Attosecond Light Pulses and Attosecond Electron Dynamics Probed using Angle-Resolved
Photoelectron Spectroscopy

Cong Chen
B.S., Nanjing University, 2010
M.S., University of Colorado Boulder, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirement for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
Attosecond Light Pulses and Attosecond Electron Dynamics Probed using Angle-Resolved
Photoelectron Spectroscopy

written by Cong Chen
has been approved for the Department of Physics

Prof. Margaret M. Murnane

Prof. Henry C. Kapteyn

Date

The final copy of this thesis has been examined by the signatories, and wefind that both the content and the form meet acceptable presentation standardsof scholarly work in the above mentioned discipline.

iii

Chen, Cong (Ph.D., Physics)
Attosecond Light Pulses and Attosecond Electron Dynamics Probed using Angle-Resolved
Photoelectron Spectroscopy
Thesis directed by Prof. Margaret M. Murnane

Recent advances in the generation and control of attosecond light pulses have opened up
new opportunities for the real-time observation of sub-femtosecond (1 fs = 10-15 s) electron
dynamics in gases and solids. Combining attosecond light pulses with angle-resolved
photoelectron spectroscopy (atto-ARPES) provides a powerful new technique to study the
influence of material band structure on attosecond electron dynamics in materials. Electron
dynamics that are only now accessible include the lifetime of far-above-bandgap excited electronic
states, as well as fundamental electron interactions such as scattering and screening. In addition,
the same atto-ARPES technique can also be used to measure the temporal structure of complex
coherent light fields. In this thesis, I present four experiments utilizing atto-ARPES to provide new
insights into the generation and characterization of attosecond light pulses, as well as theattosecond electron dynamics in transition metals. First, I describe a new method to extend
attosecond metrology techniques to the reconstruction of circularly polarized attosecond lightpulses for the first time. Second, I show that by driving high harmonics with a two-color linearlypolarized laser field, quasi-isolated attosecond pulses are generated because the phase matchingwindow is confined. Third, I present the first measurement of lifetimes for photoelectrons that areborn into free-electron-like states compared with those that are excited into unoccupied excitedstates in the band structure of a material (Ni(111)). The finite excited-state lifetime causes a ≈200
as delay in the emission of photoelectrons. Finally, I describe direct time-domain studies of

iv

electron-electron interactions in transition metals with both simple and complex Fermi surfaces.
In particular, I show the influence of electron-electron scattering and screening on the lifetime ofphotoelectrons.

Dedication

For my family

Acknowledgements

First of all, I would like to thank my advisors, Margaret and Henry for their mentoring,support and encouragement. They have provided me with incredible resources and opportunities
in my Ph. D, as well as the connections with some of the best experts in attosecond and surface
science community: Martin Aeschlimann (TU Kaiserslautern), Andreas Becker (JILA), Agnieszka
Jaron-Becker (JILA), Mark Keller (NIST), Manos Mavrikakis (UW Madison) and Dan Dessau(CU Boulder). I gained lots of invaluable perspectives on science through the collaborations withthem. I am also glad to work with the photoemission team in Kapteyn-Murnane group. Tory, thankyou for teaching me so that I am able to go on and dive deeply into photoemission. Your optimism
kept me motivational and patient during the difficult time of building a new setup. Piotr, thank you
for providing so many ideas and new perspectives to the experiment. The discussions with you on
physics upgrades my knowledge and help me grow professionally as a scientist. Zhensheng, thankyou for working together with me during so many late nights in the lab. Those moments when we
felt closer to the explanations to our data are truly unforgettable. Wenjin, thank you for theconfidence in turning over the ongoing experiments to you. I am sure there will be more scientificfindings to come.
Many thanks to our collaborators in Germany, especially Sebastian Emmerich, Markus
Rollinger, Stefan Mathias, Steffen Eich and Martin Piecuch from TU Kaiserslautern, and Moritz

vii

Ploetzing from Jülich. Thank you all for the knowledge and support I gained during my visit toGermany.
I gratefully acknowledge the sample providers and theoretical support to our experiments.
Many thanks to Mark Keller and David Miller who fabricated high-quality nickel samples for us
and helped us understand the physics of graphene/nickel system and the magnetic properties ofnickel. To Justin Shaw who provides us the permalloy sample. To Carlos for the simulations onhigh harmonic generation that constitutes tremendous theoretical support to our papers. Thanks to
Guowen Peng and Tibor Szilvasi for the DFT calculations that help to unravel the magic ofmaterials’ bandstructure.
Thanks to the JILA machine shop their technical support, especially Hans Green, David
Alchenberger, Blaine Horner, Kim Hagen, Todd Asnicar, and Kels Detra.
To JILA computing group, especially J. R. Raith who made sure the computer well-behaved
during the data-taking.
To Ofer Kfir and Oren Cohen from Technion University for the excitement in circularly
polarized harmonics.
To Yue Cao for his instructions about ARPES theory and equipment.
To Wei, for his advice on research and life and keeping me from getting lost in the difficult
days.
Many thanks to Dan Adams and his family for the wonderful Thanksgiving memories.
To group members, past and present, especially Chengyuan Ding, Bosheng Zhang, Tingting
Fan, Ming-Chang Chen, Ronny Knut, Chan La-o-vorakiat, Qing Li, Tenio Popmintchev, andDimitar Popmintchev.

viii

Finally, and most of all, I thank my parents, my brother, my grandparents and the rest of myfamily for their great love and never-ending support.

Contents

Page
Chapter
1 Introduction to Ultrafast .......................................................................................................

1

1.1 Why Ultrafast ...............................................................................................................

1

1.2 Probing into the “Invisible” .........................................................................................

4

1.3 Pump-Probe .................................................................................................................

5

1.4 Ultrafast Lasers ............................................................................................................

6

1.5 Femtosecond Solid-State Lasers ..................................................................................

7

1.6 Attosecond ...................................................................................................................

9

1.7 Beyond Attosecond ......................................................................................................

10

1.8 Conclusion ...................................................................................................................

11

2 Attosecond Science and Thesis Overview ...........................................................................

12

2.1 Attosecond Dynamics Probed by Femtosecond Pulses ...............................................

14

2.1.1 High Harmonic Spectroscopy ...........................................................................

15

2.1.2 Electron Recollision-Based Probe ....................................................................

16

2.1.3 Attoclock ...........................................................................................................

17x

2.2 How to Make Attosecond Pulses .................................................................................

18

2.3 Attosecond Pump-Probe ..............................................................................................

22

2.4 Attosecond Pulse Characterization ..............................................................................

24

2.4.1 Ex-situ Characterization of Attosecond Pulses .................................................

25

2.4.2 In-situ Characterization of Attosecond Pulses ..................................................

31

2.4.3 Applications of Sub-cycle IR Electric Field Characterization ...........................

32

2.5 Attosecond Time-Resolved Spectroscopy in Atoms ...................................................

33

2.5.1 Photoemission Time Delay in Atoms ...............................................................

34

2.6 Attosecond Time-Resolved Spectroscopy in Molecules .............................................

36

2.6.1 Born-Oppenheimer and Single Electron Approximation .................................

36

2.6.2 Photoemission Time Delay in Molecules .........................................................

38

2.7 Attosecond Time-Resolved Spectroscopy in Solids ....................................................

39

2.7.1 Photoemission Time Delay in Solids ................................................................

40

2.7.2 Photoemission Delay Reveals Intrinsic Material Properties .............................

44

2.7.3 Attosecond Control of Charge Dynamics in Solids ..........................................

46

2.8 Organization of the Thesis ...........................................................................................

49

2.9 Conclusion ...................................................................................................................

50

3 General Background .............................................................................................................

51

3.1 Photoemission Spectroscopy .......................................................................................

51

3.1.1 Photoemission: from One-Step to Three-Step Model .......................................

52

3.1.2 Direct Imaging of Electronic Bands .................................................................

55

3.1.3 The Influence of Brillouin Zone in Photoemission ...........................................

59

xi

3.1.4 The Matrix Element Effect ...............................................................................

62

3.1.5 EDC and MDC ..................................................................................................

64

3.2 Theory of High Harmonic Generation .........................................................................

66

3.2.1 Three-Step Model .............................................................................................

66

3.2.2 Phase Matching and Absorption .......................................................................

68

3.3 High Harmonics for Photoemission .............................................................................

71

3.3.1 Tailoring HHG for Photoemission Spectroscopy .............................................

72

3.3.2 1ω, 2ω or 1ω+2ω ..............................................................................................

72

4 Experimental Apparatus .......................................................................................................

76

4.1 Laser System ................................................................................................................

76

4.2 Beamline ......................................................................................................................

78

4.3 EUV Focusing Optics and Monochromator ................................................................

80

4.4 The Surface Science End Station .................................................................................

81

5 Tomographic Reconstruction of Attosecond Pulse Trains with Circular Polarization ........

85

5.1 Abstract ........................................................................................................................

86

5.2 Introduction ..................................................................................................................

86

5.2.1 Circularly Polarized High Harmonic Generation .............................................

86

5.2.2 Direct Temporal Characterization of Circularly Polarized Harmonics ............

87

5.3 Experimental Setup ......................................................................................................

88

5.4 Spectrum of Circularly Polarized Harmonics ..............................................................

91

5.5 Cu(111) Sample Preparation and Band Structure ........................................................

91

xii

5.6 Static ARPES Spectra of Cu(111) Excited by Circularly Polarized Harmonics .........

93

5.7 Measuring the Spectral Phase of Circularly Polarized Harmonics ..............................

93

5.7.1 Overview ...........................................................................................................

93

5.7.2 PROOF Algorithm ............................................................................................

95

5.7.3 Electron Interferogram ...................................................................................... 101
5.7.4 Extraction of the Interferogram Phase .............................................................. 103
5.7.5 Retrieve Harmonic Phase from Interferogram Phase ....................................... 105
5.7.6 Rotation of Circularly Polarized Harmonic Field ............................................. 107
5.8 Reconstruction of Attosecond Pulse Train with Circular Polarization ........................ 110
5.9 Numerical Simulation of Circular Harmonic Generation ............................................ 112
5.10 Ellipticity of Circularly-Polarized High Harmonic Field ............................................ 115
5.11 Conclusion ................................................................................................................... 117

6 Influence of Microscopic and Macroscopic Effects on Attosecond Pulse Generation Using
Two-Color Laser Fields ....................................................................................................... 119
6.1 Abstract ........................................................................................................................ 119
6.2 Introduction .................................................................................................................. 120
6.2.1 Generation of Isolated Attosecond Pulses ........................................................ 120
6.2.2 HHG under Two-Color Driving Fields ............................................................. 121
6.3 Experimental Setup ...................................................................................................... 123
6.4 Numerical Simulation .................................................................................................. 125
6.5 Quasi-Isolated Attosecond Pulses ................................................................................ 126
6.6 Controlling the Sub-Cycle Temporal Structure of Attosecond Pulses ........................ 131

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6.7 Conclusion ................................................................................................................... 134

7 Direct Time-Domain Measurement of Attosecond Final-State Lifetimes in Photoemission
from Solids ........................................................................................................................... 135
7.1 Abstract ........................................................................................................................ 136
7.2 Introduction .................................................................................................................. 136
7.2.1 Combine Angle-Resolved Photoemission with High Harmonic Light ............ 136
7.2.2 Direct Time-Domain Measurement of Photoelectron Lifetime ........................ 137
7.3 Ni(111) Surface Preparation ........................................................................................ 139
7.4 Experimental Setup for Attosecond Photoelectron Lifetime Measurement ................ 140
7.5 Band Structure of Ni(111) along Г-L Direction .......................................................... 142
7.6 Spectral-Domain Measurement of Final-State Lifetime in Ni(111) ............................ 146
7.7 Direct Time-Domain Measurement of Final-State Lifetime in Ni(111) ..................... 147
7.7.1 Time-Resolved Data Collection ........................................................................ 148
7.7.2 Photoemission Time Delay between 3 and 3 Bands ................................. 149
7.7.3 Photoemission Time Delay between 1 and 3 Bands .................................. 153
7.7.4 Lifetime of the Unoccupied Bulk Final Band 1B ............................................ 155
7.8 Angle-Dependent Photoemission Time Delay Caused by Final-Band Dispersion ..... 156
7.9 Classical Considerations on Photoemission Time Delay and Screening of IR Field .. 161
7.10 Semiclassical Model for RABBITT Measurements .................................................... 166
7.11 Inelastic Mean Free Path and Photoelectron Lifetime at High Energies ..................... 170
7.12 Conclusion and Outlook .............................................................................................. 172

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8 Attosecond Electron-Electron Scattering and Screening in Transition Metals ..................... 173
8.1 Abstract ........................................................................................................................ 174
8.2 Introduction .................................................................................................................. 174
8.2.1 Probing Excited-State Electron Dynamics in Materials ................................... 174
8.2.2 Distinguish Attosecond Electron-Electron Scattering and Screening .............. 175
8.3 Sample Surface Preparation ......................................................................................... 178
8.4 Static ARPES Spectra of Cu(111) and Cu(001) .......................................................... 178
8.5 Experimental Setup ...................................................................................................... 181
8.6 Static HHG Photoelectron Spectra Analysis of Cu(111) ............................................. 182
8.7 Final-State Resonance in Cu(111) ............................................................................... 187
8.8 Momentum Dispersion of Photoelectron Lifetime ...................................................... 188
8.9 Lifetime of Photoelectrons from 1 Band .................................................................. 190
8.10 Photoelectron Lifetime Difference between Cu and Ni .............................................. 192
8.11 Dynamic Screening and Inelastic Electron-Electron Scattering .................................. 193
8.11.1 Absence of Electron Screening ......................................................................... 194
8.11.2 Influence of Inelastic Electron-Electron Screening .......................................... 195
8.11.3 Spin-Dependent Electron-Electron Scattering .................................................. 203
8.12 Atto-ARPES as a Probe of High-Energy Final State and Matrix Element Effect inPhotoemission ............................................................................................................... 203
8.13 Conclusion and Outlook .............................................................................................. 206

9 Conclusion and Outlook ....................................................................................................... 207
9.1 Conclusion ................................................................................................................... 207

xv

9.2 Ongoing Experiment .................................................................................................... 209
9.2.1 Time-Domain Study of Fano Resonance in Solid-State Materials ................... 210
9.2.2 Doped Metal and Surface-Adsorbate Systems ................................................. 211
9.3 Future Opportunities .................................................................................................... 213

Bibliography ............................................................................................................................ 215

xvi

FiguresPage
Figure1.1 Events and phenomena in the universe that take place at different time scales ..........

2

1.2 Dynamical processes at time scales from 0.1 fs to 1000 fs after laser excitation ........

2

1.3 Kerr-lens-induced changes in the transmission of laser beam through an aperture ....

9

2.1 Different scenarios of electron ionization ....................................................................

14

2.2 Different techniques for generating isolated attosecond pulses ..................................

19

2.3 Different ex-situ methods of characterizing attosecond pulses ...................................

27

2.4 Photoemission time delay probed using isolated attosecond pulses ............................

41

2.5 Photoelectron lifetime probed using attosecond pulse trains ......................................

45

3.1 Illustration of one-step model and three-step model of photoemission .......................

52

3.2 “Universal Curve” .......................................................................................................

55

3.3 Illustration of the inner potential in band mapping .....................................................

59

3.4 Surface Brillouin zone for crystals with face centered cubic (fcc) structure ...............

60

3.5 Illustration of matrix element effect in photoemission ................................................

62

3.6 Photoemission spectrum of Na-intercalated graphene on Ni(111) substrate ...............

65

3.7 The on-axis intensity of HHG radiation as a function of medium length ...................

69

xvii

4.1 Schematic of Ti:sapphire oscillator and amplifier systems .........................................

76

4.2 Typical spectrum of Ti:sapphire oscillator and spectrum after chirped pulse
amplification ................................................................................................................

77

4.3 Simplified schematics of the beamline layout for tabletop EUV based photoelectron
spectroscopy system ....................................................................................................

79

4.4 EUV transmission of metal foils and EUV reflectivity of rejector mirrors .................

80

4.5 XYZ manipulator and sample holder ..........................................................................

82

5.1 Illustration of the 3D reconstruction of circularly polarized harmonics ......................

85

5.2 Characterization of circularly-polarized HHG fields ..................................................

89

5.3 Circularly-polarized HHG spectrum recorded by an EUV spectrometer ....................

90

5.4 Photoelectron spectrum of Cu(111) .............................................................................

92

5.5 Photoelectron yields depend on the HHG polarization ...............................................

95

5.6 Schematics of the quantum paths that give rise to photoelectron interferences ..........

99

5.7 Photoelectron interferogram as a function of time delay between the circular HHG andlinear IR field ............................................................................................................... 102
5.8 Reproducing photoelectron interferogram phase using retrieved HHG phase ............ 104
5.9 Rotation of the clover-leap-shaped driving field ......................................................... 108
5.10 Simulation of the EUV field rotation ........................................................................... 109
5.11 Experimental reconstruction of the circular HHG pulse train ..................................... 111
5.12 Field orientation of circular HHG from Ar .................................................................. 112
5.13 Normalized 3mth order harmonic intensity compared with numerical simulation ..... 113
5.14 Experimental reconstruction of circularly polarized harmonics in comparison withnumerical simulation ................................................................................................... 114

xviii5.15 Ellipticity of circular HHG from Ar ............................................................................ 116
6.1 Characterizing the high-order harmonics generated using interferometric laser-assisted
photoemission method ................................................................................................. 124
6.2 Measuring harmonic phases ........................................................................................ 127
6.3 Quasi-isolated attosecond pulses generated through macroscopic phase matching .... 129
6.4 Coherent control of attosecond pulses by two-color driving fields ............................. 133
7.1 Illustration of photoelectron lifetime measurement ..................................................... 135
7.2 Experimental setup of photoelectron lifetime measurement ....................................... 140
7.3 Photoemission time delay on and off resonance in the band structure ........................ 142
7.4 Static ARPES spectra of Ni(111) excited by multiple orders of HHG ........................ 143
7.5 Static spectral analysis of Ni(111) excited by HHG .................................................... 144
7.6 Band Mapping of Ni(111) ........................................................................................... 145
7.7 Photoemission time delay measurements obtained using different noble gas targets .. 150
7.8 Direct time-domain measurement of final-state lifetime ............................................. 152
7.9 Extraction of angle-dependent photoemission time delays ......................................... 156
7.10 Angle-dependent photoemission time delays .............................................................. 157
7.11 Classical model for photoemission time delay ............................................................ 162
7.12 Semiclassical simulation results .................................................................................. 168
7.13 Results of the Semiclassical model .............................................................................. 169
8.1 Illustration of the photoemission process of Cu(111) and Ni(111) surfaces ............... 173
8.2 Band structure of Cu .................................................................................................... 179
8.3 Photoemission spectrum of Cu(111) and Cu(001) ...................................................... 179
8.4 Experimental setup ...................................................................................................... 182

xix

8.5 HHG-excited photoelectron spectra of Cu(111) .......................................................... 183
8.6 Static spectral analysis of photoemission from Cu(111) surface ................................. 184
8.7 Band mapping of Cu(111) ........................................................................................... 185
8.8 Comparison of photoelectron lifetimes in Cu(111) and Ni(111) ................................. 186
8.9 Spectral intensity of 3 and 1 bands ....................................................................... 187
8.10 Extraction of angle-dependent photoemission time delay ........................................... 189
8.11 Angle-dependent photoemission time delay ................................................................ 189
8.12 Photoemission time delay of 3 and 1 bands relative to the 3 band ................... 191
8.13 Origin of different photoelectron lifetimes for photoemission into free-electron final
states for Cu and Ni ...................................................................................................... 193
8.14 Spin-dependent model of electron-electron scattering ................................................ 200
8.15 Extracting final-state effects in photoemission ............................................................ 205

xx

TablesPage
Table
3.1 Dipole-allowed initial state symmetries for transitions in normal emission geometry
from low-index cubic metals .......................................................................................

