those regions close to saturation. Consequently, those
regions are removed by masking them out. Straylight masks are computed from a single run
but applied to several runs until conditions change notably. The algorithm is based on integrals
6The mask needs to be redefined when the camera is moved between runs. It is set and verified manually.
101
5. Ultrafast dynamics in helium nano-droplets
(a) (b) (c)
102
103
104
detector signal (arb. u.)
Figure 5.5: (a) basic mask to remove areas (black) outside of the detector’s active and visible
area composed of three circles (red: edges of the detector’s active area, blue: edge of the hole in
the mirror directing the detector image toward the camera; both, blue and red, inner circles are
masked.) (b,c) Mean image of the background and weak hits (typical run, id: 3915) to identify
strong localized straylight regions for masking through a three-time standard deviation criterion
and additional filtering; (c) areas identified for masking are highlighted in green, located near the
center of the image.
over the full detector area. All shots with an integral the mean of all single shot integrals
are averaged.
7
. The mask covers those significantly brighter regions than the average pixel
(i.e., exceeding mean plus three-time standard deviation). Single pixels identified as significant
are not forming a region and hence are suppressed by further processing. Fig. 5.5 (b) shows
such example averaged image. In fig. 5.5 (c) the green area highlights the regions that the
straylight mask covers.
Diffraction detector image classification
The following describes the procedures to obtain the two image classes ’background’ and ’hit’.
These two classes are exclusive.
The run-wise identification of background images relies on computing a threshold for the
full detector integral
8
. The background signal contributing to the integral originates from
several statistical varying effects. As a direct consequence of the Poisson distribution applicable
to the probability that only one large He droplet is imaged at a time
9
, the majority of images
must only contain background signals. Hence, a histogram of the full detector integrals can be
used to identify this dominating contribution.
Fig. 5.6 (a) shows a histogram for all full detector integrals of one typical run. A broad
distribution ranging from approximately 2
·
10
8
to 2
·
10
10
can be seen. The large spread of
integrals associated with images of droplets impedes the precise identification of the background
signals. The following steps are taken to determine a threshold differentiating the data into
the classes ’hit’ and ’background’.
1.
Select all shots with below-average integral values to remove bright diffraction patterns
2. Generate a histogram of this selection (Fig. 5.6 (b))
7
Assuming the brightest diffraction images overcome the background level several times, and the majority of
images contains background data or weak hits, the selected shots also contain only background data or weak
hits
8after applying the masks (basic mask & straylight mask)
9In the experiment, the average number of detected hits was below 50 %.
102
5.1 Overview, calibration & processing of the acquired data
0.0 0.5 1.0 1.5 2.0
detector integral
(arb. u.)
1e10
101
103
counts
(a)
2 4 6 8
detector integral
(arb. u.)
1e8
0
50
100
150
counts
(b)
Figure 5.6: Identification of background images using full detector integrals. (a) histogram of all
full detector integrals (logarithmic scale) from a typical run (id: 3915). The green line is set at the
mean value of those integrals, and the red line indicates the later set threshold. (b) histogram
of the data taken into account for (a) only considering those values below the mean value (green
line). A Gaussian distribution (solid orange line, dashed: extrapolation onto data not considered
for the fitting) is fitted to a sub-selection (blue). At three times the standard deviation from the
center of the fitted Gaussian distribution, the threshold is set (red line).
3. Select only those bins close to the background peak (Fig. 5.6 (b), blue)
4. Fit a Gaussian distribution to this sub-selection (Fig. 5.6 (b), orange line)
5.
Set the threshold at three times the standard deviation from the center of the Gaussian
distribution (Fig. 5.6 (b), red line)
Those shots with full detector integrals above the defined background threshold are assumed
hits. As a result, the classification yields two types of images, which are for the presented
sample run shown in fig. 5.5. There are the classes ’background’ (fig. 5.5 (a)), ’hit’ (fig. 5.5
(b)).
Mean background image
A mean background image can be computed for each run from the shots assigned to the
’background’ class. Such a background image is plotted in fig. 5.8 (a) and (b) for two different
color scales. One can identify structures associated with shadows of apertures and other
objects that are located along the beam path between focusing mirrors and diffraction detector.
It is subtracted from single-shot data later during the analysis on demand.
5.1.3 XUV spectrum calibration
The contribution of each harmonic order (HO) can be determined from the XUV spectra
recorded with each single diffraction image. Therefore, the raw data, detector intensity as a
function of pixel position, needs to be translated to intensity per photon energy and corrected
for the diffraction efficiency of the grating. This diffraction efficiency was measured by the
Physikalisch-Technische Bundesanstalt (PTB) (see chapter 4.3 and fig. 4.8). The translation
from pixel position to photon energy has been determined as follows.
103
5. Ultrafast dynamics in helium nano-droplets
(a)
background
(b) hit
Figure 5.7: Resulting classes of diffraction detector images with sample images (row (a) and (b))
from a single run (id: 3915); The images are plotted using a logarithmic color scale.
(a)
102
103
104
detector signal (arb. u.)
(b)
102
6 × 101
2 × 102
3 × 102
4 × 102
detector signal (arb. u.)
Figure 5.8: (a) and (b) show for different color scales the mean image computed from the as no hit
/ background classified shots from a typical run (id: 3755). The optimization of background changes
over the day. Here an example from a different run than the previous examples with stronger
background has been chosen (for a mean image of background and weak hits of the previously
exemplary run see fig. 5.5 (b)). The images show a visible shadow from the ion time-of-flight
spectrometer. The bright, shadow covering feature toward the right is also visible in fig. 5.5 (b).
104
5.1 Overview, calibration & processing of the acquired data
Figure 5.9: Schematic drawing showing the geometry of the XUV spectrometer with the
dimensions and parameters relevant to the calibration. Reflected by the focusing mirrors, propagated
through the interaction region, and after a 250
µm
wide slit (not shown), the XUV light reaches
the gold-coated grating at a grazing incidence angle
ζ
. It is diffracted so that depending on the
rotation of the grating either only the diffracted first or also the zero-order reaches the MCP
detector. The distance of grating and MCP is described in terms of a coordinate system with axes
normal (y) and parallel (x) to the MCP’s surface by
dx
and
dy
. The angle of incidence of the XUV
light entering the spectrometer relative to the y-axis is called ζbase.
parameter symbol value fine-tuned value
for calibration
grating groove density N600lines/mm 600 lines/mm
angle of incident XUV light to axis
orthogonal to the MCP’s surface ζbase (24 ±2)◦25.6◦
distance of grating center to MCP
center parallel to the MCP’s surface dx(64 ±3)mm 64 mm
distance of grating center to MCP
center orthogonal to the MCP’s surface dy(390 ±5) mm 385 (390)mm
Table 5.2: Geometric parameters of the setup determining the calibration. Values as obtained
from the setup are given with errors. An additional column shows the fine-tuned parameters
used for calibration taking the absorption edge of a tin filter for which transmission spectra were
acquired into account.
dy
differs for the ’normal’ and ’calibration’-configuration. The value in
brackets is given for the latter.
105
5. Ultrafast dynamics in helium nano-droplets
0 250 500 750
x pixel index
0
200
y pixel index
(a)
0 250 500 750
x pixel index
(b)
0
5 103
1 104
detector signal
(arb. u.)
Figure 5.10: Raw data from the XUV spectrometer taken in two different angles of incidence
onto the grating. (a) In the ’calibration’-configuration, the 0-order is visible on the right edge,
while some HOs are visible on the left. (b) In the ’normal’-configuration, the first diffraction order
of all HOs is visible on the detector. The two weak peaks toward the left of the detector are due
to the second diffraction order.
The photon energy calibration results from the setup’s geometry, shown in fig. 5.9. The
geometric constraints are given in a coordinate system defined by an x-axis parallel and a
y-axis orthogonal to the MCP’s surface. The origin, also depicted in figure 5.9, is located
on the intersection of both axes, where the x-axis lies in the MCP’s plane, and the y-axis
intersects the center of the XUV illuminated area of the grating. The MCP’s center is located
at a distance
dx
from the origin. The distance can be determined from a computer aided design
(CAD) of the experiment taking into account the position of the MCP holder and the grating
to be 64
mm
. An uncertainty of ca.
±
3
mm
originates from the grating position, which was
not documented precisely. The distance of MCP and center of the XUV illuminated grating
surface along the y-axis
dy
is also given by the CAD with (390
±
5)
mm
. The uncertainty
arises from the different possible XUV illuminated areas on the grating surface. The largest
uncertainty is attached to the angle
ζbase
= (24
±
2)
◦
that describes the incident angle of the
XUV light relative to the introduced coordinate system. It depends on the XUV’s angle of
incidence (AOI) on the focusing mirror and the relative angle between the light incident into
the experiment and the here defined y-axis. The angle of incidence on the focusing mirrors is
estimated to be 12
.
0
◦±
0
.
4
◦
. The error of the experimental setup’s alignment to the incident
beam is estimated to be on the order of 1
◦
. The exact value of
ζbase
taken into account for the
calibration is fine-tuned within the specified error range so that the absorption edge of a tin
foil filter is matched with the available data.
In general, the diffraction angle
ξ
is determined from the position on the MCP
x
with
respect to the defined coordinate system.
ξ(x, ζ)=(ζbase −ζ) + arctan (︄x
dy)︄(5.1)
In the following, two different spectrometer configurations are considered, differing only
in the grating rotation and hence the angle of incidence. Fig. 5.10 shows the raw data from
both types of configuration: In the ’calibration’-configuration (fig. 5.10 (a)) the grating is
rotated such that the 0 order can be observed together with the first diffraction order of some
harmonics and the ’normal’-configuration (fig. 5.10 (b)) shows the full spectrum in the first
diffraction order. A
∼
-sign indicates parameters relating only to the ’calibration’-configuration.
106
5.1 Overview, calibration & processing of the acquired data
In ’calibration’-configuration, the grazing angle of incidence
ζ
˜
can be determined through
the position of the 0th order
x˜0
. The associated diffraction angle
ξ0
˜
must equal
ζ
˜
. Eq. 5.1
yields
ζ
˜=ξ0
˜=ξ(x˜0, ζ
˜) =⇒ζ
˜=ζbase + arctan (︂x˜0
dy)︂
2.
With the position of a HO
x˜1
and the grazing angle of incidence
ζ
˜
also it’s wavelength
λ1
is known. Now, from the position of that HO
x1
in the ’normal’-configuration, also there
the grazing angle of incidence can be determined. For grazing incidence geometry the grating
equation is given with:
m·λ=1
N·(cos(ζ) + cos(ξ)) ⇐⇒ ξ(λ, ζ) = arccos (N·m·λ−cos (ζ)) (5.2)
Combining eq. 5.1 and 5.2 for the HO with wavelength
λ1
and diffraction order
m
= 1 the
only remaining unknown in the equation is ζ, for which it can be solved numerically:
arccos(N·λ1−cos (ζ)) = (ζbase −ζ) + arctan (︄x1
dy)︄
For each position
x
on the MCP detector, for the first diffraction order, the wavelength
λ(x)is then given by
λ(x) = 1
N·[︄cos(︄(ζbase −ζ) + arctan (︄x
dy)︄)︄+ cosζ]︄
or alternatively in terms of the photon energy considering the speed of light
c
and Plank
constant h:
Ephoton(x) = c·h
λ(x)=c·h·N·[︄cos(︄(ζbase −ζ) + arctan (︄x
dy)︄)︄+ cosζ]︄−1
.
Fig. 5.11 (a) shows the resulting photon-energy calibrated spectrum
10
for the previously
in fig. 5.10 (b) shown raw data. A clear difference to the data not corrected for the grating’s
reflection efficiency (gray line) underlines the need for this correction step. As announced
above, the calibration from the assumed geometrical parameters was fine-tuned by considering
the absorption edge of a tin foil filter that lies between 23.5eV and 24.5 eV [165]. In fig. 5.11
(b), the discussed spectrum taken behind an 100
nm
thick aluminum filter is compared to
one acquired behind a 150
nm
thick tin filter. Both spectra
11
likely show contributions from
different HHG source points in the beamline. In the harmonic peaks, at least two peaks overlap
close to the resolution limit of the spectrometer. The 15
th
harmonics are close to the absorption
edge. One 15
th
harmonic photon energy is strongly absorbed (only a minor feature remaining),
and another one transmits through the tin filter at a larger fraction. Using these features,
the angle of incidence of the XUV light relative to the defined coordinate system
ζbase
was
10
Due to the non-uniform spacing of the resulting axes, the amplitudes of the signal had to be re-scaled. The
reduced peak width toward lower photon energies is (at least partially) due to the spectrometer’s entrance slit
width.
11acquired in two succeeding runs
107
5. Ultrafast dynamics in helium nano-droplets
20 30 40
photon energy (eV)
0.0
0.5
1.0
normalized intensity
(arb. u.)
1e7
(a)
20 30 40
photon energy (eV)
0.0
0.5
1.0
intensity (arb. u.)
1e7
(b)
Figure 5.11: (a) Comparison of the photon energy calibrated spectrum with (blue) and without
(gray) taking into account the diffraction efficiency of the grating. (normalized to the height of the
first visible HO for visualization purposes; for energies above 31
eV
the diffraction efficiency was
extrapolated) (b) Calibrated spectra behind an aluminum filter (blue) and a tin filter (orange); the
filled area in the background shows the relative transmission of a 150
nm
tin filter with 4
nm
oxide
layers on each side [165]. The edge around 24
eV
was used to fine-tune the geometry parameters
used for the calibration.
adjusted to 25
.
6
◦
within the bounds of the specified error. Furthermore, the grating rotation
axis that was not perfectly aligned with the XUV illuminated area moves when switching
the configurations. A better fit between the data in normal and calibration configuration
is achieved when considering
dy
= 385
mm
for the ’normal’ configuration. The last column
of table 5.2 provides an overview of all the geometric parameters used for calibration of the
spectra acquired in the experiment.
Calibrating the spectra and comparing the results from different days of the experiment
indicates that the harmonic spectra differ not only in the contributions by the harmonics
but also their position. Although, in principle, it could be an artefact of calibration, with
regards to the (fixed) NIR spectrum (see fig. 4.19 (a)) it is seems plausible to have efficient
HHG at different fundamental wavelengths. It is further supported by the already discussed
example data shown in fig. 5.10 and 5.11, where HHG apparently from different regions in
the HHG cell is produced at different fundamental wavelengths from the same NIR pulses.
The data suggests that the fundamental wavelengths varies from 770
nm
to 820
nm
. Fig. 5.12
illustrates the variation of the fundamental XUV wavelength with the spectra in ’calibration’
configuration and calibrated spectra in ’normal’ configuration acquired at two different days.
108
5.2 Laser dressing of helium droplets on the few femtosecond scale
0 200 400 600 800
0
200
400
y pixel index
(a) 814 nm / 1.52 eV
200 400 600 800
x pixel index
200
400
y pixel index
(b) 801 nm / 1.55 eV
20 25 30 35
photon energy (eV)
0
2
4
6
intensity (arb. u.)
1e6
(c)
(a) (b)
Figure 5.12: Illustration of different fundamental XUV wavelengths/photon energies retrieved
in the spectrum calibration. (a) and (b) show the ’calibration’ configuration from two (selected)
different days. The shown x-axis range was slightly adjusted (by 30 pixels) so that the 0th
diffraction orders are at approximately the same position. Gray vertical lines spanning both
plots are there to guide the eye. The distance of the harmonic orders to the 0th diffraction order
appears to be different between both days. The title indicates the retrieved fundamental XUV
wavelengths/photon energies. (c) shows the resulting calibration applied to data acquired in the
’normal’ contribution immediately before the respective data in (a) and (b). The vertical lines
indicate the expected harmonic positions according to the indicated fundamental XUV wavelengths.
They fit well to all shown harmonic peaks, suggesting a proper calibration.
5.2
Laser dressing of helium droplets on the few femtosecond
scale
The presented experiment investigates the influence of the presence of an NIR pulse during the
diffraction of XUV-light by helium nano-droplets. On the time scale of a few femtoseconds,
i.e. the scale of the NIR pulse’s and XUV APT’s duration, the observed diffraction intensity
decreases when the NIR and XUV pulse overlap temporally. This effect was already observed
in a previous work, namely, the doctoral thesis of Julian Zimmermann [35]. In this section,
the reproduction of the previous observation using considerably shorter NIR and XUV pulses
will be presented and discussed. Furthermore, the effect is studied systematically by varying
the experimental conditions, such as NIR (pump) intensities. This provides insight into the
optical response of helium droplets in the presence of those different intensities. Especially a
comparison to the results produced by a simplified model system can be made in the discussion
of both results later on in section 5.5.1.
The experimental data is analyzed in terms of the integrated signals. The integration
allows for identifying the brightest shots and is, e.g., used in pre-processing (see section 5.1.2)
to identify background images. To account for variations of XUV pulse energies, the integrals
shown here are normalized shot-wise to the single-shot integral of the spectrometer data. This
step is based on the assumption that the number of photons scattered (for a given droplet,
and XUV spectrum) scales linearly with the number photons reaching the droplet.
109
5. Ultrafast dynamics in helium nano-droplets
-30.0 -23.3 -16.7 -10.0 -3.3 3.3 10.0 16.7 23.3 30.0
Figure 5.13: The five brightest diffraction images for different time delays acquired in a single
scan are shown. On top of each column the time delay is given in femtoseconds. A decreased signal
is observed especially at the time delays ±3.3 fs.
5.2.1 General observation and representation of the acquired data
When scanning the time delay between pump (NIR) and probe (XUV) pulse, a decrease of
the scattering intensity is observed during the temporal overlap of the pulses. This general
observation is exemplified in fig. 5.13, where the five brightest diffraction images for acquired
time delays between pump and probe pulse are shown. A negative time delay corresponds to
the XUV (probe) pulse arriving before the NIR (pump) pulse. The NIR pulse exhibited a peak
intensity of approximately 2
·
10
14 W/cm2
(see also fig. 4.20)
12
. As stated in chapter 4.6, the
pulse duration of the NIR is assumed to be 8
.
3
fs
FWHM. Meanwhile the XUV pulse envelope
duration based on the temporal characterization in chapter 3.3.4 is taken in first approximation
to be 3
.
1
fs
FWHM. Accordingly, the three data points around time zero,
−
3
.
3
fs
,3
.
3
fs
and
10
fs
, show a clearly decreased brightness. Not only in the example but consistently in all
scans, such a feature is measured spanning several femtoseconds. The position of this "dip" in
the diffraction signal always coincides with a step-like decrease exhibited by the Helium ion
yield, which indicates spatio-temporal overlap (see chapter 4.7.2).
It is assumed that the brightest diffraction patterns, at a given time delay, can be associated
with the highest XUV intensities a droplet was exposed to [193]
13
. Taking the properties of
the XUV focus into account (see chapter 4.6.2), they relate further to droplets originating
from a strongly confined focal volume. Due to the non-collinear pump-probe geometry of the
experiment, spatial overlap can only be ensured for a limited volume. Further, the temporal
resolution decreases with a larger volume from which diffraction patterns may originate (see
figure 4.6). By taking only the brightest diffraction patterns into account, one can further rely
12
Basis for this value are the measurement of the NIR pulse energy behind the interaction region at a different
day in similar conditions, the beam profile measurement in a vented setup at the end of the experiment,
and the NIR pulse duration of 8
.
3
fs
measured by Kretschmar et al. [122] as a part of the OPCPA system’s
characterization.
13
Assuming a constant spectral composition of the XUV light and approximating a constant droplet size.
Although the latter is known to be statistically distributed [141].
110
5.2 Laser dressing of helium droplets on the few femtosecond scale
25 0 25
delay (fs)
0
5
10
15
normalized
mean detector
signal (arb. u.)
(a) brightest 0.5 %
He ion signal (1st scan)
He ion signal (2nd scan)
diffraction intensity (1st scan)
diffraction intensity (2nd scan)
25 0 25
delay (fs)
0
5
10
15
normalized
mean detector
signal (arb. u.)
(b) brightest 1.0 %
25 0 25
delay (fs)
0
5
10
normalized
mean detector
signal (arb. u.)
(c) brightest 2.0 %
25 0 25
delay (fs)
0.0
2.5
5.0
7.5
normalized
mean detector
signal (arb. u.)
(d) brightest 4.0 %
Figure 5.14: Typical measurements are showing a decrease of the scattering intensity during the
temporal overlap of NIR and XUV pulse in the chosen representation, namely, a given fraction
of the brightest diffraction patterns; Negative time delays correspond to the XUV pulse arriving
before the NIR pulse. The average scattering intensity for the brightest 0
.
5%,1 %,2% and 4%
of the images normalized to the single-shot spectrum integral are shown in (a), (b), (c), and (d),
respectively. Blue and orange distinguish the data from two consecutive scans, where the orange
one was carried out with an increased temporal resolution. The background shows in light blue
and light orange the respective integral of the
He1+
-signal shown by the time of flight spectrometer
scaled by a factor constant for all metrics (a-f) (including all recorded singly charged He droplet
fragments (oligomers)), that provide a reference to a time delay of zero.
on the assumption that the diffraction images acquired only within a very limited focal volume
are considered. Hence, concentrating on the brightest patterns can improve the temporal
resolution. With the spatial overlap of pump and probe pulse established in that limited
focal volume, at the same time, diffraction images acquired outside the volume, in which
spatial overlap was established, are suppressed. This approach is limited by the increase of
statistical error when taking fewer samples into account. As a quantitative representation, or
in other words, metric, to describe the diffraction images originating from the focal volume
(i.e., brightest images), the mean integral value of a fixed brightest fraction of images from a
run is chosen.
In fig. 5.14, for the data set previously illustrated with the five brightest images (see
fig. 5.13), the mean of the brightest 0
.
5%
14
(fig. 5.13 (a)), brightest 1%
15
(fig. 5.13 (b)),
149(fig. 5.13, blue) and 12 (fig. 5.13, orange) images per time delay
1518 (fig. 5.13, blue) and 24 (fig. 5.13, orange) images per time delay
111
5. Ultrafast dynamics in helium nano-droplets
brightest 2 %
16
(fig. 5.13 (c)), and brightest 4 %
17
(fig. 5.13 (d)) of the images are given
for each time delay (fig. 5.14, blue curves). The decreased diffraction signal around a time
delay of 0
fs18
is obvious for all the illustrated fractions of brightest images. In addition, a
second data set, which was obtained directly after the first data set, scanning the time delay
of the two pulses with a finer resolution (fig. 5.14, orange lines) closely reproduces the signal
decrease seen in the initial data set (fig. 5.14, blue lines) for all fractions. The appropriate
reproduction of the diffraction intensity decrease for all fractions and both measurements
occurring over several femtoseconds is not surprising. Rather, it supports the expectation
from wave-front propagation simulation predicting an error of each delay of only around
±
1
fs
(see figure 4.6). Hence, for this chapter, the data is well represented by the brightest 4% of
the images. Nevertheless, also the representation using only 1 % of the brightest images will
be used later on in the thesis for the analysis of sub-cycle dynamics (see section 5.3), since
it allows a stricter confinement of the included focal volume and thus leads to an increased
temporal resolution.
The error bars plotted in fig. 5.14 represent the 95% confidence interval obtained by
applying the bootstrap method
19
(see also [194]). The bootstrap method was chosen because
one cannot assume a Gaussian distribution for the data points. This particular method avoids
the assumption of a certain type of statistical distribution. Please note the confidence interval’s
"coverage accuracy can still be erratic for small sample sizes" [194, ch 14.1, p. 178].
5.2.2 Variation of the NIR intensity
In this section, the diffraction intensity decrease depending on the NIR intensity the droplets are
dressed with is investigated. The NIR intensity is varied by cropping the beam in addition to
the aperture mask (diameter approximately 4
mm
) using the entrance iris into the beamline’s
spectrometer chamber (see chapter 4.3.1 and figure 4.5). Due to the XUV beam being
approximately aligned to the center of the iris, it is expected that only the NIR intensity is
affected as long as the iris is set to a diameter larger than 6
.
6
mm
. For the acquired data sets,
the iris diameter was varied between approximately 6
.
6
mm
and 14
.
7
mm
. At an opening of
14
.
7
mm
, no cropping of the NIR beam in addition to the aperture mask takes place. Reducing
the iris diameter decreases the NIR pulse energy reaching the focus. At a diameter of 6
.
6
mm
no NIR should reach the focus. In addition, a decrease of the beam diameter that is focused
increases the size of the focus. Both factors contribute to a reduction of the intensity in focus.
Furthermore, due to the increased NIR focus diameter, the overlap’s quality may increase, at
least it stays the same (see also discussion in appendix B.1).
Fig. 5.15 presents the result of the measurement. In the carried time delay scans, the
entrance iris opening was set to the values 14
.
7
mm
,12
.
9
mm
,11
.
1
mm
,10
.
3
mm
and 6
.
6
mm20
.
Initially an NIR peak intensity of 2
·
10
14 W/cm2
is estimated to be present in the interaction
1636 (fig. 5.13, blue) and 48 (fig. 5.13, orange) images per time delay
1772 (fig. 5.13, blue) and 96 (fig. 5.13, orange) images per time delay
18
The zero time delay is set on the edge of the decreased ion yield. This is not necessarily corresponding to
the absolute time zero.
19
The function scipy.stats.bootstrap from scipy python package (v1.7.3) with the default method (’bias-
corrected and accelerated’-method) was used to compute the confidence interval.
20Scans were measured in the order: 14.7mm,6.6 mm,12.9mm,11.1 mm,10.3 mm.
112
5.2 Laser dressing of helium droplets on the few femtosecond scale
30 0 30
delay (fs)
0
5
10
15
(normalized)
mean detector
signal (arb. u.)
(a) 100 %
(highest NIR
intensity)
30 0 30
delay (fs)
(b) 80 %
30 0 30
delay (fs)
(c) 50 %
30 0 30
delay (fs)
0
5
10
15
(normalized)
mean detector
signal (arb. u.)