64

5.1 Phases of circular harmonics ....................................................................................... 106

Chapter 1
Introduction to Ultrafast

1.1

Why Ultrafast
Time is one of the parameters that divides the natural sciences into different fields. To

human beings, the most perceivable unit of time is the second, with one second close to the periodof the heartbeat that repeats throughout a human’s lifetime. However, our universe is filled with avariety of marvelous events and phenomena that take place at different time scales (Figure 1.1).
On the one hand, there are processes that are achingly long compared to the lifetime of a humanbeing. For example, the formation of a galaxy takes billions of years while the evolution of
biological species on earth happens in millions of years. On the other hand, modern research inphysics and chemistry is able to show us the dynamics of phenomena in the other temporal extreme
where we can visualize a world in which dynamical changes occur on an ultrafast timescale, i.e.,in the femtosecond (fs, 10-15 s) to attosecond (as, 10-18 s) range (Figure 1.2). These dynamics
usually take place in a microcosm and therefore are inherently connected by the laws of quantum
mechanics. For example, from quantum mechanics, a particle put in a superposition of two states
has a characteristic oscillation period of𝑇𝑜𝑠𝑐 = 2𝜋(ħ/∆W),(1.1)

where ∆𝑊 is the energy separation between two states. This energy separation is determined by
the potential confining the particle motion and the particle’s mass. Because the spacing between

2vibrational energy levels in molecules is in the milli-eV range, molecular vibrations occur on thetime scale of tens to hundreds of femtoseconds. Compared to these vibrational motions, the motionof an individual electron in metals, molecules, and the inner shells of atoms occurs on a faster timescale, usually in range of tens of femtoseconds to less than an attosecond (see Figure 1.1 and 1.2).

Figure 1.1: Events and phenomena in the universe that take place at different time scales.

Figure 1.2: Dynamical processes at time scales from 0.1 fs to 1000 fs after laser excitation.
Ultrafast time scales are determined through direct time-domain measurements or linewidth
analysis. After external excitation, the dynamical screening occurs on the fastest time scale. The
screening is followed by the dephasing of coherent excitations. The original nonthermal
distribution of excited electrons relaxes to a Fermi-Dirac distribution through electron-electron
scattering. The electrons then transfer energy to the lattice through electron phonon coupling.

3

To give an idea of how short one attosecond is, we know that the age of the Universe, i.e.,
the time elapsed since the Big Bang, is approximately 14 billion years. One attosecond is one
billionth of one billionth of a second. The processes of interest at the attosecond time scale aremostly related to the dynamics of electrons. The desire to understand these ephemeral electron
dynamics that are invisible to the human eye are of both scientific and practical interest. First, theinterrogation of ultrafast electronic motion will provide us answers to fundamental scientificquestions of how the microscopic behaviors of electrons and atoms of a material dictate its
macroscopic properties. For example, the well-known Ohm’s law is a macroscopic description that
relates the current J with the applied electric field E through equation J=E/ρ, where ρ is theelectrical resistivity and has the unit of time. The resistivity of copper is 1.7 μΩ·cm, which equals
to approximately 2 as. This 2 as is the time for an electronic system in metal to establish Ohm’s
law-like behavior. During the relaxation time, the electrons first accelerate in the electric field andthen slow down because of scattering. The electrons finally reach the ultimate velocity bybalancing the two competing mechanisms of acceleration and scattering. This final velocity
determines the electric current [1]. There is an abundance of similar macroscopic phenomena
dictated by the microscopic motion of electrons. These phenomena include the breakage andreformation of chemical bonds during chemical reactions, charge transfer between atoms andmolecules, and phase transitions in materials. A deep understanding of these dynamics will help
us steer specific microscopic electron motion for technological applications. Taking thesemiconductor industry as an example, during the last several decades this industry had a rapid
growth along the pathway of Moore’s Law to shrink the transistor size and increase the signalprocessing speed. However, the industry has reached a critical barrier point set by the speed atwhich the electrons can be manipulated. To overcome this speed barrier, attosecond researchers

4have discovered light field-driven carrier dynamics oscillating in the petahertz (1015 Hz) regime.
These dynamics can potentially boost transistor speed by three orders of magnitude compared tocurrent state-of-the-art devices [2,3]. In addition to semiconductor transistors, the potential
technological impact of ultrafast sciences lies in many other aspects of modern industry. For
example, improvement in the photoconversion efficiency of a solar cell relies on knowledge ofcharge movement after photoadsorption. The writing-speed limit of a computer disk depends on
how fast one can switch the spin state of ferromagnets. Moreover, an understanding of electron
dynamics in simple molecules and solids is a building block for manipulating more complicatedsystems, such as biomolecules to achieve targeted disease treatment.

1.2

Probing into the “Invisible”
Real-time observation of ultrafast electron dynamics requires metrology or spectroscopy

tools with sufficient time resolution. The response time of the human eye is one twentieth of asecond, precluding us from directly observing many physical and chemical dynamics without theassistance of special techniques. Everything below the limit of naked eyes is “invisibly fast”.
Fortunately, major advance into the “temporal invisible” was made with flow tubes in 1923 by H.
Hartridge and F. Roughton for resolving solution reactions [4]. In their experiments, the productsof chemical reactions were measured at different positions in flow tubes, which could translate
into time if the flow speed was known. After that advance, millisecond precision was reached in1940 by B. Chance with the invention of the stopped-flow method, which is still used in the studyof biological kinetics [4].
Around 1950, the invention of flash photolysis enabled microsecond resolution in
observing radical intermediates during a chemical reaction via a change in the absorption spectrum

5[6]. This technique was later recognized with a Nobel Prize in 1967 that was awarded to R. Norrishand G. Porter and shared with M. Eigen [7]. M. Eigen also developed a series of techniques toinvestigate chemical reactions with microsecond-to-nanosecond resolution. These experiments
were achieved by first disturbing the equilibrium of a solution with energy pulses, either a soundwave or an electric field, and then following the kinetics of the solution. Eigen’s techniques
indicated that there are two ingredients for overcoming the temporal resolution limit: first, we need
the instrumentation to produce ultrafast light pulses or other particle sources, and second, we apply
these light and particle sources to create, investigate, and manipulate the short-lived transient statesin atoms, molecules, or materials.

1.3

Pump-ProbeTo use ultrafast light pulses to create and investigate the evolution of transient states, mostof ultrafast experiments employ a pump-probe scheme. In pump-probe experiments, the firstenergy pulse is injected into the system to induce a dynamical change and the second pulse with acontrolled time delay relative to the first one follows the temporal evolution of the system. The
pump-probe idea can be traced back to 1864 when Toepler applied photography to investigate
microscopic dynamics of a sound wave [8]. In Toepler’s experiment, two light sparks weregenerated with their relative delay controlled electronically. The first one initiated the sound wave,
and the second one recorded its motion by taking pictures of the wave. Abraham and Lemoine
later (1899) improved Toepler’s method by splitting the same spark into the pump and probe armwith an adjustable optical-path length between them. With the pump-probe concept established,subsequent development of time-resolved studies are based upon shorting temporal duration ofultrafast sources, and developing advanced spectroscopy techniques that unravel different aspects

6of the dynamical change. In terms of ultrafast light sources, the invention of the pulsed lasers mademajor breakthroughs in the accessible temporal realm.

1.4

Ultrafast Lasers
The 1960s witnessed several milestones in laser development. In particular, the inventionof laser modelocking triggered the advancement of ultrafast optics and brought time-resolved
studies into the picosecond regime. The first generation of picosecond lasers used solid statematerials such as ruby and Nd:glass as the gain media. Modelocking was achieved either actively
through an external electrical signal, or passively through loss modulation introduced by asaturable absorber. The latter approach typically generated shorter picosecond pulses. Withpicosecond resolution achieved, the rotational motion and vibrational modes of polyatomic
molecules became accessible in direct time-domain study. Direct temporal study of phonon
relaxation in solids and many other nonradiative processes in chemistry were also achieved withpicosecond resolution. Pioneering discoveries regarding ultrafast electron motion were made,including studies of photo-induced intramolecular electron transfer. Nevertheless, the responsetime of organic saturable absorbers limited the further reduction of laser-pulse duration towards
the femtosecond regime. Therefore, the mechanisms of many femtosecond excited-state processes,
such as electron relaxation in semiconductors, electron transfer in molecules, bond-breaking
reactions and photoisomerization, remained out of reach.
The femtosecond barrier was overcome with the invention of organic dye lasers. Organic
dye lasers were invented in 1966 by Sorokin et al. [9]. Ten years later, subpicosecond pulses fromdye lasers were generated, heralding the dawn of the “femtosecond era”. Compared to previous
solid-state lasers, the gain saturation in dye lasers can be significant during pulse formation, and7

the response time of the saturable absorber no longer constitutes a limitation to the achievablepulse duration. The momentous development of sub-100-fs pulsed dye lasers revolutionized thefield of femtosecond laser spectroscopy in the 1980s, thanks to the discovery of the dispersioncontrolled colliding-pulse-modelocking [10] and prismatic pulse-compression techniques [11–13].
Using the latter method, Fork et al. demonstrated six femtosecond pulses in 1987 [14].

1.5Femtosecond Solid-State Lasers
The development femtosecond lasers initially occurred alongside dye lasers until 1990

because it was assumed that passive modelocking in solid state lasers was difficult and usuallyunsteady because of Q-switching instability. However, breakthrough techniques including
semiconductor saturable absorber mirrors (SESAMs) and Kerr lens modelocking (KLM) gave riseto the emergence of femtosecond solid-state lasers.
SESAMs are Bragg reflector mirrors with semiconductor saturable absorbers integrated
into them [15]. The light absorption of the mirrors becomes higher for more intense incoming
pulses and finally saturates when the available initial states in saturable absorbers are depleted. In
these saturable absorbers, different relaxation mechanism in semiconductors with varied timescales from nanosecond (carrier recombination and trapping) to femtosecond (carrier
thermalization) can be used for pulse shaping. SESAMs are the essential components to modelock
a variety of solid-state lasers and fiber lasers, which now have become the building blocks formany scientific instruments as well as industrial and medical photonic systems. Nevertheless,current SESAMs cannot reach a pulse duration as short as that of Ti:sapphire lasers.
In 1999, the modelocked Ti:sapphire laser surpassed the six femtosecond record set by dyelasers. Along with its other merits [16], this laser is recognized as the best ultrafast laser today.

8

Historically, the titanium-doped sapphire had already been discovered in the late 1980s as a laser
gain medium that possessed a broad bandwidth to support femtosecond pulses. However,researchers assumed that it would be difficult to find a saturable absorber to support stablefemtosecond modelocking in such systems. This is why it was very surprising when Sibbett’sgroup demonstrated 60-femtosecond pulses with a Ti:sapphire laser that didn’t even use a saturable
absorber [17]. Subsequent theoretical and experimental research efforts revealed the responsible
mechanism [18,19]. Under the radiation of a laser beam with a radial intensity profile, the Kerr
nonlinearity of the Ti:sapphire crystal turns the crystal itself into a focusing lens. This “virtual lens”
allows the laser cavity to be designed with a certain aperture to favor the pulsed-lasing mode thathas a larger lensing effect as compared to the rivalling continuous-wave mode (Figure 1.3). In
other words, the laser crystal itself is an ultrafast saturable absorber for modelocking. This kind ofmodelocking method is named Kerr lens modelocking (KLM). The KLM Ti:sapphire laser allowed
routine generation of pulses with < 20 fs duration. It was quickly used for many kinds offemtosecond time-resolved spectroscopies. Zewail won the Nobel Prize in 1999 forfemtochemistry, i.e., the real-time observation of the breakage and formation of chemical bondsby using the short pulse duration of the femtosecond lasers [20].
In addition to its short pulse duration, Ti:sapphire as a solid-state gain medium can achieve
high peak power after amplification because of its unique properties including high saturation
fluence, excellent thermal conductivity, and broad emission bandwidth. These properties enabled
efficient laser-pulse amplification with the development of high-power semiconductor diodepump lasers and the ingenious invention of the chirp-pulse-amplification technique. The state-ofthe-art Ti:sapphire laser systems today can obtain an intensity of >1021 W/cm2 at focus, exceeding
the atomic Coulomb field by more than four orders of magnitude [21]. This high intensity field9

brings an unprecedented regime of nonlinear optics and strong field physics. The development ofstrong field physics also brought our temporal resolution into the attosecond regime.

Figure 1.3: Kerr-lens-induced changes in the transmission of laser beam through an aperture.
The Ti:sapphire crystal forms a Kerr lens that serves as a saturable absorber with ultrafast response
time. The Kerr lens results in a self-focusing effect for the laser field in the pulsed mode. The selffocusing effect results in a higher transmission of the pulsed-laser light through the aperture ascompared to CW laser mode and causes mode locking.

1.6

Attosecond
Attosecond science was initiated with the invention of phase-stabilized high power

Ti:sapphire lasers. In nonlinear laser-matter interactions, electronic motion can be sensitive to thesub-cycle slopes of the oscillating field rather than exhibiting cycle-averaged dynamics. This
dynamical sensitivity to electric field not only gives a way for the direct probing of attosecond
charge dynamics but also allows the generation of attosecond light pulses through the process
called high harmonic generation (HHG). HHG is the heart of attosecond science. For the past 20
years, the field of attosecond has witnessed substantial progress including pulse generation, pulse
characterization, and attosecond spectroscopy of atoms, molecules and solid-state materials. I will
briefly discuss the current developments in attosecond science and then give an overview of theattosecond studies performed in this thesis work in chapter 2.

10

1.7

Beyond Attosecond
In the pursuit of temporal resolution beyond attoseconds, zeptosecond (zs, 10-21 second) is

the time scale related to the proton and neutron motion inside the nuclei of atoms. Therefore,
zeptosecond pulses will grant us the capability to capture nuclear processes in real time andinterrogate nuclear dynamics such as quasi-fusion/fission decay. It was demonstrated
experimentally that high harmonic generation driven by mid-IR lasers (3.9 μm) could support ahuge spectral bandwidth (> 0.7 keV) [22]. Although this broad bandwidth corresponds to singlecycle attosecond pulses in the Fourier transform limit, compensation of the “atto-chip” to actuallyachieve these short pulses was experimentally challenging. As a result, the exploration of
zeptosecond pulse generation remains mainly theoretical. Theory predicted that few-cycle longwavelength driving lasers (9 μm) could generate x-ray pulses with sub-attosecond temporal
structure because of the interference of x-ray radiation produced in multiple rescattering of theelectron wave packet with the ion [23]. Other theoretical proposals of zeptosecond pulse
generation included using nonlinear Thomson back-scattering [24], using the interaction between
a muonic hydrogen atom and high power chirped laser pulses [25], and using the reflection of a
relativistically intense femtosecond laser from an over-dense plasma [26].
Yoctosecond (10-24 second) is a factor of a million faster than the shortest attosecond pulse
achieved experimentally this far. In the pursuit of such extreme photon sources, we can still hopefor a surprise from highly nonlinear laser-matter interactions. At the same time, we must
investigate in systems involving motion at the same time scale. Yoctosecond is the time taken fora quark to emit a gluon. Ipp et al. theoretically worked out that the quark-gluon plasma createdthrough heavy-ion collisions could emit zeptosecond pulses when the plasma cools on [27]. A11delayed pulse pair is also possible by taking advantage of the evolution of isotropy during theplasma expansion.

1.8

Conclusion
The expected future evolution of ultrafast science will keep overcoming the limits of

temporal resolution and pushing the frontiers of science and technology. This thesis focuses on the
dynamics at the attosecond time scale. Therefore, in Chapter 2, I will discuss the current
developments of attosecond science and present the overview of the thesis.

Chapter 2
Attosecond Science and Thesis Overview

Electrons bind atoms into molecules and solids. They are carriers of electrical current.
Electrons also radiate photons that we can harness to interrogate the microscopic world. The
understanding and steering of electron dynamics in the microscopic world are the driving forcesof the technological and industrial revolution in the 21st century. One effort of modern computerand information technology is to increase the speed at which electrons can be controlled. To realize
this appealing vision of future electronics, we need to achieve the required temporal resolution tocapture the real-time evolution of an electronic system during a dynamical change. Attosecond (as)is the fundamental time scale of electron motion and electron-electron interactions. For example,
the atomic unit of time is 24.2 as. The period of electron orbiting the nucleus in a hydrogen atom
in its ground state is approximately 150 as. The time needed for an electron in the conduction bandof a metal with Fermi velocity to travel through a distance equal to the metal lattice constant isnear 220 as. These attosecond electron dynamics will impact the subsequent femtosecond-topicosecond dynamics in materials. Such dynamics include molecular vibration, photodissociation,and phase transitions in solids. The direct time-domain investigation of these extremely fast
dynamics relies on the development of attosecond science.
Attosecond science can be categorized into different subfields including attosecond pulse
generation [28,29], pulse characterization [30–34], attosecond spectroscopy [33,35–38], and13attosecond control [2,39–41]. These subfields are deeply intertwined. For example, attosecond
pulse generation with tailored-temporal structures relies on our knowledge of electron ionization
and recombination in a strong laser field, which can be acquired from spectroscopy. Thisconnection extends to attosecond spectroscopy and attosecond control because most time-resolved
measurements require the change of a dynamical system in a controlled and reproducible manneron the related time scale. In more than two decades of development, attosecond science hasadvanced to investigating electron dynamics in a variety of systems ranging from atoms topolyatomic molecules [42,43], to dielectric, metallic, and semiconducting solid-state materials[33,36,44], as well as to nanostructures and quantum dots [45,46]. The spectroscopy methods usedin attosecond science include high-harmonic spectroscopy [47–50], above-threshold ionization[51], recollision-based imaging [52,53], transient absorption spectroscopy [2,36], andphotoelectron spectroscopy [35,44,54–56]. Each spectroscopy technique interrogates the systemfrom its unique perspective, using different light sources. Therefore, attosecond spectroscopies are
performed using waveform-controlled femtosecond pulses [57,58], isolated attosecond pulses[33,40,42,44,56], and attosecond pulse trains [35,54,59,60]. The information we gained fromattosecond spectroscopy opens the door to attosecond technology. For example, controlled
femtosecond and attosecond light fields permit the manipulation of electric current on the atomic
scale at petahertz frequencies, in the same way radio-frequency fields control the mesoscopic
currents in semiconductor chips [2,3,40,61].
The work presented in my thesis includes attosecond pulse characterization and attosecond
spectroscopy. This chapter first reviews various advances and future developments of attosecond
science. Then I introduce how the thesis is organized and briefly describe the content of eachchapter.

14

Figure 2.1: Different scenarios of election ionization. When the photon energy is larger than theionization potential, the electron goes through direct ionization. If the photon energy is smaller
than the ionization potential, the electron can be removed through either multiphoton ionization ortunnel ionization, depending on the Keldysh parameter 𝛾.

2.1

Attosecond Dynamics Probed by Femtosecond Pulses
In conventional pump-probe experiments, the temporal resolution is limited by the pulse

duration of the pump and probe beams. Therefore, one could assume that attosecond experiments
must be performed with attosecond pulses. However, this is not the case. Waveform-controlled
femtosecond IR pulses allow the use of sub-cycle gradients of the electric field to initiate and probe
subsequent dynamics with attosecond-timing precision. One example is probing the electron
ionization in a strong field. Since a strong IR field rivals the interatomic Coulomb force, electrons
can be freed from the ion core either through multiphoton ionization or tunneling ionization. These
two regimes are categorized via the Keldysh parameter 𝛾 = √𝐼𝑃 /2𝑈𝑃 , where IP is the ionization
potential, and UP is the ponderomotive energy of electrons in the laser field, defined as 𝑈𝑃 =𝐸02 /4𝜔02 , where E0 is the peak-laser electric field, and 𝜔0 is the laser carrier frequency [62]. For
γ >1, the electron goes through multiphoton ionization, with the final kinetic-energy distribution
depending on the waveform of the driving laser, including the carrier-envelope phase. For the15

Ti:sapphire laser systems that are usually used in these experiments, the peak intensity isapproximately 1013-1014 W/cm2. In this regime, the laser filled bends the atomic Coulomb potential
trapping the electrons and creates a penetrable potential barrier. The electron tunnels through thisbarrier and starts to accelerate in the laser field. Depending on when the tunnel ionization happens
during a laser cycle, the electron undergoes different acceleration, deceleration, recombination,and scattering processes. Studying these sub-cycle dynamics at the attosecond time scale provides
information on strong-field electron dynamics, including the process of high harmonic generation(HHG).

2.1.1

High Harmonic Spectroscopy
HHG is a highly nonlinear process caused by electron motion in a strong field. After strong

field ionization, a tunneled electron could recombine with the parent hole it left upon ionizationand emits an attosecond burst of radiation [63,64]. This unique attosecond EUV/X-ray light source
promises many important spectroscopic applications such as resolving the emission time ofelectrons upon photoionization. In addition, HHG from laser-atom or laser-molecule interactions
carries a wealth of information on both the ionization process and the recollision dynamics. For
example, HHG is sensitive to electron trajectories and molecular orbital changes as well as the
structural evolution of the ionized target [47,49,65]. We can obtain these strong-field electron andion dynamics by analyzing the properties of the emitted harmonics, e.g., the intensity, phase, andpolarization [66]. Since the HHG process is triggered by ionization and terminated byrecombination, the time between the two events can be viewed as a time delay. This time delay
can be controlled by selecting the electron trajectories or adjusting the driving laser wavelength[48,67].

16

Most of the experiments on high harmonic spectroscopy only probe the intensity andpolarization of the HHG emission. In Chapter 5 and 6, I show that attosecond interferometer
technique gives access to the phase of HHG radiation [30]. For manipulating the electron
trajectories in HHS, two-color circular or linear high harmonic generation scheme discussed in
Chapter 5 and 6 allow us to explore additional degrees of freedom [30,41,48,50,68–74].

2.1.2

Electron Recollision-Based Probe
HHG happens when the ionized electron can recombine with the parent ion. However, inmany cases, the recolliding electron diffracts or scatters off the parent ion instead of recombining
with it. Since the ionization prior to the recollision creates a nonequilibrium charge distributionthat evolves on the attosecond time scale, the diffraction patterns resulting from electrons
recolliding at different times can follow the temporal rearrangements of the ion [75–77]. In anotherintriguing scenario, the interference between the diffracted electrons and undiffracted
photoelectron will form a hologram [78,79]. Because of the coherence of the electron wave packets
and the subcycle temporal confinement of the process, ångström spatial resolution and attosecond
temporal resolution can be achieved.
The laser-induced electron diffraction described above requires the recollision between
electron and ion to be a coherent process. In a general term, the electron scatters off the parent ion
upon recollision. The scattering can occur either elastically or inelastically. In elastic scattering,
the electron’s final momentum is sensitive to the IR vector potential at the instant of ionizationand the moment of recollision. This dependence has been exploited to trace the subcycle fieldevolution [80]. For example, this dependence permitted measurement of the carrier-envelope phaseof an ultrafast laser pulse [81,82]. In contrast, the laser-induced electron scattering becomes an17inelastic process when back-scattered electrons lose energy via interaction with other particles or
quasiparticles. For example, if a back-scattered electron possesses higher energy than the binding
energy of the valence electrons, the back-scattered electron may knock off a valence electron uponrecollision with the parent ion [83]. This process is called nonsequential double ionization. It
provides a spectroscopy method for studying the multielectron correlation [84] and theionization/recollision-induced structural changes in the molecular ion.

2.1.3

Attoclock
Both electron recollision-based probes and HHG begin with the photoionization ofelectrons. How long photoionization takes is one of the most intriguing problems in strong field
physics. One of the techniques used to measure the photoionization time with attosecond resolution
is attoclock. Attoclock uses only one femtosecond laser beam. It measures the angle- and energyresolved spectra of photoelectrons that tunnel out of the rotating barrier created by a nearlycircularly polarized laser field in combination with the atomic Coulomb potential [57,58]. In thistechnique, the rotation of the laser electric field provides a temporal ruler that can be used todetermine the tunneling time of the electron by measuring the direction of maximum photoelectron
distribution compared to the direction of maximum electric field [58,85]. The temporal resolutionof attoclock is determined by the angular resolution of the photoelectron detection i.e., δτ = δθ/ωL,
where ωL is the angular frequency of the laser field. Experimentally with attoclock, an accuracy ofδθ ~ 1o yields δτ ~ 10 as for 800 nm radiation.
Attoclock provides straightforward access to directly probing the ionization process in thetime domain. However, it has not settled the long-standing debate regarding electron-tunnelingtime because of ambiguities in interpreting the measurement. First, the interpretation of an

18

attoclock measurement requires three assumptions [57,58,85–87]: 1) the highest ionization
probability coincides with the electric field peak, 2) the ionization is completed once electrons
tunnel through the barrier, and 3) an electron is treated classically after tunneling. Moreover, it isdifficult to disentangle the real tunneling delay from the additional delay introduced by the ionic
potential, strong IR field, and multielectron effects. For example, Pfeiffer et al. found that theinteraction with the ionic potential during streaking would cause an angular offset for the freed
electrons; the offset is independent of laser intensity in He, but monotonically decreases in Ar
because of multielectron and stark-shift effects [57]. A theoretical paper by Torlina et al. indicatesno tunneling delays associated with one-electron hydrogen atom, while additional delays may beintroduced by multielectron excitation by an IR field [87]. One way to better quantify the timedelay introduced by multielectron excitation is to perform attosecond chronoscopy using
attosecond pules.

2.2

How to Make Attosecond Pulses
Although HHG and attoclock can probe attosecond electron dynamics using only

femtosecond pulses, most of attosecond spectroscopy methods require attosecond pulses. A
common way to generate attosecond pulses is through HHG. For example, HHG driven bymulticycle laser pulse leads to the generation of attosecond pulse trains because the electron repeats
the three-step sequence of tunneling-acceleration-recombination every half period of the laser field[63,64]. If the HHG process is phase-matched over the gas medium, the single atom response will
add up, leading to an amplified harmonic radiation [88]. The radiated-high-harmonic spectrum iscomposed of a frequency comb characterized by a flat plateau region extending from below the19atom’s ionization threshold to the high harmonic cutoff [89]. This comb structure in the frequency
domain corresponds to attosecond pulse trains in the time domain.

Figure 2.2: Different techniques
for generating isolated
attosecond pulses. (a) Intensity
gating. (b) Polarization gating. (c)Phase matching gating. (d)Attosecond lighthouse effect.

20

An attosecond pulse train paired with its multicycle driver constitutes a powerful tool for
attosecond spectroscopy. When using attosecond pulse trains in spectroscopy to probe electron
dynamics, one needs to make sure that the electron dynamics terminate within a half period of theIR field. This termination avoids multiple starts and exposures of the undergoing process andguarantees that each burst in the attosecond pulse train probe the interrogated system in its groundstate.
Attosecond pulse trains offer several advantages compared to isolated attosecond pulses inspectroscopy applications such as high energy resolution, easy experimental implementation, andlow dressing-field intensity in streaking-type experiments [35]. However, some attosecond
experiment requires isolated attosecond pulses. On example of this type of experiment is excitinga coherent superposition of multiple final states in attosecond-beating experiments [2,43]. This
excitation is made possible by the super-continuum bandwidth associated with single attosecond
pulses. The most intuitive method for achieving a single attosecond pulse is limiting the numberof electric field cycles of the driving laser pulse to reduce events of electron recombination.
However, this method is difficult to realize because harmonic emission from multiple half cycles
cannot be entirely suppressed even with a pulse duration as short as 2.1 fs (corresponding to 0.9
optical cycles). Another difficulty for single attosecond-pulse generation is dispersion control. The
dispersion of HHG radiation is different for emission in the plateau and cutoff regions. Highharmonic photons in the plateau region are emitted by electrons returning to the core along both
short and long trajectories. These two trajectories have different frequency dispersions and impose
a strong chirp on the HHG radiation. The emission in the cutoff region, on the other hand, can be
confined within the most intense half cycle of the driving pulse (i.e., intensity gating, see Figure2.2a) and is chirp-free. Therefore, intense waveform-controlled few-cycle laser pulses, combined

21with carefully designed bandpass multilayer mirrors to select only the cutoff harmonics, haveallowed reproducible generation of isolated attosecond pulses. In this scenario, the harmonic
spectrum is usually centered at high energy (> 80 eV). Also, the ultimate pulse duration is limitedby the bandwidth of the harmonic cutoff [90].
The generation of isolated attosecond pulses at low EUV photon energies or withultrabroad bandwidth requires other gating techniques to restrict the HHG process to one single
half-cycle of the driving field. For example, the polarization gating technique uses two counterrotating elliptically polarized driving-laser pulses that are properly delayed so that only one halfcycle of the two fields overlaps and creates a temporal window with linear polarization [31,91–
93]. Outside this window, the tunneled electron cannot return to the parent ion and the harmonic
emission is eliminated (Figure 2.2b). Thanks to the polarization-gating approach, isolated singlecycle attosecond pulses with a duration of 130 as and a central photon energy of 36 eV wasdemonstrated [31,92,93]. Compared to polarization gating, the double optical gating techniqueintroduces a second harmonic field in addition to two counter-rotating, elliptically polarized,
driving-laser pulses [94–97]. The second harmonic field breaks the symmetry of two consecutive
half-cycles of the IR field. This broken-symmetry results in one attosecond pulse being generated
within one IR optical cycle. In this way, double optical gating widens the temporal gate by a factorof two compared to polarization gating.
While polarization gating and double optical gating use the microscopic single-atom
response to the driving laser fields, ionization gating takes advantage of the macroscopic phasematching effect to confine the HHG emission to the leading half cycle of the driving pulse [98,99].
Therefore, ionization gating can be also called phase matching gating (Figure 2.2c). In ionization
gating, phase matching ceases when the time-dependent ionization of the gas medium reaches the22critical ionization level after the first half-cycle of the laser field. This cessation occurs because
once the critical ionization level is reached, the negative dispersion of the overly dense freeelectron in the medium cannot be compensated by the neutral gas atoms any more. Therefore, thiscritical ionization level sets a temporal window for phase matching. Because the temporal window
for phase matching shrinks rapidly with longer driving laser wavelengths [89,100], HHG driven
by mid-infrared lasers offers a great advantage for generating isolated attosecond pulses. Anotheradvantage of implementing this phase matching gating using mid-infrared lasers is that because
the harmonic cutoff energy increase with increasing λ2 [89], using long wavelength driving laser
fields in HHG can push the isolated attosecond pulse into the soft X-ray regime.
Compared to gating techniques that confine HHG to a single half-cycle of the driving field,
the attosecond lighthouse technique generates isolated attosecond pulses by emitting each burst in
the attosecond pulse train at a different angle and spatially separated in the far field (Figure 2.2d)[101]. This spatial separation is achieved by using a driving field with a rotating wavefront.

2.3

Attosecond Pump-Probe
Attosecond pulses can be used in pump-probe experiments to directly observe attosecond

processes in the time domain. In pump-probe experiments, the pump pulse arrives first to initiate
the dynamics of interest, the subsequent probe pulse arriving with a variable time delay produces
an observable from the interrogated system. The temporal resolution of a pump-probe experimentis determined by the duration of the pump and probe pulses. Because of this requirement, one
could decide to use two isolated attosecond pulses for the attosecond pump-probe experiments.
However, the low photon fluxes of existing attosecond light sources encumber such a measurement
scheme [1,102]. In addition to the flux problem, the cross section of two-photon processes is low

23compared to that of a one-photon process. As a result, the EUV/X-ray intensity needs to be > 1015
W/cm2 to obtain a time-resolved signal. This intensity is difficult to realize.
To overcome these difficulties, most of the current attosecond pump-probe experiments
use one attosecond pulse and one IR pulse. These two pulses are perfectly synchronized relative
to each other because HHG under IR driving field is a coherent process. The temporal-resolution
limit imposed by the IR pulse duration can be circumvented by probing dynamics that are sensitive
to the instantaneous value of the IR electric field. In this case, the IR carrier frequency imprints a
temporal ruler on the dynamical evolution of the system, giving access to the sub-cycle temporal
resolution. Although attosecond temporal resolution is obtained, the IR pulses used in experiments
add certain difficulties to interpreting the measurement results. For example, intrinsic dynamicalchanges of the system are coupled with strong-field, light-matter interaction induced by intense IR
pulses. To decouple this interaction from intrinsic material dynamics, high-level theoretical work
is required to model this interaction. For example, in the measurement of attosecond photoemissiontime, the Wigner time delay is convolved with a measurement-induced time delay [42,103,104].
Although this measurement-induced delay can be calculated, it requires complex and difficultmodeling. Another difficulty of this pump-probe scheme is that the broad bandwidth of attosecond
pulses excites a manifold of final states. The excitation is further complicated by state-dependent
cross sections and subsequent IR dressing of the final states. Such complications appear in
attosecond quantum control of solid-state materials and large polyatomic molecules [43,44,105].
Nevertheless, these complications in attosecond pump-probe experiments provide a richframework for ultrafast dynamics and strong-field physics.
There are two types of attosecond pump-probe experiments that are categorized by theorder in which the attosecond and IR pulses interact with the interrogated system. The order also

24determines the information extracted from the system [93,106]. The first type is performed with aconventional pulse sequence, i.e., IR excitation followed with attosecond an EUV/X-ray probe.
This scheme can be used to investigate sub-cycle population dynamics of a system excited by theIR pump pulse [107]. The second type employs a less intuitive pulse sequence, i.e., the attosecond
pulse precedes the IR pulse. This scheme has found extensive application in attosecond-pulse
characterization [28,30,32], the timing of the birth of a photoelectron [42,44,104], and measuring
a high-energy photoelectron lifetime [35]. This scheme has also been used to study quantumpathway interference. For example, quantum beats of multiple bound-excited states launched byisolated attosecond pulses have been observed in neon using the pump-probe scheme just described[108].

2.4

Attosecond Pulse Characterization
The application of attosecond pulses in pump-probe spectroscopy to investigate attosecond

dynamics requires knowledge of the time-domain structure of attosecond pulses. Before the adventof attosecond science, the full temporal characterization of ultrafast pulses was achieved primarily
by using frequency-resolved optical gating (FROG) technique [109]. FROG applied a temporalgate G(t) with a controlled delay τ to decompose the field to be characterized E(t) into temporalslices, and then measured the spectrum of each slice. This procedure yielded the so-called FROG
trace
2

+∞

𝑆(𝜔, 𝜏) = |∫𝑑𝑡𝐺(𝑡)𝐸(𝑡 − 𝜏)𝑒

𝑖𝜔𝜏

| .

(2.1)

−∞

Now, various iterative algorithms can be applied to extract G(t) and E(t) from S(ω, τ). In practice,most implementations of FROG use the self-gating method where G(t)=E(t), and the auto-25

correlation is achieved via the nonlinear interaction of the pulses with a crystal, e.g., secondharmonic generation. The FROG method cannot be straightforwardly extended to measuring
attosecond pulses because of several difficulties. These difficulties include the broad bandwidthof attosecond pulses, difficulty in beam-splitting, and the low nonlinearity of materials in EUV /Xray regime. Instead, attosecond pulse characterization relies on the combination of an attosecond
pulse with a delayed-laser field that acts as a temporal gate. The nonlinear interaction is achieved
either using the photoionization process, or the HHG process itself. These two approaches are
categorized as ex-situ and in-situ characterization, depending on whether the characterization is
performed at the same location as the attosecond pulse generation [31,41].