(d) 30 %
(lowest NIR
intensity)
30 0 30
delay (fs)
(e) 0 %
(no NIR)
30 0 30
delay (fs)
4.8
5.2
5.6
1e8
(f)
Figure 5.15: (a-e) Time delay-dependent diffraction intensity for different NIR pump intensities;
The solid line shows the average detector signal for the brightest 4 % (72) of the images normalized
to the single-shot spectrum integral. The NIR intensity in the interaction region, manipulated
using the motorized iris in the spectrometer chamber (see chapter 4.3.1), is not necessarily evenly
varied. Due to the changed shape of the incident NIR beam, the position of the NIR focal spot is
shifted within the XUV’s focal plane. Spatial overlap correction may induce a slight shift of zero
time delay position between the plots. The time zero indicated by the x-axis is set with regards
to (a). The light-colored filled area in the background shows the scaled integrated single charged
helium droplet fragment signal acquired with the ion time of flight spectrometer. This provides an
orientation for time zero. Plots (a-e) use the same scaling factor for the ion time of flight data.
During the experiment, the scans were acquired in the order (a), (e), (b-d). (f) Stability of the
XUV signal represented by the integral of the acquired spectra for each scan point color-coded
to the colors of the plots (a-e). Please note that the y-axis in the plot starts not at zero. This
clipping of the y-axis is to visualize also smaller variations and the difference between the XUV
signal strength available to measurements shown in (a-e). Overall the variations are on the order
of 10%.
113
5. Ultrafast dynamics in helium nano-droplets
region. According to (idealized) NIR wavefront propagation simulations the relative peak
intensities associated to the figures are 100% for fig. 5.15 (a), 80% for fig. 5.15 (b), 50%
for fig. 5.15 (c) and 30 % for fig. 5.15 (d) and 0 % for fig. 5.15 (e). Reducing the NIR
intensity from 100 % to 30 % yields a barely significant reduction of the depth of the diffraction
intensity decrease feature from 40 % (fig. 5.15 lower dash-dotted line) to 30 % (fig. 5.15 higher
dash-dotted line) below the assumed baseline (fig. 5.15 dashed line). The width of the decrease
on the baseline stays approximately constant around 28
fs
. The full width half minimum
relative to the assumed baseline is very similar between fig. 5.15 (a-c) at (18
±
1)
fs
but reduces
significantly to 12
fs
in fig. 5.15 (d). As expected, with no NIR present in the interaction
region, as shown in fig. 5.15 (e), the decrease does not occur at all. The generation of helium
ions and ionic fragments appears to be more sensitive to the reduction of NIR intensity: From
fig. 5.15 (a) to (b) the peak of the average ion signal from the brightest (normalized) 4% of
the diffraction patterns reduces by a factor of 2
.
2, from (b) to (c) it reduces further by a factor
of 2
.
5and from (c) to (d) by a factor of 1
.
3. Beginning with fig. 5.15 (b), the ion signal hints a
plateau-like behaviour towards larger positive time delays (XUV pulse hits the droplet before
the NIR). The level of the two most positive delay times acquired during the scans reduces by
a factor of 2
.
2(b-c) and 1
.
7(c-d). Meanwhile the ion signal at negative delays (NIR hits the
droplet before the XUV) decreases by factors of approximately 1
.
6(a-b), 1
.
2(b-c), and 1
.
3
(c-d). The XUV spectrometer integrals (see fig. 5.15 (f)) indicating a stable XUV intensity
with drifts in the mean XUV signal between the scans of in total approximately 10 %, which
is within typical values. A decrease of the spectrometer signal due to closing the iris can be
excluded. The assumption of an uncut XUV is further confirmed by the measurement present
in fig. 5.15 (e) with no NIR present that was taken immediately after the data-set shown in
fig. 5.15 (a). The closing of the iris to ca. 6
.
6
mm
shows a decrease of the spectrometer signal
by only 2 % and the diffraction intensity is at the same baseline as the other measurements in
fig. 5.15 (a-d).
5.2.3 Limitations of the acquired data
The measurements carried out, concerning their duration and number of scanned time delays,
are a compromise between the amount of data at a given delay (statistical error), delay range,
and resolution. The total duration of the measurements is limited by the temporal stability of
the experimental conditions. Those include the provided XUV and NIR pulses in the sense of
pointing (e.g., influencing the spatial overlap), pulse energy and other properties, the helium
nano-droplets in terms of their size, their amount, and their timing relative to the pump and
probe pulses, as well as the mechanical stability of the setup thus among others the stability
of the delay between pump and probe pulse. Those properties vary to some extent on a
shot-to-shot level, which introduces an error to each time delay step in the measurement.
Slower drifts of parameters cause a change of the experimental conditions throughout a scan
or between scans. In fact, few-femtosecond spanning delay scans (see appendix B.1) show
that the slight sub-millimeter variation of the position of the overlap along the beam axis
influences some properties of the scattering intensity decrease significantly. Hence, particularly,
the spatial overlap is confirmed to be a fragile component of the experiment. By experience,
good stability can be typically achieved over a scan duration of 30 −45 minutes.
114
5.2 Laser dressing of helium droplets on the few femtosecond scale
While the short-term variation of the droplet source’s performance cannot be well controlled,
operation conditions like stable valve temperature and backing pressure improve the stability
over the day. Despite several stabilization systems (such as the pointing stabilization of the
OPCPA), Laser and XUV pulse parameters delivered by the beamline exhibit drifts limiting
the scan duration. Several times per day, the pointing and the output power of the OPCPA
and the HHG conditions need to be readjusted to correct mainly for pointing and pulse energy
drifts.
To suppress features in the scans that may arise from those slow drifts, the order of the
investigated time delays was randomized for the data presented above. To avoid any bias, the
randomization was automatized
21
, and it is improbable that two scans were carried out in the
same order. In fact, this is the case for none of the presented scans.
The presented time delay scans show each the significant decrease in diffraction intensity
during the temporal overlap of NIR and XUV pulses. The 95 %-confidence intervals of the data
points during the scattering decrease are not overlapping with the baseline given by the values
towards the minimal and maximal delays during the scans. Near those, the scans show very
different behaviors. Some form a clear baseline (see e.g. fig B.2 (c) or fig. 5.15 (d)). Others
show more variation in the data points, whose 95%-confidence intervals not necessarily overlap
with the apparent baseline (see e.g. fig 5.14 or fig 5.15 (b)). During the analysis performed
here, no systematic behavior of those features could be uncovered. Possible systematics behind
this behavior remain to be investigated in future experiments.
21relying on the numpy.random.shuffle-function implemented with the python package numpy
115
5. Ultrafast dynamics in helium nano-droplets
20 10 0 10 20
delay (fs)
0
2
4
6
8
10
12
14
16
normalized mean detector
signal (arb. u.)
(a)
1 0 1
delay (fs)
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
(b)
Figure 5.16: Time-dependent sub-femtosecond resolved diffraction intensity in the context of
the observed few femtosecond dynamics; (a) A time delay scan with a step size of 1
.
7
fs
shows the
diffraction intensity decrease (blue) around temporal overlap indicated by the step-like decrease of
the ion signal plotted in light-gray. (b) Two subsequent time delay scans (blue & orange) with
a step size of 330
as
taken around the center of the diffraction intensity decrease in (a) showing
an oscillation behavior in diffraction intensity. (a & b) The diffraction intensity is shown as the
normalized mean diffraction intensity of the brightest 1% of the images. In light gray, the mean
integral of the Helium ion signal for the 1% brightest diffraction images is plotted to indicate
spatio-temporal overlap.
5.3
Observation of sub-cycle dynamics in helium nano-droplets
Previous to this work, researchers have succeeded on revealing attosecond dynamics in laser
dressed gaseous [66, 195, 196] or solid systems [163]. Now, the goal is to extend the field to
laser dressed nano-particles, specifically helium nano-droplets. The in this thesis presented
experimental setup allowed a first observation of such dynamics in the diffraction response
oscillating faster than one period of the NIR laser field. In fact, a switch from minimal to
maximal diffraction response occurs over time delays in the attosecond domain.
In this section, the made observations will be described. The observed oscillation are only
just above the stability and resolution limits of the experiment. To convincingly demonstrate
their existence, null hypotheses are formulated, tested against the data, and rejected.
As described in the previous section, when the droplet is illuminated simultaneously with
NIR and XUV pulses, a significantly reduced diffraction signal is observed, as can also be seen
in fig. 5.16 (a). Now, the signal in the center of the observed several femtosecond "dip" was
examined more closely in the region indicated by the gray box in fig. 5.16 (a). For this, the
time delay between NIR and XUV was sampled at high resolution, more precisely in 330
as
steps. Fig. 5.16 (b) shows two consecutive measurements with this high resolution. The
116
5.3 Observation of sub-cycle dynamics in helium nano-droplets
measurements shown reproduce each other well. It can be seen that the integrated diffraction
signal oscillates around a mean value, with a period corresponding to about half an NIR cycle,
which measures ca. 1
.
3
fs22
. Further, both measured oscillations have a comparable amplitude
corresponding to approximately 13% of the signal away from temporal overlap plotted in fig.
5.16 (a).
5.3.1 Setting null hypotheses
The data will be tested against two plausible null hypotheses:
1.
There is no dynamic on the sub-cycle level. Data acquired over one or two laser cycles
can be approximated by its mean value.
2.
There is no dynamic on the sub-cycle level. However, due to slow drifts over the duration
of the scan and the random order of the time delays, the impression of an oscillation
occurs. Hence, when sorted by their acquisition time, the data is well approximated by
a constant drift (linear slope).
The significance of data points deviating from the behavior implied by the null hypotheses
is judged using confidence intervals. As explained previously in section 5.2.1, the confidence
intervals are computed through the bootstrap approach (see [194]). This way, no underlying
probability distribution needs to be assumed. A significant deviation is given when the
confidence interval of a data point does not contain the value suggested by the hypothesis. To
be more specific, in this section, the following vocabulary is used:
•
’possibly significant’: significance with regard to an interval with 68
.
1% confidence
23
•
’marginally significant’: significance with regard to an interval with 90 % confidence
•’significant’: significance with regard to an interval with 95 % confidence
Note the labels are non-exclusive: every marginally significant data point is also possibly
significant, and every significant data point is also marginally and possibly significant. It is
expected that for a confidence of 95 %,90 % and 68
.
1% the true value is not included by the
confidence intervals in 5 %,10 % and 31
.
9% of cases, respectively. If the fraction of confidence
intervals not including the value suggested by the null hypothesis is consistently higher, the
null hypothesis must be rejected.
5.3.2 Testing of sub-cycle oscillations on null hypotheses
The measured sub-cycle signal is shown in fig. 5.17 (a) together with the associated mean
Helium ion yield. The diffraction signal’s data points oscillate around their mean value
(horizontal line in fig. 5.17 (a)). The ion signal seems to follow that oscillation. To judge
the significance of the observed deviation of the diffraction signal’s data points from their
mean, i.e., from null-hypothesis 1, the three different plotted error bars represent the three
22
The spectrally broad NIR pump pulses are expected to have a cycle duration between 2
.
3
fs
(700
nm
) and
3.2fs (950 nm).
23This is the fraction covered by plus-minus one standard deviation in a Gaussian distribution.
117
5. Ultrafast dynamics in helium nano-droplets
1 0 1
delay (fs)
4
5
6
7
8
9
10
11
12
normalized mean detector signal (arb. u.)
(a)
not significant
possibly significant
marginally significant
significant
null hypothesis
confidence interval 95 %
confidence interval 90 %
confidence interval 68.1 %
3772.5 3775.0 3777.5 3780.0 3782.5
run id
4
5
6
7
8
9
10
11
12
(b)
Figure 5.17: Time-dependent sub-femtosecond resolved diffraction intensity in the context of
the observed few femtosecond dynamics; (a) The first time delay scan with a step size of 330
as
taken around the center of the diffraction intensity decrease in fig. 5.16 (a) showing an oscillation
behavior in diffraction intensity as well as integrated ion signal. The plot further shows the data’s
testing against the first null hypothesis. (b) Shows the same data as (a), but now sorted by run id
(i.e., acquisition time) to compare it against the second null hypothesis. (a & b) The diffraction
intensity (colored and gray data points) is shown as the normalized mean diffraction intensity of the
brightest 1 % of the images with error bars representing confidence intervals for 68
.
1% confidence
in light green, 90 % confidence in light orange and 95% confidence in light blue. The black line
represents the hypothesis against which the data is tested. The color of each data point indicates
its significance with respect to the null hypothesis, i.e., by which confidence interval the hypothesis
is not included anymore: blue for 95% confidence, orange for 90% confidence, green for 68
.
1%
confidence and gray if included by all three confidence intervals. In light gray, the mean integral
of the Helium ion signal for the 1 % brightest diffraction images is shown with 95% confidence
intervals.
118
5.3 Observation of sub-cycle dynamics in helium nano-droplets
1 0 1
delay (fs)
4
5
6
7
8
9
10
11
normalized mean detector signal (arb. u.)
(a)
3787.5 3790.0 3792.5 3795.0 3797.5
run id
4
5
6
7
8
9
10
11
(b)
Figure 5.18: Subsequent measurement reproducing the oscillating behaviour of the time-dependent
sub-femtosecond resolved diffraction intensity during temporal overlap between pump and probe
where the diffraction intensity decrease occurs. (a) & (b) show the testing of the measurement
against the first and second null-hypothesis, respectively. This is analog to fig. 5.17 (a & b). For a
description of the color scheme see caption of fig. 5.17.
different confidence intervals
24
of interest: 95% (light blue), 90% (light orange), and 68
.
1%
(light green). In the sub-cycle signal, 1data point is significant, 3data points are marginally
significant, and 4data points are possibly significant with respect to the mean value. However,
according to the confidence values, only 0
.
55,1
.
1, and 3
.
5would be expected, respectively.
Hence, the first null hypothesis can be rejected. In fig. 5.17 (b), the data points are sorted
according to their run id (i.e., time of acquisition) and compared against the alternative,
the second null-hypothesis. Here, 1data point is significant, 2data points are marginally
significant, and 5data points are possibly significant. Although barely, all values exceed the
expectation value, and the second null hypothesis can be rejected.
So far, the claim of sub-cycle dynamics could be made barely rejecting the null hypotheses.
It is strengthened through the second sub-cycle resolving measurement, which was already
shown in fig. 5.16 (b) (orange curve) in direct comparison to the first measurement. Now
in fig. 5.18 (a) also the ion signal is plotted, which seems to follow the oscillation shown
by the diffraction signal. Thus, this measurement reproduces the qualitative features of the
first measurement with respect to diffraction and ion signal. Again, the diffraction signal
data can be tested against both null hypotheses, as shown in fig. 5.18 (a) and (b). There
are 3significant (expected 0
.
55) and 3marginally significant (expected 1
.
1) and 5
/
4possibly
significant (expected 3
.
5) deviations from each hypothesis, which leads to the rejection of both
null hypotheses.
24obtained through the bootstrap-method;
119
5. Ultrafast dynamics in helium nano-droplets
2 1 0 1
delay (fs)
8
10
12
14
normalized
mean detector
signal (arb. u.)
(a)
2 1 0 1
delay (fs)
8
10
12
14
(b)
2 1 0 1
delay (fs)
8
10
12
14
normalized
mean detector
signal (arb. u.)
(c)
2 1 0 1
delay (fs)
8
10
12
14
(d)
(a) (b)
Figure 5.19: Series of succeeding measurements with sub-cycle resolution reproducing an
oscillating behavior of the diffraction intensity that was observed previously in two succeeding
measurements shown in fig 5.18 (a). (a-c) testing of the first null hypothesis against the measured
data (for color scheme see caption fig. 5.17) only in (a) and (b) the null hypothesis can be rejected.
Both are directly compared in (d): In blue, the data from (a) and in orange the data from (b). The
data at different delays was acquired in linear order. Thus the second null hypothesis is obsolete.
Due to the null hypothesis rejection in both measurements and the result’s reproducibility,
it is concluded that sub-cycle dynamics were resolved.
Nevertheless, the reproduction of the oscillation signal has proven to be very challenging.
Fig. 5.19 shows such a reproduction acquired 8days after the data shown in fig. 5.17 and
5.18. Opposed to the already presented measurements, the time delays were measured in
(reversed) order and not randomized. Skipping the randomization allowed for a longer scan
duration since the loss of spatial overlap and hence loss of signal or features would only affect
the smallest delays (i.e., latest measurements). It would not be randomly scattered across the
scan. It also implies that the smallest delays are most unreliable. The second null hypothesis
does not apply anymore.
The same scan was carried out three times in a row. A single scan takes approximately
45
min
plus 15
min
for verification of overlap and beam parameters. Only, the first (fig. 5.19
(a)) and the second (fig. 5.19 (b)) scan show significant deviations from the null hypothesis
120
5.3 Observation of sub-cycle dynamics in helium nano-droplets
(1). Accordingly, similar to the measurements discussed before, the data sets, except for the
third measurement (fig. 5.19 (c)) qualify for rejection of the null hypothesis.
Fig 5.19 (d) shows a direct comparison of the significant data sets (fig. 5.19 (a/b)) and
reveals that, except for the last few data points, they are closely reproducing each other. Here,
the oscillation has a period of approximately 1
.
7
fs
. This period is longer than expected from
half a cycle but still clearly below the duration of a full laser cycle. Compared to the initially
measured oscillations, shown in fig. 5.16 (b), the reproduction has only approximately half the
amplitude with respect to the diffraction intensity away from temporal overlap (see fig. B.11
(a) in the appendix), specifically ca. 6%.
The shown data illustrates a central difficulty experienced throughout the experiment: The
experimental conditions cannot be fully controlled and can change so that sub-cycle oscillations
can be either resolved or not. In appendix B.2 all potentially meaningful sub-cycle resolved
measurements are shown and tested against the first null hypothesis. It can be rejected for
many of them. Thus the presence of sub-cycle dynamics is indicated. However, none of the
additionally shown scans were well reproduced by a subsequent scan. This hints towards
insufficient stability of experimental conditions. Direct reproducibility was only demonstrated
for the traces discussed above. Another oscillation with an approximately half laser cycle
period can be seen in fig. B.6 (b). Unfortunately, no subsequent time delay scan that could
indicate its reproducibility was made.
5.3.3 Limitations
The observed sub-cycle dynamics are weak compared to the seen change in diffraction intensity
on the few-femtosecond time scale. The normalized mean of the brightest one percent was
shown for all presented data. While taking into account more images makes the observed
oscillation look more smooth, the significance of the result concerning the confidence intervals
decreases. A reduction of significance can be expected since images originating, in tendency,
from a larger focal volume that corresponds to a worsened temporal resolution contribute to
the mean value.
Reproducibility plays a crucial role in the observation’s credibility for those weak signal
amplitudes. Sub-cycle measurements are susceptible to vibrations coupled into the setup,
drifts of the delay stage, and other drifts that may shift the oscillation’s phase during the time
delay scan. The occurrence of those drifts throughout a delay scan could be confirmed during
delay stage calibration measurements in a vented setup. While the scan step of the delay stage
produces, on average, a consistent change in delay, when averaging the difference between
the mean delays over the entire duration of a measurement at a given delay stage position,
it deviated notably. This deviation of the average delay step from the driven one is due to
long-term drifts, which are further observable when measuring over a longer duration at a
fixed delay. Hence, long term drifts are one possible explanation for the observed oscillation
period of 1
.
7
fs
in the reproduction of the oscillation several days after measuring for the first
time with a 30 % shorter period.
121
5. Ultrafast dynamics in helium nano-droplets
5.3.4 Summary
In summary, the analysis of the time-delay scans in the attosecond domain shows:
•
Sub-cycle oscillations are observed, when the helium nanodroplets are exposed to NIR
and XUV simultaneously.
•On each of two different days, two subsequent measurements show such an oscillation.
•
Depending on the day, the data shows an oscillation period of 1
.
3
fs
or 1
.
7
fs
, which
matches approximately the duration of half an NIR laser cycle.
•
depending on the day, the amplitude of the oscillations is at 13 % or 6 % with respect to
the diffraction signal away from temporal overlap.
122
5.4 Modeling of the delay-dependent diffraction response
5.4 Modeling of the delay-dependent diffraction response
To explore the underlying physics of the observations, i.e., the diffraction intensity decrease
and sub-cycle oscillations, simulations that model the performed experiment are carried out.
These simulations are based on a helium atom’s theoretical, numerical description, and the
results are extended to the bulk domain. In this work, the diffraction image generated from
spherical, sub-micrometer-sized Helium droplets exposed to a superposition of ultra-short NIR
and XUV pulses has to be modeled. The modeling must allow for the description of sub-cycle
effects, limiting the choice of approximations and assumptions.
Previously, in the scope of understanding few-femtosecond dynamics, as presented in the
doctoral thesis of Julian Zimmermann [35], the optical response was modeled based on solving
the TDSE for a single helium atom exposed to NIR and XUV pulses in temporal overlap at
different NIR peak intensities. There, the single-atom response forms the basis for the optical
response of an effective bulk. Knowing the optical response of the bulk, the diffraction patterns
were computed with Mie-theory scattering simulations. The time delay between NIR and
XUV pulse was modeled through effective intensities of NIR and XUV determined by their
envelope. Since this approach yields results independent of the NIR’s phase and attosecond
pulse trains structure of the XUV pulse, it neglects effects on a timescale below an optical
cycle of the NIR (sub-cycle). The present work primarily uses the method and code from
[35]. However, it adapts the implementation of time delays between NIR and XUV pulse by
directly simulating a delay scan and therefore does take into account the attosecond pulse
train structure of the XUV pulse and the NIR’s phase. Furthermore, this work extends the
previous work by considering non-linear effects involving up to three photons.
In this section, first, the full modeling approach is explained. Then it is applied in two
steps: in linear approximation, resolving effects on the time scale of a few NIR cycles and then
including non-linear effects, which will allow the inclusion of sub-cycle dynamics. Moreover, to
uncover the underlying effects, the contributions by the different harmonic orders are extracted.
In section 5.4.4, the model’s underlying most significant assumptions, approximations, and
limitations are discussed.
5.4.1 Modelling the scattering response of helium droplets
In this thesis, the time-delay-dependent diffraction intensity of a laser-dressed helium nano-
droplet is modeled. Following the approach in [35], this modeling can be divided into three
steps:
1. computing the atomic response
2. constructing the effective optical response of bulk helium
3. simulating the scattering response
Before a detailed description of each step is provided in the following paragraphs first, an
overview of the whole process is given. It is illustrated by fig. 5.20. In the first step, for a given
time delay
td
between pump and probe (NIR and XUV) pulses and XUV/NIR pulse properties,
the atomic dipole response is retrieved by solving the TDSE. The second step constructs a bulk
123
5. Ultrafast dynamics in helium nano-droplets
Figure 5.20: Modelling of the scattering response of a helium droplet at different time delays
td
; orange box: at a given delay first, the atomic response is computed by solving the TDSE
numerically. Blue box: then bulk properties are added onto the obtained dipole response. First,
a dephasing time is imprinted. Then, via the dynamic polarizability, the response is fed into
the Clausius-Mossotti-relation that accounts for the local field due to polarization in the bulk
and yields a complex refractive index representing the optical response. In the case of non-linear
response, multiple atomic response computations are required to retrieve the dynamic polarizability.
Green box: knowing the optical response, together with XUV spectrum, temporal pulse structure,
and droplet size distribution, a mean diffraction image and integral can be computed applying
Mie-theory.
124
5.4 Modeling of the delay-dependent diffraction response
accounting for dephasing by introducing a dephasing time and models the local electric field due
to the collective polarization of the helium atoms using the Clausius-Mossotti-relation. Where
necessary, a non-linear extension yielding the complex refractive index is applied. The third
step uses the photon-energy-dependent optical response combined with the XUV spectrum
and pulse train structure to compute the diffraction image and finally diffraction intensity for
a spherical droplet with a given radius. The droplet size distribution is included by a weighted
average of results for different droplet radii. This full process has to be repeated for each
time delay of NIR and XUV pulse
td
, so that the time-delay-dependent scattering response is
obtained.
If not declared otherwise, the description of steps one and two use atomic units. For the
third step, SI units are used.
Computing the atomic dipole response
The atomic dipole response is computed by numerically solving the TDSE based on a code
developed by H.G. Muller (for a description of the algorithm see [197])
25
. This code has been
used in the past to model laser-atom interactions, e.g., closely reproducing an experimental
measurement of above threshold ionization [198]. Since in the investigated scenario, the
wavelengths of the light pulses are much longer than the extension of an atom, the variation
of their electric field over the spatial extend of the atom can be neglected, i.e., the dipole
approximation is applied [13]. The TDSE is treated in velocity gauge (see eq. 2.6). It assumes
cylindrical symmetry and cannot support m quantum numbers other than 0. Furthermore, it
assumes that only a SAE is interacting with the laser field [13]. Hence, the interaction between
both electrons in the helium atom and double ionization is omitted.26
The Hamiltonian is composed of the kinetic energy
1
2·
p
ˆ2
, the static spherically symmetric
potential of the Helium atom
V
ˆ
(r)and the time-dependent vector potential of the incident
light A(t)multiplied with the momentum operator p
ˆ:
H
ˆ=1
2·p
ˆ2+V
ˆ(r) + A(t)·p
ˆ[197].
The A(
t
)
2
term that would result from eq. 2.6 is omitted. It has no effect on observables ([13],
p. 259). The momentum operator pˆis defined as
pˆ =−i·∇r[13].
The incident NIR pump (A
NIR
(
t
)) and XUV probe (A
XUV
(
t
)) pulses at a given time delay
td
are represented by their superposition:
A(t) = ANIR(t) + AXUV (t).(5.3)
25
For this work, the definition of the two (NIR & XUV) fields was modified. The used definition is given
below.
26
In the discussed CDI experiment, for NIR intensities of up to 2
·
10
14
, no or only a minor amount of doubly
ionized helium was detected.
125
5. Ultrafast dynamics in helium nano-droplets
500 0 500
time (a.u.)
1.0
0.5
0.0
0.5
1.0
1.5
field strength (arb. u.)
(a)
full
SVEA
error
02468
radial distance (a.u.)
10 1
100
101
potential (a.u.)
(b)
used potential
1/r-potential
Figure 5.21: (a) comparison of the electric field given in eq. 5.6 considering the full expression
(solid blue) and the expression in slow varying envelope approximation (SVEA) (dotted orange,
neglecting the cosine term). Although visually both seem identical, their difference (solid green)
reveals a local field strength deviation of up to 5 % of the peak field strength. (b) potential of the
ionic core (helium core plus one inactive electron) used within the TDSE-solving code (solid blue)
compared to a simple 1/r-potential (dashed orange).