2.4.1

Ex-situ Characterization of Attosecond Pulses
Ex-situ characterization of attosecond pulses is essentially a pump-probe measurement.

Attosecond pulses generated through HHG is focused onto a target to produce photoelectrons. The
kinetic momentum spectrum of these photoelectrons is an electron-replica of the attosecond pulse.
Also, the spectral phase of each frequency component ωX of the attosecond pulse is mapped into
the time of ionization for photoelectrons with kinetic energy 𝐸𝑘 = ħ𝜔𝑋 − 𝐼𝑝 , where Ip is theionization potential. To measure the ionization time of the photoelectrons, an IR laser pulse isfocused on the target with a controlled time delay relative to the attosecond pulse. The IR field
induces perturbations to the liberated photoelectrons. The perturbations depend on thephotoelectrons’ time of ionization relative to the laser field. Within the strong field approximation,
the electron’s transition amplitude to the final state with momentum v is given by [34]+∞𝑎(𝒗, 𝜏) = −𝑖 ∫−∞𝑑𝑡𝑒 𝑖𝜑(𝑡) 𝒅𝒑(𝑡) 𝑬𝑋 (𝑡 − 𝜏)𝑒 𝑖(𝐸𝑘+𝐼𝑃 )𝑡26∞

𝑨2 (𝑡 ′ )𝜑(𝑡) = − ∫ 𝑑𝑡 [𝒗 ∙ 𝑨(𝑡 +
],
2
𝑡
′

′)

(2.2)

where A(t) is the vector potential of the laser field, 𝒑(𝑡) = 𝒗 + 𝑨(𝑡) is the canonical momentumof a free electron dressed by the laser field, dp is the electron transition dipole matrix from thebound-state to the continuum-state, and Ek is the kinetic energy of free electron Ek=v2/2. Equation
2.2 indicates that a phase gate 𝐺(𝑡) = 𝑒 𝑖𝜑(𝑡) 𝑒 𝑖(𝐸𝑘 +𝐼𝑃 )𝑡 is applied to the electron wave packet
replica of the attosecond pulse 𝒅𝒑(𝑡) 𝑬𝑋 (𝑡 − 𝜏) . Equation 2.2 constitutes the basis for thefrequency-resolved optical gating for complete reconstruction of attosecond bursts (FROG-CRAB)
technique. Within this general picture, two special schemes are utilized for two specific temporal
structures, i.e., the RABBITT technique for attosecond pulse trains and the attosecond streak
camera for isolated attosecond pulses. To understand the RABBITT technique, it is easier to startwith laser-assisted photoemission.
Laser-assisted photoelectric effect (LAPE). The laser-assisted photoelectric effect is thebasis of many advanced attosecond pulse-characterization techniques such as RABBITT and Phase
retrieval by omega oscillation filtering (PROOF). The process can be described as [110]:
𝑨 + 𝛾 → 𝑨+ + (𝑒 − + 𝑛𝜔)𝑛 = ±1, ±2, ±3 ∙∙∙∙∙∙∙,(2.3)

where the system A absorbs a high-energy photon 𝛾 and emits a bound electron into the continuum(Figure 2.3a). In the presence of a laser field, the continuum electron 𝑒 − can exchange an integer
number of photons 𝑛𝜔 with the field, forming sidebands corresponding to the absorption or
stimulated emission of photons. Since free electrons in vacuum cannot absorb or emit photons,
LAPE should also be assisted by the ionic core or the surface to satisfy the both energy andmomentum conservation. This conservation can only happen when electron is in close proximitywith the ion or surface. This restriction can be exploited to study the evolution of an electron wave
function moving away from the ion or surface. The wave function evolution and attosecond pulse

27

structure can be obtained by studying the intensities of different orders of sidebands as a functionof laser wavelength, intensity, pump-probe delay etc.

Figure 2.3: Different ex-situ methods of characterizing attosecond pulses. (a) Laser-assisted
photoemission (LAPE). (b) Reconstruction of attosecond beating by interference of two-photontransitions (RABBITT). (c) Phase retrieval by omega oscillation filtering (PROOF), adapted from[111].

28

For simplicity, we can just consider the first and second orders of sidebands. In this case,the LAPE response function can be written as [55,112,113]𝑓(𝐸 − 𝐸0 ) =1 − 2𝐴1 − 2𝐴2√2𝜋𝜎 2
𝐴1

𝑒

−

+∑(
𝑒
2
√2𝜋𝜎
±

(𝐸−𝐸0 )22𝜎2−

(𝐸−𝐸0 ±ħ𝜔)2
2𝜎2
+

𝐴2
√2𝜋𝜎 2𝑒−(𝐸−𝐸0 ±2ħ𝜔)2
2𝜎2
),

(2.4)

where A1 and A2 represent the first- and second-order sideband intensities, E0 is the electron kinetic
energy of the photoemission peak, σ is the peak width, and ħω is the peak separation. Both theory
and experiment have shown that the sideband intensity for small values of argument 𝑥 can bemodeled by the square of a Bessel function:
𝐴1 = 𝐽𝑛2 (𝑥)𝑥=√16𝜋𝛼 𝐼𝐸0.
𝑚𝑒 ħ 𝜔 4(2.5)

Here, I is the dressing-field intensity. For small 𝑥, A1 can be approximated by 𝐴1 ≈ 𝑥 2 /4 leading
to𝐼𝐸

𝐴1 ∝ 𝜔40 .

(2.6)

In this regime, the sideband height depends linearly on the IR laser intensity. This linear
relationship forms the basis for EUV+IR cross-correlation measurements with the LAPE technique.
Reconstruction of attosecond beating by interference of two-photon transitions(RABBITT). The LAPE technique measures the overall pulse duration of the attosecond pulse.
However, detailed temporal information about attosecond pulses is required for certain
experiments. The temporal structure of attosecond pulse trains corresponds to combs in thefrequency domain. Therefore, the photoelectron spectrum produced by an attosecond pulse train

29

consists of discrete peaks with kinetic energies separated by 2ħω. The photoelectron can absorb or
emit integer numbers of IR photons in the presence of the laser field, forming sidebands aroundthe main photoelectron peaks (Figure 2.3b). If the IR intensity is properly adjusted so that only
first-order LAPE sidebands exists, there are two different quantum paths exciting electrons fromthe same initial state to the same sideband, nħω: (i) absorbing a lower-order harmonic photon and
an IR photon (n-1)ħω+ħω; (ii) absorbing a higher-order harmonic photon while emitting an IR
photon (n+1)ħω-ħω. Because of the inference between two quantum paths, the sideband intensity
oscillates as a function of delay between the attosecond pulse train and the IR pulse 𝑆𝑛 ∝𝑠𝑖𝑛 [2𝜔 (𝜏 −φ𝑛+1 −φ𝑛−12𝜔

− 𝜏𝑞𝑎𝑡𝑜𝑚 )]. The atomic delay 𝜏𝑞𝑎𝑡𝑜𝑚 can be calculated or ignored when𝜏𝑞𝑎𝑡𝑜𝑚 ≪ 𝜏𝑞𝐺𝐷 . This gives access to phase information of harmonics that can be combined with theamplitude information to reconstruct the temporal profile of the attosecond pulse train [32]. This
reconstruction technique is called RABBITT.
Attosecond streak camera (atto-streaking). While the RABBITT technique is used tocharacterize attosecond pulse trains, the attosecond streak camera was developed to characterize
isolated attosecond pulses. Similar to RABBITT, the isolated attosecond pulse photoemits theelectrons to the continuum in the laser field. The electron ejected into the laser field at the moment𝑒ti with initial velocity vi has a final drift velocity of 𝒗𝑓 = 𝒗𝑖 + 𝑚 𝑨(𝑡𝑖 ). In other words, the laser𝑒field modulates (“streaks”) the momentum of the photoelectrons in proportion to the laser vector
potential at the instant of ionization. Therefore, the laser-modulated electron-momentum
distribution can be changed by adjusting the delay between the attosecond pulse and the laser pulse.
This time-dependent electron-momentum distribution forms the streaking spectrogram and allows
us to retrieve the temporal shape of both fields from the spectrogram [34].

30

Phase retrieval by omega oscillation filtering (PROOF). The FROG-CRAB techniqueneeds to assume a narrow spectral bandwidth of the attosecond pulse for the retrieval algorithm tobe accurately and reliably used. For attosecond pulses with ultrabroad bandwidth, phase retrieval
by omega oscillation filtering (PROOF) was developed [31,111]. The PROOF method employs aquantum-path interference scheme similar to RABBITT. However, because of the continuous
spectrum, there is an additional quantum path of one-photon direct excitation besides two twophoton pathways to a given photoelectron final state (Figure 2.3c). The quantum-path interference
results in three components in the photoelectron momentum spectrum: 𝐼(𝒗, 𝜏) = 𝐼0 (𝒗) +𝐼𝜔 (𝒗, 𝜏) + 𝐼2𝜔 (𝒗, 𝜏), where ω is the frequency of the dressing laser field. 𝐼0 (𝒗) is the combinedtransition probability of three paths. 𝐼𝜔 (𝒗, 𝜏) and 𝐼2𝜔 (𝒗, 𝜏) oscillates with frequency ω and 2ω
respectively as a function of delay τ between the attosecond pulse and the laser pulse. The spectralphase of the attosecond pulse can be extracted from 𝐼𝜔 with proper algorithm.
(Ex-situ) tomographic reconstruction of attosecond pulses (TRAP). Most of theexisting methods of attosecond pulse characterization are for linearly polarized light. Circularly
polarized coherent EUV/X-ray has been demonstrated through several generation schemes
including using two counter-rotating circularly polarized driving light [68–70], using belowthreshold resonant HHG driven by elliptical laser pulse [114], and using bi-chromatic crosspolarized linearly polarized drivers [115]. Direct reconstruction of the temporal structure of thesethree-dimensional light pulses not only facilitates their applications to study magnetic matters andchiral materials, but also serves as a sensitive probe of the HHG medium under multi-color
ionization or below-threshold tunneling and recombination.

I demonstrate the first ex-situ

tomographic reconstruction of attosecond pulses in chapter 5 [30].

31

2.4.2

In-situ Characterization of Attosecond Pulses
Different from the ex-situ techniques that rely on the photoelectric effect in a downstream

setup, the in-situ characterization of attosecond pulses employs an additional second harmonic
field to directly probe the process of attosecond pulse generation [31,41,116,117]. The added field
is too weak to cause electron ionization but can perturb the HHG process. The perturbation
modifies the spectral patterns of the HHG radiation. As a result, the HHG pulse duration can be
retrieved by measuring the radiation modification as a function of time delay between the second
harmonic and fundamental IR field. For HHG under a single-color fundamental-driving field, theelectron trajectories in two consecutive subcycles are symmetric with respect to the atom, resulting
in a HHG spectrum with only odd-order harmonics. The addition of an orthogonally polarized
second harmonic field collinear with the fundamental field breaks the half-cycle symmetry,causing the emergence of even-order harmonics. In the microscopic single-atom picture, a given
even-order harmonic appears when the second harmonic field perturbs the electron trajectoryduring the temporal window between ionization and recombination, and maximizes when second
harmonic peak overlaps with the excursion of the electron in continuum. This relationship between
electron-trajectory modulation at second harmonic field strength allows the recombination time ofthe electrons to be extracted, corresponding to the emission time of attosecond pulses. Therefore,
the second harmonic field acts as a temporal gate in the in-situ characterization of attosecond
pulses.
(In situ) space-time reconstruction of attosecond pulses (STRAP). The in-situ
characterization method in principle cannot be applied to an isolated attosecond pulse, because thegeneration process in this case is confined in only one half-cycle. This difficulty is overcome byintroducing the second harmonic field at a small angle relative to the propagation direction of the32

fundamental field [31,41]. Because of the noncollinearity, the second-harmonic perturbation
modifies the electron trajectories not only in time, but also in space. Spatially, the second-harmonic
field changes the near-field wave front of the driving pulse and thus the far-field propagation
direction of the attosecond EUV/X-ray pulses. Measurement of the space-dependent EUV/X-ray
spectrum as a function of delay between two fields gives the space-time reconstruction of theattosecond pulses. Although in-situ characterization of isolated attosecond pulses including
STRAP is an all optical approach and sensitive to the quantum trajectories of electrons, it is also
an invasive probing technique and destroys the attosecond pulse structure for downstream
applications. On the other hand, it doesn’t require a bright attosecond pulse to produce adequate
photoelectrons, and offers several advantages compared to the attosecond streak camera.

2.4.3

Applications of Sub-cycle IR Electric Field Characterization
Attosecond pulse-characterization techniques, such as CRAB, not only provide temporal

information about attosecond pulses, but also give the full time-domain structure of the ultrafast
IR probe pulses. This capability of sub-cycle laser-field characterization is beyond the resolution
limit of the conventional FROG method based on nonlinear crystals. In addition, CRAB is not
limited by the bandwidth of nonlinear materials. This capability can help us acquire insights into
the propagation and modulation of an ultrafast laser field in materials on attosecond time scales
and atomic length scales [56,60,118]. For example, the sub-cycle temporal resolution was used to
study the attosecond nonlinear polarization and light-matter energy transfer in solids [119,120].
The nonlinear polarization, or light-matter interaction, changes the sub-cycle laser field structure
after the laser passes through the sample. The change was measured in the downstream attosecond

33

streak-camera setup. A similar method was applied to study the near field’s temporal response to
the ultrafast laser excitation in nanostructured materials [45].
The screening of laser field on material surfaces is another intriguing problem in near field
physics. In particular, screening at metal surfaces is a dynamical dielectric response of electrons
in materials to the incident laser field. Additionally, the combination of the incident and reflected
laser fields on a metal surface can create a dynamical 2D transient grating on the surface. The
effect of this grating on electron motion is no longer negligible at attosecond time scales. By using
the RABBITT method to measure the photoemission time delay of electrons emitted at different
angles into this transient grating, Lucchini et al. discovered that, surprisingly, the macroscopic
Fresnel equation can still reproduce the spatiotemporal profile of the electromagnetic field underultrafast electron screening [60,118]. In contrast, atto-streaking results claims that Fresnel
equations based on the macroscopic properties of a target cannot be applied at the near-field ofmetal-vacuum interface [56]. To fully understand the effect of an IR field at a metal-vacuum
interface on photoelectrons, we need to achieve enough time, angle and energy resolution to
distinguish the laser-field-induced and band-structure-related photoemission time delay asfunctions of electron kinetic energy and emission angle. This problem is discussed in Chapter 8.

2.5

Attosecond Time-Resolved Spectroscopy in Atoms
The full temporal information of attosecond pulses and IR pulses facilitates theirapplications in pump-probe spectroscopy to investigate attosecond processes in atoms, moleculesand solid-state materials. Atom represents the simplest form of matter. Attosecond spectroscopy
starts with atoms. Besides the importance of directly probing atoms, the knowledge gained fromstudying atoms also guides us in applying similar methods to molecules and solids. One example

34

is the study of photoemission time delay in atoms, molecules and solids. In atoms, a large
contribution to photoemission time delay comes from the scattering of photoelectrons with the
Coulomb potential, which is dominant over the time delay caused by electron-electron interactions.
For molecules, such scattering-induced delay is observed to be angle-dependent due to theasymmetric stereopotential [121]. For metallic solids, the Coulomb potential-induced delay is verysmall for valence electrons, allowing us to study the state-specific photoelectron lifetime and theeffect of many-body electron-electron interactions [35,122].

2.5.1

Photoemission Time Delay in Atoms
Attosecond streak camera was first used to measure the lifetime of 3d-1 core hole in krypton

during Auger decay [37]. A 3d-1 core hole was created in a krypton atom by an EUV pulse andsubsequently filled by a 4s electron. The released energy from core-hole filling was transferred tothe emission of a 4p Auger electron. The difference between the emission time of the Auger
electron and direct-photoemitted electron is the core-hole lifetime. To measure the core-hole
lifetime, an IR pulse was introduced and dressed both the direct-photoemitted 3d electron and the
4p Auger electron. The evolution of the electron kinetic-energy spectra as a function of relative
delay between two pulses revealed a 3d-1 core-hole lifetime of 7.9 fs. The localization of the corehole provides an element-specific probe to investigate the local electronic structure at a given
atomic site. However, when a core-hole is created by resonant excitation of a core-level electron
to an unoccupied electronic level, the core-hole lifetime can be influenced by localization of theexcited core electron. This dependence of core-hole lifetime and core-electron localization can beused to study the charge transfer between adsorbed atoms or molecules and the metal surface. The
direct time-domain study of core-hole lifetime of atoms on a metal surface was pioneered by

35

Miaja-Avila et al. through measuring the 4d-1 core-hole lifetime of xenon on Pt(111). The
measurement was enabled by their first observing LAPE on a metal surface [112]. By comparing
the LAPE sideband intensity as a function of pump-probe delay for the Pt d-band and Xe Auger
photoelectrons, the researchers measured the lifetime of the Xe 4d-1 core-hole to be 7.1 fs. Here,although the measured core-hole lifetime is in the femtosecond regime, the experiments usedpulses with attosecond structure.
Attosecond pulses can also be used to investigate attosecond excited-state lifetimes suchas the autoionizing states of noble atoms [59,123,124]. Autoionization results from the presenceof discrete states above the ionization threshold in atomic or molecular systems. These discrete
states are embedded in the continuum and leads to two quantum pathways for electron ionization,i.e., 1) direction ionization to continuum and 2) excitation to discrete states then coupled to thecontinuum. The interference between two quantum pathways lead to an asymmetric Fano intensity
profile as a function of the exciting-photon energy [125]. Direct time-domain study of Fano
resonances is a test bed for theories to address many electron interactions.
Fano resonance induces strong phase variations in an electron wave packet. This phase canbe regarded as a timing shift. Now the question is: does direct photoionization into the continuum
take time? To answer this question, one of the first experiments to measure the attosecond
photoionization delay was done in gas-phase neon atoms [42]. An isolated attosecond pulse < 200as centered at 106 eV photoionized neon atoms and the few-cycle IR pulse streaked thephotoelectrons. The streaking trace showed that 2p electrons were emitted at a 20 as delay ascompared to the 2s electrons. These attosecond delays shined light on the scattering ofphotoelectrons in the ionic potential. The spherical symmetry of the atomic potential facilitated
the theoretical calculations to understand the origin this delay. However, multiple efforts were36

performed [126–128], and so far they has underpredicted the delay by a factor of two. This
discrepancy between experiment and present theory indicates the possible influence of laser-atominteraction, shakeup and many-electron correlation in attosecond photoionization dynamics.