To avoid artifacts from zero-frequency components in the electric field that can occur easily
for ultra-short pulses27, the pulses are defined through their vector potential ([13], p.259):
ANIR(t) = E0,NIR
ωNIR ·exp{︄(−2 ln 2 t2
τ2
NIR
)}︄·cos(ωNIRt+ϕNIR)(5.4)
AXUV (t) = E0,XUV
ωXUV ·exp{︄(−2 ln 2(t−td)2
τ2
XUV
)}︄·cos(ωXUV (t−td)) + ϕXUV ).(5.5)
The pulse durations
τNIR
and
τXUV
is given by the full width at half maximum (FWHM) of
the intensity envelope of the respective pulse. The peak field strengths are given by
E0,NIR
and
E0,XUV
, and their central frequencies by
ωNIR
and
ωXUV
. The phases of the NIR and
XUV pulses are defined by
ϕNIR
and
ϕXUV
respectively. The time delay between both pulses
is implemented with the XUV pulse definition and given by
td
. The electric field of the pulses
applied in the TDSE is then given by the first derivative of the vector potential:
E(t) = −d
dtA(t) ([13],p.259).
Assuming linear polarization, the vector notation of the electric field and vector potentials is
dropped. Now, for example, the electric field of the NIR pulse is given as:
ENIR(t) = −E0,NIR·exp{︄(−2 ln 2 t2
τ2
NIR
)}︄·{︄sin (ωNIRt+ϕNIR)−4 ln 2 ·t
ωNIR ·τ2
NIR
cos(ωNIRt+ϕNIR)}︄.
(5.6)
27
Zero frequency components in the electric field occur when the integral over the electric field is non-zero.
For ultra-short pulses, such non-zero integral may occur due to the short pulse envelope.
126
5.4 Modeling of the delay-dependent diffraction response
excited state TDSE result in eV published values in eV
2s 20.27 20.62 [190]
2p 21.15 21.22 [190]
3s 22.83 22.92 [190]
3p 23.07 23.09 [190]
3d 23.08 23.07 [190]
4s 23.64 23.67 [199]
4p 23.73 23.74 [190]
5s 23.99 24.01 [199]
5p 24.04 24.05 [199]
6s 24.18 24.19 [199]
6p 24.21 24.21 [199]
Table 5.3: Excitation energies from the 1s ground state and associated orbitals resulting from the
TDSE calculation compared to published values; for
n
= 4
−
6higher angular momentum states
than p are mostly, at the shown precision, energetically indistinguishable from the p orbital and
therefore not listed explicitly.
As illustrated for an exemplary pulse with a duration of
τNIR
= 350
a.u.
(
≈
8
.
5
fs
) and a
central photon energy of 0
.
057
a.u.
(
≈
1
.
55
eV28
) in figure 5.21 (a), the cosine term in equation
5.6 has only a minor influence. Hence despite the pulse definition in terms of the vector
potential
A
(
t
), it behaves approximately like expected from the intuitive definition in terms of
the electric field
E
(i.e., a sine oscillation with a Gaussian envelope). In fact, for pulses, where
the envelope slowly varies over a single oscillation, the cosine term may be neglected. This
approximation is referred to in the literature as SVEA ([37], eq. 1.14). However, in this work
the SVEA is only applied for the qualitative discussion of the results. The further numerical
treatment does not benefit from this approximation.
The properties of the helium atom are represented through the element-specific potential
V
(
r
). As shown in fig. 5.21 (b), the potential matches a simple 1
/r
-potential towards large
radial distances, however close to the ionic core, notable corrections are made. It was included
with the original code and yields improved energy levels that reproduce the known excitation
energies of helium more accurately. The result is shown in table 5.3. Except for the 2s state, a
close reproduction of the expected excitation energies (from the ground state) is achieved.
The computation yields an electron spectrum that was used to extract the energies of the
orbitals shown in table 5.3 and the atomic dipole response. Fig. 5.22 (c) shows a resulting
time-dependent atomic dipole moment for the NIR and XUV vector potentials shown in
fig. 5.22 (a) and (b). In fig. 5.22 (c), one can see that as soon as the XUV light (here,
an IAP) contributes to the excitation of the dipole, the dipole moment starts oscillating at
corresponding high frequencies. The Fourier transform of the dipole moment shown in fig.
5.22 (d) reveals resonant excitation at energies close to the expected ones shown in table 5.3.
Deviations are expected since the mutual interaction of NIR and XUV with the atom can
affect the observed resonances (see chapter 2.1.5).
28corresponding to approximately 800 nm;
127
5. Ultrafast dynamics in helium nano-droplets
500 0 500
time (a. u.)
0.2
0.0
0.2
vector
potential
(a.u)
(a)
50 25 0 25 50
time (a. u.)
0.02
0.00
0.02
(b)
600 400 200 0 200 400 600
time (a. u.)
0.00
0.05
dipole
moment
(a. u.)
(c)
20 21 22 23 24 25
energy (eV)
0
100
dipole
moment
(arb. u.)
(d)
Figure 5.22: (a & b) shows exemplary contributions to the vector potential of the incident fields.
Thereby the contribution by the NIR field is shown in red (see eq. 5.4, peak field strength 0
.
012
a.u.
,
i.e., peak intensity 5
·
10
12 W/cm2
) and by the XUV field in purple (see eq. 5.5, peak field strength
0
.
012
a.u.
, i.e., peak intensity 5
·
10
12 W/cm2
). The fields are shown for a delay of
t0
= 0. (c)
plots the time-dependent dipole response that results from the code numerical solving the TDSE:
initially the NIR field gives rise slow oscillations, where the dipole moment follows the field. When
at a time of 0the attosecond XUV pulse excites the atom, the dipole moment oscillates rapidly. At
the same time, due to ionization, in average the dipole moment continously increases. (d) shows
frequency composition of this rapid oscillation induced by the XUV (specifically the absolute value
of the dipole responses Fourier transform is shown). The dipole moment is most responsive, where
the helium atom’s resonances are expected according to table 5.3.
128
5.4 Modeling of the delay-dependent diffraction response
Constructing an effective optical bulk response
The dipole response obtained from solving the TDSE for a single atom does not account for any
relaxation, i.e., the lifetime of the excited states, or dephasing, i.e., loss of coherence between
the induced time-dependent dipole moments in an ensemble of atoms, occurring due to the
atom’s environment. Both manifest as a decay of the induced dipole moment [200], though
the dephasing time is typically much shorter than the relaxation time [37]. Consequently,
the latter will be neglected. Dephasing already plays a role in gaseous media. For example,
Chen et al. [200], use a dephasing time of 2687
a.u.
(65
fs
) with helium to account for its
gaseous environment [200]. To mimic a dephasing time in a simulation (see e.g., [54] or [200]),
a continuous windowing function
W
(
t
)is multiplied with the the time-dependent single-atom
single active electron dipole response patom,SAE(t)to enforce its decay:
pbulk,SAE(t) = patom,SAE(t)·W(t).(5.7)
Also here, for helium nano-droplets, following the method described in [35], the bulk single
active electron dipole response
pbulk,SAE
(
t
)is obtained from the atomic dipole response by
applying an appropriate windowing function. In the presented case, a nuttall4c window [201]
is used. This specific windowing function is chosen for its minimized artifacts due to the
frequency response of the window.
29
Later on, the width of the window, i.e., dephasing time,
is adjusted such that the refractive index obtained from an XUV pulse interacting with helium
(no NIR present) yields a refractive index fitting to the known one of liquid helium.
Now, to obtain the spectral atomic dipole response, it is Fourier transformed, yielding the
frequency-dependent single active electron dipole moment
pbulk,SAE
(
ω
). Furthermore, when
solving the TDSE in single active electron approximation, the dipole moment was computed
only for a single electron. To account for both electrons, the response is doubled [64]:
pbulk(ω)=2·pbulk,SAE(ω).
In the linear regime, at a given frequency
ω
, the dipole moment
pbulk
(
ω
)and electric field
E
(
ω
)acting locally onto the atom are linked through a factor
α
(
ω
)known as the dynamic
polarizability [88]:
pbulk(ω) = α(ω)·E(ω)⇔α(ω) = pbulk(ω)
E(ω).(5.8)
In the case of non-linear effects, the dipole moment does not depend linearly on the electric
field and higher order terms must be considered [202]:
pbulk(ω) = α1(ω)·E(ω) + α2(ω)·(E(ω)∗E(ω)) + α3(ω)·(E(ω)∗E(ω)∗E(ω)) + ... . (5.9)
The non-linear treatment becomes necessary to ensure that the obtained spectral response
in terms of the dynamic polarizability at a given delay is also valid for NIR/XUV pulses with
29
In [35] a Hann window was used. In the context of this work, it was found that it yields a ringing in the
vicinity of the 2p resonance that is not present with a nuttall4c window.
129
5. Ultrafast dynamics in helium nano-droplets
different spectral intensities (i.e., a different spectrum) than used with the TDSE. This is a
requirement to translate between microscopic and macroscopic properties.
While for the linear case, as stated in equation 5.8, one combination of electric field
and dipole moment is sufficient, taking into account non-linearities up to the n-th order n
combinations of electric field and dipole response need to be available to solve for the dynamic
polarizability coefficients using eq. 5.9. The problem becomes a system of linear equations and
thus has a straightforward solution. By using the result from more than the required three
dipole moment computations, artifacts can be reduced.
The dynamic polarizability describes microscopic properties within the nano-droplets. It
must be linked to the macroscopic response, which is characterized through the dielectric
susceptibility
χ
(
ω
)linking the external electric field acting upon the bulk helium
E
(
ω
)
(superpositioned XUV and NIR) to the induced polarization
P
(
ω
), which describes the
dipole moment density in bulk. It is crucial to point out that the field acting upon the bulk
E
(
ω
)is not equal to the field acting locally onto the atom
E
(
ω
). The induced polarization of
the surrounding atoms changes the local field at a single atom [88]:
E(ω) = E(ω) + 4π
3P(ω) [88].(5.10)
A link between
χ
(
ω
)and
α
(
ω
)is facilitated by the Clausius-Mossotti relation that accounts
for the relation of local and external field in eq. 5.10. For the case of linear response it is well
known and given as:
χ(ω) = N·α(ω)
1−4π
3·N·α(ω)[88].
Here,
N
is the density of atoms (number density) of the bulk. It relates the atomic dipole
moment
p
to the dipole moment density, i.e., the polarization
P
. The dielectric susceptibility
χ
(
ω
)relates to the refractive index
n
(
ω
)by
n
(
ω
)
2
= 1 + 4
πχ
(
ω
)[88]. As a direct consequence
the refractive index is given in terms of the dynamic polarizability by:
n(ω) = √︄3·N·α(ω)
3
4π−N·α(ω)+ 1 .(5.11)
In the case of non-linear response, analogous to the dynamic polarizability, the dielectric
susceptibility
χ
(
ω
)is expanded into higher-order terms, so that the macroscopic polarization
Pis given by:
P(ω) = χ1(ω)·E(ω) + χ2(ω)·(E(ω)∗E(ω)) + χ3(ω)·(E(ω)∗E(ω)∗E(ω)) + ... . (5.12)
Comparable to the linear case, equations linking the orders of the dynamic polarizability to
the dielectric susceptibility can be derived. Here the first three orders are taken into account.
The full derivation is given in the appendix C.1 (following [202]).
χ1=N·α1
1−4π
3N·α1
χ2=ϑNα2
(ϑE)∗(ϑE)
E∗E
130
5.4 Modeling of the delay-dependent diffraction response
χ3=ϑNα3
(ϑE)∗(ϑE)∗(ϑE)
E∗E∗E+ 24π
3ϑN2α2
(ϑ(α2(ϑE)∗(ϑE))) ∗(ϑE)
E∗E∗E
with
ϑ(ω) = (︃4π
3χ1(ω)+1)︃
The terms for the dielectric susceptibilities (linear and non-linear) can be summarized into
an effective dielectric susceptibility:
χeff (ω, E) = χ1(ω) + χ2(ω)·E(ω)∗E(ω)
E(ω)+χ3(ω)·E(ω)∗E(ω)∗E(ω)
E(ω).
Hence the polarization Pis by
P(ω) = χeff (ω, E)·E(ω)
and consequently the refractive index is given as [88, 203]:
n(ω, E) = √︂1+4πχeff (ω, E)(5.13)
.
The obtained refractive index is complex. For convenience it is represented through
δ
(
ω
)
showing the deviation from 1in its real part and
β
(
ω
), the imaginary part, revealing the
absorption properties [204, 38]:
n(ω)=1−δ(ω) + i·β(ω).(5.14)
Now with the ability to obtain the complex refractive index from the atomic dipole response,
still following the modeling approach in [35], the window’s width applied in eq. 5.7 can be
optimized. As a reference serves the published refractive index of liquid helium retrieved
through the interpretation of experimental data in the spectral range of 19
.
5
−
24
eV
[87]. The
dipole moment is computed using a 300
as
(FWHM)
30
XUV pulse with Gaussian envelope
that is spectrally centered at 22
eV
with a peak intensity of 5
·
10
12 W/cm2
. No NIR pulse is
present. The result involves only linear processes. Fig. 5.23 shows the computed refractive
index for two scenarios: A long dephasing time of 30
fs
and the manually optimized dephasing
time of 12
.
7
fs
to fit the shape of the published data for liquid helium [87]. In the context of
the carried out modeling, the dephasing time refers to the time between excitation of the atom
with the XUV and the dipole moment having decayed to 0.
While for the optimized dephasing time (fig. 5.23 (b/d) the qualitative shape of
δ
(
ω
)and
β
(
ω
)around the 2p-resonance is well reproduced, there is a 0
.
5
eV
redshift of the computed
result along the energy axis. This shift is due to solving the TDSE for an atomic potential
that aims to reproduce the energy levels of atomic helium. However, the energy levels of bulk
helium are blue-shifted relative to the atomic ones. Experimental data for the 1s2p transition
shows a shift of 0
.
43
eV
between nano-droplet and atomic resonance [75], together with the
mismatch of the TDSE result & published atomic data of 0
.
07
eV
(see table 5.3) it matches
the in fig. 5.23 observed 0
.
5
eV
shift. As the shift is not the same for all resonances, it cannot
30As long as the pulse has sufficient bandwidth, the result does not depend on the pulse duration.
131
5. Ultrafast dynamics in helium nano-droplets
20 22 24
photon energy (eV)
1
0
(a) 30.0 fs dephasing time
Lucas et al. Lucas et al. shifted Modelling result
20 22 24
photon energy (eV)
0
1
(c) 30.0 fs dephasing time
20 22 24
photon energy (eV)
0.5
0.0
(b) 12.7 fs dephasing time
20 22 24
photon energy (eV)
0.0
0.5
(d) 12.7 fs dephasing time
Figure 5.23: Modelled refractive index for bulk helium compared to published data considering
different dephasing times. For (a/c) a long dephasing time of 30
fs
and for (b/d) the optimized
dephasing time of 12
.
7
fs
is shown. The top row (a/b) shows
δ
and the bottom row (c/d)
β
that
make up the refractive index as given in eq. 5.14. The modelling result (orange) is compared to
refractive index data of bulk helium based on experimental data [87] (blue). For better comparison
also a curve with the reference data shifted by −0.5 eV is shown.
be properly corrected by post-processing the dipole response. Furthermore, experimental data
suggest that there is a small resonance related to a 1s to 2s transition [75], which is forbidden
in the atomic model and hence not included with the modeling result. The harmonic spectrum
from the experiment (see e.g. fig. 4.21) does not overlap with the 2p resonance. Off-resonance
the error due to the shift and excluded transitions is smaller than on-resonance. Nevertheless,
the presented model will work for qualitative comparison with the experiment but may not be
sufficient for a quantitative comparison or for fitting to experimental data.
Simulating the scattering response
From the optical response, i.e., the refractive index, the wavelength-dependent scattering
response can be obtained. Here this is realized by only including spherical helium droplets
that make up most of the experimental data (see, e.g., fig. 5.13, showing mostly rotational
symmetric patterns). Consequently, the scattering response can be computed by applying
Mie theory. From the results, the diffraction pattern and its integral can be constructed. The
integral can then be used to compare the modeling result to the experimental data.
The Mie theory applied for the scattering was already introduced with theoretical concepts
in chapter 2.2.2. Now, the focus lies on its application together with the modeled refractive
index. Computing a diffraction pattern by applying Mie theory depends on the complex
refractive index at a given photon energy and the spherical particle’s size, i.e., droplet radius.
To reconstruct the experimental measurement, a framework around it is constructed to account
for:
132
5.4 Modeling of the delay-dependent diffraction response
1. analysis of the integrated and averaged signal
2. the statistical distribution of droplet sizes imaged
3. the temporal and spectral structure of the imaging XUV pulse (APT)
The conducted analysis (see sections 5.2 and 5.3) yields the time-delay-dependent integrated
diffraction intensity normalized to the spectrometer signal (i.e., XUV pulse energy) averaged
on the number of included hits. The analysis result is now modeled by integrating the obtained
patterns and using weighted averages of those integrals accounting for the statistic distribution
of the droplet sizes and different spectral intensities of the XUV pulse. Due to the normalization
of the experimental data to the integrated XUV spectrum, fluctuations in the XUV pulse
energies do not need to be modeled.
The size distribution of droplets produced by the employed helium droplet source was
characterized for similar operation conditions in [141] and yielded an average radius of 273
nm
.
From the diffraction images observed in the presented experiment (see, e.g., fig. 5.2), the
mean radius is estimated to be also around 300
nm
. For the modeling, a Gaussian distribution
with a standard deviation of 70
nm
is assumed for the radius. It matches approximately the
distribution towards larger droplet radii retrieved in [141]. To compromise between an exact
reproduction of the distribution and computational effort, the distribution is sampled in 20
nm
steps so that 15 droplets of different radii are considered. Hence, for each requested diffraction
integral computation, a pattern must be computed for each of the 15 droplet sizes. Then the
requested diffraction integral is obtained from a weighted average according to the sampled
distribution of droplet sizes.
Before taking the size distribution into account, the spectral and temporal properties of
the XUV must be modeled to calculate (in the non-linear case) the electric-field-dependent
refractive index and to include the properties in the Mie-theory simulations. The results of
the RABITT experiment (see chapter 3.3.4) provide the basic properties. Two aspects are to
be taken into account:
1. a spectrum composed of multiple harmonic peaks
2. the temporal structure of the attosecond pulse train
The spectrum for the modeling is artificiality computed because the experimental acquired
ones lack spectral resolution. The spectral intensities extracted from the experiment (see fig
4.21) are used to approximate the spectrum with a single photon energy per harmonic (see
figure 5.24 (a)). For the linear case, the following construction of the spectrum to retrieve
the refractive index depending on the electric field
31
is unnecessary since the refractive index
does not depend explicitly on the electric field (see eq. 5.11). However, in the non-linear case,
the specific XUV spectrum with the spectral widths of the harmonics has to be known. As
illustrated with figure 5.24 (b), on the time axis, a Gaussian pulse envelope with a FWHM of
3.1fs (as retrieved through the RABITT experiment discussed in chapter 3.3.4) is applied to
the otherwise infinite pulse train. A Fourier transform yields the spectrum (see figure 5.24
(c))
32
. This spectrum can now be used to compute the refractive index depending on the
31according to the formalism introduced with the construction of an effective bulk
32now harmonic peaks with a spectral width, but the same spectral contributions
133
5. Ultrafast dynamics in helium nano-droplets
20 30
photon energy (eV)
0
2
4
intensity (arb. u.)
(a)
5 0 5
time (fs)
0.00
0.25
0.50
0.75
1.00
intensity (arb. u.)
(b)
20 30
photon energy (eV)
0.0
0.2
0.4
intensity (arb. u.)
(c)
Figure 5.24: Construction of a realistic pulse train structure and its spectrum from the relative
contribution of each HO. (a) spectrum made from discrete HOs and their relative contribution; (b)
pulse train structure resulting from (a) in gray and after applying the dotted (blue) XUV pulse
envelope in blue; (c) spectrum of the XUV APT with applied pulse envelope. All y-axes give a
scale for the intensities but are also arb. scaled between each other.
electric field (eq. 5.13) from the non-linear dynamic polarizabilities known from the atomic
TDSE computations according to the previously introduced routine. In both the linear and
non-linear case, for a reduced computational effort, diffraction integrals are only computed
with Mie theory at the central photon energy of each harmonic.33 A weighted average of the
resulting patterns or integrals accounts for the spectral intensities.
While the so far described procedure accounts for the spectral response at an XUV pulse
at a given delay, the temporal structure, specifically the many attosecond pulses spaced by
half an NIR cycle, have to be considered as well. Each pulse in the attosecond pulse train
effectively interacts with the sample at a different delay relative to the NIR pulse. Hence,
as illustrated in fig. 5.25, an artificial pulse train can be built and is applied to the result
by a weighted average of computations from multiple delays spaced by half an NIR cycle.
The structure of the artificial pulse train follows the one already constructed based on the
result of the RABITT experiment in the previous step. Attosecond pulses with an intensity
of more than 1% of the train’s peak intensity, i.e., the strongest five pulses in the train are
considered. By modeling the APT temporal structure through superposition of results from
an IAP, possible fast-acting modifications of the helium droplets properties lasting longer than
the attosecond pulses are neglected for and thus not experienced by the succeeding pulses (see
also discussions in sections 5.4.4 and 5.5.2).
When including non-linear effects, processes like wave mixing may yield emission of light
[203]. The emission is indicated through a negative
β
. Since the here employed Mie scattering
code
34
does not support this, the emission of light is disregarded, and
β
is set to zero for
those cases. The simulation results presented in the following sections will show that negative
absorption
β
values occur only in confined regions (concerning photon energy and delay). Thus
the qualitative result should not be affected.
33
regarding the computational effort: since for each photon energy 15 Mie patterns need to be computed
accounting for the nano-droplet size distribution, evaluating 5photon energies per pulse means that per data
point 75 diffraction patterns have to be computed.
34Here the python package miepython is used. (see https://github.com/scottprahl/miepython)
134
5.4 Modeling of the delay-dependent diffraction response
0.2
0.0
0.2
(a-e)
0.2
0.0
0.2
0.2
0.0
0.2
field strength (arb. u.)
0.2
0.0
0.2
20 10 0 10 20
time (fs)
0.2
0.0
0.2
20 10 0 10 20
time (fs)
0.2
0.0
0.2
field strength
(arb. u.)
(f)
20 10 0 10 20
time (fs)
0.00
0.02
0.04
intensity
(arb. u.)
(g)
Figure 5.25: Visualization of the construction of an attosecond pulse train to account for its
temporal structure. Analog to this illustration, simulation results from different time delays are
combined. A fixed factor scales the relative amplitudes of the XUV and NIR fields for illustration
purposes. (a-e) The attosecond pulse train structure is assembled from isolated attosecond pulses
at half NIR cycle spaced delays. The pulses (and analog the integrated Mie scattering results)
are scaled according to their relative contribution in the pulse train. In (f), their superposition is
shown, and (g) gives the resulting attosecond pulse train structure in terms of the intensity.
Simulating the scattering response requires the computation of many diffraction patterns
by applying Mie theory to obtain a single data point in a delay scan. Since the applied method
needs to be able to account for sub-cycle features, the attosecond pulse structure of the XUV
pulse needs to be taken into account. To cover moreover also non-linear effects, including
multi-photon couplings as is, e.g., the case with quantum path interference (see chapter 2.1.4
or 2.1.5), the refractive index used for computing the diffraction patterns depends on the
specific XUV spectrum.
5.4.2 Few-cycle resolved simulations
The atomic dipole response yielded by the solution to the TDSE inherently contains a response
of the atom at sub-cycle resolution. The same is valid for the few-femtosecond resolved
experiment carried out. However, to investigate the few-femtosecond response in a first step,
features varying faster than one cycle of the NIR pulse are omitted. Therefore a low-pass filter
is applied to the simulated refractive index along the delay axis.
For the modeling presented in this subsection, first, it will be shown that when looking at
the low-pass filtered refractive index retrieved under the linear assumption stated previously,
the optical response (i.e., refractive index and thus effective dielectric susceptibility) does not
depend on the XUV’s spectral intensities and thus indeed can be obtained applying the linear
assumption concerning the XUV pulse. The refractive index will be modeled depending on
the NIR intensity, also investigated experimentally. After simulating the Mie response, a data
set comparable to one of the experimental investigations is produced from the modeling.
135
5. Ultrafast dynamics in helium nano-droplets
NIR pulse
parameter value
peak intensity up to 2·1014 W/cm2
central wavelength λNIR 800 nm
pulse duration (FWHM) τNIR 8.3 fs
CEP ϕNIR 0
XUV IAP (TDSE)
parameter value
peak intensity 5·1012 W/cm2
central photon energy ωXUV 19 /22 /25 eV
pulse duration (FWHM) τXUV 300 as
CEP ϕXUV 0
XUV APT (diffraction intensity)
parameter value
fundamental wavelength λXUV,0770 −830nm
pulse duration (FWHM, envelope) τXUV,AP T 3.1fs
harmonic contributions
(HO 11 / 13 / 15 / 17 / 19 / 21) 0.174 /0.320 /0.203 /0.173 /0.085 /0.044
relative pulse intensities 0.127 /0.605 /1/0.605 /0.127
Table 5.4: NIR and XUV pulse parameters used for the modelling of the diffraction intensity
An overview on the NIR and XUV pulse parameters used for the modelling of the few-cycle
resolved dynamics is given in table 5.4. The experiment is modelled with a transform limited
NIR pulse of FWHM pulse duration (with respect to its intensity) of 8
.
3
fs
(see chapter 3.1)
and a central wavelength of 800 nm (approximately matching the mean wavelength retrieved
in a measurement behind the interaction region, see fig. 4.19 (a)). Matching the measured
beam profile in fig. 4.20 (b), intensities up to 2
·
10
14 W/cm2
are used in the modelling. The
TDSE simulations are carried out with an XUV IAP. For the proof of linearity its central
photon energy is varied, however otherwise a central photon energy of 22
eV
in combination
with a FWHM duration (with regards to intensity) of 300
as
is used. A peak intensity of
5
·
10
12 W/cm2
, similar to the experimental estimated value of up to 10
13 W/cm2
is used (see
chapter 4.6.2). Then when simulating the diffraction intensity by applying Mie theory, the
XUV pulse train is build for a variable fundamental wavelength around 800
nm
and a pulse
duration (envelope) of 3
.
1
fs
(as reconstructed with the RABITT measurement, see chapter
3.3.4). The five brightest pulse from the pulse train are included. Exemplary the harmonic
contributions shown in fig. 4.21 are used.
Linearity of the few-cycle resolved macroscopic response
The assumption of linearity is verified by looking at three optical responses obtained from a
TDSE result for a different IAP each. More specifically, the pulses are centered around 19,
22, and 25
eV
, but otherwise are of the same duration and produce the same peak intensity.
With this method, the spectral contributions of each harmonic change differently from pulse
to pulse; thus, the method is also sensitive to possible non-linearities, such as wave mixing
processes. Fig. 5.26 shows the refractive index computed with the linear Clausius-Mossotti
136
5.4 Modeling of the delay-dependent diffraction response
20 30
photon energy (eV)
0.00
0.05
0.10
(a) at t0= 0.0 fs
20 30
photon energy (eV)
0.05
0.00
(b) at t0= 0.0 fs
20 30
photon energy (eV)
0.00
0.05
0.10
0.15
(c) at t0= 5.0 fs
20 30
photon energy (eV)
0.10
0.05
0.00
(d) at t0= 5.0 fs
20 30
photon energy (eV)
0.0
0.2
0.4
(e) at t0= 10.0 fs
20 30
photon energy (eV)
0.4
0.2
0.0
0.2
(f) at t0= 10.0 fs
Figure 5.26: Line-outs of the low-pass filtered (along delay axis) refractive index at three delays
of NIR and XUV pulse for three different XUV spectrally shifted XUV pulses centered around 19
(blue, solid curves), 22 (orange, dashed curves) and 25
eV
(green, dotted curves). A NIR pulse
with a high peak intensity of 2
·
10
14 W/cm2
was chosen to maximize possible non-linearities. Only
minor deviations between the results obtained with different XUV pulses indicate that linearity
with respect to the XUV may be assumed.
137
5. Ultrafast dynamics in helium nano-droplets
20 25
photon energy (eV)
20
10
0
10
20
delay (fs)
(a) 2 1014 W/cm2,
0.5
0.0
0.5
20 25
photon energy (eV)
20
10
0
10
20
(b) 2 1014 W/cm2,
0.5
0.0
0.5
20 25
photon energy (eV)
20
10
0
10
20
delay (fs)
(c) 5 1012 W/cm2,
0.5
0.0
0.5
20 25
photon energy (eV)
20
10
0
10
20
(d) 5 1012 W/cm2,
0.5
0.0
0.5
16 18 20
photon energy (eV)
20
10
0
10
20
delay (fs)
(e) 2 1014 W/cm2,
0.050
0.025
0.000
0.025
0.050
16 18 20
photon energy (eV)
20
10
0
10
20
(f) 5 1012 W/cm2,
0.050
0.025
0.000
0.025
0.050
Figure 5.27: Refractive index in terms of
β
and
δ
obtained with the described modelling method
(low-pass filtered along delay axis) for a two NIR peak intensities, namely (a/b/e) 2
·
10
14 W/cm2
and (c/d/f) 5
·
10
12 W/cm2
. (a-d) overview for photon energies between 16
eV
and 30
eV
; (e/f)
details of the absorption (
β
) with cropped color-scale below the 2p resonance for both intensities;
relation after a low-pass filter was applied
35
at three different delays: when the XUV arrives
together with the NIR (delay 0
fs
), when it arrives on the slope of the NIR pulse (delay 5
fs
)
and when it arrives when the NIR pulse has almost completely passed the sample ahead of the
XUV (delay 10
fs
). The refractive indices at a given delay show only minor deviations between
the different XUV pulses and hence can be treated with this linear approximation.
Intensity depend refractive index
The refractive index at the experiment’s peak NIR intensity (2
·
10
14 W/cm2
) in comparison to
a lower more moderate intensity of 5
·
10
12 W/cm2
is shown in figure 5.27. Comparable to the
atomic case, one observes a modification of the resonance positions during the present NIR
electric field. For the atomic case, these shifts are attributed to the AC Stark effect (below
35along the delay scan axis
138
5.4 Modeling of the delay-dependent diffraction response
20 10 0 10 20
delay (fs)
0.0
0.5
1.0
1.5
normalized diffraction intensity
770 nm
780 nm
790 nm
800 nm
810 nm
Figure 5.28: Illustration of the different delay-dependent normalized diffraction intensity signals
obtained from modelling different HHG fundamental wavelengths while keeping the NIR wavelength
constant at 800
nm
for an NIR peak intensity of 2
·
10
14 W/cm2
. Each signal is normalized to its
diffraction intensity away from temporal overlap.
the ionization threshold), and ponderomotive shifts (in the continuum) [205] (see also chapter
2.1.5). Furthermore, new resonances emerge. Those resonances that only exist during the
presence of the NIR pulse are known as LISs [200]. The emission features, e.g., observable
for negative delays around 20
eV
, are also seen in atomic investigations and associated with
PFID [54, 206]. For the lower, more moderate intensity, these features are distinguishable.
This is why such low intensity is provided for comparison. For the experiment’s peak NIR
intensity, i.e. in the top panels (a) & (b) and the panel (e) in fig. 5.27, these features are also
visible, but overlapping and merging. In particular, the separation of different LIS and LIS
overlapping with AC Stark shifted resonances is not obvious.
Time-dependent diffraction intensity
Now the diffraction signal can be obtained with the Mie scattering computation. While the NIR
pulse in the modeling has a constant central wavelength of 800
nm36
, the fundamental XUV
wavelength is varied. Not only does the spectrum calibration data in this experiment indicate
different fundamental XUV wavelengths at different days of the experimental campaign (see
section 5.1.3), also previously published data on the beamline shows that the fundamental
wavelength with regards to the high harmonic spectrum does not match with the central NIR
wavelength [36, 122]. For an NIR peak intensity of 2
·
10
14 W/cm2
, a temporary decrease in
diffraction intensity during the temporal overlap of NIR and XUV pulse can be seen with the
modeling. In addition, various structures occur at the edge of the decrease, depending on the
XUV’s fundamental wavelength. Fig. 5.28 shows the simulation result for selected fundamental
wavelengths to highlight the various possible shapes. Besides the generally observed temporary
decrease of the diffraction intensity during temporal overlap, for an early arriving XUV pulse,
partially a secondary decrease can be seen or even an increase of the signal, higher than the
36
Due to the pulse duration of 8
.
3
fs
and thus its associated broad bandwidth, small variations of the central
wavelength are not expected to affect the modeling result significantly.
139
5. Ultrafast dynamics in helium nano-droplets
25 0 25
delay (fs)
0.0
0.5
1.0
normalized diffraction
intensity
(a)
770 nm
HO 11
HO 13
HO 15
HO 17
HO 19
HO 21
25 0 25
delay (fs)
0
1
2
(b)
780 nm
25 0 25
delay (fs)
0.0
0.5
1.0
(c)
790 nm
25 0 25
delay (fs)
0.0
0.5
1.0
(d)
800 nm
25 0 25
delay (fs)
0.0
0.5
1.0
(e)
810 nm
Figure 5.29: Separation of the superimposed diffraction signals shown in fig. 5.28 by single HOs.
The decrease in diffraction intensity predominantly originates from HO 11 and 13. The shown
signals are normalized to the diffraction intensity of HO 13 away from temporal overlap between
NIR and XUV pulses.
subsequent decrease. This coincides with the PFID features visible at negative delays in the
modeled refractive index (see fig. 5.27). Although the different dynamics towards negative
delays can yield an overall very asymmetric signal, the main scattering decrease itself shows
only a minor asymmetry. The center of this "dip" is only shifted approximately 2
fs
from
time zero. Furthermore, the modeling yields a "dip" in the diffraction signal that sometimes
becomes nearly flat around time zero.
The chosen modeling method allows for the separation of the influence of the single
harmonics. Thus to reveal the origin of the decrease in diffraction intensity and other features
in fig. 5.28, the decomposition of the there shown modeled delay scans is depicted in fig. 5.29 .
It reveals that the 13
th
and partially the 11
th
harmonic are mostly responsible for the decrease
in diffraction intensity. The 15
th
harmonic either reduces the depth of the feature or adds some
slowly oscillating structure to it. The influence of higher HOs is almost negligible. The strong
domination of the 13
th
and 11
th
harmonic, i.e., photon energies below the 2p resonance
37
,
suggests that with regards to the few-femtosecond dynamics, more realistic modeling can be
achieved by approximately correcting the energetic shift of the predominantly involved states.
Since, due to the laser dressing, the AC Stark shift and LIS known from atomic physics appear
to play a significant role for the simulated dynamics, the dominating effects can be expected
for the AC Stark shift of the 2p state and the LIS that yields a coupling of the XUV through
one NIR photon into the 1s state. Other LIS involving higher-order non-linearities are also
possible but expected to be less significant. As discussed in section 5.4.1 the 2p resonance in
the modeling is shifted by
−
0
.
5
eV
compared to experimental measurements in bulk helium.
According to fluorescence measurements in helium nano-droplets the 2s state is located at
approximately 21
eV
[75], but the modelling yields the 2s state at 20
.
3
eV
(see table 5.3). Hence
it is shifted by −0.7eV. To compromise between both, the optical response data is corrected
37
For fundamental wavelengths above 765
nm
, the 13
th
harmonic has a photon energy below the 2p-resonance
(without a present NIR field) in the modeling.
140
5.4 Modeling of the delay-dependent diffraction response
20 10 0 10 20
delay (fs)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
normalized diffraction intensity
770 nm
780 nm
790 nm
800 nm
810 nm
Figure 5.30: Reproduction of the diffraction intensity signals shown in fig. 5.28, now including
the refractive indices shifted by 0
.
6
eV
to correct for the red-shift of 2p and 2s resonances due to
the single atom potential considered in the TDSE.
through an artificial shift of 0
.
6
eV
for the following modeling of experimental data. The
influence of this decision on the simulation results is in the focus of the next paragraph.
Energy shift-corrected time-dependent diffraction intensity
To show the influence of applying an energetically shifted optical response, in fig. 5.30 the
results of fig. 5.28 are recreated, but now corrected for the shift. The result for a central XUV
wavelength of 770
nm
in the un-shifted modeling (fig. 5.28) is comparable to the one shown in
fig. 5.30 for 790
nm
. The strong resemblance of the data shifted by 20
nm
can be explained by
the dominance of the 13
th
(and 11
th
) harmonic. A shift of 0
.
6
eV
corresponds to a shift of around
2
nm
at the harmonic wavelengths corresponding to 20
−
30
nm
in their fundamental wavelength.
Notably, the diffraction intensity decrease between different fundamental wavelengths becomes
very similar. This concerns the width as well as the depth. Only the signal between
−
10 and
−
20
fs
deviates more strongly. This feature can be again attributed to the PFID signatures in
the optical response. The further the 13
th
harmonic is away from the 2p resonance (i.e., the
larger the HHG’s fundamental wavelength), the weaker are the perturbed free-induction decay
features.
NIR intensity dependency of the diffraction intensity
One novelty compared to a former experiment [35] in this work is the experimental investigation
of the NIR intensity dependence of the observed diffraction intensity decrease. While
experimentally, the intensity decrease is accessible down to an estimated 30% of the topmost
NIR peak intensity at a fixed XUV fundamental wavelength, the modeling has further reach.
Therefore, fig. 5.31 shows a comprehensive scan of peak intensities ranging from 2
·
10
14 W/cm2
down to 5
·
10
12 W/cm2
. For each intensity, delay scans of the diffraction intensity decrease
are shown for a range of fundamental HHG wavelengths. While for the peak intensities most
relevant to this experiment (fig. 5.31 (a-d)), only slight changes can be observed, the modeling
suggests a different picture for lower intensities. E.g., for 1
·
10
13 W/cm2
, the modeling predicts
141
5. Ultrafast dynamics in helium nano-droplets
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(a) 2.0 1014 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(b) 1.5 1014 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(c) 1.0 1014 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(d) 5.0 1013 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(e) 1.0 1013 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
20 0 20
delay (fs)
780
800
820
840
XUV fundamental
wavelength (nm)
(f) 5.0 1012 W/cm2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Figure 5.31: Influence of the NIR peak intensity (stated in the sub-figure titles) onto the delay
dependent normalized diffraction intensity for different XUV fundamental wavelengths, while the
central wavelength of the NIR and other properties are held constant at 800
nm
. Each row in (a-f)
is normalized to its diffraction intensity away from temporal overlap.
142
5.4 Modeling of the delay-dependent diffraction response
20 10 0 10 20
delay (fs)
0.0
0.2
0.4
0.6
0.8
1.0
normalized diffraction
intensity
2.0 1014 W/cm2
1.5 1014 W/cm2
1.0 1014 W/cm2
5.0 1013 W/cm2
Figure 5.32: Approximate recreation of the NIR intensity scan carried out in the experiment
(see fig. 5.15). For a central XUV fundamental wavelength of 800
nm
, a delay scan was modeled
for each of the NIR intensities 2
·
10
14 W/cm2
(100%), 1
.
5
·
10
14 W/cm2
(75%), 1
·
10
14 W/cm2
(50%), 5
·
10
13 W/cm2
(25%). The shown diffraction intensity is normalized to its signal away
from temporal overlap.
a decrease of diffraction intensity by only 20 %, while for higher intensities or other fundamental
XUV wavelengths a decrease of over 60 % results. For a lower peak intensity of 5
·
10
12 W/cm2
the delay-dependent diffraction intensity depends even stronger on the choice of fundamental
wavelength. A systematic investigation at these lower intensities might uncover a more detailed
response of the helium nano-droplets - also in comparison to this model.
A more specific recreation of the experimental intensity scan is presented in fig. 5.32 . In the
modeling the intensity was decreased in regular steps of 25%. With this the intensity decrease
steps from the experiment
38
of 100%,80 %,50 % and 30 % are approximately matched. In the
simulated data-set, the three highest intensities show very similar behaviour. Only the lowest
intensity clearly deviates. Still its qualitative shape is consistent with the higher intensities.
The lowest intensity curve shows a less strong decrease, which in addition is narrower.
In summary, the modeling of few-cycle resolved dynamics using an effective medium
constructed from the atomic response of helium is able to reproduce a decrease in the diffraction
intensity during the temporal overlap of XUV and NIR. Around time zero, two effects known
from atomic physics appear to play a role:
1. AC Stark shifts / ponderomotive shifts
2. light-induced states
Moreover, PFID introduces for some XUV fundamental wavelengths a strong asymmetry
at negative delays. A separation of the harmonic contribution reveals that in the modeling,
38that were estimated with wave-front propagation simulations
143
5. Ultrafast dynamics in helium nano-droplets
primarily the 13
th
harmonic and, to some extent, also the 11
th
harmonic are responsible for
the decrease in diffraction intensity. The high NIR peak intensities in the modeled experiment
enforce the decrease in diffraction intensity. However, the simulations suggest that systematic
investigation at lower intensity could show more interesting structures, possibly providing a
more detailed picture of the underlying processes.
Simulating a scenario where linear assumptions are applicable is a first step in the presented
modeling. The following subsection will show that including sub-cycle features impacts the
resulting scattering response beyond the sub-cycle time scales, also on the discussed few-
femtosecond domain. Nevertheless, the observations and conclusions from the simulations in
this subsection aid the understanding of the simulation results for the non-linear, more general
scenario.
5.4.3 Sub-cycle resolved simulations
After discussing effects occurring on the time scale of a few femtoseconds and reproducing
a decrease in the diffraction intensity that can be explained with effects known from atomic
physics, this subsection now includes sub-cycle effects. Opposed to the few-femtosecond
response, the sub-cycle response will be shown to rely on non-linear effects. Therefore, it will
be shown that by implementing non-linear orders in the dynamic polarizability (see eq. 5.9),
the dipole response of other IAPs can be obtained approximately. Indeed, the so constructed
dipole response outperforms any result under the linear assumption. It will be shown that the
optical response predicts sub-cycle features and how those change when a harmonic spectrum
is changed. Through Mie-theory scattering simulations, a modeled delay-dependent diffraction
intensity will be produced.
In the following, simulations for NIR peak intensities of 2
·
10
14 W/cm2
and 5
·
10
12 W/cm2
are discussed. A NIR peak intensity of 2
·
10
14 W/cm2
approximately models the experiment,
and the lower peak intensity of 5
·
10
12 W/cm2
can be used to evaluate the performance of the
simulation and gain more insight into the underlying physics. Again, different fundamental
wavelengths for high harmonics are discussed, as those are expected to play a role in the
experimental observations and to influence the diffraction signal.
Linearity of the sub-cycle response
Analog to the modeling of the few-femtosecond diffraction response, the linearity is first
investigated by applying the linear Clausius-Mossotti relation to linear polarizabilities obtained
for XUV IAP’s of different central XUV photon energies. The resulting refractive index is
shown in figure 5.33. Now, opposed to the conclusions drawn from the figure 5.26 in the
previous section, the retrieved refractive indices deviate from each other. Particular strong
deviations are observed in figure 5.33 (a-d), i.e., closer to time zero. Hence, the assumption of
linearity breaks down when including sub-cycle features.
Evaluation of the simulated microscopic response
When switching to the non-linear case (here, up to third order), the computation of the dynamic
polarizabilities (
α1
,
α2
&
α3
) involves the results of multiple atomic TDSE simulations for
144
5.4 Modeling of the delay-dependent diffraction response
20 30
photon energy (eV)
0.1
0.0
0.1
0.2
(a) at t0= 0.0 fs
20 30
photon energy (eV)
0.50
0.25
0.00
(b) at t0= 0.0 fs
20 30
photon energy (eV)
0.2
0.0
0.2
(c) at t0= 5.0 fs
20 30
photon energy (eV)
0.5
0.0
(d) at t0= 5.0 fs
20 30
photon energy (eV)
0.0
0.2
0.4
0.6
(e) at t0= 10.0 fs
20 30
photon energy (eV)
0.4
0.2
0.0
0.2
(f) at t0= 10.0 fs
Figure 5.33: Line-outs of the refractive index at three delays of NIR and XUV pulse for three
different XUV spectrally shifted XUV pulses centered around 19 (blue, solid curves), 22 (orange,
dashed curves) and 25
eV
(green, dotted curves). A NIR pulse with a peak intensity of 2
·
10
14 W/cm2
was chosen to maximize possible non-linearities. The partially strong deviations between the results
obtained with different XUV pulses indicate that linearity with respect to the XUV may not be
assumed.
different central XUV photon energies at each delay between XUV and NIR pulses. Therefore,
the electric field and dipole moment computed for XUV IAPs centered around 16
eV
,19
eV
,
20
.
5
eV
,22
eV
,23
.
5
eV
and 25
eV
are used. With known dynamic polarizabilities (
α1
,
α2
&
α3
) one should be able to compute the dipole response for any given incident XUV spectrum
at a defined time delay. Thus to check the accuracy for spectral intensities not directly used
with the fitting, in addition, the dipole response for an XUV IAP centered around 17
.
5
eV
is computed. A comparison of the frequency-dependent dipole moments computed from the
TDSE results, from the fitted non-linear dynamic polarizabilities, and from the linear dynamic
polarizability obtained for an XUV IAP centered around 22
eV
is depicted in figure 5.34.
For energies below 22
eV
, the response is generally well reproduced through the non-linear
approach, however for energies from around 22
eV
and upwards, stronger deviations even
for data sets included with the fitting are observed. Still, including the non-linear dynamic
polarizabilities yield a dipole response that deviates much less than one only obtained from
the linear response of a single TDSE simulation.
Evaluation of the simulated macroscopic response
Having the performance evaluated with regard to the dipole moment, i.e. on the microscopic
scale, an evaluation of the macroscopic behavior utilizing the refractive index, in particular
β
,
is now carried out. Therefore the lower NIR intensity of 5
·
10
12 W/cm2
is used, because, as
illustrated for the cycle averaged result shown in figure 5.27, features like LIS and AC Stark
145
5. Ultrafast dynamics in helium nano-droplets
20 25 30
energy (eV)
0
5
dipole moment
(arb. u.)
(a) 17.5 eV (reference)
TDSE non-linear dynamic pol. linear dynamic pol.
20 25 30
energy (eV)
0
5
dipole moment
(arb. u.)
(b) 19.0 eV
20 25 30
energy (eV)
0
5
dipole moment
(arb. u.)
(c) 22.0 eV
20 25 30
energy (eV)
0
5
dipole moment
(arb. u.)
(d) 25.0 eV
Figure 5.34: Frequency dependent dipole moment retrieved from atomic TDSE computations
compared to those computed from dynamic polarizabilities for an NIR peak intensity of 2
·
10
14 W/cm2
at a time delay of 0
fs
. (a) shows the results for an IAP centered around 17
.
5
eV
that
was not included in the fitting of the non-linear dynamic polarizabilities. The linear dynamic
polarizabilty was computed from the TDSE result of an IAP centered around 22
eV
(see (c)). (b-d)
show data for IAPs centered around 19
eV
,22
eV
and 25
eV
. All three were used together with
not shown TDSE results for IAPs centered around 16
eV
,20
.
5
eV
, and 23
.
5
eV
to fit the non-linear
dynamic polarizability.
shift are at this intensity still spectrally localized. For the atomic case, investigations by other
groups using IAPs, experimentally and theoretically [200, 196, 54, 207], observe sub-cycle
oscillations with two times the NIR frequency and attribute those to which-way interferences,
and two-photon resonant coupling of 2 LISs coupling into the same state (e.g., 2s) of the atom
[53]39.
Now, also for the simulations carried out here, it must be ensured that resulting sub-cycle
oscillations are properly modelled as this kind of coupling. Specifically, they must reflect that
an oscillation only occurs, if indeed XUV photons are present to couple into both LISs and
thus finally into the same state. Otherwise, physically, no interference should occur.
To do so, special XUV spectra are constructed that consist of only 2 harmonics, each spaced
by approximately one fundamental photon energy around the 2s state resulting from the TDSE
simulations
40
. Thus these two harmonic orders can couple either through the
LIS2s−1
and the
LIS2s+1
into the 2s state, which should give rise to interference effects. Now, by varying the
relative intensities of the two harmonics, it can be investigated how the spectral intensity of
one harmonic impacts sub-cycle oscillations occurring at the LIS at the other harmonic order.
Consequently, when increasing the XUV intensity, which couples into the
LIS2s+1
, and
decreasing the XUV intensity, which couples into the
LIS2s−1
, the oscillation on the
LIS2s−1
should increase and decrease on the
LIS2s+1
. When there is no XUV coupling into the
LIS2s+1
,
39The provided reference identifies such coupling around the 3d state.
40
When one of the intensities for the two harmonics is set to zero instead another harmonic order below or
above the remaining one is applied.
146
5.4 Modeling of the delay-dependent diffraction response
18 20 22
energy (eV)
20
10
0
delay (fs)
(a)
18 20 22
energy (eV)
(b)
18 20 22
energy (eV)
(c)
18 20 22
energy (eV)
(d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.1
0.0
0.1
(e) 18.9 eV LIS 2s-
(a) (b) (c) (d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.0
0.2
(f) 22.0 eV LIS 2s+
0.2
0.0
0.2
Figure 5.35: Benchmarking the qualitative behavior of the optical response mapped by the
fitted non-linear dynamic polarizabilities, where sub-cycle features are allowed for all considered
dynamic polarizability coefficients (
α1, α2, α3
), for an NIR peak intensity of 5
·
10
12 W/cm2
; For the
benchmarking the spectra of artificial pulse trains (envelope FWHM 3
.
1
fs
) from two XUV photon
energies ((a-c) 18
.
9
eV
and 22
.
0
eV
/ (d) 18
.
9
eV
and 15
.
8
eV
) and different relative contributions
were computed and applied. The photon energies are chosen originally to approximately match
the positions of the
LIS2s−1
and
LIS2s+1
. The resulting absorption
β
is depicted in (a-d), the
respective spectral contributions are indicated by offset and arb. scaled spectra shown by in black
on top of the two dimensional absorption. Lineouts at the photon energies 18
.
9
eV
and 22
.
0
eV
(dotted lines in (a) - (d)) are shown in (e) and (f), respectively. It is expected that the lower the
contribution in the higher photon energy, the lower the oscillation in the lower photon energy and
vice versa.
the
LIS2s−1
should not show any oscillation. This investigation is visualized by fig. 5.35. There,
the sample’s optical response was computed for four different XUV spectra that are shown as
an overlay at the top and the optical response is plotted in terms of the absorption
β
in fig.
5.35 (a-d). Looking at the line-out for the
LIS2s−1
in fig. 5.35 (e), the behavior looks very
much like expected. First of all, indeed, the oscillation known from the atomic case translates
into the here constructed bulk response. Moreover, second, the oscillation at the
LIS2s−1
has
the strongest response for the highest amount of XUV coupled into the
LIS2s+1
(fig. 5.35 (b) /
(e) orange curve) and for the case of no XUV light coupled into the
LIS2s+1
the oscillation
is almost gone (fig. 5.35 (d) / (e) black curve). Due to the transfer of those properties, it
is claimed that the non-linear modeling works for energies around the
LIS2s−1
rather well.
However, problematic is the observation made on the lineout of the
LIS2s+1
(fig. 5.35 (f)).
There no change for the spectral variations in fig. 5.35 (a-c) is observed at all. Hence, the
fitting does not work everywhere. A possible explanation for this is given in appendix C.2.1.
In short, overlapping couplings are likely not to be adequately represented by the chosen
147
5. Ultrafast dynamics in helium nano-droplets
non-linear modeling approach. Here, for example both the
LIS2s+1
and
LIS3d−1
are exepected
to be located around 22 eV and thus overlapping [54].41
While in general, oscillations do not have to originate from such coupling, having one
quantum path showing a response to the spectral composition of the XUV and one quantum
path not is unrealistic. For this work, it is important to exclude any false positives. This
means, it is acceptable to miss out on parts of the sub-cycle oscillations, but not to observe
sub-cycle oscillations, where the spectral composition of the XUV does not allow for it or in
short where there is none.