2.6

Attosecond Time-Resolved Spectroscopy in Molecules

2.6.1

Born-Oppenheimer and Single Electron Approximations
Compared to simple atoms, attosecond-photoinduced dynamics in molecular systems havea crucial influence in chemical or biological processes such as photosynthesis, vision and radiation
damage of biomolecules. It is well-accepted that the explanation of these ultrafast molecular
dynamics involves the breakdown of two conventional approximations [105].
The first one is the Born-Oppenheimer approximation, in which the electronic
configuration is assumed to adapt instantaneously to the nuclear motion since the nuclei have muchlarger masses than the electrons. However, this is true only when the energy difference between
electronic states of interest is sufficiently large compared to the energy contained in the nuclear
degrees of freedom. For example, it has been accepted in scientific community as a general rulerather than the exception that the Born-Oppenheimer approximation fails during nonadiabatic
dynamics when crossing conical intersections where the nuclear and electronic degrees of freedom
are strongly coupled and evolve at a few femtosecond-to-attosecond time scale [129]. This ultrafast
relaxation at conical intersections is responsible for the extraordinary photo stability of the DNA
double helix structure [130].
The second approximation is the independent electron approximation, i.e., the electrons aretreated as individual particles in a mean field generated by nuclei plus all other electrons. Thisapproximation is valid if we assume that electron-electron coupling is negligible compared to the37energy difference between relevant electronic states. However, this assumption can become invalid
during photoemission [105,131]. In photoemission, the sudden removal of an electron may leave
the cation in a non-eigenstate, i.e., a non-stationary state that is a coherent superposition of multiple
cationic eigenstates. These superposition states can lead to charge redistribution across themolecule on an attosecond-to-femtosecond time scale. This charge migration can be terminated byan additional IR pulse or the molecular nuclear dynamics. The termination will trap (localize) thehole and determine the subsequent reaction pathways [43]. This hole-localization-determined
reaction pathway is an example of how attosecond techniques manipulate the chemical reactions
through coherent control. Chemical reaction pathways can also be affected by molecular vibrations.
It has been experimentally demonstrated that certain electronic superposition states can beprepared to circumvent the intramolecular vibrational energy dissipation. Thus, coherent control
of attosecond charge dynamics is opening the door to engineering nonequilibrium reaction
pathways. These pathways can be investigated by pump-probe spectroscopy.
The first attosecond pump-probe experiment studied the intramolecular charge dynamics
in the simplest form of molecule, a hydrogen molecule [132]. An EUV pulse ionized H2 andpromoted it into an excited state. An intense few-cycle IR pulse with a controlled delay interactedwith the excited molecule and changed the localization of the remaining hole, thereby influencing
the emission direction of the neutral and charged fragments (H and H+) after molecular dissociation.
Moving to more complicated large molecules, Calegari et al. applied isolated attosecond and IR
pulses in a pump-probe geometry to track ultrafast hole migration in the amino acid phenylalanine
in the gas phase [43]. In the experiment, a single sub-300-as pulse with a spectrum covering 15 to
35 eV prompted sudden ionization of phenylalanine. A waveform-controlled 4-fs visible/IR pulse
ionized the molecule for the second time, causing it to dissociate into –COOH and doubly charged

38

immonium (++NH2-CH-R) fragments. The yield of ++NH2-CH-R oscillated as a function of pumpprobe delay with evolving frequencies. This pattern emerged because several beating processes
were initiated by the attosecond broad-band excitation. It showed that the attosecond charge
dynamics in large molecular systems are more complicated than simple atoms because of the
intricate energy levels that are intertwined with the vibrational and rotational motion of molecules.

2.6.2

Photoemission Time Delay in Molecules
Compared to atoms, the photoemission time in molecules has additional complexity

originating from the dependence of the photoionization matrix element on the orientation of themolecule. For hetero nuclear molecules, the photoelectron wave packet will accumulate a highly
structured scattering phase after travelling through an anisotropic scattering potential, leading to a𝜕𝜑Wigner delay of 𝜏𝑊 = ħ 𝜕𝐸𝑊 depending on the electron emission angle. Cattaneo and coworkers
observed this effect in the photoionization time delay of carbon monoxide (CO) by using the
RABBITT technique combined with a Cold Target Recoil Ion Momentum Spectroscopy(COLTRIMS) apparatus [121]. In this experiment, the electrons were expected to escape faster on
the O-Side compared to C-Side, in accordance with the steeper Coulomb potential on the O side.
However, this expectation is only true in experiments at low photon energies, a reversed behavior
is observed at higher photon energies. Catraneo et al. ascribe this observation to photoemission
from different dissociative states of CO at different photon energies [121]. This work by Catraneo
et al. shows that electron photoionization time in a molecule is influenced by both its electronicand structural properties.
Photoionization time delay can also be influenced by molecular resonances. These
resonances were only characterized in the frequency domain before the advent of attosecond

39

spectroscopy. For example, shape resonance occurs when the outgoing electron wave packet
matches the shape of the potential barrier formed by combining the molecular and centrifugal
potential felt by photoelectrons. Recently, this shape resonance was studied in time domain by
Huppert et al. They discovered through RABBITT measurements that the shape resonance in N2O
molecules can trap the photoelectron for a duration up to 110 as [133].

2.7

Attosecond Time-Resolved Spectroscopy in Solids
As with molecules, the Born-Oppenheimer approximation and the single-electron

approximation have been standards for describing the electrons in solid. Based on these two
approximations, the well-celebrated Sommerfeld model and Drude model are developed. In spiteof the effective determination of band structure and phonon frequencies in a wide range of metallic
systems, these approximated models are more a compromise rather than accurate. First, thebandgap is zero in metal. Therefore, the change in nuclear coordination can strongly influence theelectronic properties, and vice versa. This coupling between nuclei and electron is manifested inphonon-mediated superconductivity and phonon-induced renormalization of the electronic
dispersion. Second, phenomena beyond the single-electron picture occur in metal-to-insulatortransitions, charge screening, ultrafast demagnetization, and in high-temperature superconductors.
In these material systems, electron-electron many-body interactions play a pivotal role [134]. To
directly investigate the these interaction in time domain, attosecond spectroscopies performed
above and below transition temperatures may disentangle phonon, charge, and spin, as well as
orbital-assisted dynamics and temporally study the process of electron-electron interactions
responsible for the exotic electronic properties in solids.

40

2.7.1

Photoemission Time Delay in Solids
Although attosecond spectroscopy of solids is still in its infancy, impressive discoveries

have been made. The first attosecond-streaking experiment in solids was an attempt to determine
the escape time of electrons from tungsten (Figure 2.4) [44]. In the experiment, isolated EUV
pulses with a duration of ~300 as centered at 91eV were used to initiate electron emission from
W(110) surface. In the photoelectron spectrum, the spectral lines from 5d and 6s conduction-band
electrons are congested because of the broad bandwidth of isolated attosecond pulses (6-eV
FWHM). As a result, the photoelectron spectrum consists of two peaks: one at the kinetic energyof 56 eV corresponding to the emission from 4f core level electrons, and the other at 83 eVcorresponding to emission from integrated conduction bands. To time the electron emission, a fewcycle waveform-controlled 5-fs pulse at 750 nm was used to streak the photoelectrons. The
measured streaking spectrogram revealed a delay of 110 ± 70 as for the emission of 4f core-level
photoelectrons relative to the conduction-band photoelectrons from the W(110) surface. The paper
explained this delay through the variation in transport time of 4f (~150 as) and conduction-band
photoelectrons (~ 60 as) to the surface on the basis of two considerations. First, the mean groupvelocity of the conduction-band photoelectrons is approximately twice than that of the 4f corelevel photoelectrons. Second, 4f photoelectrons have a longer inelastic mean free path, originating
1 Å deeper than conduction-band photoelectrons.
This paper triggered multiple theoretical efforts to explain the origin of the measured delay[93,103,135–139]. Two main contradictory interpretations have been proposed. The first
explanation suggested that the delay is caused by propagation effects, and is supported bysemiclassical calculations [140] and subsequent quantum-mechanical calculations including theinterference effects of electron wave packets emerging from different layers [138]. The second

41

explanation ascribes the observed photoemission delay to the influence of initial-state electron
localization, i.e., strongly bound 4f core-level electrons are photoemitted later than delocalized
conduction-band electrons [103,141].

Figure 2.4: Photoemission time delay probed using isolated attosecond pulses. Schematic of
the photoemission delay measurements performed by Cavalieri et al. [44]. In the experiment,
isolated EUV pulses were used to initiate electron emission from solid surface. The broad
bandwidth of the EUV pules congested the spectral lines from different conduction-band electrons.
A few-cycle IR pulse was used to streak the photoelectrons. They measured the streaking
spectrograms for the emission of core-level photoelectrons relative to the conduction-band
photoelectrons. The streaking spectrograms revealed a delay for photoemission from core levels
compared to conduction bands.

42

The follow-up experiment was performed in magnesium (Mg) [142]. Two advantageous
properties of Mg rendered it a good material to discern different mechanisms responsible forphotoemission time delay. First, Mg is a free-electron metal, with the applied IR field effectively
screened at the surface, ensuring that the photoelectrons are only streaked after being liberated
from the bulk. Second, the conduction-band electrons are much more delocalized than the 2p corelevel states in Mg, creating a greater contrast between conduction-band vs core-level localization
than W. One would expect that similar to W(110), the photoemission delay should also be in the~100 as range. Surprisingly, no discernable relative delay within the experimental error wasmeasured between photoelectrons from the conduction band and from the core level. The paper
suggested this result was due to the offsetting effects of escape depth (mean free path) and escapevelocity. In Mg, higher (lower) kinetic-energy conduction-band (core-level) electrons are emitted
from larger (smaller) depth inside the bulk. Thus, the results of Mg favor the interpretation thatphotoemission delay originates from the propagation effects rather than initial-state localization[93,142]. In contrast, other theoretical work suggests a difference in photoemission time forresonant interband and nonresonant surface-emission processes [143]. To measure this difference,
we need to achieve enough energy resolution to distinguish various valence bands.
The energy resolution in attostreaking experiments is limited by the broad bandwidth
associated with isolated-single attosecond pulses. Therefore, multiple conduction bands are treatedas a block emission and referenced to core level states [44,142]. The other disadvantage is thatphotoemission time delay is mostly measured at one photon energy. Both implementations areproblematic given that the mean free path of photoelectrons is both band-specific and photonenergy dependent [144]. Finally, the extraction of time delay requires complicated algorithms that
may have errors or uncertainties in the retrieval processes [145]. This problem is less severe for43

gas atoms, because the photoelectron wave packet is a perfect replica of the incident EUV pulse
shifted by the photoemission time [42]. However, in solid-state materials, in addition tophotoemission time, the temporal structure of the photoelectron wave packet is modified becauseof electron transport and dispersion in solids [146]. This influence needs to be taken into accountin the retrieval algorithm of attostreaking.
In addition to attostreaking, RABBITT techinique can be used to investigate photoemission
time delay. Locher et al. first extended the RABBITT technique to study energy-dependent
photoemission delays from the noble metal surfaces Ag(111) and Au(111) [54]. The delays were
measured at four photon energies from sideband 16 to 22. Emissions from different conduction
bands, however, were convolved because the energy resolution was comparable to previous
attostreaking experiments. To reference the release time of this convolved photoelectron, Locher
et al. conducted the RABBITT measurement simultaneously on argon and metal surfaces in two
interaction regions. Co-propagating EUV and IR pulses were focused onto the argon gas at thefirst interaction region where a time-of-flight spectrometer recorded the RABBITT race. The
diverging beams were then refocused onto a metal surface at the second interaction region where
the hemispherical electron analyzer measured the electron kinetic-energy spectrum. The
photoionization time delay of Ar that had been widely investigated before served as a reference
for calibrating the spectral phase of the attosecond pulse train. However, the measured delay in Ag
and Au cannot be fully accounted for with initial state localization and electron transport. Locher
et al postulated that a final-state effect may play an important role.
This result has several limitations because of the measurement scheme. First, uncertainty
comes from the error bar (~100 as) presented in the time delay results. First uncertainty ocuurs
because the results comprise two RABBITT measurements on two different targets (Ar and metal

44

surfaces). Second uncertainty comes from the complicated-photoemission time delay from Ar.
Using Ar gas as the benchmark does not guarantee that the absolute photoemission time delay canbe extracted from the results. Recent experiments have shown that in the energy range probed by
Locher et al., the photoemission time delay from Ar can have very complicated energy-dependent
structures due to various resonance effects [104,147]. Third uncertainty comes from the systematic
errors introduced by additional EUV optics, even though the authors attempted to correct the timedelay caused by EUV dispersion and Ar through additional experiments and theoretical
calculations. Finally, the authors did not specify the momentum range and the integrated angle for
their results, which are extremely important for assigning band-structure effects.

2.7.2

Photoemission Delay Reveals Intrinsic Material Properties
The problems of previous attostreaking and RABBITT measurements are carefully

considered in Chapter 7 of this thesis work where I measure the time delay using “self-referencing”
strategy, i.e., referring the timing of the resonant photoemission to the off-resonant emission fromthe same metal surface (Figure 2.5) [35]. Using this technique, I unambiguously present the first
direct observation of the influence of the final-state band structure on the lifetime of high-energy
photoelectrons from multiple valence bands of Ni(111). We carefully investigated the energy-,
angle-, and polarization-dependence of measured photoemission time delays. Our results showthat photoelectrons experience an abrupt increase in lifetime by ≈ 212 as, when the ultraviolet(EUV)-photon-induced direct transition coincides with a final-state resonance in the Ni band
structure. Moreover, large angle- (momentum-) dependent variations in photoemission time delay
were also observed for the first time. This angle-dependent photoemission time delay are directly

45

related to the final-state band dispersion. This result represents the shortest final-state lifetime
measured directly in the time domain to date.

Figure 2.5: Photoelectron lifetime probed using attosecond pulses trains. Schematic of thephotoemission delay measurements performed in Chapter 7-8 of this thesis. In the experiment,
attosecond EUV pulse trains were used to initiate electron emission from solid surface. The narrow
bandwidth of the EUV pulse trains is able to distinguish spectral lines from different conductionband electrons. A IR pulse was used to dress the photoelectrons. We measured the RABBITT
spectrograms for the emission of multiple conduction-band photoelectrons. The measurement
revealed the influence on photoelectron lifetime from both unoccupied final states and occupied
initial states.

46

After the final-state effects are taken into account, we can investigate the influence ofoccupied-band dynamics to photoemission time delay. These dynamics include electron-electroninteractions. Electron-electron interactions are among the fastest processes in materials, playing
prominent roles in strongly correlated electron systems and quantum materials. In recent years,
these interactions have become accessible to direct time-domain studies by using femtosecond
lasers in techniques such as time-resolved two-photon photoemission experiments. However,measurements using infrared or visible wavelength light are limited to the electron-electroninteractions at the low-energy excited states with femtosecond time scales, while many electronelectron interactions can happen on faster attosecond time scales.
Chapter 8 of the thesis presents the measurement of attosecond electron-electroninteractions in metals with time-, energy-, angle-, and polarization-resolved photoemission. In
particular, we extract the time delays associated with photoemission from occupied bands inNi(111) and Cu(111) into free-electron final states. This strategy allows us to demonstrate thatphotoemission from the occupied d band of Cu into free-electron final states is delayed by ~100
as compared to photoemission from the same band of Ni. We attribute this difference to the factthat the d band in Ni is half-filled, resulting in enhanced electron-electron scattering during
photoemission. Moreover, we present a unified understanding of electron-electron interactions intransition metals across a broad energy range (0.5~40eV), by comparing attosecond photoemission
measurements to the results from previous two-photon photoemission experiments.

2.7.3

Attosecond Control of Charge Dynamics in Solids
Photoemission is a surface-sensitive technique because of the low escape depth of electrons.

Complementarily, attosecond transient-absorption spectroscopy is implemented to study the field-

47

induced subcycle interband transitions in bulk solids. The first experiment was performed on
silicon dioxide (SiO2) where a 72-as isolated-EUV pulse tracks the change in silicon’s L-edge
absorption (2p to conduction band) after excitation by <4 fs 780 nm waveform-controlled IR pulse[40]. The experiment reveals both a transient reduction in the L-edge absorption and a red shift ofthe conduction-band edge. Both changes are driven by the instantaneous electric field and therefore
oscillate at twice the frequency of the driving field. The variation in L-edge absorption follows thetransient change in conduction-band population. The shift of the conduction-band edge isidentified as a dynamic stark shift resulting from the field-induced polarization in the material. The
Stark shift of the bands brings interband transition into resonance with the IR laser excitation andamplified the increase in the conduction-band population.
A similar attosecond transient-absorption experiment was performed on the semiconductor
Si, which is different from the dielectric SiO2 [36]. In Si, an IR pulse excited the valence-band
electrons across the bandgap and caused a change in the Si L-edge absorption. This change survives
the duration of the pulse, indicating that the lifetime of the excited electron is longer than thesubcycle duration of the IR pulse. The most fascinating result is the step-like increase in the
conduction band population following the doubling of the frequency of the driving field, indicating
a carrier-carrier interaction faster than the half period of the driving pulse. These results
demonstrated that electron dynamics can be manipulated by a strong field at a frequency
corresponding to a half-cycle period (~1.3 fs) up to ~800 THz. In the quest of manipulating theelectron dynamics at a faster speed, Mashiko et al. later broke the 1-PHz barrier in the ultrafast
control of electron motion in solid-state material by using the third-order nonlinearity of a wideband-gap semiconductor GaN. The band gap of GaN is 3.35 eV, requiring three IR photons (ħωL
= ~1.6 eV) to complete the first resonant interband transition. Therefore, the third-order interband

48

polarization dominates the IR excitation, leading to a modulation in the absorption of the isolated
attosecond pulse with a periodicity of TL/3=860 as, corresponding to triple frequency of the IR
field. This process can be described as a quantum beating between two channels coupling valenceband electrons to continuum states. These two channels include 1) direct excitation by isolated
attosecond pulse and 2) absorption of three IR photons to complete the interband transition
followed by an additional EUV photon absorption.
In addition to interband transitions, attosecond-intraband motion of charges in dielectrics
was studied using attosecond transient-absorption technique [33]. The system studied ispolycrystalline diamond. First, a waveform controlled 5-fs IR pulse pumped the sample. Second,an EUV single attosecond pulse spanning 30 to 55 eV probed the dynamics through absorption.
The absorption features oscillated with twice the frequency of the driving field over the whole
bandwidth of the single attosecond pulse, for small values of time delay. The phase of theoscillation was photon-energy dependent and assumed a V-shape with a vertex at 43 eV. TimeDependent Density Functional Theory (TD-DFT) calculations showed that the oscillation resulted
from the pump-induced change in the imaginary part of the dielectric function and hence in the
EUV absorption of diamond. Since the V-shaped energy dispersion of oscillation can be
reproduced by using two IR-dressed sub-bands, the main IR-induced dynamic in diamond is likely
due to intraband rather than interband transitions.
These experiments show that attosecond spectroscopy can be applied to investigate
electron dynamics in metals, dielectrics, and semiconductors. These applications provide us
valuable insights into electron-electron and electron-phonon interactions as well as allowing us toinvestigate tunneling or multiphoton processes during IR-excitation.