Here, only sub-cycle oscillations that are independent of the XUV spectrum are considered
potential false positives. As a response independent of the XUV spectrum can only occur
due to the linear component of the dynamic polarizability, by applying a low pass filter to it
(
α1
), those can be removed. However, removing the oscillation artifacts in the linear response
comes at the cost of ignoring oscillations due to the medium’s instantaneous response, which
is here an acceptable error. A secondary fit of the dynamic polarizability to the TDSE results
is performed to avoid artifacts, but
α1
is held constant at the filtered value from the initial fit.
As stated before, doing so, the here performed modeling will only prove the existence of
oscillations due to non-linear couplings. Some of the coupling and thus oscillating features as
well as sub-cycle oscillations due to the instantaneous response of the medium (e.g., a sub-cycle
stark effect) are not covered by the modeling. Thus, even qualitative features such as the
strength of the sub-cycle feature due to different harmonics bear an increased uncertainty.
The previous investigation into the dependence of the sub-cycle response of the refractive
index on the relative harmonic order’s strength can now be repeated. Fig. 5.36 shows the
modelling analog to fig. 5.35, but now with the low-pass filtered linear dynamic polarizability
component. Like motivated, by low pass filtering
α1
, the oscillations that were not adapting
to the different spectral contributions (fig. 5.36 (f)) as well as the remaining oscillation with
disabled coupling (fig. 5.36 (e) black line) vanish, but the oscillations reacting to the coupling
remain. The findings for the higher NIR intensity are comparable, although observed oscillation
is more complicated (see appendix fig. C.1 and C.2). Furthermore, figures C.3 and C.4 in the
appendix show the impact of the modification in the modelling, by comparing the spectrum
of the induced dipole moment retrieved via TDSE and the modified fitting. The dominating
qualitative features up to approximately 25
eV
remain intact. However as predicted some
non-dominating, but visible features oscillating along the delay axis, mostly on and above the
2p resonance, are not represented and cause an additional error compared to the direct fitting
initial presented.
The evaluation of the modeling with respect to the macroscopic response, revealed a flaw.
More specifically, the method yielded false positives regarding the existence and strength of sub-
cycle oscillations for discrete harmonic spectra. To prevent such from causing misinterpretation,
low-pass filtering of the linear component of the dynamic polarizability removes these false
41
Be aware that most references model the position of the 2s state more precise than in this work. Consequently,
the
LIS2s+1
in this work should be shifted by approximately
−
0
.
35
eV
(see table 5.3) compared to the simulation
results in the references. E.g. in [54], the
LIS3d−1
is located approximately 0
.
35
eV
below the
LIS2s+1
. Hence,
in this work’s simulation, both LIS would overlap.
148
5.4 Modeling of the delay-dependent diffraction response
18 20 22
energy (eV)
20
10
0
delay (fs)
(a)
18 20 22
energy (eV)
(b)
18 20 22
energy (eV)
(c)
18 20 22
energy (eV)
(d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.1
0.0
0.1
(e) 18.9 eV LIS 2s-
(a) (b) (c) (d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.0
0.1
0.2
(f) 22.0 eV LIS 2s+
0.2
0.0
0.2
Figure 5.36: Benchmarking the qualitative behavior of the optical response mapped by the fitted
non-linear dynamic polarizabilities, where for
α1
the low-pass filtered response from a preliminary
fitting step is assumed, for an NIR peak intensity of 5
·
10
12 W/cm2
; For the benchmarking the
spectra of artificial pulse trains (envelope FWHM 3
.
1
fs
) from two XUV photon energies ((a-c)
18
.
9
eV
and 22
.
0
eV
/ (d) 18
.
9
eV
and 15
.
8
eV
) and different relative contributions were computed
and applied. The photon energies are chosen originally to approximately match the positions of
the
LIS2s−1
and
LIS2s+1
. The resulting absorption
β
is depicted in (a-d), the respective spectral
contributions are indicated by offset and arb. scaled spectra shown by in black on top of the two
dimensional absorption. Lineouts at the photon energies 18
.
9
eV
and 22
.
0
eV
(dotted lines in (a) -
(d)) are shown in (e) and (f), respectively. It is expected that the lower the contribution in the
higher photon energy, the lower the oscillation in the lower photon energy and vice versa.
positives. This comes at the cost of other instantaneous responses that, e.g., could be due to
the sub-cycle AC-Stark shift are removed as well.
The simulated sub-cycle response
After understanding the limitations of the modeling and removing contributions that are
at least partially artifacts, the sub-cycle response to realistic harmonic spectra will now be
discussed.
42
Fig. 5.37 shows the computed refractive index for the three different central XUV
wavelengths 790
nm
,800
nm
, and 810
nm
. Analog to the previous discussions, the NIR central
wavelength is held constant at 800
nm
. In general, pulse parameters were chosen according
to table 5.4. However, HOs are modeled only up to order 19 for the refractive index and
up HO 17 in the subsequent Mie-theory simulations. Indeed, the modeled refractive index
shows sub-cycle oscillations. Only small changes are observed between refractive indices for
the different fundamental wavelengths of the high harmonics. (see also line-outs in fig. C.7 in
the appendix). Overall, the qualitative sub-cycle response at time delays ranging from
−
5
fs
to
42
Be aware that although realistic spectra are used, the temporal pulse train structure is only included with
the modeling of the scattering response and consequently not with the refractive index to be discussed first.
149
5. Ultrafast dynamics in helium nano-droplets
20 30
energy (eV)
5
0
5
delay (fs)
(a) , 790 nm
0.2
0.0
0.2
20 30
energy (eV)
5
0
5
delay (fs)
(d) , 790 nm
0.2
0.0
0.2
20 30
energy (eV)
(b) , 800 nm
0.2
0.0
0.2
20 30
energy (eV)
(e) , 800 nm
0.2
0.0
0.2
20 30
energy (eV)
(c) , 810 nm
0.2
0.0
0.2
20 30
energy (eV)
(f) , 810 nm
0.2
0.0
0.2
Figure 5.37: Delay-dependent refractive index for an NIR pulse with a peak intensity of
2
·
10
14 W/cm2
and a realistic XUV spectrum composed of the HOs 11
−
19. The refractive index
is shown for different fundamental XUV wavelengths, namely 790
nm
(a/d), 800
nm
(b/e) and
790
nm
(c/f), determining the spectral position of the harmonics. Regions between the harmonics
are not shown (replaced by white stripes). Line-outs at the central harmonic photon energies are
shown in the appendix in fig. C.7
5fs is the same for all three XUV pulses. All observed oscillations must be due to non-linear
couplings because the introduced simulation methods suppress other effects in the sub-cycle
domain.
Finally, the last modeling step, namely Mie-theory simulation, is carried out to simulate
the scattering response accounting for nano-droplet size distribution, XUV spectrum, and
temporal APT structure. And indeed, as shown by figure 5.38, sub-cycle oscillations are
predicted. Predominantly the simulation predicts oscillations at approximately double the NIR
frequency, but also higher-order components, such as four times the NIR frequency, appear to
be contributing. The simulated oscillations are not sinusoidal. A limited temporal resolution
of the experiment is emulated by convolving the simulation result shown in fig. 5.38 (a-c)
with a Gaussian function having FWHM of 200
as
to 800
as
. As shown by 5.38 (d-f) the
oscillation becomes approximately sinusoidal at approximately twice the NIR frequency for a
time resolution of 500
as
or worse. The unconvoluted oscillations measure peak to peak up to
approximately 100 % of the signal far away from temporal overlap, i.e., effectively the signal for
only the XUV. Their amplitudes are, e.g., influenced by the temporal resolution, but certainly
also the strength of the sub-cycle oscillations of the refractive index.
Analog to the discussion of the few-cycle resolved simulations, the contributions of the
different HOs can be extracted. Those are shown for the three discussed fundamental XUV
wavelengths in fig. 5.39. Though the strongest contribution is still due to the 13
th
harmonic,
opposite to the few-cycle resolved simulations, it is not sufficiently dominant to apply the
150
5.4 Modeling of the delay-dependent diffraction response
Figure 5.38: Simulated sub-cycle resolved delay-dependent diffraction intensity for different XUV
fundamental wavelengths, which determine the spectral position of the high harmonics (XUV),
around a time delay of 0
fs
. Harmonics up to order 17 are taken into account for the Mie-theory
simulations. Otherwise the pulse parameters are according to table 5.4. The NIR is at a constant
central wavelength of 800
nm
and has a peak intensity of 2
·
10
14 W/cm2
. (d-f) show the respective
signals from the top row ((a-c)) convoluted with Gaussian distributions of different FWHM to
imitate a limited time resolution. The diffraction intensity is normalized to its value at large
positive delays (i.e. away from temporal overlap of NIR and XUV).
151
5. Ultrafast dynamics in helium nano-droplets
2.5 0.0 2.5
0
1
normalized
diffraction
intensity
(a) 790 nm, HO 11
2.5 0.0 2.5
0
1
normalized
diffraction
intensity
(d) 790 nm, HO 13
2.5 0.0 2.5
0
1
normalized
diffraction
intensity
(g) 790 nm, HO 15
2.5 0.0 2.5
delay (fs)
0
1
normalized
diffraction
intensity
(j) 790 nm, HO 17
2.5 0.0 2.5
0
1
(b) 800 nm, HO 11
2.5 0.0 2.5
0
1
(e) 800 nm, HO 13
2.5 0.0 2.5
0
1
(h) 800 nm, HO 15
2.5 0.0 2.5
delay (fs)
0
1
(k) 800 nm, HO 17
2.5 0.0 2.5
0
1
(c) 810 nm, HO 11
2.5 0.0 2.5
0
1
(f) 810 nm, HO 13
2.5 0.0 2.5
0
1
(i) 810 nm, HO 15
2.5 0.0 2.5
delay (fs)
0
1
(l) 810 nm, HO 17
Figure 5.39: Simulated sub-cycle resolved delay-dependent diffraction intensity for different XUV
fundamental wavelengths from fig. 5.38 separated into the contributions made by each HO. The
NIR is at a constant central wavelength of 800
nm
and has a peak intensity of 2
·
10
14 W/cm2
.
The diffraction intensity is normalized to the value of the 13
th
HO’s contribution at large positive
delays (i.e. away from temporal overlap of NIR and XUV).
152
5.4 Modeling of the delay-dependent diffraction response
20 15 10 5 0 5 10 15 20
delay (fs)
0.0
0.5
1.0
1.5
normalized
diffraction
intensity
(a) 790 nm
sub-cycle resolved response
sub-cycle resolved response (low pass filtered)
few-cycle resolved response
20 15 10 5 0 5 10 15 20
delay (fs)
0.0
0.5
1.0
1.5
normalized
diffraction
intensity
(b) 800 nm
20 15 10 5 0 5 10 15 20
delay (fs)
0.0
0.5
1.0
1.5
normalized
diffraction
intensity
(c) 810 nm
Figure 5.40: Sub-cycle resolved diffraction intensity over the full time delay range in comparison
to the result of the few-cycle resolved simulation for three different fundamental HHG wavelengths
((a) 790
nm
, (b) 800
nm
and (c) 810
nm
). The NIR is at a constant central wavelength of 800
nm
and has a peak intensity of 2·1014 W/cm2.
shift to correction for the energetic offset of the resonances between the atomic (simulated)
ones and those of helium droplets or bulk liquid helium. The dominant contributions at twice
the NIR frequency come from the HOs 13 and 15. Both HOs, 11 and 17, make a smaller but
substantial contribution at predominantly higher frequencies.
153
5. Ultrafast dynamics in helium nano-droplets
Revisiting the simulation results on the few femtosecond time scale
Being now able to simulate the helium droplets’ scattering response, including sub-cycle effects,
the few-femtosecond time scale can be revisited. The question of to what extent the sub-cycle
dynamics have a net impact on the few-cycle dynamics can now be answered. Certainly not all
sub-cycle features are included with the modelling of the refractive index, nevertheless the low
pass filtered refractive index from this modelling - though not in full agreement - reproduces
many of the features of the one from the few-cycle resolved modeling in section 5.4.2 (see fig.
C.8 in the appendix). Since the shift of the resonance is not corrected, the comparison is made
to the unshifted few-cycle resolved simulation of the diffraction intensity decrease (see previous
section fig. 5.28). Those are shown together with the sub-cycle resolved simulation results in
fig. 5.40. The results of both simulations deviate significantly from each other. Despite that,
both also share basic qualitative features: Both show for all shown fundamental wavelengths
a reduced diffraction intensity around a delay of 0
fs
. Also, both have a similar asymmetry
around time zero. The few-cycle resolved simulation yields a secondary minimum between
−
10
fs
and
−
20
fs
that was attributed to PFID. The (low-pass filtered) sub-cycle resolved
simulation also shows an additional (weak) minimum in this range. However, the depth of the
diffraction intensity "dip" for the simulation, including sub-cycle effects, behaves very different
from the few-cycle case. It is not as deep, it does not show the otherwise seen long and flat
plateau, and it partially shows humps towards its edge for positive and negative delays.
Deeper insight into the differences between both simulation results can again be gained
by discussing the contributions made by each harmonic. Those are shown in fig. 5.41. It
reveals that still (i.e., in both cases) HO 13 is most responsible for the decreasing diffraction
intensity. However, the decrease in diffraction intensity due to the 13
th
harmonic is not as
strong as the few-cycle resolved simulation predicted. In fact, during the temporal overlap of
NIR and XUV, all harmonic contributions from the sub-cycle resolved simulation exceed the
contributions yielded by the few-cycle approach. The humps toward the diffraction intensity
decrease’s edges in the sub-cycle resolved simulation can be attributed to the contributions of
HO 15 and 17. Around a delay of 0
fs
(time zero), the (low-pass filtered) refractive indices
from the both simulation approaches are very similar for HO 13 and 15 (see also fig. C.8), but
the diffraction responses are very different. Remember, for the sub-cycle resolved simulation,
those are computed from the sub-cycle resolved refractive index, not the low-pass filtered one.
Hence, including sub-cycle features in the Mie-theory makes a substantial difference. This
difference applies even when the simulated sub-cycle features are removed by a low-pass filter
(experimentally corresponding, e.g., to an inferior time resolution).
The sub-cycle resolved simulations suggest many possible shapes that the diffraction
intensity response may show for a helium nano-droplet dressed by an 800
nm
,8
.
3
fs
NIR
pulse. While in fig. 5.40 three examples are given, more can be found in the appendix in fig.
C.9, where the different diffraction intensity responses are computed by varying the XUV’s
fundamental wavelength as well.
Indeed, the sub-cycle dynamics make a net contribution also to the slower varying diffraction
signal. While the few-cycle resolved simulations are a good first step that also aids the
understanding of the occurring effects, the sub-cycle resolved simulations provide a more
154
5.4 Modeling of the delay-dependent diffraction response
20 0 20
0
1
normalized
diffraction
intensity
(a) 790 nm, HO 11
20 0 20
0
1
normalized
diffraction
intensity
(d) 790 nm, HO 13
20 0 20
0
1
normalized
diffraction
intensity
(g) 790 nm, HO 15
20 0 20
delay (fs)
0
1
normalized
diffraction
intensity
(j) 790 nm, HO 17
20 0 20
0
1
(b) 800 nm, HO 11
20 0 20
0
1
(e) 800 nm, HO 13
20 0 20
0
1
(h) 800 nm, HO 15
20 0 20
delay (fs)
0
1
(k) 800 nm, HO 17
20 0 20
0
1
(c) 810 nm, HO 11
20 0 20
0
1
(f) 810 nm, HO 13
20 0 20
0
1
(i) 810 nm, HO 15
20 0 20
delay (fs)
0
1
(l) 810 nm, HO 17
Figure 5.41: Comparison of the contributions made by each harmonic to the delay-dependent
diffraction intensity (shown in fig. 5.40) between the few-cycle resolved simulation (light colors)
and the low-pass filtered result of the sub-cycle resolved simulation (dark colors). Again results
are shown for three different fundamental HHG wavelengths. The NIR is at a constant central
wavelength of 800
nm
and has a peak intensity of 2
·
10
14 W/cm2
. The diffraction intensity is
normalized to the value of the 13
th
HO’s contribution at large positive delays (i.e. away from
temporal overlap of NIR and XUV).
comprehensive picture for the slowly varying dynamics and thus will be used for the comparison
between simulation results and experiment.
5.4.4
Applicability and limitations of the simulated optical response and
diffraction intensity
So far, a theoretical modeling method of the diffraction intensity produced by helium nano-
droplets was introduced and applied. A response on the attosecond time scale predicts
diffraction intensity oscillations by the delay with at least double the NIR frequency. On longer
timescales, i.e., a few to some tens of femtoseconds, a decrease of the diffraction intensity
with varying width and features toward the edges of the decrease was produced. A direct
comparison with experimental data shall be made in this chapter’s discussion (section 5.5) to
draw conclusions on the physical effects yielding the experimental observation. Before, it must
be clear to what extent the modeling results may reproduce the experiment. Therefore the
major assumptions and approximations this modeling makes have to be discussed concerning
their impact on the simulation result. Those are:
155
5. Ultrafast dynamics in helium nano-droplets
1.
The bulk liquid helium’s optical response is described only by accumulated responses of
single atoms. The impact of the bulk environment on the helium potential experienced
by a single electron and (except for local field corrections) the interaction between the
atoms are neglected.
2.
Fitting non-linear dipole response of a single atom (at a given delay and NIR peak
intensity) to an analytical expression depending on the electric field neglecting non-
linearities higher than third order. Any sub-cycle response due to the instantaneous
response of helium to the electric field oscillations is neglected.
3.
With the application of Mie-theory for the scattering response, a homogeneous sphere
interacting with homogeneous fields is assumed. Even if computed for non-linear
conditions, the refractive index is included as a linear quantity. Mie-theory simulations
are carried out only for the central harmonic photon energies.
4.
Emission of light, as predicted for certain non-linear processes, is disregarded. A negative
absorption βis set to 0during Mie-theory scattering simulations.
5.
The temporal pulse train structure is constructed by superpositioning the scattering
responses due to the single attosecond pulses in the train. This approach neglects any
changes introduced by the preceding pulses that may affect the response of the helium
droplet experienced by subsequent pulses.
1. The electronic states, i.e., resonances of a system, described with the TDSE are determined
by the used potential. Modeling the response of a helium nano-droplet using a single atom
potential in the TDSE sets the resonance energies to the atomic ones. Compared to resonances
in liquid helium or helium nano-droplets, those are shifted [87, 75]. However, different
resonances experience different shifts [85]. A correction of the shift is only feasible when all
effects but those from very similar shifted resonances can be neglected (as it is done in section
5.4.2). Nevertheless, for the sub-cycle resolved simulations, this is not the case. In addition,
the potential implemented with the TDSE shows a significant deviation in the 2s state position
due to which the detuning between the NIR wavelength and the 2s-2p transition is reduced
by approximately 0
.
3
eV
(ca. 30%). This means the simulation overestimates the 2p/2s AC
Stark shifts (see chapter 2.1.5), which are most responsible for the decrease of the diffraction
signal (see e.g. fig. 5.41). Hence, the amount of diffraction intensity decrease is overestimated.
In addition, due to the discussed shifts in the simulations, HO 13 is located closer to the 2p
resonance of the undressed system. The impact of these shifts is difficult to estimate. Overall
using the single atom potential makes the model a qualitative one. The simulated behavior
will show qualitative features that can be seen experimentally. Varying the fundamental XUV
wavelength, as it is done for the discussed simulations, will shift the harmonic positions relative
to the dressed optical response and, therefore, can reveal possible features.
As the bulk response is constructed, including the dephasing due to the accumulation of
atoms and local field corrections, interactions between atoms are neglected. The atomic-like
electronic excitation behavior of helium nano-droplets described in [85], their weak inter-atomic
interactions [77] and successfully applied theoretical models based on a single atom approach
[94, 35] motivate the applicability of such approach for this work.
156
5.4 Modeling of the delay-dependent diffraction response
2. The fitting of the non-linear atomic (i.e., microscopic) dipole response using dynamic
polarizability coefficients including non-linearities scaling with the electric field up to the power
of three obviously neglects any higher orders. As two NIR photons approximately give the
distance between two HOs, the coupling from one HO into the region of the neighbored HO
can be realized (i.e.,
XUV photon ±two NIR photons
). Higher-order processes may also play a
role in the highest NIR peak intensities considered in the simulation. Still, those are expected
to be less significant and are neglected. As discussed in the appendix C.2.1, a significant
drawback of the fitting method is the inability to provide individual weights to different wave
mixing processes affecting the frequency response of the medium at the same frequency. As a
result, not all couplings and thus sub-cycle features are adequately described. Since features
due to couplings would otherwise at least partially be reflected by linear contributions, those
were low-pass filtered along the delay axis. Thereby also, near-instantaneous responses (e.g.,
due to a potential sub-cycle AC Stark effect [208]) are neglected. Consequently, the modeling,
including sub-cycle effects, can show the existence of sub-cycle features due to non-linear
couplings relevant for LIS [53]. It further reveals possible features and, therefore, can be used
to make qualitative comparisons with experimental results. Quantitative comparisons are
unlikely to match. Because not all couplings are correctly modeled, this applies especially to
the amplitudes of simulated oscillations.
3. The scattering response of the helium droplets is computed by applying Mie-theory
for a homogeneous sphere exposed to homogeneous light fields. A constant, i.e., linear
optical response given by an effective refractive index (see eq. 5.13, for sub-cycle resolved
simulations computed under non-linear assumption) is assumed. Since helium droplets are
almost transparent for NIR light [77] and in focus, the NIR’s beam diameter measures more
than 30
µm
(FWHM, see chapter 4.6.1) compared to the sub-micrometer sized droplets, this
appears to be a valid assumption for the NIR pulse. However, the XUV beam is significantly
smaller (
<
5
µm
FWHM, see chapter 4.6.2), and thus the error due to assuming a homogeneous
intensity distribution in the droplet increases. However, the XUV’s absorption likely introduces
a larger error due to the harmonic spectrum being close to resonances. As also predicted by
the modeling, the absorption for different harmonics is of different strength (see e.g. absorption
β
in fig. C.8). Hence during propagation of the XUV through the nano-droplet, its spectral
composition changes, which due to the non-linear couplings is known to affect the effective
refractive index
n
. However, this change of the XUV’s spectral composition is not reflected by
the effective refractive index assumed to be constant across the droplet. The amplitude of
non-linear contributions, especially sub-cycle features, will be distorted. Again conclusions
drawn by comparing experiment and simulation may be only qualitative.
Furthermore, the scattering response is only computed at the central harmonic photon
energies (mostly to reduce the computational effort). However, as can be seen from the
computed energy- and time-delay-dependent refractive index in fig. 5.37, the sub-cycle
oscillations along the time-delay axis partially show a phase-shift (i.e. tilt) across a single
harmonic. In practice, including those features likely would yield a reduced amplitude of the
sub-cycle oscillations reflected by Mie-theory simulations.
157
5. Ultrafast dynamics in helium nano-droplets
4. The theoretical modeling predicts partially negative values for the absorption
β
. In the
case of gaseous helium, such values were not only theoretically predicted [54] but also confirmed
experimentally [209]. It remains an open question to what extent they play a role in helium
droplets. This work sets negative absorption values to zero for the Mie-theory simulations.
Again a primarily quantitative impact is expected since except for features attributed to PFID,
the relevant observed behavior of absorption
β
is characterized by oscillations that include a
crossing of the zero.
5. The temporal pulse structure of the XUV APT is simulated as an incoherent superposition
of the scattering signals produced by the IAPs in the train at the central harmonic photon
energies. Such a superposition approach may be valid if the excitation by a preceding XUV
pulse of the ’average’ helium atom in the nano-droplet is negligible. Such negligible pre-
excitation can be shown with a simple, rough estimation: By computing the average number
of XUV photons
nphotons,XUV,atom
interacting with a single helium atom
43
within the droplet
at an intensity of 5
·
10
12 W/cm2
(pulse duration 300
as
and a photon energy of 20
eV
and
combining it with the absorption length
labs
for an absorption
β
= 0
.
4(a value that is typically
not exceeded by the simulated absorption β).
nphotons,XUV,atom =5·1012 W
cm2·((2Å)2·π)·300 as
20eV ≈1.2
According to [38], the absorption length
labs
depends on the XUV wavelength
λXUV
and is
given by:
labs =λXUV
4πβ ≈24.7nm .
Thus according to the average inter-nuclear spacing in large helium nano-droplets of below 4 Å
[85], the average number of helium atoms along an absorption length of 24
.
7
nm
is at least 61.
Hence, if in average 1of 50 atoms is excited by each of the 5pulses in the simulated pulse
train, for the last of the pulses
44
only 5% of the atoms would be pre-excited. Consequently,
for the present XUV intensities, simulating a pulse train by superposition of the responses of
isolated attosecond pulses is a valid approximation. It should not affect the results significantly.
In addition to this simplistic picture, see also the discussion in the context of experimental
and theoretical work by others in section 5.5.2.
In summary, many fundamental approximations have been made to simulate the time-
delay-dependent diffraction intensity. As a consequence, the simulation results can serve as
a qualitative reference. However, a quantitative comparison will not be possible. Though
maybe a resulting signal can be quantitatively matched, such a match would not allow
deducting quantitative parameters present in the experiment from the parameters present in
the simulation.
43
The area in which the atom interacts with the photon is assumed to be given by a radius not exceeding
half of the average inter-nuclear separation of helium atoms in a large helium-4 nano-droplet, which is given to
be below 4 Å[85]. Hence the upper limit for the area is computed for a radius of 2 Å.
44assuming relative contributions of the other pulses in the train according to table 5.4;
158
5.5 Discussion of experimental and modeling results
5.5 Discussion of experimental and modeling results
In this section, the experimental observations made on the time scale of a few optical cycles of
the NIR and in the sub-cycle (attosecond) domain are combined with the qualitative results
from simulations. Doing so, a picture of the dynamics yielding the experimental observations
is formed in the context of published work by other researchers.
5.5.1 Dynamics on the time scale of few NIR cycles
Experimentally three fundamental observations are made regarding the delay-dependent
diffraction intensity (see section 5.2):
1.
During or close to the temporal overlap of XUV and NIR pulses, the diffraction intensity
decreases. It recovers towards both, positive and negative, time delay directions.
2.
A decrease is consistently observed. However, its specific shape varies. E.g., partially
humps are observed towards the decrease’s edge (onset). It could be shown that a
variation in the quality of overlap between XUV and NIR or the NIR intensity can lead
to a change of the shape.
3.
Both depth and width of the reduced diffraction intensity feature decrease with a
decreasing NIR intensity.
The theoretical model identifies three effects that may affect the diffraction intensity on
the few cycle time scale:
•The laser field induces shifts of transitions, which is termed AC Stark effect.
•The emergence of new transitions, so called light-induced states (LISs)
•
The modification of the optical response by the laser field, when following after the XUV,
namely perturbed free induction decay (PFID).
In this subsection, first, these three effects identified by the theoretical modeling are discussed
concerning the general experimental observation of a decrease in diffraction intensity during
the temporal overlap (observation 1). Then to illustrate the qualitative correspondence
of theoretical modeling and experimental results, selected experimental measurements are
assigned to a possible counterpart retrieved from theoretical modeling (observation 2). Finally,
the modeling’s qualitative reproduction of the NIR intensity dependence is demonstrated
(observation 3).
Intensity decrease due to harmonic order 13
The theoretical modeling in both, linear (few-cycle resolved) and non-linear (sub-cycle resolved),
approximations reproduces the decrease in diffraction intensity (see e.g. figure 5.40). According
to the modeling, the decrease in diffraction intensity is mainly
45
caused by HO 13 (see fig.
5.41), which is located below the 2p-like and 2s-like resonances in helium nano-droplets (and
also in atomic helium).
45according to the non-linear approach even completely
159
5. Ultrafast dynamics in helium nano-droplets
In the single-atom picture, the coupling of both levels through the NIR pulse leads to
the AC Stark effect (see theory section 2.1.5), i.e., an NIR-intensity-dependent shift of the
resonances. Since the NIR photon energy (ca. 1
.
55
eV
) exceeds the energy difference between
the 2p and 2s states (0
.
6
eV
)
46
, the 2p resonance shifts to lower excitation energies. This
understanding can be transferred to a helium nano-droplet by describing it in a simplified
manner as an accumulation of non-interacting helium atoms (see modeling, section 5.4.1). Due
to the dephasing of the many atoms, the resonances are spectrally homogeneously broadened
[37]. A small amount of absorption due to the 2p resonance is possible more than one electron
volt below the center of the resonance [75]. A shift of the resonance compared to the photon
energy of the HO 13 that effectively moves the resonance closer will be measurable as a change
of the optical properties experience by that HO. Among others, the absorption will increase.
The modeling indicates a second effect that may affect the optical properties for HO 13
(and others) around time zero - namely the occurrences of LISs. Within the single atom
picture, a combination of an odd number of NIR photons and one XUV photon can access
dark states47. With an even number of NIR photons and one XUV photon, bright states are
accessible. Consequently, additional resonance-like absorption features occur at odd and even
multiples of the NIR photon energy from dark and bright states, respectively. Like, e.g., the
resonances themselves are affected by the AC Stark shift, also the LIS show a bending along
the delay axis (see fig. 5.27). Again, this picture is transferable to a helium nano-droplet
behaving like accumulated non-interacting helium atoms. As in helium nano-droplets a direct
excitation from the ground state into the 2s-like state is allowed
48
[75], additional LISs might
occur. The stronger the NIR pulse, the stronger the AC Stark shifts and thus the stronger
the bending of the LISs. Furthermore, the intensity of the NIR pulse also affects the spectral
region in which the LISs make a substantial contribution to the atom’s optical response [210].
For the experiment’s NIR peak intensity, the theoretical modeling predicts strong shifts and
spectrally broad features, which are, as a consequence, not anymore distinguishable for photon
energies below the 2p resonance (see fig. 5.27). Only a general increased absorption
β
and an
increase in δare modeled.
An increase in absorption may yield a decrease the amount of scattered light. This is
exactly what may occur to the scattered light of the 13
th
HO. It makes up a dominating
fraction of the scattered light when no NIR is present (according to the modeling approximately
70
−
80%. However, the 13
th
harmonic makes up only around 30% of the XUV
49
, see fig.
5.41). Hence, when averaging with the other harmonic’s contributions, the decreased diffracted
intensity of the 13
th
HO translates to an only slightly less pronounced decrease of the total
amount of diffracted light.
In addition to the AC Stark shift and LISs that affect the optical response around time
zero, the theoretical modeling further predicts PFID. It is an effect that occurs for the NIR
(pump) pulse following the XUV (probe) pulse (i.e., at negative time delays) [211] potentially
yielding to the emission of XUV light (see also theory in chapter 2.1.5). Such emission features
46and the 2p resonance is at higher energies than the 2s resonance
47i.e., such not permitted to be excited directly like s and d states.
48not included with the modeling
49
The modeled dominant diffraction intensity due to photon energies around 20
eV
is in qualitative agreement
with recent experimental data measuring the diffraction of helium nano-droplets at different XUV wavelengths
(unpublished, experiment by Linos Hecht (ETH Zürich) et al.)
160
5.5 Discussion of experimental and modeling results
were also experimentally measured in attosecond transient absorption spectroscopy (ATAS)
experiments using gaseous neon [209]. When the XUV light interacts with the nano-droplet,
it introduces a polarization
P
. Due to dephasing, the polarization decays rapidly (in the
modeling within 12
.
7
fs
). If the NIR interacts with the decaying polarization, it may modify it
and yield to the emission of light. The modeling predicts that it leads to a secondary diffraction
intensity decrease feature
50
for HO 13 (see fig. 5.41). However it also predicts that a distinct
feature is only produced for particular spectral positions of the harmonics
51
(see e.g. figures
5.29 and C.9). Nevertheless, generally, it seems plausible that such features could be and have
been observed, as the emission is predicted to occur while the microscopically induced dipole
moments oscillate coherently. The agreement between simulation results and experiment, as
discussed in the next paragraph, points also in this direction. Still, conclusive experimental
proof of distinct features is missing. It remains an open question to what extent such effect
plays a role in CDI experiments.
Qualitative agreement of experiment and simulations
The theoretical modeling produces time-delay-dependent diffraction intensities in remarkable
agreement with the experiment. This can be well seen from fig. 5.42, where experimental data
is plotted with matched (low-pass filtered) simulation results. Theoretical data, computed with
the sub-cycle resolved approach, was matched manually (no algorithm). Only a scaling factor
to match the y-axis, a shift of the relative time delay between experiment & simulation data,
and the fundamental HHG wavelength in the simulation (in 2
nm
steps) was adjusted. Other
parameters like the contributions of the different HOs are taken from experimental data. As
expected from the modeling’s limitations (see section 5.4.4), the fundamental HHG wavelength
determined from experimental data is typically not matching the one of the matched simulated
case. The fact that the matching simulated fundamental HHG wavelengths (except fig. 5.42
(f)) are varying only by 2
nm
, while the experimental data indicates a range spanning 46
nm
may have two possible reasons: first, as discussed in section 5.1.3, the error of the spectral
calibration is not well known, already an error of approximately 5% could lead to such deviation
in the fundamental HHG wavelength; second, a depth of the diffraction intensity decrease
similar to the experiment, can be seen in the simulation results only around (818
±
8)
nm
. In
a real helium droplet, where the resonances of at slightly different energies, there could be
multiple wavelength ranges in which such a depth can be achieved.
A special case is shown in fig. 5.42 (f). There, in the experiment, a tin filter (instead of
aluminum) was used on the XUV arm. Consequently, only HO 11, 13, and 15 contribute to
the XUV signal. While the, compared to the other shown measurements, slimmer diffraction
intensity decrease around time 0 is well reproduced, a second decrease-feature at negative time
delays attributed to PFID is not. Nevertheless, the experimental data shows a change in the
slope of the decrease over the range of the simulated PFID feature. Overall, the results remain
inconclusive regarding the contribution of PFID to the diffraction process.
Also, the dependency of the diffraction intensity decrease on the NIR intensity was
successfully modeled. Fig. 5.43 (a) - (d) shows the comparison of modeling and experiment,
50
In the modeling, the emission is neglected. It is still visible in the modelled diffraction intensity as the
absorption otherwise is not zero and therefore effectively decrease due to this effect.
51Corresponding in the modeling to a variation of the XUV’s fundamental wavelength
161
5. Ultrafast dynamics in helium nano-droplets
25 0
delay (fs)
0
5
10
normalized diffraction
intensity (arb. u.)
(a)
sim. 818 nm
exp. 771 nm
25 0 25
delay (fs)
0
10
(b)
sim. 820 nm
exp. 817 nm
25 0
delay (fs)
0
10
(c)
sim. 820 nm
exp. 801 nm
25 0
delay (fs)
0
10
(d)
sim. 818 nm
exp. 812 nm
25 0 25
delay (fs)
0
10
20
normalized diffraction
intensity (arb. u.)
(e)
sim. 820 nm
exp. 800 nm
25 0
delay (fs)
0
10
20
(f)
sim. 794 nm
exp. 794 nm
50 0
delay (fs)
0
10
20
(g)
sim. 818 nm
exp. 805 nm
25 0
delay (fs)
0
10
20
(h)
sim. 818 nm
exp. 802 nm
25 0
delay (fs)
0
10
normalized diffraction
intensity (arb. u.)
(i)
sim. 818 nm
exp. 802 nm
25 0
delay (fs)
0
5
10
(j)
sim. 818 nm
exp. 792 nm
simulation
experiment
Figure 5.42: Comparison of experimental measurements and matched theoretical simulations
at an NIR peak intensity of approximately 2
·
10
14 W/cm2
; Overall, a good agreement can be
demonstrated. To match the experimental y-scale, the normalized theoretical data is up-scaled by
a factor. Due to the experimentally only approximate determination of time-zero, the relative delay
between theory and experiment is adjusted for the best fit. The fundamental HHG wavelengths
of the simulation are chosen (manually) to achieve the best qualitative agreement in the data,
sampled in 2
nm
steps. The experimental and simulated data’s fundamental wavelength is stated
in the title. For all measurements except (f), an aluminum filter was in the XUV arm. A tin filter
was present for measurement (f), thus suppressing all HOs other than 11, 13, and 15. (a-j) are
shown together with associated measurements in the attosecond domain in appendix B.2 fig. B.4
to B.13.
from highest to lowest intensity from left to right. As is the case for many of the comparisons
in fig. 5.42, simulations with a fundamental HHG wavelength of 820
nm
make the best match
with the experimental data. Though the variation of the features at the edges of the delay
scans in the experiment and the significant experimentally observed width reduction of the
decrease in 5.43 (d) compared to 5.43 (a) - (c) are not captured by the modeling, overall a
good agreement between simulation and experiment is achieved.
162
5.5 Discussion of experimental and modeling results
25 0 25
delay (fs)
0
5
10
15
20
normalized diffraction
intensity (arb. u.)
(a)
sim. 820 nm
exp. 799 nm
simulation
exp., NIR intensity = 100 %
exp., NIR intensity 80 %
exp., NIR intensity 50 %
exp., NIR intensity 30 %
25 0 25
delay (fs)
0
5
10
15
20
(b)
sim. 820 nm
exp. 800 nm
0 50
delay (fs)
0
5
10
15
20
(c)
sim. 820 nm
exp. 800 nm
0 50
delay (fs)
0
5
10
15
20
(d)
sim. 820 nm
exp. 799 nm
Figure 5.43: Comparison of experimental measurements and matched theoretical simulations for
scanning the NIR intensity. The reduced intensities present in the experiment were estimated with
wave-front propagation simulations (see section 5.2 and fig. 5.15). For the theoretical simulations,
the intensities are only approximately matched. There the intensity is reduced in equal steps of
25% each. Similar to the results shown in fig. 5.42, a good match is produced for a simulated
fundamental HHG wavelength of 820
nm
. Again the simulation results are up-scaled, here by a
factor of 20 for all subplots, and the relative time-shift between experimental data and simulation
results is manually adjusted. The wavelength computed from experimental data and the wavelength
used with the simulation is also denoted in the title of each subplot. Except for the features near
the edges of the delay scan and the observed width of the decrease in (d), here, the simulation also
provides a good qualitative reproduction of the experimental results.
Summary
In summary, the routinely observed time-delay-dependent decrease in diffraction intensity
can be qualitatively reproduced with a simplified model partially accounting for non-linear
couplings on sub-cycle timescales. Widely consistent modeling parameters (not necessarily
matching the experimental ones) can accurately reproduce temporal width and relative depth.
In particular, the experimentally observed intensity dependency of the depth and width of the
decrease around time zero is well matched by simulation. The modeling, which assumes the
helium droplet to be an accumulation of single atoms that are not interacting, but dephasing
and forming a medium with a polarization, agrees well with the experimental results. Its main
conclusions are transferred. Accordingly, the decrease in diffraction intensity is attributed
to the AC Stark shift of an atomic-like electronic structure in the Helium droplet and the
in combination appearing LISs during the presence of the NIR. That is in agreement with a
previous experimental, and theoretical investigation of the effect [35]. Furthermore, PFID was
identified to be an additional contributing effect. However, no conclusive identification of this
effect in experimental results could be achieved.
163
5. Ultrafast dynamics in helium nano-droplets
2 0 2
delay (fs)
0.0
0.5
1.0
normalized
diffraction
intensity
1.3 fs
(a) sim. 800 nm
2 0 2
delay (fs)
0.0
0.5
1.0
normalized
diffraction
intensity
1.4 fs
(c) sim. 820 nm
1 0 1
delay (fs)
6
8
10
normalized
diffraction
intensity (arb. u.)
1.3 fs
(b) exp. 816 nm
2 0
delay (fs)
8
10
12
normalized
diffraction
intensity (arb. u.)
1.7 fs
(d) exp. 803 nm
Figure 5.44: Comparison of the modeled sub-cycle resolved signal (left, (a) and (c)) with
experimental measurements (right, (b) and (d)) at an NIR peak intensity of approximately
2·1014 W/cm2. Both show oscillations at a frequency of approximately twice the NIR’s.
5.5.2 Dynamics in the attosecond domain
As a result of the experimental investigation of the behavior of helium nano-droplets on
attosecond time scales, two key observations could be made:
1.
Reproducible oscillations with a period shorter than one NIR laser cycle (i.e., sub-cycle)
occur. In fact, the oscillation period measures approximately half a laser cycle. The
transition from minimal to maximal diffraction signal and vice versa spans approximately
half an oscillation period and thus occurs in the attosecond domain.
2.
The amplitude of the oscillations is less than 13% relative to the diffraction intensity
away from temporal overlap52.
The main results of the modeling of sub-cycle dynamics are:
•
Non-sinusoidal sub-cycle oscillations in the diffraction intensity are predicted and caused
by non-linear couplings, i.e., which-way interferences due to the laser-dressing of the
helium atoms.
•
Opposed to the decrease of the diffraction intensity on the few femtosecond timescale,
which could be mainly assigned to the contribution of HO13, several high harmonics
substantially contribute to the oscillation (see fig. 5.39).
52
The diffraction intensity away from temporal overlap also serves in the simulations also as a reference for
the normalization of the diffraction intensity.
164
5.5 Discussion of experimental and modeling results
•
The dominating frequency of the sub-cycle oscillations changes according to the
fundamental HHG wavelength, i.e., the temporal spacing of the attosecond pulses
in the pulse train. Hence, the attosecond pulse train promotes frequency components of
the signal at multiples of twice its fundamental frequency and suppresses others.
Comparison of measured and simulated sub-cycle oscillations
In principle, the modeling reproduces the presence of sub-cycle oscillations that were
observed experimentally. Fig. 5.44 shows side-by-side simulation results (left) and the
two subsequent experimental observation of the oscillation. The period of the oscillations
matches approximately
53
. In fact, fig. 5.44 (a) and (b) show a particularly well matched
period.
However, the oscillation amplitudes are not well matched: While experimentally the
measured amplitude does not exceed 13% of the signal away from temporal overlap, the
simulation indicate amplitudes of 25% and more. Certainly, the time delay stability
54
and
resolution
55
in the experiment may play a role. In fact, the impact of temporal blurring was
investigated and indeed similar amplitudes can be reached (see fig. 5.38). Nevertheless, at
ideal time resolution, one widely seen feature in the simulation results is an oscillation signal
that exceeds the diffraction signal outside of temporal overlap (see e.g. fig. 5.38 or 5.40). From
the measurements on the timescales of a few femtoseconds, it is, however, known that the
brightest images during the scattering decrease do not exceed the brightness of those further
away from temporal overlap
56
(see, e.g., fig 5.13). Consequently, the oscillation amplitude
yielded by the simulation, especially as seen in fig. 5.44 (a), seems unrealistic. Of course, as
only non-linear couplings, i.e., which-way interferences are included with the modeling on a
sub-cycle time scale and further strong assumptions are made, a full quantitative match to
the experimental result cannot be expected. Still this poses a qualitative discrepancy to the
simulation that may be addressed in future work.
Interpretation of the sub-cycle oscillations regarding the underlying physical effects
For the interpretation of the sub-cycle oscillations, the modeling can provide a physical insight,
although limited, because it reproduces the presence of sub-cycle oscillations due to the
laser-dressed atom only where non-linear couplings, i.e., which-way interferences, take place.
Thereby it obviously suggests that the observed and reproduced oscillations can be due to
such which-way interferences.
To also bring up the eventual influence of other effects, a discussion based on the available
literature for such sub-cycle effects in gaseous helium follows.
In the literature, where sub-cycle effects induced by coinciding NIR and XUV pulses in
gaseous helium were investigated experimentally utilizing absorption [195, 212], as well as
electron and ion [66, 67] spectroscopy (including VMI [66, 67]), for experiments with APTs, two
possible effects are identified to produce an oscillating (at twice the NIR frequency) behavior
53Reasons for the mismatch between fig. 5.44 (b) and (d) were discussed in section 5.3.
54e.g., due to instabilities of the delay unit.
55e.g., due to the non-colinear NIR-XUV geometry (see fig. 4.6 (d)).
56
Although with a bad temporal resolution according to the simulations the average diffraction signal would
not exceed the brightness away from temporal overlap, there would be single images that do.
165
5. Ultrafast dynamics in helium nano-droplets
of the respective observed signals. On one hand-side, oscillations could be produced by the
interference of electron wavepackets (EWPs) excited by different attosecond pulses from the
APT with a trajectory controlled by the NIR or, on the other hand, due to the laser-dressing
of the helium atom by the NIR [66, 195]. As the simulations carried out in this thesis are
modeling the APT based on incoherent superposition of IAPs, only the latter is included.
Nevertheless, the possible contributions of both need to be discussed.
To the knowledge of the author, sub-cycle oscillations due to the interaction of an NIR
pulse and XUV APTs were first observed by Johnsson et al. [66] as an oscillation in the helium
ion yield. In the scope of their investigation, based on modeling employing the TDSE in SAE
and experimentally measured electron momentum distributions, the interference of EWPs was
suggested to exceed the contributions from the laser dressing of the helium atoms by one order
of magnitude [66]. According to their calculations, the peak-to-peak span of the oscillations
shows a complex dependence on the NIR intensity and increases significantly with the number
of pulses in the APT [66]. A first experiment employing absorption spectroscopy confirms the
presence of sub-cycle oscillations in the absorbance and agrees with the findings by Johnsson
et al. on EWP interference to be the dominant mechanism yielding these oscillations [195].
However, later experimental work employing absorption spectroscopy aiming for a systematic
comparison to the expectations due to EWP interference cannot confirm a dependence of
the oscillation strength on the number of pulses in the APT [212]. Moreover, the same work
directly compares oscillations along the delay axis produced by a short APT and an IAP. No
obvious change in the delay-dependent absorption, including its oscillation strength, is found
[212]. Lucchini et al. conclude that not the EWP interference, but the which-way interferences
due to the laser dressing of the atom make up the dominant contribution to the sub-cycle
oscillations of the absorption [212].
The laser dressing picture of atoms pumped by NIR pulses and probed with XUV has been
researched extensively, experimentally and theoretically [207, 54]. Sub-cycle oscillations are
attributed to which-way interferences [53, 196], i.e. the interference of multiple quantum paths
to the same resonance, sub-cycle AC Stark shifts [208] and sub-cycle ponderomotive shifts
[196]. Chini et al. find that sub-cycle AC Stark and ponderomotive shifts are most responsible
for dynamics close to the ionization threshold of helium [196].
Opposite to the here presented work, all the discussed, published work was carried out
using gaseous helium and NIR peak intensities below 10
14 W/cm2
. The transferability of
atomic physics results to the helium nano-droplet at the ultra-short timescales relevant to
this work was already discussed. The modeling of the APT as an incoherent superposition of
IAPs appears based on the conclusion of a dominating contribution from the laser dressed
atom picture in [212] reasonable. The TDSE-based simulations carried out in the scope of
the presented work suggest that at the high intensities (up to 2
·
10
14 W/cm2
), the basic
oscillation features are preserved but become accessible less dependent on the fundamental
HHG wavelength (see fig. C.6 and C.7). Therefore the associated LISs are no longer obvious
(see fig. C.4 (a) in comparison to C.3 (a)). Still, the underlying mechanisms identified at lower
NIR intensities in atomic targets are assumed to be applicable.
166
5.5 Discussion of experimental and modeling results
Summary
In summary, the observed oscillation of the diffraction intensity at approximately twice the
NIR frequency is predominately attributed to ’which-way’ interferences as known from atomic
physics. For most of the XUV light, sub-cycle AC Stark and ponderomotive shifts may only
play a secondary role (as they are only dominant close to the ionization threshold [196]). To
what extent interference between excitations from different pulses in the APT play a role could
not be determined, but according to [212] an only non-dominating contribution is expected
to be of relevance. While the high NIR intensity conditions from this experiment certainly
are too high to separate the different LIS and other features contributing to the effect, they
ensure the existence of sub-cycle features within the available harmonic spectrum. Therefore
those are ideal conditions for a proof of principle that shows the experimental accessibility of
dynamics in the attosecond domain taking place in isolated nanoparticles through CDI. Likely
future experiments will investigate the effect at more moderate NIR intensities, where also the
difference between gaseous helium and helium nano-droplets could become more apparent.
167
6
Summary & outlook
This work demonstrates the accessibility of attosecond dynamics occurring in isolated
nanoparticles by coherent diffractive imaging. In the presented experiments, it was found that
the amount of XUV light coherently diffracted by helium nano-droplets is altered as a function
of the relative time delay to a dynamics-inducing NIR pulse to which the droplets are also
subjected. As a result, an oscillation at twice the NIR frequency was observed for scans over
ultrashort time delays. This experimental observation was made possible by the collaborative
effort of multiple groups at Max Born Institute to develop an HHG-based light source that can
generate particularly intense XUV attosecond pulse trains phase-coupled to the NIR pulse.
The performed coherent diffractive imaging experiment was based on an HHG beamline
setup optimized to generate high XUV intensities instead of relying on high XUV pulse energies
alone. In that context, considerations were presented for an experiment-specific optimization
of the beamline layout. For CDI, this means finding a compromise between homogeneous
illumination of the object to be imaged and the peak intensity that can be achieved. In
accordance with the optimization results, an asymmetric beamline layout is used in practice.
Here, the gas cell (i.e., the HHG medium) is placed much closer to the beginning of the
beamline so that the XUV light can propagate a long distance behind this cell before reaching
the experiment. It is then focused there to achieve the expected high intensities. The success
of such an asymmetric arrangement could be confirmed with a measurement done by Bernd
Schütte and Martin Kretschmar, where particularly highly ionized argon - up to
Ar5+
- could
be generated [36]. For this work, the XUV light, generated with sub-10-fs NIR pulses, was
temporally characterized using a RABITT or CRAB scheme. The attosecond pulse train
structure could be verified, and its envelope could be determined, though probably slightly
underestimated, to measure a FWHM of 3.1fs.
For this work, a novel experimental setup was developed and characterized utilizing some
components from previous CDI experiments in which helium nano-droplets were imaged at the
Max Born Institute [34, 35]. However, in contrast to previous experimental setups, the NIR
pump pulse is derived directly from the NIR pulse which leaves the HHG medium together
with the XUV pulse. In addition, a compact delay unit compatible with planar and focusing
169
6. Summary & outlook
mirrors in near-normal incidence geometry has been designed to allow precise attosecond time
delays between the NIR and XUV pulses. Furthermore, diagnostic tools behind the interaction
region grant access to the NIR pulse energy and the NIR spectrum in addition to the XUV
spectrum. Moreover, specifically for this experiment, the focusing optics were designed from
scratch. In the process, ray-tracing calculations revealed that for this HHG beamline with
the expected short focal length of about 18
cm
, a focusing toroidal mirror produces higher
intensity at near-normal incidence than at grazing incidence. The so produced XUV focus
was characterized with a phosphor screen to a beam waist radius of less than 4
µm
(1
/e2
) and
imaged, spectrally resolved, with fluence mapping gratings.
The high XUV intensities in the experiment, made possible by the optimized beamline
layout and focusing optics design, produced bright diffraction patterns of intercepted helium
droplets with diameters of several hundred nanometers. The patterns were analyzed in terms
of their integral, i.e., diffraction intensity. Measurements in which the NIR and XUV pulses
were delayed on the order of their duration, i.e., in the range of several femtoseconds, routinely
reproduced a decrease in diffraction intensity while the pulses overlapped in time. Here, the
duration of the decrease is also on the order of the NIR and XUV pulse durations, corroborating
the observation from a previous measurement of this phenomenon with longer NIR and XUV
pulses [35]. Furthermore, the measurements show that the lower the NIR intensity present,
the smaller the observed decrease. It proved to be very difficult to advance into the attosecond
regime by resolving sub-cycle features. Nevertheless, in two consecutive measurements, a
signal oscillating at approximately twice the frequency of the NIR pulse could be observed
and reproduced, very close to the resolution limit of the experimental setup. Moreover, the
observation could be reproduced again in further measurements a few days after the first
measurement, both confirming the existence of the observed dynamics.