49

2.8

Organization of the Thesis
In Chapter 3, the relevant theoretical background will be presented regarding thetechniques used in this thesis work: photoelectron spectroscopy and high harmonic generation.
Chapter 4 shows the experimental apparatus of attosecond photoelectron spectroscopy. It describes
the laser system, EUV/soft x-ray generation setup, monochromator design, and end station. The
endstationisequippedwithavarietyofspectroscopic

detectors

andsamplepreparation/characterization tools.
Chapters 5-6 present the application of attosecond metrology in characterizing the temporal
structure of attosecond pulses. In Chapter 5, 3D attosecond pulse trains of circularly polarizedharmonic field are reconstructed by combining the interferometric laser-assisted photoemission
technique with the ability to rotate the 3D attosecond field via controlling the high harmonic
generation process. Chapter 6 describes the measurement of a linearly polarized attosecond pulse
train generated by two-color laser pulses. By controlling the phase delay between the two drivers,
the sub-attosecond pulse structure can be manipulated. More importantly, the phase-matchingeffect can confine most of the EUV emission to one burst, giving rise to quasi-isolated single
attosecond pulses.
Chapters 7-8 then discuss the investigations of the attosecond electron dynamics intransition metals including nickel and copper. Here I want to emphasize that even though this is
an attosecond experiment, the strong field influence is minimal. The goal is to unravel several
central mysteries of photoemission identified since the invention of this technique. What is the
lifetime of the highly excited states (>20 eV above Fermi) in solids and how does the resonant
excitation of these states influence the photoelectron lifetime? What is the velocity ofphotoelectrons when they are travelling inside solids before being liberated into space? How does

50

the many-body interactions like electron screening and scattering influence the photoelectron
lifetime? Are the interactions spin dependent? We will consider the first two questions in Chapter
7 with measurements of the dependence of photoelectron lifetime on incident photon energy innickel. Chapter 8 answers the latter two questions by comparing the photoelectron lifetime innickel and copper. Chapter 8 also shows how attosecond measurement can inform us about the
coupling between initial and final states during photoemission.
Chapter 9 summaries the thesis and describes future work that would take the attosecond
metrology to another level.

2.9

Conclusion
The rapid progress in attosecond science over the past decades provide us with access tothe fastest electronic dynamics. Understanding the microscopic motion of electrons advances thecontrol over tabletop EUV/X-ray generation and attosecond light sources. Advanced attosecondsources coupled with state-of-the-art spectroscopy techniques reveal electron transport, thelifetime of highly-excited states, and electron-correlation dynamics. Extending attosecond studies
to more complicated systems like surface adsorbates and strongly correlated systems may shedlight on the central mysteries of photovoltaic and superconductivity. The development of waterwindow attosecond sources will grant us the capability to study carbon-based organic molecules
and ultrafast biology such as DNA damage and repair.

Chapter 3
General Background

This chapter outlines the theoretical and technical background for two common techniques
used in this thesis: angle-resolved photoemission (ARPES) and high harmonic generation (HHG).

3.1

Photoemission Spectroscopy
The photoelectric effect was discovered by Hertz in 1887 during experiments with a spark-gap generator (the earliest version of radio receiver). Although the initial observation that theelectric sparks were easier to create when electrodes were illuminated with ultraviolet light did not
attract much interest, later findings of the so called “threshold frequency” failed the classicaltheory of electromagnetic radiation. In 1905, Einstein provided his revolutionary explanation at
the dawn of quantum mechanics. Quantum mechanics, on the other hand, provides the theoreticalfoundation for the invention of lasers, which are used for time-resolved pump-probe photoemission
spectroscopy to study the laser-matter interaction. Again, laser-matter interaction can produce
attosecond pulse, which is recently ultilized to interrogate the fundamental timescale ofphotoemission [35,42,44].

52

3.1.1

Photoemission: from One-Step to Three-Step Model

Figure 3.1: Illustration of one-step model and three-step model of photoemission. (a) The onestep model directly calculates the transition dipole from the initial state to final state. The final
state wavefunction include all the possible interactions with the rest of the system. (b) In threestep model, the photoemission process is separated into three steps of (1) optical excitation, (2)electron transport to surface, and (3) escape of photoelectron into the vacuum. The E axis
represents the electron energy and z axis denotes the depth into the bulk. Adapted from [148].

The complete description of photoemission in the language of many-body physics should𝑁
be a one-step process [148]: electrons are excited from the initial state ψ𝑁𝑖 to the final state ψ𝑓 =ψ𝑁𝛋,𝑠 with the transition probability given by Fermi’s golden rule as a result of perturbation theory
in the first order:
2𝜋

2

𝑁
𝑁
𝑁
𝜔𝑓𝑖 (𝒌, ω) = ħ ∑𝑠|⟨ψ𝑁𝐤,𝑠 |𝐻𝑖𝑛𝑡 |ψ𝑖 ⟩| 𝛿(𝐸𝐤 − 𝐸𝑖 − ħω),

(3.1)

where ψ𝑁𝑖 is one of the possible initial N-electron eigenstates characterized by the eigenvalue𝑁
𝐸𝑖𝑁 while the final state ψ𝑁𝛋,𝑠 is one of the eigenstates with eigenvalue 𝐸𝐤 containing the escaping

photoelectron and the remain (N-1)-electron system; the index s represents a set of quantum
numbers that incorporates all possible interactions in the final state of the system, including phonon

53

absorption, plasmons, electron-hole pair, scattering and multiple excitations. Apparently, this ideal“one step model” (Figure 3.1a) matrix element is inherently difficult to solve and an alternative
simpler model divides the process into three steps, isolating each event experienced by theelectrons [148,149]:(1) Optical excitation of the electron within the bulk.
(2) Travel of the excited electron to the surface.
(3) Escape of the photoelectron into the vacuum.
Under the three step model (Figure 3.1b), the photoexcitation step (1) is isolated from thescattering and transmission processes and the transition probability is calculated between the initialand final Bloch states within the crystal. For the final state, a very central simplification, known
as the sudden approximation, has to be made. It means the photoexcited electron doesn’t interact
with the remaining (N-1)-electron solid. With this simplification, Equation 3.1 can be written as
2𝜋2𝜔𝑓𝑖 = ħ |∆𝑓𝑖 | 𝐴< (𝒌, 𝜔),

(3.2)

2

where ∆𝑓𝑖 = |⟨ψ𝑓 |𝑨 ∙ 𝒑|ψ𝑖 ⟩| is the photoemission matrix element and 𝐴< (𝒌, 𝜔) is the one
electron spectral function, which is connected to imaginary part of the single-electron Green’s
function by
1

𝐴< (𝒌, 𝜔) = − 𝜋 𝐼𝑚{𝐺𝒌 (𝐸𝒌 − 𝑖0+ )} ∙ 𝑓(𝐸, 𝑇),(3.3)

with 𝑓(𝐸, 𝑇) being the Fermi-Dirac distribution. The Green function is
𝐺𝒌 (𝐸𝒌 ) =1,

𝐸𝒌 −𝜖𝒌 −∑(𝒌,𝐸𝒌 )(3.4)

𝜖𝒌 is the electronic dispersion in the absence of many-body interactions and is often termed as the“bare dispersion”. ∑(𝒌, 𝐸𝒌 ) is the complex electron self-energy including all the contributions

54

from many-body processes like electron-electron, electron-phonon or electron-impurity
interactions that determine the intrinsic quasi-particle spectrum or photoemission line shape:∑(𝒌, 𝐸𝒌 ) = 𝑅𝑒{∑(𝒌, 𝐸𝒌 )} + 𝐼𝑚{∑(𝒌, 𝐸𝒌 )},(3.5)

By plugging Equation 3.4 and 3.5 into Equation 3.3, we get the spectral function[𝐼𝑚{∑(𝒌,𝐸 )}]1𝒌
𝐴< (𝒌, 𝜔) = 𝜋 |𝐸 −𝜖 −𝑅𝑒{∑(𝒌,𝐸 )}|2 +|𝐼𝑚{∑(𝒌,𝐸
,
)}|2
𝒌

𝒌

𝒌

𝒌

(3.6)

Step (2) represents the scattering of photoexcited electrons on their way to the surface. As
a result, most of the electrons suffer from an energy or momentum loss and cannot make it to thesurface. The electrons have an exponentially decaying probability of escaping the solids dependingon their distance to the surface
𝑃(𝑑) = 𝑒 −𝑑/λ𝑀𝐹𝑃 ,(3.7)

The characteristic length λMFP is called the escape depth of the electrons, which is also referred toas electron’s inelastic mean free path mainly dictated by electron-electron scattering. The
dependence of λMFP on electron’s kinetic energy is named the “universal curve” [150] and plottedin Figure 3.2. For the high harmonic photons used in this thesis work, the electron kinetic energy
is about 20~100 eV and the escape depth is around 5Å. In this case, photoemission is said to be
surface-sensitive. On one hand, this can serve as an advantage for probing atomic layer structures
like graphene or surface adsorbate system. On the other hand, the photoemission measurements
for bulk materials might be a mere a projection of the band structure on the surface and may notreflect the intrinsic bulk properties.

55

Figure 3.2: “Universal Curve”. The curve indicates the relation between the electron mean freepath and the kinetic energy of the photoelectrons in the solid. Reproduced from [151], and data
originally from [150].

3.1.2

Direct Imaging of Electronic Bands
Energy Conversion. According to energy conservation (i.e. the Dirac function intransition probability), the kinetic energy of a photoelectron is related to the photon energy ħʋ via
the following equation
𝐸𝑘 = ħʋ − 𝐸𝐵 − Φ𝑠 ,(3.8)

where Φ𝑠 is the sample work function and 𝐸𝐵 is the binding energy, i.e., the energy of the electron
in solid relative to the Fermi level. However, the determination of work functions in photoemission
spectra is more complicated.
Measurement of the Sample Work Function. During the last two stages of the “threestep model”, the majority of the generated electrons suffer from inelastic collisions, which resultin a decrease in their kinetic energy. Hence photoemission spectra are composed of two main
components: (1) the primary electrons without suffering inelastic collisions, which is of main56

interest in the spectra because they show spectral features directly reflecting the density of statesof the sample, and (2) secondary electrons, which lose varying amount of energy in collisions andconstitute a continuous energy background down to zero kinetic energy. As shown below, acomplete characterization of the work function requires the interplay between both features [152].
Since the sample and spectrometer are in good electrical contact hence equilibrates theirFermi levels in practical measurements of electrons’ kinetic energy, thus, emitted photoelectrons
suffer from the resultant “contact potential (∆Φ = Φ𝑠 − Φ𝑑 , Φ𝑑 is the detector work function)”and accelerate (or decelerate) on their way to the detector. Therefore, the measured kinetic energy
for primary electrons is:
𝐸𝑘𝑚𝑒𝑎𝑠 = ħʋ − 𝐸𝐵 − Φ𝑠 + (Φ𝑠 − Φ𝑑 ) = ħʋ − 𝐸𝐵 − Φ𝑑 .

(3.9)

The deducted work function for the primary electrons is the one of the electron detector. The high𝑚𝑒𝑎𝑠
cutoff of the spectra (assuming a metallic sample here) is 𝐸𝑘,𝐻𝑐= ħʋ − Φ𝑑 from electrons with

𝐸𝐵 = 0. For the secondary electron continuum, the slowest electrons are those barely made it to
the sample surface with 𝐸𝑘𝑖𝑛 = 0, and subsequently gain/lose their energy through the “contact
potential”. Applying a bias voltage between sample and detector allows for the detection of these
low-edge electrons even when they are repelled by the contact potential. These electrons form the𝑚𝑒𝑎𝑠
low cutoff of the spectra with 𝐸𝑘,𝐿𝑐= Φ𝑠 − Φ𝑑 . The sample work function can be calculated

from the high and low energy cutoff of the spectra:
𝑚𝑒𝑎𝑠
𝑚𝑒𝑎𝑠
Φ𝑠 = ħʋ − (𝐸𝑘,𝐻𝑐− 𝐸𝑘,𝐿𝑐).

(3.10)

Momentum Conversion. Angle-resolved Photoemission (ARPES) is a leading tool inpushing the frontier of complex phenomenon in solids like High-Tc superconductors and chargedensity-wave materials. The addition of angular detection of photoelectrons provides a way of
studying the electronic dispersion or taking a “snapshot” of the bandstructure in the solid. Since

57

the momentum parallel to the sample surface is conserved during photoemission because oftranslational symmetry:𝒌∥ + 𝒌ℎʋ = 𝑲∥ ,(3.11)

where 𝒌∥ and 𝑲∥ are momentum component parallel to the sample surface for bound andphotoemitted electron, respectively. 𝒌ℎʋ is the momentum of the incident photon, which can beneglected for low photon energies used in most of ARPES experiments (ℎʋ < 100 𝑒𝑉) . Then wehave:𝒌∥ = 𝑲∥ .

(3.12)The photoelectron momentum 𝑲 is usually obtained in terms of the polar (𝜃) and azimuthal(𝜑) emission angles:𝑲∥ = 𝑲x + 𝑲y =

𝐾x =

𝐾𝑦 =

√2𝑚𝑒 𝐸𝑘sin𝜃ħ√2𝑚𝑒 𝐸𝑘sin𝜃cos𝜑ħ(3.13)√2𝑚𝑒 𝐸𝑘sin𝜃sin𝜑ħ𝐾z =√2𝑚𝑒 𝐸𝑘ħcos𝜃.

Determination of 𝒌⊥ (Band Mapping). Electron momentum perpendicular to the sample
surface k⊥ is not conserved because the potential change across the surface breaks translational
symmetry. However, the value of k⊥ is required for a complete 3D band mapping of the electronic
dispersion 𝐸(𝒌) [153]. There are several specific experimental methods developed for this purpose,
which are mostly complicated and/or need other complementary measurements. Alternatively, 𝒌⊥
can be calculated by knowing the dispersion of the final bulk Bloch states, which can be obtained

58

either from bandstructure calculations, or more commonly, by just assuming a simple free electron
dispersion:
ħ2 𝒌2

𝐸𝑓 (𝒌) = 2𝑚 − |𝐸0 | =𝑒ħ2 (𝒌2∥ +𝒌2⊥ )2𝑚𝑒− |𝐸0 |.

(3.14)

𝐸0 corresponds to the bottom of the valence band as indicated in Figure 3.3. Note that both 𝐸0 and𝐸𝑓 are referenced to the Fermi level 𝐸𝐹 , while 𝐸𝑘 is referenced to the vacuum level 𝐸𝑉 . Given 𝐸𝑓 =𝐸𝑘 + Φ𝑠 and ħ2 𝒌∥ 2 /2𝑚𝑒 = 𝐸𝑘 sin2 𝜃, we have:1𝒌⊥ = ħ √2𝑚𝑒 (𝐸𝑘 cos2 𝜃 + 𝑉0 ) ,(3.15)

where 𝑉0 = |𝐸0 | + Φ𝑠 is the inner potential, which corresponds to the bottom of the valance bandreferenced to the vacuum level. At normal emission geometry where 𝜃 = 0 thus 𝒌⊥ =1
ħ√2𝑚𝑒 (𝐸𝑘 + 𝑉0 ) , the mapping of 𝐸(𝒌⊥ ) can be achieved by measuring the photoelectrons as afunction of incident photon energy. There are three commonly used methods to determine 𝑉0: (1)optimize 𝑉0 to match the experimental measurements with theoretical calculations; (2) set 𝑉0 as
the theoretical zero of the muffin tin potential applied in band structure calculations; (3) infer 𝑉0from the experimentally observed periodicity of the dispersion 𝐸(𝒌⊥ ) [153].
There are several cases where 𝒌⊥ is less relevant. (1) 2D materials like graphene has avanishing 𝒌⊥ component, which can be ignored. (2) Surface state, as opposed to bulk-related states,shows no dispersion in 𝐸𝑘 (𝒌⊥ ) because one can think of the surface state as being present for all
values of 𝒌⊥ . (3) For 3D materials with large c-axis vector, 𝒌⊥ is almost independent of 𝐸𝐵 or 𝐸𝑘
and it is also reasonable to assume 𝒌⊥ is constant and we can focus only on the in-plane electronic
behavior.

59

Figure 3.3: Illustration of the inner potential in band mapping. In direct optical excitationfrom initial state to free-electron final state in vacuum, the lattice supplies the required momentum
G. The inner potential V0 corresponds to the bottom of the valence band E0 referenced to thevacuum level EV. Adapted from [153].

3.1.3

The Influence of Brillouin Zone in Photoemission
Except amorphous solids and glasses, most solids have periodic arrays of atoms which

form a crystal lattice. The existence of the crystal lattice implies a characteristic symmetry undera combination of one or more translation-, rotation- and inversion- operations. This symmetry canbe represented by the Bravais lattice, which is an infinite array of discrete points in 3D real space
generated by a set of discrete translation operations described by:
𝑹 = 𝑛1 𝒂𝟏 + 𝑛2 𝒂𝟐 + 𝑛3 𝒂𝟑 .

(3.16)

where ni are integers and ai are primitive vectors. {R} should be closed under vector addition andsubtraction. All points of lattice defined by R(n1, n2, n3) are in the same environment.