To aid the interpretation of the observed ultrafast dynamics, a theoretical modeling
approach for the helium nano-droplets was applied, which is based on the dipole response of a
single electron of a helium atom. An initial similar approach was presented in the doctoral
thesis of Julian Zimmermann [35], and extended in the present work to account for dynamics
on sub-cycle time scales by taking into account a nonlinear polarizability of helium and the
attosecond pulse train structure of the XUV pulses.
For dynamics occurring on a time scale of a few femtoseconds, the modeling qualitatively
reproduces the observed decrease in diffraction intensity. It even closely reproduces the decrease
in width and relative depth. Since this modeling is founded on an atomic picture, atomic
physics effects are employed to provide an explanation. Following this concept, in agreement
with previous work [35], the observed dynamical behavior is attributed to an AC Stark shift
and the occurrence of light-induced states (LISs) within a helium droplet. Whether, in addition,
perturbed free induction decay (PFID) plays a role remains inconclusive.
The modeling also shows time-delay-dependent sub-cycle oscillations in the diffraction
intensity. The calculated oscillations occur predominantly at twice the NIR frequency in
agreement with experimental observation. Again, applying the results of previous studies in
atomic physics [53] to helium nano-droplets, the sub-cycle oscillations can be attributed to
which-way interference or quantum path interference. These interference phenomena occur in
light-induced states when multiple quantum paths lead to the same final state. Because of the
170
assumptions made in the modeling, the question of to what extent other effects contribute to
the oscillation cannot be answered.
The work presented here is one of the first to demonstrate the feasibility of time-resolved
studies of isolated nanoparticles in the attosecond range using CDI. Future work could therefore:
1. realize immediate improvements to the here developed setup,
2.
follow a pathway that emerges from this work and thus may resolve question that could
not be answered here,
3. utilize the here investigated effects for a novel experiment.
1. For future work building upon the here developed setup, an increased temporal resolution
could be achieved by implementing active delay stabilization with the delay unit, which
have been already envisioned in the design. In addition, the repetition rate limitations
could be removed by replacing the pulsed droplet source used with a source that generates a
continuous stream of helium nano-droplets. Moreover, since these sources are characterized by
the generation of more similar droplets [213], the analysis could be simplified because fewer
variations in shape or size need to be considered.
2. In any case, future experiments taking this work further would benefit from a more
sophisticated analysis capable of reconstructing the complex refractive index of the nano-
droplets at all XUV photon energies directly from the diffraction patterns obtained. A first
step for developing such analysis algorithms that preserve size- and shape-dependent optical
properties at all superimposed harmonic orders could be an experiment with monochromatized
light, i.e., with a single harmonic order. Since this approach eliminates the attosecond pulse
train structure of the XUV pulse, no attosecond dynamics can be resolved. Nevertheless, such
an analysis would also be fascinating for the observed decrease in diffraction intensity on the
femtosecond time scale. The high losses typically associated with monochromatization of XUV
light can be overcome by using multilayer mirror coatings to suppress unwanted harmonic
orders. Those coatings can realize reflectivities similar to the
B4C
-coated mirror used in this
work at near-normal incidence [214, 159]. Such experiments could be also realized with an
FEL source, given that beamtime is granted.
Then, in a next step, time-resolved CDI studies of isolated nanoparticles in the attosecond
range could become possible with broadband isolated attosecond XUV pulses. For this, one
must either gain the ability to generate these pulses at high intensities or the ability to
bring highly repetitive nanoparticles into the interaction region so that averaging over many
individual diffraction patterns is applicable. Of course, the broadband XUV light makes the
analysis of the diffraction patterns much more difficult. Nevertheless, algorithms are currently
being developed to solve this problem [215]. Indeed, it is already possible to monochromatize
diffraction patterns for a two-dimensional binary sample structure [114].
This work used rather high NIR intensities to ensure strong electronic dynamics in the
sub-cycle region. However, as the simulations in this work show, this also leads to overlapping
features of different origins in the altered electronic landscape. It is, therefore, practically
171
6. Summary & outlook
impossible to separate them in the analysis. Therefore, to suppress overlapping features, one
should use more moderate NIR intensities on the order of 10
12 W/cm2
in follow-up experiments,
regardless of the XUV pulse duration suggestions.
(a) reference object / volume within a droplet (b) transient incoupling window on the surface
NIR
NIR
large
helium droplet
large
helium droplet
Figure 6.1: Concept to introduce localized changes in the optical properties with an extremely
tightly focused NIR pulse in a several micrometer helium droplet: (a) by focusing an NIR pulse
into a helium droplet it changes the diffraction response in a well-defined volume, which then may
serve as a reference and (b) by focusing an NIR pulse onto the surface a helium droplet, it will
transiently change the optical properties there, such as spectral reflectivity, which could be used to
"open a window" into the droplet.
3. In an extended view, the NIR-induced change in diffraction response as a function of
intensity could be used to produce localized, transient structures in micrometer-sized particles,
e.g., in large helium droplets. The dressing NIR pulse would need to be focused extremely
tight, to spot-sizes close to the diffraction limit. This concept is illustrated in Fig. 6.1 for two
examples concerning the dynamics on the timescale of a few femtoseconds: (a) If tight-focused
NIR pulses are introduced sideways into a helium droplet of several micrometers in size,
with the focus contained within the droplet, one could create a reference object/volume of
well-defined size, potentially enabling holographic imaging techniques. (b) Instead the NIR
could also be focused to the surface of a large helium droplet, modify the optical properties
there for a well controlled duration and, e.g., thereby opening a short lived window into the
droplet by strongly reducing the reflectance at the surface for a particular angle of incidence.
In this context, it is noteworthy that large helium droplets have been proposed to be used as
an optical cavity for quantum optics experiments [216].
The experimental results obtained in this work represent an important step towards a
better understanding of ultra-fast dynamics in isolated single nanoparticles. The observation
of a changing scattering response of helium nano-droplets on sub-cycle timescales demonstrates
the accessibility of the attosecond domain with coherent diffractive imaging and paves the way
for observing and controlling electron dynamics in condensed-phase nanoscale matter.
172
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188
A
Data processing
A.1 Additions to pre-processing of diffraction detector data
(a) (b) (c)
(d) (e) (f)
104
2 × 104
3 × 104
4 × 104
detector signal (arb. u.)
Figure A.1: Each row shows the three brightest images of a reference run in which the droplet
source was operated at a repetition rate of 10 Hz. (a-c) The reference run (id: 3702) shows an
overall comparably low amount of signal, still rings in the diffraction pattern are visible, but show a
weak contrast. (d-f) Taken after improving the XUV pulses reaching the experiment, this reference
run (id: 3706) shows clear and intense patterns with a notable better contrast of the diffraction
rings. (logarithmic color scale, for visualization purposes the color scale clips a detector signal
below 10000 (arb. u.))
189
A. Data processing
A.1.1 Validation of the diffraction image quality
Fig. A.1 shows such diffraction data during conditions that are either insufficient (first line) or
produce images of a suitable quality (second line). The contrast difference of the diffraction
rings is visible.
A.1.2 Robustness of the background classification algorithm
0
100
normalized
counts
(arb. u.)
(a) 99.87 % (b) 87.13 % (c) 77.20 % (d) 68.40 % (e) 68.10 %
2.5 5.0
integral
(arb. u.)
1e8
0
100
normalized
counts
(arb. u.)
(f) 60.10 %
2.5 5.0
integral
(arb. u.)
1e8
(g) 58.63 %
2.5 5.0
integral
(arb. u.)
1e8
(h) 62.70 %
2.5 5.0
integral
(arb. u.)
1e8
(i) 49.57 %
2.5 5.0
integral
(arb. u.)
1e8
(j) 44.07 %
Figure A.2: Behaviour of the background identifying algorithm for different amounts of helium
nanodroplets introduced into the experiment ranging from none (a) to the maximal possible amount
in (j). Dark gray bars indicate the selected range of data below the average integral value. From
those, the blue bars are selected so that they likely only contain background data. The orange
Gaussian function is fitted using the blue bars’ values (dashed line indicates regions not included
in the fit). The threshold (red line) is set at three times the standard deviation from the central
position of the fitted Gaussian function. The title shows the fraction of the respective run’s shots
contained in the identified background.
The robustness of the method is demonstrated by looking a at series of XUV only data
in which the amount of helium droplets in the interaction region was varied. Fig. A.2 (a) -
(j) plots the data comparable to fig. 5.6 (b)1. The amount of Helium droplets increases from
(a) to (j). In fig. A.2 (a) no droplets reach the interaction region. The detected background
makes up 99
.
87% of the signal. In accordance with the previously specified error, 0
.
13%
are wrongfully labeled. Increasing amount of helium droplets introduced by the source, the
background level and width of the Gaussian distribution increases and with it the computed
threshold.
A.2 Toward spatial analysis of the diffraction patterns
Although, in this work a spatial analysis of the diffraction patterns is not included. Steps
toward such interpretation were developed and are shown in the following sections.
1The data from 5.6 (b) is also shown in fig. A.2 (h)
190
A.2 Toward spatial analysis of the diffraction patterns
A.2.1 Identifying patterns among hits
mean
background
integral
analysis
structural
analysis
find
center
background
hit
pattern
masking image
classification
Figure A.3: Adapted pre-processing procedure for coherent diffraction imaging data, now
including structural analysis; The algorithms and steps described in this section are represented by
rectangles. Circles are the resulting classes/labels from the classification that are treated separately.
Finally, octagons represent the basic types of analysis to be performed.
region
1
region 2full detector
Figure A.4: Visualization of the region taken into account for the full integral (left), as well as the
defined integral regions 1 and 2 (right) on the diffraction detector. In the background the masks
(basic mask and exemplary straylight mask) applied prior to integration are shown in black/gray.
Images of the diffraction pattern classified as non-background, i.e. hits, contain, among
others, very bright ones, sufficient to analyze their spatial structure. These give rise to the
sub-classification of ’hits’ as ’patterns’. Patterns become apparent when their contribution to
the image overcomes contributions from background noise. A threshold above which ’hits’ can
also be labeled as ’patterns’ is determined through the following procedure illustrated for a
typical run in fig. A.5:
1.
Compute the ratio of the integrals in region 2 and 1 for each shot (fig. A.5, blue dots).
Here, two clearly different domains of the distribution can be differentiated.
2.
Fit the ratio depending on the integral in region 1 with two linear functions (fig. A.5,
orange lines)
3.
Set a threshold with regards to the integral in region 1 at the intersection of the linear
functions (fig. A.5, red line)
The power of this method can be seen, e.g., by visually comparing fig. A.5 (a) and (b),
whose region 1 integrals are close to the threshold. However, fig. A.5 (b) starts to form a
visible pattern consisting of multiple diffraction rings, while in fig. A.5 (a) the signal seems
191
A. Data processing
01234
detector integral
in region 1 (arb. u.)
1e9
1.0
1.5
2.0
2.5
3.0
ratio of detector integrals
in region 2 and 1 (arb. u.)
(a)
(b)
(c)
(d)
(a) (b)
(c) (d)
Figure A.5: (a) Plotting the ratio of the detector integrals in region 2 and 1 over the detector
integral in region 1 (blue dots) of the single shots from a typical run (id: 3915) allows the
differentiation between two ranges with inclining and declining slopes. Fitting linear functions
(orange lines) to those slopes yields an intersection where a threshold (red line) differentiating
between hits showing a pattern and not showing a pattern is defined. Magenta dots, labeled with
(a) - (d), indicate the data points whose respective images are shown exemplarily. Dotted circles
indicate the integral regions introduced in fig. A.4.
to be too weak to do so. More obvious patterns can be seen in fig. A.5 (c) and (d), whose
integrals significantly overcome the threshold.
A.2.2 Routine to find the center of diffraction patterns
In the acquired data, the center of the diffraction patterns appears to vary slightly between
different shots. For optimal structural analysis results, individual centers should be fitted.
When the confidence in the individually fitted result decreases for weak diffraction signals, a
mean center, derived from the brightest hits’ centers in the run, can be used instead.
Here, two different center finding algorithms may be used: a self-developed algorithm
and an algorithm developed by Alessandro Colombo (NUX Group, ETH Zurich) that is
publicly available within the python module ’nux-utils’
2
. For each shot, the results of both are
compared, the better center
3
is then used within further analysis. Most patterns are expected
to be spherical, and those are only of interest; the two algorithms are optimized for spherical
symmetry.
The self-developed algorithm’s routine, that is also visualized in fig. A.6, consists of the
following steps:
1. From the diffraction pattern (fig. A.6 (a)) extract a binary ring pattern (fig. A.6 (b))
2https://gitlab.ethz.ch/nux/essential-tools/nux-utils/; function: nux_utils.pattern.find_center
3i.e., such that produces a higher contrast of between diffraction rings in a radial integral
192
A.2 Toward spatial analysis of the diffraction patterns
(a) (b) (c)
Figure A.6: Illustration of the algorithm to identify the center of a spherical diffraction pattern;
(a) the raw diffraction image (b) extracted binary diffraction rings, the green overlay indicates the
fitted diffraction rings (dashed lines) and center (dot) (c) the raw diffraction image shown in (a)
with the overlayed fit results also shown in (b)
2.
Make an initial guess on the center of the pattern, the number of rings, ring radii and
ring width
3. Fit concentric rings to a downscaled binary ring pattern and obtain their center
4.
Repeat the above procedure six times and select the best center (fig. A.6 (b) and (c),
green dot)
It follows a detailed description of the steps:
1. A ring pattern is extracted by effectively applying a band pass filter to the image through
subtraction of a strongly blurred diffraction pattern from a slightly blurred diffraction pattern.
Weak signal regions are masked out by default as well as potentially saturated areas in the
center. While the ratio of the blurring strengths is fixed, the absolute magnitude is adapted to
each individual pattern. The resulting ring pattern is converted to a binary image by setting a
threshold at the mean value of the unmasked areas within the ring pattern.
2. The guess on the pattern center consist of the center of the detector provided with the
basic mask and a randomized variation by up to
±
15
px
. A guess on the number of rings, ring
radii and ring width is computed from the binary ring pattern using the guessed center. No
more than four rings are taken into consideration.
3. The binary ring pattern is down scaled, allowing also values between 0 and 1. A concentric
ring pattern is generated through superposition of single concentric rings, whose radial
component is given by Gaussian distribution defined through ring position and width, for a
given center. A fit is performed through the ’Trust Region Reflective algorithm’ using the
’curve_fit’-function that is implemented with the widely-used python module ’scipy’. The
resulting center is then converted to the coordinate system of the diffraction pattern at original
scale.
193
A. Data processing
4. The procedure given by steps 1 to 3 is repeated in this case 6 times. Each time the
initial center guess and consequently the other initial parameters may vary slightly (due to
randomization of the center guess).
194
B
Additional experimental observations
B.1
Influence of the spatial overlap on time-scale of few-
femtoseconds
500 0 500
distance from
XUV's focal plane
(m)
20
0
20
beam horizontal
axis ( m)
(a) 480 m
500 0 500
distance from
XUV's focal plane
(m)
20
0
20
(b) 240 m
500 0 500
distance from
XUV's focal plane
(m)
20
0
20
(c) 0 m
NIR intensity
XUV intensity
Figure B.1: (a-c) Top view of the XUV and NIR foci according to wavefront propagation
simulations of idealized Gaussian beams. The intensity distributions for spatial overlap established
at different distances from the XUV’s focal planes are shown. Vertical dotted lines indicate the
planes in which the spatial overlap is established. Their positions relative to the ideal situation
depicted in (c) are given in the title of the subfigures. The slight difference in the relative angles
between NIR and XUV beams, due to the intersection of their beam axis at different positions, is
neglected. The nano-droplet beam is expected to be aligned for a distance of 0µmfor all cases.
A fundamental requirement for any pump-probe experiment is establishing a proper spatial
overlap between the pump beam, the probe beam, and the sample (here, helium nano-droplets).
In this experiment, the helium nano-droplet beam is aligned onto the XUV focus. The
procedure to establish spatial overlap was explained in chapter 4.7, here the position along the
XUV beam’s propagation axis of the (orthogonal) plane in which the overlap between pump
and probe beam is established is shifted. Its effect on the measured signal is investigated.
1
In
1
Since to establish the overlap between pump and probe beams, only the NIR (pump) beam is steered, the
overlap of the XUV focus and nano-droplet beam is preserved.
195
B. Additional experimental observations
fig. B.1, the expected situation in focus is shown for overlap between both beams established
in three different planes. The shown wavefront propagation results are based on idealized
beam properties. In reality, the XUV beam waist radius may be somewhat larger, and so is
the Rayleigh length, i.e., the length of the most intense region, which is scaling with the square
of the beam waist radius. According to the characterization results presented in chapter 4.6,
the Rayleigh range in the experiment compared to the idealized propagation results shown in
fig. B.1 could be larger by a factor of up to 4.
0 25
delay (fs)
0
5
10
(normalized)
mean detector
signal (arb. u.)
(a) -480.0 m
0 25
delay (fs)
(b) -240.0 m
20 0 20
delay (fs)
(c) 0.0 m
25 0 25
delay (fs)
6.0
6.2
6.4
1e8
(d)
Figure B.2: Influence of the plane in which spatial overlap is established onto the time delay
dependent diffraction intensity; (a-c) scan of the time delay for overlap established in different
planes spaced by 240
µm
. The solid line shows the average detector signal for the brightest 4%
(72) of the images normalized to the single-shot spectrum integral. The different planes are set
by translating the phosphor screen on the focus tool (used to establish overlap) along the beam
propagation axis. Reestablishing overlap requires tilting the NIR focusing mirror, which causes the
shift of time zero between the measurements. The time zero indicated by the x-axis is set with
regards to (c). The light-colored filled area in the background shows the scaled integrated single
charged helium droplet fragment signal acquired with the ion time of flight spectrometer. Plots
(a-c) use the same scaling factor for the ion time of flight data. (d) The stability of the XUV signal
is represented by the integral of the acquired spectra for each scan point color-coded to the colors
of the plots (a-c).
An experimental pump-probe measurement for spatial overlap established at three planes
spaced by 240
µm
is shown in fig. B.2.
2
One observes from fig. B.2 (a) to (c) an increase of the
width of the decreased diffraction intensity feature when measured on the dotted line from 13
fs
(a) via 16
fs
(b) up to 18
fs
(c). At the same time, the width of the feature measured on the
approximate baseline (dashed line) is minimal in fig. B.2 (c) with 27
fs
(otherwise 35
fs
). These
two behaviors of the widths yield the impression of improved clarity of the feature from fig.
B.2 (a) to (c). Changes in the depth of the feature do not exceed the confidence intervals and
hence are not significant. In the presented data, the depth of the diffraction intensity decrease
measures approximately 50 % (dashed-dotted line) of the diffraction intensity observed for
the approximate baseline (dashed line). Also, the feature’s position relative to the time zero
signature of the singly charged Helium fragments and the time zero signature itself does not
change notably. During and between the scans, the XUV source was very stable as indicated
by fig. B.2 (d), showing the integral of the XUV spectra over all three measurements to vary
within a scan on the order of 10 % (peak-to-peak) exceeding its variation between the scans.
2The absolute positions of the planes relative to the situation depicted in fig. B.1 (c) is not known.
196
B.2 Sub-cycle dynamics
When shifting the plane of spatial overlap, the brightest diffraction patterns still in tendency
originate from the volume with the highest XUV intensities. However, this is, as illustrated
by figure B.1 not necessarily equivalent to fully overlapping with the highest NIR intensities.
According to the wavefront propagation results of idealized beams (see fig. B.1), the NIR
intensity in the center of the XUV’s focus may be decreased by up to 40 % when the overlap is
established in a plane offset to the ideal one by 480
µm
. Furthermore, with the XUV focus
located at the slope of the NIR beam profile in focus, the jitter of the NIR’s focus position
relative to the XUV’s corresponds to a much larger jitter of NIR intensity than within the
center of the NIR focus. With this, a decreased clarity of the feature’s shape could be explained.
Even with the presented explanation, the reason for the not significantly changing helium ion
signals has not been identified.
This data set shows that the slight sub-millimeter variation of the overlap plane influences
some properties of the scattering intensity decrease significantly. Hence, the spatial overlap
appears to be a fragile component of the experiment.
B.2 Sub-cycle dynamics
In this section, plots of all potentially relevant acquired time delay scans with sub-cycle
resolution are shown.
0 10 20
delay (fs)
0
10
20
normalized
mean detector
signal (arb. u.)
(a)
7.5 10.0
delay (fs)
10
15
normalized
mean detector
signal (arb. u.)
(d)
7.5 10.0
delay (fs)
10
15
20
(b)
6 8 10
delay (fs)
10
15
20
(c)
Figure B.3: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scans.
(a) Few femtosecond resolved delay scan (run id 3573 to 3593) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b,c) Sub-cycle resolved delay scans with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a-c) The shown errorbars represent a
confidence interval of 90 %. (d) All sub-cycle resolved scans in direct comparison.
197
B. Additional experimental observations
20 0 20
delay (fs)
0
5
10
15
normalized
mean detector
signal (arb. u.)
(a)
1 0 1
delay (fs)
8
10
normalized
mean detector
signal (arb. u.)
(d)
1 0 1
delay (fs)
6
8
10
12
(b)
1 0 1
delay (fs)
8
10
(c)
1 0
delay (fs)
6
8
10
(e)
Figure B.4: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scans.
(a) Few femtosecond resolved delay scan (run id 3755 to 3769) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield in
light blue. (b,c,e) Sub-cycle resolved delay scans with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a-c, e) The shown errorbars represent a
confidence interval of 90 %. (d) All sub-cycle resolved scans in direct comparison.
20 0 20
delay (fs)
0
5
10
15
normalized
mean detector
signal (arb. u.)
(a)
2 0
delay (fs)
5
6
7
8
9
(b)
Figure B.5: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scan.
(a) Few femtosecond resolved delay scan (run id 3836 to 3850) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b) Sub-cycle resolved delay scan with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a,b) The shown errorbars represent a
confidence interval of 90 %.
198
B.2 Sub-cycle dynamics
20 0 20
delay (fs)
0
5
10
15
normalized
mean detector
signal (arb. u.)
(a)
2 0
delay (fs)
7
8
9
10
(b)
Figure B.6: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scan.
(a) Few femtosecond resolved delay scan (run id 3977 to 3991) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b) Sub-cycle resolved delay scan with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a,b) The shown errorbars represent a
confidence interval of 90 %.
20 0 20
delay (fs)
0
10
20
normalized
mean detector
signal (arb. u.)
(a)
0 5
delay (fs)
8
10
12
normalized
mean detector
signal (arb. u.)
(d)
2 0
delay (fs)
8
10
12
(b)
2 4
delay (fs)
8
10
12
(c)
Figure B.7: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scans.
(a) Few femtosecond resolved delay scan (run id 4013 to 4027) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b,c) Sub-cycle resolved delay scans with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a-c) The shown errorbars represent a
confidence interval of 90 %. (d) All sub-cycle resolved scans in direct comparison.
199
B. Additional experimental observations
20 0 20
delay (fs)
0
10
20
30
normalized
mean detector
signal (arb. u.)
(a)
4 2 0
delay (fs)
8
10
12
normalized
mean detector
signal (arb. u.)
(d)
4 2 0
delay (fs)
10
12
14
(b)
4 2 0
delay (fs)
10
12
14
(c)
4 2 0
delay (fs)
7
8
9
10
(e)
Figure B.8: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scans.
(a) Few femtosecond resolved delay scan (run id 4250 to 4264) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield in
light blue. (b,c,e) Sub-cycle resolved delay scans with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a-c, e) The shown errorbars represent a
confidence interval of 90 %. (d) All sub-cycle resolved scans in direct comparison.
20 0 20
delay (fs)
0
10
20
normalized
mean detector
signal (arb. u.)
(a)
4 2 0
delay (fs)
12
14
16
(b)
Figure B.9: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scan.
(a) Few femtosecond resolved delay scan (run id 4313 to 4327) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b) Sub-cycle resolved delay scan with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a,b) The shown errorbars represent a
confidence interval of 90 %.
200
B.2 Sub-cycle dynamics
25 0 25
delay (fs)
0
10
20
30
normalized
mean detector
signal (arb. u.)
(a)
0 5
delay (fs)
7.5
10.0
12.5
15.0
normalized
mean detector
signal (arb. u.)
(d)
2.5 5.0
delay (fs)
12
14
16
(b)
0.0 2.5
delay (fs)
12
14
16
(c)
0.0 2.5
delay (fs)
10
12
14
(e)
0 5
delay (fs)
10
15
(f)
Figure B.10: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay
scans. (a) Few femtosecond resolved delay scan (run id 4473 to 4493) with its normalized mean
diffraction intensity of the brightest 1 % of the images in solid blue and the corresponding helium
ion yield in light blue. (b,c,e,f) Sub-cycle resolved delay scans with colored dots indicating their
significance in respect to the first null hypothesis (see section 5.3). (a-c, e, f) The shown errorbars
represent a confidence interval of 90%. (d) All sub-cycle resolved scans in direct comparison.
201
B. Additional experimental observations
0 20
delay (fs)
0
10
20
normalized
mean detector
signal (arb. u.)
(a)
22 24
delay (fs)
9
10
11
12
normalized
mean detector
signal (arb. u.)
(d)
22 24
delay (fs)
10
12
14
(b)
22 24
delay (fs)
10
12
14
(c)
22 24
delay (fs)
10
12
(e)
Figure B.11: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay
scans. (a) Few femtosecond resolved delay scan (run id 4697 to 4711) with its normalized mean
diffraction intensity of the brightest 1 % of the images in solid blue and the corresponding helium
ion yield in light blue. (b,c,e) Sub-cycle resolved delay scans with colored dots indicating their
significance in respect to the first null hypothesis (see section 5.3). (a-c, e) The shown errorbars
represent a confidence interval of 90%. (d) All sub-cycle resolved scans in direct comparison.
0 20
delay (fs)
0
5
10
15
20
normalized
mean detector
signal (arb. u.)
(a)
7.5 10.0 12.5
delay (fs)
8
10
12
(b)
Figure B.12: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay scan.
(a) Few femtosecond resolved delay scan (run id 4787 to 4801) with its normalized mean diffraction
intensity of the brightest 1 % of the images in solid blue and the corresponding helium ion yield
in light blue. (b) Sub-cycle resolved delay scan with colored dots indicating their significance
in respect to the first null hypothesis (see section 5.3). (a,b) The shown errorbars represent a
confidence interval of 90 %.