60

Figure 3.4: Surface Brillouin zone for crystals with face centered cubic (fcc) structure. From[154].

The reciprocal lattice is a construction with significant importance for condensed matterphysics. It represents the Fourier transform of the real-space lattice to the reciprocal space (ormomentum space). Starting with the Bravais lattice, the reciprocal lattice is a group of wave
vectors G that give plan waves 𝑒 𝑖𝑮∙𝒓 with the same translation symmetry as the Bravais lattice:𝑒 𝑖𝑮∙(𝒓+𝑹) = 𝑒 𝑖𝑮∙𝒓.

(3.17)The reciprocal lattice can be determined through the formula: 𝑮 = 𝑚1 𝒃𝟏 + 𝑚2 𝒃𝟐 +𝑚3 𝒃𝟑 , where mi are integers and bi are reciprocal primitive vectors defined as:
𝒃𝟏 = 2𝜋𝒂𝟐 × 𝒂𝟑
𝒂𝟏 ∙ (𝒂𝟐 × 𝒂𝟑 )𝒃𝟐 = 2𝜋

𝒂𝟑 × 𝒂𝟏
𝒂𝟏 ∙ (𝒂𝟐 × 𝒂𝟑 )𝒂𝟏 × 𝒂𝟐
𝒃𝟏 = 2𝜋.
𝒂𝟏 ∙ (𝒂𝟐 × 𝒂𝟑 )(3.18)The Wigner-Seitz cell of the reciprocal lattice (constructed as the set of points enclosed by
the nearest Bragg planes, i.e. the planes perpendicular to a connection line from the origin to each

61

lattice point and passing through the midpoint) is called the Brillouin zone. Photoemission with afixed photon energy probes a projection of the band structure at the surface Brillouin zone with
fixed 𝒌⊥. The surface Brillouin zone for fcc (face-centered-cubic) lattice structure, which is the
crystal structure of the materials mostly used in this thesis work [Cu(111) and Ni(111)], is plottedin Figure 3.4 [154].
Note that no k-conversing optical transition along the free-electron parabola is possible in
the limit 𝒌ℎʋ = 0. The periodic lattice can provide the external momentum in bulk transitions. In
the extended Brillouin zone scheme, the transition is connected by the reciprocal-lattice vector𝒌𝑓 − 𝒌𝑖 = 𝑮 , which can be simplified as a vertical transition in the reduced Brillouin zone scheme𝒌𝑓 − 𝒌𝑖 = 0. The problem is different when the surface is taken into account in the semi-infinite
crystal picture. While the 𝒌∥ is conserved within a reciprocal-lattice vector because of thetranslational symmetry of the surface, 𝒌⊥ is not conversed across the surface due to the potentialchange. Thus, the surface acts like a momentum reservoir delivering the necessary momentum forindirect transitions even in the absence of the crystal potential (i.e., the so-called surface
photoelectric effect) [153].

62

3.1.4

The Matrix Element EffectFigure 3.5: Illustration of matrix element effect in photoemission. 𝒌𝑖 and 𝒌𝑓 are the initial andfinal state momentum. The pink color indicates the mirror plane that contains 𝒌𝑖 , 𝒌𝑓 , and the
Poynting vector of the light. The initial-state wavefunction is even relative to the mirror plane.
Therefore, (a) when the incident light field is s-polarized, the light field is odd with respective to2
the mirror plane. In this case, the matrix element |⟨ψ𝑓 |𝑨 ∙ 𝒑|ψ𝑖 ⟩| vanishes. (b) p-polarized lightfield is even to the mirror plane and leads to a non-zeros matrix element. Adapted from [155].

63
2

The matrix element ∆𝑓𝑖 = |⟨ψ𝑓 |𝑨 ∙ 𝒑|ψ𝑖 ⟩| represents the transition probability between
the electron wavefunction in solid |ψ𝑖 ⟩ (or “the initial state”) and the photoexcited electron |ψ𝑓 ⟩(or “the final state”) [155,156]. 𝒑 is the electron momentum and 𝑨 is the electromagnetic gauge,
which is related to the electric field 𝑬 through 𝑬 = −𝜕𝑨/𝜕𝑡, thus shares the same spatial mirror
symmetry as 𝑬. The initial state wavefunction carries information about the electron’s orbital. The
final sate wavefunction is usually approximated as a plane wave propagating into vacuum with thephotoelectron momentum 𝒌𝑓 . This approximation is usually valid for sufficiently high photon
energies at or above tens of eV.
The matrix element is so complicated to calculate that it is mostly treated as a mysteries elf
controlling the amount of electrons released into vacuum upon the arrival of light. There are,however, special cases where we can use symmetry analysis to determine whether certain
electronic states will contribute to the photoelectron spectra under the illumination of polarized
light. Since the matrix element includes integration over all spatial dimensions, it goes to zero if
the product ⟨ψ𝑓 |𝑨 ∙ 𝒑|ψ𝑖 ⟩ is odd with respect to a particular mirror plane. We demonstrate thissymmetry analysis by studying two examples in Figure 3.5. To simply the problem, we deliberately
put the momentum of the initial (𝒌𝑖 ) and final (𝒌𝑓 ) states in the plane of incidence, which is alsochosen as the mirror plane here. The in-plane momentum 𝒌∥ = 𝒌𝑖 and the final state wavefunction|ψ𝑓 ⟩ are all even to the mirror plane. For s-polarized light, the electric field E (and thus the gauge
field A) is odd with respect to the mirror plane. if the orbital wavefunction |ψ𝑖 ⟩ is odd to the mirror
plane, the overall mirror symmetry of (ψ𝑓 ∙ 𝑨 ∙ 𝒑 ∙ ψ𝑖 ) is odd and the matrix element vanishes.
For p polarized light, the electric field E is even with respect to the mirror plane and matrix element
is nonzero only when the orbital wavefunction |ψ𝑖 ⟩ is even regarding the mirror plane. To sum up:

64
2|⟨ψ𝑓 |𝑨 ∙ 𝒑|ψ𝑖 ⟩| = 0 {|ψ𝑖 ⟩ even ⟨+|−|+⟩ → 𝑨 odd|ψ𝑖 ⟩ odd ⟨+|−|+⟩ → 𝑨 even

Using the symmetry analysis, the dipole-allowed initial states for normal emission from
low-index faces of cubic metal are listed in Table 3.1 [157]. The spin-orbit coupling is ignored forsimplicity. The z axis the normal to the sample surface. Note that each polarization x, y or z excites
only one initial symmetry. This selective excitation of the initial states with different symmetry bychanging the light polarization (the so called “photoemission selection rule”) is heavily used inthis thesis work.
CrystalCoordinate Axes

Irreducible

Allowed Initial Symmetries

Face

x

y

z

Representations

𝐄∥x

𝐄∥y

𝐄∥z

(001)

<100>

<010>

<001>

∆1 ∆1′ ∆2 ∆′2 ∆5

∆5

∆5

∆1

(110)

<001>

<11̅0><110>

Σ1 Σ2 Σ3 Σ4

Σ3

Σ4

Σ1

(111)

<1̅10><1̅1̅2><111>

Λ1 Λ 2 Λ 3

Λ3

Λ3

Λ1

Table 3.1: Dipole-allowed initial state symmetries for transitions in normal emission
geometry from low-index cubic metals. Adapted from [157].

3.1.5

EDC and MDC
An example of photoemission spectrum of Na-intercalated graphene on Ni(111) substrateis shown in Figure 3.6a, where the well-celebrated Dirac-cone is shown. The intensity map
indicates the normalized (measured) density of states as a function of k∥ and binding energy. Figure
3.6b shows the bandstructure calculated by density function theory (DFT). This is an illustrationthat photoemission spectra are not a mere “shift” of the electron density of states in the solid.
Matrix element effect can generate an intensity modulation that may change with photon energy
and polarization.

65

There are two useful ways of slicing the 2D photoemission spectra up for analysis. (1) The
ARPES spectrum as a function of energy at a fixed momentum or integrated over certain
momentum range is called energy distribution curve (EDC). The EDCs of the photoemission
spectra of Na-intercalated Graphene/Ni(111) are shown in Figure 3.6c. (2) Similar to EDC,
Momentum distribution curve (MDC) is ARPES spectrum as a function of momentum at a constantkinetic energy or over a certain kinetic energy range.

Figure 3.6: Photoemission spectrum of Na-intercalated graphene on Ni(111) substrate. The
spectrum is measured along Γ-K direction. (a) Hybridization of the π state near the K-point ingraphene/Ni(111). Black vertical line (dashed) indicates position of the K-point, as establishedfrom the minimum of the 3 state of graphene; a manifold of states arising from the hybridizationof the Ni 3d and graphene π states is indicated with the bracket. (b)DFT calculated states near the
K-point. 2pz states of graphene are highlighted in red. (c) Corresponding EDC lineouts of (a).
Graphene states are indicated with symbols and the EDC lineout at the K-point is indicated as red.
Weak feature near the Fermi level is a hybridized state between Ni and graphene, which is alsovisualized in (b).

66

3.2

Theory of High Harmonic Generation
High harmonic generation (HHG) takes place when intense linearly polarized laser pulse

is focused onto a target of noble gas. During the process, the driving laser light at a given frequency
is converted into odd integer multiples of this fundamental frequency. In HHG spectrum, following
the rapidly decreasing low-order harmonics, there exists a plateau region where higher orderharmonics are produced with almost equal intensity extending to the so called cut-off region, where
harmonic intensity drops sharply again [158].
The occurrence of a “plateau region” was not immediately understood with the birth of
HHG [159,160] because under the scope of the established perturbation theory that well described
the low-order harmonic generation, the probability of a n-photon excitation decreases
exponentially with its order n. This unexpected highly nonlinear behavior was realized later to berelated to the enormous applied laser strength comparable to the binding energy of electrons inside
the atom. A generic intensity scale for laser-atom interactions is the so called atomic unit ofintensity: Iatom=3.5×1016 W/cm2. The laser source commonly used to drive HHG are Ti:sapphirelaser system operating at IL≈ 1014 W/cm2 therefore cannot be treated as small perturbation to thesystem [161].

3.2.1

Three-Step Model
Although a complete quantum description of HHG is based on numerically solving thetime dependent Schrödinger equation (TDSE), a more convenient semi-classical theory proposedby Kulander and Corkum explained the mechanism of the process via the celebrated “three stepmodel” [63,64]:

67(1) Ionization. A bound electron is driven away from the atomic Coulomb potential of thenucleus when the electric field is close to its peak during an optical cycle, and tunnel
into the continuum.
(2) Acceleration in the laser field: once freed, the electron is first accelerated away fromthe nucleus in the laser field like a classical particle. When the oscillating electric fieldswitches its direction about a quarter of a period after electron ionization, the electron
decelerates and then re-accelerates back towards its parent ion.
(3) Recombination: There is a small probability for the ionized electron to recombine with
its parent ion. Upon relaxation to the ground state, the kinetic energy the electron gains
during its excursion in the laser field plus the ionization potential can be released in theform of high energy photons, much larger than the fundamental photon energy.
A detailed analysis of the electron trajectory in the laser field shows that the maximum
energy 3.17Up can be transferred from the laser field to the electrons when they are ionized at
ωt=17o after the field peak, where Up is the ponderomotive potential of the laser field, which is𝑒 2 𝐸2𝑈𝑝 = 4𝑚 𝜔0 2,𝑒(3.19)

𝐿

where e and me are the electron charge and mass separately and E0 and ωL are the amplitude andangular frequency of the electric field. This dictates the highest photon energy, i.e. the harmonic
cut-off, generated when electron reencounters its parent ion:ℎυ𝑚𝑎𝑥 = 𝐼𝑝 + 3.17𝑈𝑝 .

(3.20)

Here 𝐼𝑝 is the ionization potential of the atom. For harmonics in the plateau region, each kinetic
energy of electrons can be produced by two different electron trajectories, differentiated the timethey spent in continuum. The one in which electrons are ionized at a smaller phase of the electric

68

field and spend longer time in continuum is called long trajectory and the second one is called
short trajectory. Macroscopic phase-matching is able to eliminate the long-trajectory contribution
and this is important for the generation of attosecond pulses [162].

3.2.2

Phase Matching and Absorption
The coherent buildup of the harmonic field requires that the light generated in different

portions of the gas across the interaction region are in phase with each other, i.e., phase-matched.
This can be accomplished by forcing the phase velocities 𝑐𝑝 =𝜔𝑘of both two participating fields(the fundamental driving field and generated harmonic field) to be equalized, where ω is theangular frequency and k is the magnitude of the wavevector. This corresponds to:
𝜔𝑓
𝑘𝑓

𝜔𝑞

𝑞𝜔

𝑞

𝑞

= 𝑘 = 𝑘 𝑓.

(3.21)

where f refers to the fundamental driving laser and q is the harmonic order. The best-case scenario
of complete phase matching is:∆𝑘 = 𝑞𝑘𝑓 − 𝑘𝑞 = 0.

(3.22)For nonzero ∆𝑘, it is convenient to define coherence length 𝐿𝑐 = 𝜋/∆𝑘, the characteristic
length in which a π phase shift is developed for harmonics generated at two positions along the
propagation direction. Beyond π phase shift, the energy starts to flow back to the driving field.
Another characteristic parameter is the absorption La length of medium, in which the harmonic
intensity drops by a factor of e. The on-axis harmonic intensity as a function of interaction length
Li is given by [163]:
𝐼𝑞 ∝ 𝐿2𝑎𝐿
𝐿
𝜋𝐿
1+exp(− 𝑖 )−2exp(− 𝑖 )cos( 𝑖 )
𝐿𝑎

2𝐿𝑎

𝐿 2
(2𝜋 𝑎 ) +1
𝐿𝑐

𝐿𝑐

.

(3.23)

69

The intuition of how La and Lc influence the build-up of the high-harmonic beam can begained by reviewing several limiting cases: (1) the ideal case is when there is neither absorption
nor phase mismatch (La, Lc→∞). In this case, the harmonic field Iq grows continuously over theinteraction region:lim

𝐿𝑎 , 𝐿𝑐 →∞𝐼𝑞 ∝ 𝐿2𝑖 .

(3.24)

Figure 3.7: The on-axis intensity of HHG radiation as a function of medium length. The
curves are calculated from Equation 3.23. It is shown that the radiation intensity oscillation as afunction of medium length is suppressed when Lc > La, leading to an enhanced HHG emission
intensity. When Lc >> La, the HHG radiation intensity can be maximized by increasing the
medium length to longer than 10 times the absorption length, approximately. Phase-matched
growth in the absence of absorption is plotted in blue dash line for reference. Figure adapted from[163].

70(2) When there is no absorption (La→∞), Equation 3.23 reduces to:
𝐿

𝜋𝐿

2

lim 𝐼𝑞 ∝ [ 𝜋𝑐 𝑠𝑖𝑛 (2𝐿 𝑖 )] .

𝐿𝑎 →∞(3.25)𝑐Equation 3.25 means the HHG output oscillates into and out of constructive interference during
the propagation, with the maximum achievable intensity proportional to the square of the coherent
length.
(3) When the coherence length is long (Lc→∞), Equation 3.23 can be written as:𝐿𝐿

lim 𝐼𝑞 ∝ 𝐿2𝑎 [1 + 𝑒𝑥𝑝 (− 𝐿 𝑖 ) − 2𝑒𝑥𝑝 (− 2𝐿𝑖 )].

𝐿𝑐 →∞

𝑎

𝑎

(3.26)The harmonic intensity saturates with sufficiently long interaction length and the maximum is
proportional to the square of the absorption length. The intensity dependence on medium lengthfor different cases is also illustrated in Figure 3.7. It is clear that the key to generate bright harmonic
field is to maximize both the absorption length and coherence length. Since the absorption isentirely medium-dependent can be barely controlled in experiment, the harmonic emission
efficiency is usually improved by optimizing the phase matching and medium length. According
to Equation 3.26, the following conditions typically ensure the macroscopic response is more thanhalf the maximum response:
𝐿𝑖 > 3𝐿𝑎(3.27)
𝐿𝑐 > 5𝐿𝑎 .
Phase Matching in Capillary Waveguide. HHG by propagating the driving lasers in agas filled hollow waveguide offers several advantages for efficient phase matching. First, a nondiverging plane wave geometry is maintained over the entire interaction region, eliminating the
Gouy phase shift introduced by focusing the beam into a gas jet/cell [164]. Second, it isstraightforward to tailor the gas pressure profile inside the waveguide for optimal phase matching.
The phase mismatch, △k, for the qth harmonic can be written as [165,166]:

71
𝑢2 λ

2𝜋

11 𝐿
(∆𝑛 + 𝑛2 ) − η𝑁𝑎𝑡𝑚 𝑟𝑒 𝜆𝐿 ]},∆𝑘 ≈ 𝑞 { 4𝜋𝑎2 − 𝑃 [(1 − η) λ𝐿

(3.28)

where q is the harmonic order, u11 is the lowest-order waveguide mode factor, λL is the central
wavelength of the fundamental driving laser, a is the inner radius of the hollow waveguide, P is
the gas pressure, η is the ionization fraction, △n is the difference in indices of refraction of theneural gas between the fundamental and harmonic wavelength, n2 is the nonlinear index ofrefraction per atmosphere at λL. Natm is the number density of the gas at one atmosphere and re isthe classical electron radius. Since the phase velocity of EUV can be assumed to be the speed oflight in vacuum, the phase matching condition, or △k→0, is achieved by varying the gas pressure
inside the waveguide, because neutral gas and laser ionized plasma have opposite dispersions.

3.3

High Harmonics for Photoemission
There exists a variety of light sources for photoemission spectroscopy. Compared to

traditional ones like plasma discharge lamp and 10-100 ps pulse-length synchrotron facilities, theclearest advantage of implementing HHG in photoemission spectroscopy is the temporally short
EUV beam coexisting with a phase locked IR beam, which can also go through frequency
conversion in nonlinear crystal to change color. This allows us to investigate the femtosecond toeven attosecond dynamics in materials. Although the newly-developed Free Electron Laser (FEL)
facilities offers shorter pulse down to ~10 fs and has done successful work with it, tremendous
work is needed to suppress temporal jitter in pump-probe and it is also not quite available forphotoemission application because of its low repetition rate resulting in extensively bright single
pulses causing severe space charge effect [167,168]. Tunable 6 eV radiation generated byquadrupled Ti: Sapphire laser beam started the era of high resolution ARPES with bulk sensitivity
and has also demonstrated its time-resolved capability [169]. In comparison, besides the72complementary surface sensitivity, harmonic radiation holds the advantage of possessing higher
energy to reach deep core levels and the larger Brillouin zone in solids. Furthermore, by using
higher energy incident photons, primary electrons are better separated from the secondary
electrons and IR-induced above threshold ionization (ATI) background. This spectral separation
allows for a more intense pump beam to promote the electrons to a far non-equilibrium state inphase transition and photochemistry investigations [170].