202
B.2 Sub-cycle dynamics
0 20
delay (fs)
0
5
10
normalized
mean detector
signal (arb. u.)
(a)
10.0 12.5 15.0
delay (fs)
4
5
6
7
normalized
mean detector
signal (arb. u.)
(d)
10.0 12.5 15.0
delay (fs)
4
6
8
(b)
10.0 12.5 15.0
delay (fs)
4
5
6
7
(c)
Figure B.13: Few femtosecond resolved delay scan with subsequent sub-cycle resolved delay
scans. (a) Few femtosecond resolved delay scan (run id 4876 to 4890) with its normalized mean
diffraction intensity of the brightest 1 % of the images in solid blue and the corresponding helium
ion yield in light blue. (b,c) Sub-cycle resolved delay scans with colored dots indicating their
significance in respect to the first null hypothesis (see section 5.3). (a-c) The shown errorbars
represent a confidence interval of 90%. (d) All sub-cycle resolved scans in direct comparison.
203
C
Non-linear microscopic & macroscopic
response
C.1 Derivation of the non-linear macroscopic response
Here the derivation of the nonlinear macroscopic response (polarization and dielectric
susceptibility) is shown using atomic units. Non-linearities higher than third order are
identified and neglected. The derivation follows the approach presented in [202]. It is carried
out for the one dimensional case and photon energy dependent dynamic polarizabilities.
Note: ∗denotes a convolution
In case of a non-linear relationship between dipole moment
p
(
ω
)and the local electric field
E(ω), the dipole moment is given by the linear term and non-linear terms contained in a(E):
p(ω) = α1(ω)E(ω) + a(E).
Thereby the dynamical polarizability is indicated by
αorder
(
ω
). The subscript identifies the
associated order on non-linearity. The non-linear terms are given as:
a(E) = α2(ω)(E(ω)∗E(ω)) + α3(ω)(E(ω)∗E(ω)∗E(ω)) + higher orderterms .(C.1)
Terms with a non-linearity above third order are dropped. The macroscopic polarization
P
gives the dipole moment density through the number density N:
P(ω) = N·p(ω) = N·α1(ω)·E(ω) + N·a(E).(C.2)
Furthermore it relates to the external electric field via the dielectric susceptibilities
χorder
(
ω
)
(here given including non-linear terms, where the order of non-linearity is again indicated by
the subscript):
P(ω) = χ1(ω)E(ω) + χ2(ω)(E(ω)∗E(ω)) + χ3(ω)(E(ω)∗E(ω)∗E(ω)) .(C.3)
205
C. Non-linear microscopic & macroscopic response
Again, terms with a non-linearity above third order are dropped. The local field acting upon
the single atom
E
contains contributions from the external field
E
and induced Polarization
P
(see e.g. [88] for a derivation):
E(ω) = E(ω) + 4π
3P(ω).
This expression can be directly inserted into eq. C.2:
P(ω) = N·p(ω) = N·α1(ω)·[︃E(ω) + 4π
3P(ω)]︃+N·a(E(ω) + 4π
3P(ω)) .
The resulting equation can now be transformed into an expression of the same format as eq.
C.3:
(1 −4π
3N·α1(ω)) ·P(ω) = N·α1(ω)·E(ω) + N·a(E(ω) + 4π
3P(ω))
P(ω) = N·α1(ω)
1−4π
3N·α1(ω)
⏞ ⏟⏟ ⏞
χ1
·E(ω) + N
1−4π
3N·α1(ω)·a(E(ω) + 4π
3P(ω)) .(C.4)
By equating coefficents of eq. C.4 and C.3, the χ1is determined to be given by:
χ1=N·α1(ω)
1−4π
3N·α1(ω)
This is the Clausius-Mossotti relation. It can be also written as follows:
1
1−4π
3N·α1(ω)=(︃4π
3χ1(ω)+1)︃.
This equation can now be resubstituted into the non-linear component in eq. C.4:
P(ω) = χ1(ω)E(ω) + (︃4π
3χ1(ω)+1)︃·N·a(︃E(ω) + 4π
3P(ω))︃.(C.5)
For convenience ϑis introduced:
ϑ(ω) = (︃4π
3χ1(ω)+1)︃
Equation C.5 becomes:
P(ω) = χ1(ω)E(ω) + ϑ(ω)·N·a(︃E(ω) + 4π
3P(ω))︃(C.6)
From now on the
ω
dependence will not be explicitly stated anymore to shorten the expressions.
All involved parameters were introduced and their
ω
dependence is known. By recursively
substituting equation C.6 into itself, it becomes:
P=χ1E+ϑ·N·a(︃E+4π
3(χ1E+ϑ·N·a(E)))︃.
206
C.1 Derivation of the non-linear macroscopic response
This can be further simplified to:
P=χ1E+ϑNa (︃ϑE +4π
3ϑNa(E))︃.
Now, the non-linear terms contained in aas given by eq. C.1 are explicitly written:
P=χ1E+ϑN
α2
1
⏟ ⏞⏞ ⏟
[︃(︃ϑE +4π
3ϑNa(E))︃∗(︃ϑE +4π
3ϑNa(E))︃]︃
+α3[︃(︃ϑE +4π
3ϑNa(E))︃∗(︃ϑE +4π
3ϑNa(E))︃∗(︃ϑE +4π
3ϑNa(E))︃]︃
⏞ ⏟⏟ ⏞
2
.(C.7)
As the expression is becoming lengthy the parts, 1
and 2
are now rewritten seperately.
1
can be rewritten in the following way and higher order terms can be dropped:
(ϑE)∗(ϑE)
⏞ ⏟⏟ ⏞
2nd order
+24π
3N(ϑa(E)) ∗(ϑE)
⏞ ⏟⏟ ⏞
3rd order and higher
+16π2
9N2(ϑa(E)) ∗(ϑa(E))
⏞ ⏟⏟ ⏞
4th order and higher
≈(ϑE)∗(ϑE)+24π
3N(ϑa(E)) ∗(ϑE).
2
can be rewritten and higher order terms can be dropped:
(ϑE)∗(ϑE)∗(ϑE)
⏞ ⏟⏟ ⏞
3rd order
+34π
3N(ϑa(E)) ∗(ϑE)∗(ϑE)
⏞ ⏟⏟ ⏞
4th order and higher
+316π2
9N2(ϑa(E)) ∗(ϑa(E)) ∗(ϑE)
⏞ ⏟⏟ ⏞
5th order and higher
+64π3
27 N3(ϑa(E)) ∗(ϑa(E)) ∗(ϑa(E))
⏞ ⏟⏟ ⏞
6th order and higher
≈(ϑE)∗(ϑE)∗(ϑE).
Consequently, when dropping higher order terms, the polarization given in eq. C.7 is given by:
P=χ1E+ϑN {︃α2[︃(ϑE)∗(ϑE)+24π
3N(︃ϑa (︃E+4π
3P)︃)︃∗(ϑE)]︃+α3(ϑE)∗(ϑE)∗(ϑE)}︃.
Again eq. C.6 is substituted and higher order terms are dropped:
P=χ1E+ϑN {︃α2[︃(ϑE)∗(ϑE)+24π
3N(ϑ(α2(ϑE)∗(ϑE))) ∗(ϑE)]︃
+α3(ϑE)∗(ϑE)∗(ϑE)}(C.8)
By equating the coefficients of eq. C.8 and C.3, the non-linear orders of the dielectric
susceptibility can be extracted:
χ2=ϑNα2
(ϑE)∗(ϑE)
E∗E
207
C. Non-linear microscopic & macroscopic response
χ3=ϑNα3
(ϑE)∗(ϑE)∗(ϑE)
E∗E∗E+ 24π
3ϑN2α2
(ϑ(α2(ϑE)∗(ϑE))) ∗(ϑE)
E∗E∗E
Since
χ2
and
χ3
are used to compute an effective dielectric susceptibility
χeff
that depends
on the electric field, no further simplification is necessary for this work.
C.2
Limitations of the non-linear dynamic polarizability-
approach on the microscopic level
C.2.1 Analytical discussion
The in this thesis applied non-linear approach has the ability to include non-linear couplings
into the response at a given frequency. This can be illustrated with a very simple example:
Looking at an (cw) electric field
E
(
t
)composed of 2 colors
ω1
and
ω2
in the time-domain (for
a simple illustration of capabilities and limitations all colors of the field are defined to be in
phase) :
E(t) = A·sin(ω1t) + B·sin(ω2t).
In the linear case, the induced dipole moment can be described only with
α1
multiplied with
the Fourier transformed electric field, i.e., now in the frequency domain:
p(ω) = α1(ω)·E(ω).
As the electric field has frequency components for
ω
=
ω1
or
ω
=
ω2
(otherwise
E
(
ω
) = 0), the
dipole moment can also only show a response at these frequencies, but only will show one, if
α1(ω)is non-zero there.
In the non-linear domain (for now only up to the second order), the dipole moment is given
by the relation:
p(ω) = α1(ω)·E(ω) + α2(ω)·(E(ω)∗E(ω)) .
Since the convolution of the electric fields in the frequency domain corresponds to their
product in the time domain E(t)2contains the frequency response:
E(t)2=1
2[︂−A2cos((2ω1)t)+2ABcos((ω1−ω2)t)−2ABcos((ω1+ω2)t)−B2cos(2ω2t) + B2+A2]︂.
Hence, the Fourier transform
F
(
E
(
t
)
2
)(
ω
)will be non-zero at the frequencies 2
ω1
,2
ω2
,
ω1−ω2
, and
ω1
+
ω2
and through
α2
(
ω
)each physical process that yields these additional
frequencies can be given an individual weight. For the theoretical modeling in this thesis these
weights are retrieved through fitting TDSE simulation results.
It is important to point out that with this, depending on the real (or simulated) atoms
response sum frequency generation (
ω1
+
ω2
) works much better and thus induces a much
stronger dipole moment than difference frequency generation (
ω1−ω2
), i.e.
α2
(
ω1
+
ω2
)
>>
α2
(
ω1−ω2
). This different weighting is possible, because the processes yield a response at
different frequencies.
Now consider a different electric field E
˜(t)composed of three colors ω1,ω2and ω3:
208
C.2 Limitations of the non-linear dynamic polarizability-approach on the microscopic level
E
˜(t) = A·sin(ω1t) + B·sin(ω2t) + C·sin(ω3t).
Further also the third order non-linearity is included:
p(ω) = α1(ω)·E(ω) + α2(ω)·(E(ω)∗E(ω)) + α3(ω)·(E(ω)∗E(ω)∗E(ω)) .
Analog to the previous description the possible frequency response due to the third order
is contained in E(t)3. In the following only the most relevant terms are written explicitly:
E
˜(t)3=... −3
4A2Bsin((ω2+ 2ω1)t)−3
4A2Csin((ω3−2ω1)t) + ... . (C.9)
This indicates now that among others a frequency response at
ω2
+ 2
ω1
and
ω3−
2
ω1
is
possible, each through 2
ω1
photons and one photon of the second and third color respectively.
Now suppose
ω1
is the NIR frequency,
ω2
is the 11th and
ω3
is the 15th harmonic. Then,
ω2
+ 2
ω1
=
ω3−
2
ω1
, i.e. two non-linear wave mixing processes yield a response at the same
frequency. With the presented modelling, although two processes yield a response at the same
frequency, only one scaling factor (
α3
(
ω2
+ 2
ω1
) =
α3
(
ω3−
2
ω1
)) is introduced. Although
the sum-frequency mixing (
ω2
+ 2
ω1
) might be much stronger than the differential mixing
(
ω3−
2
ω1
), a change of the field amplitude
B
in eq. C.9 has the same effect on to the response
as a change of the field amplitude C.
In conclusion, non-linear processes overlapping in the frequency domain cause a systematic
error in the presented modelling approach. In helium, for energies above the 2p resonance,
light induced states occur in close vicinity to each other and partially overlap [54]. This could
also be one reason why the non-linear fits of the induced dipole moment, e.g. shown in fig.
5.34, deviate toward higher energies.
C.2.2 Numerical investigation
209
C. Non-linear microscopic & macroscopic response
18 20 22
energy (eV)
20
10
0
delay (fs)
(a)
18 20 22
energy (eV)
(b)
18 20 22
energy (eV)
(c)
18 20 22
energy (eV)
(d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.5
0.0
0.5
(e) 18.9 eV
(a) (b) (c) (d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.0
0.5
(f) 22.0 eV
0.5
0.0
0.5
Figure C.1: Benchmarking the qualitative behavior of the optical response mapped by the
fitted non-linear dynamic polarizabilities, where sub-cycle features are allowed for all considered
dynamic polarizability coefficients (
α1, α2, α3
), for an NIR peak intensity of 2
·
10
14 W/cm2
; For the
benchmarking the spectra of artificial pulse trains (envelope FWHM 3
.
1
fs
) from two XUV photon
energies ((a-c) 18
.
9
eV
and 22
.
0
eV
/ (d) 18
.
9
eV
and 15
.
8
eV
) and different relative contributions
were computed and applied. The photon energies were chosen originally to approximately match
the positions of the LIS 2s- and 2s+. However, likely doe to a strong AC stark shift of the 2s
resonance, here, this might be only the case for a small range of delays. The resulting absorption
β
is depicted in (a-d), the respective spectral contributions are indicated by offset and arb. scaled
spectra shown by in black on top of the two dimensional absorption. Lineouts at the photon
energies 18
.
9
eV
and 22
.
0
eV
(dotted lines in (a) - (d)) are shown in (e) and (f), respectively. It is
expected that the lower the contribution in the higher photon energy, the lower the oscillation in
the lower photon energy and vice versa.
210
C.2 Limitations of the non-linear dynamic polarizability-approach on the microscopic level
18 20 22
energy (eV)
20
10
0
delay (fs)
(a)
18 20 22
energy (eV)
(b)
18 20 22
energy (eV)
(c)
18 20 22
energy (eV)
(d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.5
0.0
0.5
(e) 18.9 eV
(a) (b) (c) (d)
10.0 7.5 5.0 2.5 0.0
delay (fs)
0.0
0.1
0.2
(f) 22.0 eV
0.5
0.0
0.5
Figure C.2: Benchmarking the qualitative behavior of the optical response mapped by the
fitted non-linear dynamic polarizabilities, where for
alpha1
the low-pass filtered response from
a preliminary fitting step is assumed, for an NIR peak intensity of 2
·
10
14 W/cm2
; For the
benchmarking the spectra of artificial pulse trains (envelope FWHM 3
.
1
fs
) from two XUV photon
energies ((a-c) 18
.
9
eV
and 22
.
0
eV
/ (d) 18
.
9
eV
and 15
.
8
eV
) and different relative contributions
were computed and applied. The photon energies were chosen originally to approximately match
the positions of the LIS 2s- and 2s+. However, likely doe to a strong AC stark shift of the 2s
resonance, here, this might be only the case for a small range of delays. The resulting absorption
β
is depicted in (a-d), the respective spectral contributions are indicated by offset and arb. scaled
spectra shown by in black on top of the two dimensional absorption. Lineouts at the photon
energies 18
.
9
eV
and 22
.
0
eV
(dotted lines in (a) - (d)) are shown in (e) and (f), respectively. It is
expected that the lower the contribution in the higher photon energy, the lower the oscillation in
the lower photon energy and vice versa.
211
C. Non-linear microscopic & macroscopic response
20 30
energy (eV)
20
10
0
10
20
delay (fs)
(a)
TDSE result
20 30
energy (eV)
(b)
non-linear
dyn. pol.
simple fit
20 30
energy (eV)
(c)
non-linear
dyn. pol.
1 const.
0
2
4
6
8
10
dipole moment (arb. u.)
16 18 20 22 24 26 28 30
energy (eV)
20
10
0
10
20
delay (fs)
(d)
relative deviation of (c) from TDSE result in (a)
1.0
0.5
0.0
0.5
1.0
relative error
16 18 20 22 24 26 28 30
energy (eV)
20
10
0
10
20
delay (fs)
(e)
additional relative deviation from TDSE result in (a)
due to (c) compared to (b)
1.0
0.5
0.0
0.5
1.0
relative error
Figure C.3: Error in the dipole moment introduced by the fitted non-linear dynamic polarizability
with suppressed sub-cycle features in the linear term compared to a numerical TDSE result. The
dipole moments were computed for an XUV IAP centered around 17
.
5
eV
, which was not used to
fit the non-linear dynamic polarizability. Thus it serves as a reference. The NIR pulse is centered
around 800
nm
and has peak intensity of 5
·
10
12 W/cm2
. (a) shows the numerical TDSE result,
(b) shows the result computed from directly fitted dynamic polarizabilities (’simple’ fit) and (c)
shows the results computed from fitted dynamic polarizabilities for which the linear coefficient
α1
is set to the low-pass filtered result for α1from (b).
212
C.2 Limitations of the non-linear dynamic polarizability-approach on the microscopic level
20 30
energy (eV)
20
10
0
10
20
delay (fs)
(a)
TDSE result
20 30
energy (eV)
(b)
non-linear
dyn. pol.
simple fit
20 30
energy (eV)
(c)
non-linear
dyn. pol.
1 const.
0
2
4
6
8
10
dipole moment (arb. u.)
16 18 20 22 24 26 28 30
energy (eV)
20
10
0
10
20
delay (fs)
(d)
relative deviation of (c) from TDSE result in (a)
1.0
0.5
0.0
0.5
1.0
relative error
16 18 20 22 24 26 28 30
energy (eV)
20
10
0
10
20
delay (fs)
(e)
additional relative deviation from TDSE result in (a)
due to (c) compared to (b)
1.0
0.5
0.0
0.5
1.0
relative error
Figure C.4: Error in the dipole moment introduced by the fitted non-linear dynamic polarizability
with suppressed sub-cycle features in the linear term compared to a numerical TDSE result. The
dipole moments were computed for an XUV IAP centered around 17
.
5
eV
, which was not used to
fit the non-linear dynamic polarizability. Thus it serves as a reference. In fact, it is the same pulse
used for reference in fig. 5.33. The NIR pulse is centered around 800
nm
and has peak intensity
of 2
·
10
14 W/cm2
. (a) shows the numerical TDSE result, (b) shows the result computed from
directly fitted dynamic polarizabilities (’simple’ fit) and (c) shows the results computed from fitted
dynamic polarizabilities for which the linear coefficient
α1
is set to the low-pass filtered result for
α1from (b).
213
C. Non-linear microscopic & macroscopic response
C.3 Simulated response to realistic spectra
20 30
energy (eV)
5
0
5
delay (fs)
(a) , 790 nm
0.2
0.0
0.2
20 30
energy (eV)
5
0
5
delay (fs)
(d) , 790 nm
0.2
0.0
0.2
20 30
energy (eV)
(b) , 800 nm
0.2
0.0
0.2
20 30
energy (eV)
(e) , 800 nm
0.2
0.0
0.2
20 30
energy (eV)
(c) , 810 nm
0.2
0.0
0.2
20 30
energy (eV)
(f) , 810 nm
0.2
0.0
0.2
Figure C.5: Spectrum and delay-dependent refractive index at central harmonic photon energies
for different fundamental wavelengths, specifically 790
nm
,800
nm
, and 810
nm
. The NIR has peak
intensity of 5
·
10
12 W/cm2
and a central wavelength of 800
nm
. The effective refractive index
under non-linear assumptions was computed for an XUV spectrum according to table 5.4 including
harmonics up to harmonic order (HO) 19. For line-outs at the central harmonic energies (dotted
vertical lines, see fig. C.6. Spectral regions minor or no contribution to the total XUV signal are
covered by white boxes for a better visibility of the relevant spectral contributions.
214
C.3 Simulated response to realistic spectra
5 0 5
0.2
0.1
0.0
0.1
0.2
(a) HO 11 (790 nm)
5 0 5
0.3
0.2
0.1
0.0
0.1
(d) HO 13 (790 nm)
5 0 5
0.1
0.0
0.1
0.2
(g) HO 15 (790 nm)
5 0 5
delay (fs)
0.2
0.1
0.0
0.1
0.2
(j) HO 17 (790 nm)
5 0 5
(b) HO 11 (800 nm)
5 0 5
(e) HO 13 (800 nm)
5 0 5
(h) HO 15 (800 nm)
5 0 5
delay (fs)
(k) HO 17 (800 nm)
5 0 5
(c) HO 11 (810 nm)
5 0 5
(f) HO 13 (810 nm)
5 0 5
(i) HO 15 (810 nm)
5 0 5
delay (fs)
(l) HO 17 (810 nm)
Figure C.6: Line-outs (for figure C.5) of the refractive index at central harmonic photon energies
for different fundamental wavelengths, specifically 790
nm
,800
nm
, and 810
nm
. The NIR has peak
intensity of 5
·
10
12 W/cm2
and a central wavelength of 800
nm
. The effective refractive index
under non-linear assumptions was computed for an XUV spectrum according to table 5.4 including
harmonics up to harmonic order (HO) 19.
215
C. Non-linear microscopic & macroscopic response
5 0 5
0.2
0.1
0.0
0.1
0.2
(a) HO 11 (790 nm)
5 0 5
0.2
0.1
0.0
0.1
0.2
(d) HO 13 (790 nm)
5 0 5
0.1
0.0
0.1
0.2
(g) HO 15 (790 nm)
5 0 5
delay (fs)
0.2
0.1
0.0
0.1
0.2
(j) HO 17 (790 nm)
5 0 5
(b) HO 11 (800 nm)
5 0 5
(e) HO 13 (800 nm)
5 0 5
(h) HO 15 (800 nm)
5 0 5
delay (fs)
(k) HO 17 (800 nm)
5 0 5
(c) HO 11 (810 nm)
5 0 5
(f) HO 13 (810 nm)
5 0 5
(i) HO 15 (810 nm)
5 0 5
delay (fs)
(l) HO 17 (810 nm)
Figure C.7: Line-outs (for figure 5.37) of the refractive index at central harmonic photon energies
for different fundamental wavelengths, specifically 790
nm
,800
nm
, and 810
nm
. The NIR has peak
intensity of 2
·
10
14 W/cm2
and a central wavelength of 800
nm
. The effective refractive index
under non-linear assumptions was computed for an XUV spectrum according to table 5.4 including
harmonics up to harmonic order (HO) 19.
216
C.3 Simulated response to realistic spectra
20 0 20
0.2
0.0
0.2
/
(a) HO 11 (790 nm)
(non-linear, low-pass filtered)
(linear)
(non-linear, low-pass filtered)
(linear)
20 0 20
0.2
0.0
0.2
/
(d) HO 13 (790 nm)
20 0 20
0.2
0.0
0.2
/
(g) HO 15 (790 nm)
20 0 20
delay (fs)
0.2
0.0
0.2
/
(j) HO 17 (790 nm)
20 0 20
(b) HO 11 (800 nm)
20 0 20
(e) HO 13 (800 nm)
20 0 20
(h) HO 15 (800 nm)
20 0 20
delay (fs)
(k) HO 17 (800 nm)
20 0 20
(c) HO 11 (810 nm)
20 0 20
(f) HO 13 (810 nm)
20 0 20
(i) HO 15 (810 nm)
20 0 20
delay (fs)
(l) HO 17 (810 nm)
Figure C.8: Line-outs of the refractive index at central harmonic photon energies for different
fundamental wavelengths comparing the low-pass filtered delay dependent response retrieved under
linear and non-linear assumptions. The NIR has peak intensity of 2
·
10
14 W/cm2
and a central
wavelength of 800
nm
. The effective refractive index under non-linear assumptions was computed
for an XUV spectrum according to table 5.4 including harmonics up to harmonic order (HO) 19.
217
C. Non-linear microscopic & macroscopic response
20 0 20
0.5
1.0
normalized
diffraction
intensity
768 nm
20 0 20
0.5
1.0
770 nm
20 0 20
0.5
1.0
772 nm
20 0 20
0.5
1.0
774 nm
20 0 20
0.5
1.0
776 nm
20 0 20
0.5
1.0
normalized
diffraction
intensity
778 nm
20 0 20
0.5
1.0
780 nm
20 0 20
0.5
1.0
782 nm
20 0 20
0.5
1.0
784 nm
20 0 20
0.5
1.0
786 nm
20 0 20
0.5
1.0
normalized
diffraction
intensity
788 nm
20 0 20
0.5
1.0
790 nm
20 0 20
0.5
1.0
792 nm
20 0 20
0.5
1.0
794 nm
20 0 20
0.5
1.0
796 nm
20 0 20
0.5
1.0
normalized
diffraction
intensity
798 nm
20 0 20
0.5
1.0
800 nm
20 0 20
0.5
1.0
802 nm
20 0 20
0.5
1.0
804 nm
20 0 20
0.5
1.0
806 nm
20 0 20
0.5
1.0
normalized
diffraction
intensity
808 nm
20 0 20
0.5
1.0
810 nm
20 0 20
0.5
1.0
812 nm
20 0 20
0.5
1.0
814 nm
20 0 20
0.5
1.0
816 nm
20 0 20
0.5
1.0
normalized
diffraction
intensity
818 nm
20 0 20
0.5
1.0
820 nm
20 0 20
0.5
1.0
822 nm
20 0 20
0.5
1.0
824 nm
20 0 20
0.5
1.0
826 nm
20 0 20
delay (fs)
0.5
1.0
normalized
diffraction
intensity
828 nm
20 0 20
delay (fs)
0.5
1.0
830 nm
20 0 20
delay (fs)
0.5
1.0
832 nm
20 0 20
delay (fs)
0.5
1.0
834 nm
20 0 20
delay (fs)
0.5
1.0
836 nm
Figure C.9: Simulation of the few-femtosecond response based on sub-cycle resolved modeling
at an NIR peak intensity of 2
·
10
14 W/cm2
for a realistic XUV spectrum for different HHG
fundamental wavelengths; The variation of the XUV fundamental wavelength in 2
nm
steps from
768nm to 836 nm illustrates what for shapes the response may be possible. The NIR pulse has a
constant central wavelength of 800
nm
. After computing the sub-cycle resolved scattering response
with Mie-theory, the obtained signal was low-pass filtered along the delay axis. The diffraction
intensity was normalized to its value at large positive delays.
218