3.3.1

Tailoring HHG for Photoemission Spectroscopy
Depending on the interrogated sample and dynamics, photoemission spectroscopy benefits

from a light source that is easily tunable in bandwidth, energy and polarization state. For example,
typical photoemission experiments require monochromated photons while atto-streaking and
RABITT measurements requires broadband harmonics to support attosecond temporal structure[35,44]. Although photons with higher energy are able to access larger Brillion zone in k spaceand deeper core-level states, consideration of other factors such as photoemission cross section,
resonant excitation and final state effect add the complication of selecting the optimal wavelengthfor specific materials [144]. Recently developed circularly polarized harmonics have the advantage
to unravel the spin and orbital related dynamics in magnetic materials and graphene [30,68–70].
Here we present our capability of tailoring the high harmonic source to offer the highest
experimental power possible.

3.3.2

1ω, 2ω or 1ω+2ω
The cutoff-law (equation 3.20) dictates that harmonics with higher photon energy can begenerated by increasing the wavelength of the driving laser. It was demonstrated in experiment

73

that keV harmonics can be achieved using 4μm lasers [22]. However, the laser system used within
the work of this thesis (2mJ, 4 kHz, <30fs) doesn’t reached the point at which mid-infrared (1.3 or2 μm) pulses of sufficient energy are practical through optical parametric amplifier (OPA). Thus,
the driving light fields implemented in our EUV beamline are fundamental Ti: sapphire beam - 1ω(centered at 780 nm), its frequency-doubled beam - 2ω (centered at 390 nm) or a mix of both in
two-color linear HHG (Chapter 6) and circularly polarized HHG (Chapter 5).
1ω (780nm) driven HHG. 1ω-driven harmonic emission is the most straightforward way
to obtain EUV light. In the spectral domain, the harmonic peaks are separated by ~3eV from theneighboring ones, allowing us to design the multilayer mirrors with certain bandwidth to selectparticular harmonic while maintaining moderate efficiency. In attosecond RABBITT experiment,
the entire bandwidth is ultilized by using a toroidal mirror with carefully designed coating.
2ω (390nm) driven HHG. High harmonic radiation generated by frequency doubled
Ti:Sapphire laser possesses several advantages in photoemission spectroscopy compared to 1ω
(780nm) drivers [171,172]. (1) It eliminates the use of EUV multilayer mirrors or grating based
monochromators. (2) The photon flux is higher because of (1) and higher quantum efficiency in
HHG, which scales with driving wavelength at λ-5 to λ-9. [63,89,173,174]. (3) The intensity of thedriving field is moderate and avoids the damage of the waveguide and EUV optics. It also makes
the harmonics more stable. (4) It offers better energy resolution (<150 meV) while preserving the
temporal resolution (<30 fs) because the blue drivers are longer in duration after the BBO crystal
but still preserve sufficient peak intensity to produce bright harmonic radiation [171].
Although bright and narrow harmonic of 60 eV has been demonstrated using 2ω driver in
Neon, the laser system used in this thesis work only allows the 7th harmonics in Krypton (~22 eV)with flux high enough for photoemission use.

74

Two color (1ω+2ω) HHG. Two-color laser field can increase the spectral coverage andminimize the low-flux regions between odd harmonics. This is because the addition of the 2ω field
breaks the inversion symmetry and allows both the odd the even harmonics to be generated. This
spectral shape is found to be beneficial for transient absorption experiments [175]. In terms ofattosecond photoemission spectroscopy, combined two-color field reduces the requirement on
driving pulse duration in the generation of supercontinuum or isolated attosecond pulse [176].
Circular HHG. It is not straightforward to generate circularly polarized harmonics
because the liberated electron cannot recombine with its parent ion under circularly polarized
driving field. There are several ways to overcome this limitation. The brute force method isconverting the linear HHG to circular using EUV optics (usually achieved using multiple
reflections on surfaces, which possess a different complex reflectivity/transmission for s and ppolarized light thus result in a phase shift between the two) [177,178]. The disadvantage is that the
EUV optics are difficult to fabricate, low in efficiency (~1%) and limited in bandwidth. Therefore,
there are multiple efforts in the direct generation of circular harmonics with high efficiency.
One way to obtain circular harmonic is through molecular HHG. It was reported that theharmonics generated from N2 molecules can reach an ellipticity of 40% when the molecules arealigned at 60o from the polarization direction of the linear driving field [179]. However, thisscheme is cumbersome to implement due to the pre-alignment of molecules and complicatedrevival dynamics.
An alternative way is using resonant HHG in elliptical laser fields. Different types of
below- or above-threshold resonances in atoms and molecules can be utilized for this purpose,covering a broad spectral range. For example, Ferré et al. achieved highly-elliptical (0.6~0.8)
harmonic radiation up to 49 eV by using the shape resonance in the ionization continuum of SF6

75[114]. Nevertheless, this method cannot generate a continuously tunable EUV source because
circular harmonic emission is only emitted under the influence of resonance.
A universal way of circular harmonic generation is using bi-chromatic counter-rotating
circularly polarized driving lasers in a collinear geometry. The harmonic spectrum consists ofdouble peaks with alternating left and right circular polarization [68,69]. The first experimental
demonstration of this method is by combing 0.8 μm Ti:sapphire laser and its frequency doubled
beam at 0.41 μm. In a later work, photon energies >160 eV is obtained by mixing 0.79 μm and 1.3
μm beams in this counter-rotating scheme [70]. Another work shows that efficient generation of
elliptical high harmonics can be achieved by simply using orthogonally polarized two-color(800nm+400nm) laser field [115].
One particularly interesting scheme for circular harmonic generation is by using counterrotating circularly polarized lasers in non-collinear geometry [180]. The advantage of this scheme
is that harmonics of different orders are angularly separated. Therefore, it is easy to select one
harmonic for photoemission spectroscopy.

Chapter 4
Experimental Apparatus

This chapter focuses on discussion of the laser system, beamline layouts, EUV focusing
optics and monochromators, as well as the UHV end station used in the thesis work.

4.1

Laser System

Figure 4.1: Schematic of Ti:sapphire oscillator and amplifier systems. (a) Ti:sapphireoscillator. M1 and M2 are concave mirrors with 10-cm radius of curvature, between M1and M2 isa Ti:sapphire crystal, M3 is a fold mirror, M4 is an end mirror, M5 is an output coupler, P1 and P2
are fused silica prisms. A green pump laser beam (532 nm) is focused onto the Ti:sapphire toinduce population inversion for lasing. (b) Layout of a Ti:sapphire amplifier system using chirp
pulse amplification technique, figure adapted from [161].

77

Figure 4.2: Typical spectrum of Ti:sapphire oscillator (red) and spectrum after chirped pulse
amplification (blue).
Efficient HHG in noble gas requires the laser field strength to be >1013 W/cm2. This isoften achieved by Ti:Sapphire ultrafast laser system operating at a wavelength ~0.8 μm. To reachthe necessary intensity, ultrafast laser pulses produced in Ti:Sapphire oscillator (~80 MHz with 6
nJ, ~10 fs, see Figure 4.1a) are amplified at a reduced repetition rate (3-5 kHz) using chirped pulse
amplification (CPA) techinique. Figure 4.1b shows the schematic layout of CPA. The output fromthe oscillator is first sent through a stretcher with positive-dispersion before amplification to avoid
damage or nonlinear distortions of optical components. The stretcher chirps the pulses to roughly
150 ps via two reflections on the same grating. The pulses are then picked at the selected repetition
rate in the Pockels cell and amplified in a Ti:Sapphire crystal. The crystal is pumped by afrequency-doubled Nd:YAG laser at 532nm (Lee Laser LDP-200MQG-HP) with the repetition
rate and output timing synched to the Pockels cell. To reduce the thermal lensing effect, the crystal
is cryo-cooled with a closed-loop liquid helium system (Cryomech PT90). Laser pulses are

78

amplified according to the number of passes through the crystal. They are then re-compressed in
the grating compressor with negative dispersion to compensate for the chirp introduced in thestretcher and crystal. A typical laser spectrum before and after amplification is shown in Figure
4.2. The spectrum is narrower after amplification due to gain narrowing, which corresponds to alonger pulse. Higher order dispersion that cannot be compensated by compressor also contributes
to the elongation of laser pulses.
The laser system used in this thesis work is a single stage multi-pass amplifier. The output
pulse duration is in the range of 25 fs to 30 fs. Although the repetition rate and the corresponding
energy per pulse of the laser can be varied according to the ionization potential of the noble gas
used in HHG, the typical setting we use is 4 kHz in repetition rate and 2 mJ/pulse. If we assume
a Gaussian spatial and temporal distribution and the beam is focused to 100 μm in diameter (~60%of the fiber diameter) to couple into a 150 μm diameter fiber, the resulting electric field amplitudeis ~1.2×1011 V/m, which is enough to distort the Coulomb potential trapping the electrons in noble
gas atoms to allow tunnel ionization.

4.2

Beamline
After the laser system, the IR pulses are guided to the beam splitter and divided into two

beams. The majority of the pulse energy (~95%) goes through the HHG capillary waveguide to
generate the EUV beam and the rest is used to pump the material or dress the photoelectrons(Figure 4.3). The pump beam is focused by a concave silver mirror that has a radius of curvatureof 2 meters. A computer interfaced delay stage (Aerotech ANT130-1110-L-25DU-MP) is used tocontrol the relative pump-probe delay with a resolution approaching 100 as. The EUV probe beam
is focused and monochromated according to the requirements in experiments.

79

Figure 4.3: Simplified schematics of the beamline layout for tabletop EUV based
photoelectron spectroscopy system. (a) the laser beam from a Ti:sapphire laser system isseparated into two arms. The majority of the beam is focused into a noble gas-filled capillary
waveguide to produce EUV light. The EUV light is then focused by the EUV optics in the mirror
chamber onto a sample in the UHV chamber to generate photoelectrons. The rest of the Ti:sapphire
beam that transmits through the beam splitter is used to pump the sample or to dress thephotoelectrons. (b) Zoom-in view of the mirror chamber. The silicon rejector reduces the intensityof the residual laser beam after HHG. The toroidal mirror focuses the EUV light onto the sample
surface. (c) Layout of the mirror chamber when multilayer mirrors are used.

80

4.3

EUV Focusing Optics and Monochromator

Figure 4.4: EUV transmission of metal foils and EUV reflectivity of rejector mirrors. (a)Transmission curves of aluminum and zirconium filters with 200 nm thickness. (b) Reflectivity
curves for bare and ZrO2 coated rejector mirrors under grazing incidence angle of 10 degrees.
Reflectivity and transmission data are acquired from CXRO database.

Spectral selection and beam focusing are essential during the propagation of generated
harmonics to the sample surface. These are achieved with the EUV optics. Firstly, because of the
low conversion efficiency of the high harmonic process, the residual IR light co-propagating with
the EUV beam needs to be filtered to avoid its effect (electron excitation and dressing) on sample.
Different metal thin films (thickness=200~300 nm) are chosen depending on their transmission at
the desired photon energy. Aluminum (Al) filters are commonly used for low photon energies (20eV < hυ < 70 eV), and zirconium (Zr) filters for higher photon energies (70 eV < hυ < 150 eV).
Their transmission curves are plotted in Figure 4.4a. Additionally, these filters maintain a pressure
differential between the UHV chamber (10-10 Torr) and the HV environment (10-6-10-5 Torr) after
HHG capillary. When residual IR intensity exceeds the damage threshold of the filters, anadditional “rejector” mirror (superpolished silicon substrate) is placed after the HHG fiber at a

81

grazing angle to reflect the EUV light and absorb the majority of the IR light. A 200 nm ZrO2 layer
is coated on silicon for its use at high photon energies. Their difference in reflectivity is plotted in
Figure 4.4b. The “rejecter” mirror also protects the downstream EUV optics if they are sensitive
to intense IR. It also maintains the beam propagating direction when paired with the toroidal
mirror in beam focusing (as shown below).
The design of the focusing geometry in EUV beamline is to construct a 1:1 imaging systemfrom the end of the HHG capillary to the sample surface. For blue (390nm) driven harmonics, notonly the 7th harmonic (22.26 eV) is dominant over the entire harmonic spectra but also the energy
spacing between adjacent harmonics is 6.36 eV, which is large enough to avoid the overlap ofspectral features for most materials. Therefore, a focusing toroidal mirror after the rejecter mirror
at grazing incidence (Figure 4.3a-b) is used to focus the EUV beam onto the sample. All theattosecond metrology experiments presented in the thesis require the maximum bandwidth of theharmonic pulse train to be utilized. Therefore, the same focusing geometry is used. For
experiments requiring monochromated high energy harmonics (42 eV, 65 eV, and 90 eV) driven
by IR light, one pair of multilayer mirrors (one flat, one with radius of curvature R=1 m) with
precisely manufactured multilayer coatings are used to isolate the desired harmonics (Figure 4.3c).
Mg/SiC coatings are used for low photon energies <45 eV; Al/Zr in middle-range from 45 eV to
65 eV and Mo/Si for high energies from 65meV to 95 eV. Further details for designing the coatings
for these multilayers can be found in reference.

4.4

The Surface Science End Station
The end station is a μ-metal chamber pumped with 700 l/s magnetically-levitated turbo

pump (Pfieffer Hi Pac 700 M) and 270 l/s ion pump (Varian VacIon Plus). The chamber is also

82

equipped with a titanium sublimation pump (TSP). The chamber base pressure is 2.5×10-10 Torrbut operates at 3.5×10-10 Torr during measurements with harmonics because of small pinholes in
the pressure differential aluminum filter. All the sample preparation, characterization andmeasurement tools are integrated in the same chamber, offering great convenience in experiment.
We divide the vacuum ports of the chamber into three categories: vacuum pumps, surface
preparation tools and surface characterization tools. Surface preparation tools include: an ion
sputter gun, an alkali metal deposition system (SAES Getter source) and a gas dosing systemcomposed of a capillary doser attached to a gas leak value for flow control.

Figure 4.5: XYZ manipulator and sample holder. (a) The blue square highlights the manipulator
in the photoelectron spectroscopy setup. (b) schematics of the sample holder.

83

The solid-state samples are mounted on a screw-fastened transferable sample plateattached to the head of a xyz-manipulator (Figure 4.5a). A differentially pumped feedthrough at
the top of the manipulator enables 360o rotation of the sample around the z-axis. The transfer ofthe sample from atmosphere to UHV chamber is achieved by a load-lock system. The sample
temperature can be controlled continuously from 50 K to 1250 K by thermal contact of the sample
holder to the cold finger of an open loop continuous flow cryostat integrated inside the manipulator,and by tungsten wire radiative heating plate, respectively. Type-N thermocouple mounted to theside of the sample plate monitors the temperature profile and provides a feedback to a homedeveloped proportional-integral-derivative (PID) software driving the heater power supply. A
Residual Gas Analyzer (RGA, Leybold Inficon Transpector 2) was used to monitor the species
and content of gas molecules inside the chamber during sample preparation.
The surface characterization level of the chamber accommodates a Low Energy Electron
Diffraction (LEED) system (OCI BDL800IR), home-built Time-Of-Flight (TOF) spectrometer, aHemispherical Electron Analyzer (HEA, SPECS PHOIBOS 100) equipped with a CCD detector,
a EUV source of duo plasmatron discharge lamp (SPECS UVS300) that generates bright plasma
radiation (He I: 21.2 eV or He II: 40.8eV) for standard static photoemission measurement.
The IR and EUV beam enter the end station collinearly through one fixed port on thesample characterization level (Figure 4.3a). The collinear geometry is achieved by an annularmirror located right before the UHV port: the EUV beam passes through the central hole in the
mirror and the IR beam reflects off it at 45o incidence angle with beam centered on the mirror.
Temporally and spatially overlapping the EUV and the IR pulses on the surface of thesample is crucial for successful pump-probe (especially attosecond metrology) experiments. First,
a gross adjustment is obtained by finding the overlap between the IR pump beam and residual red

84

beam from HHG in the EUV beamline (referred to as IR probe). For this purpose, the Al foil isretracted and a sapphire window is inserted to pass the IR and maintain the pressure differential.
The sample is translated up from the measurement position and a BBO crystal underneath is liftedto the beam height. By focusing both IR beams on the BBO crystal, overlap is determined byobserving the optical interference between two frequency-doubled beams. Pump-probe
collinearity is achieved by optimizing the beam pointing to make the interference fringes as
symmetric and uniform as possible. A more accurate IR-pump and IR-probe overlap is determinedthrough the enhancement in multiphoton photoemission in cross correlation of two beams on
sample surface.
Fine tuning of the spatial overlap between IR pump and EUV probe is by shooting bothbeams on the phosphor screen where EUV beam can induce a visible fluorescence. The phosphor
screen is right underneath the sample plate (Figure 4.5b). Zero-of-time between EUV and IR isdetermined with Laser Assisted Photoemission (LAPE).

Chapter 5
Tomographic Reconstruction of Attosecond Pulse Trains with Circular Polarization

This chapter is adapted, with permission, from:
C. Chen, Z. Tao, C. Hernández-García, P. Matyba, A. Carr, R. Knut, O. Kfir, D. Zusin, C.
Gentry, P. Grychtol, O. Cohen, L. Plaja, A. Becker, A. Jaron-Becker, H. Kapteyn, M. Murnane,
Tomographic reconstruction of circularly polarized high-harmonic fields: 3D attosecond
metrology. Sci. Adv. 2, e1501333 (2016).

Figure 5.1: Illustration of the 3D reconstruction of circularly polarized harmonics. (a) By
simultaneously illuminating a copper surface with circularly polarized extreme ultraviolet (EUV)
light and an infrared laser beam that is perfectly synchronized with the EUV light, we are able toreconstruct the most complex light field to date. Art by Steve Burrows, JILA.

86

5.1

Abstract
Bright, circularly polarized, extreme ultraviolet (EUV) and soft X-ray high harmonic

beams can now be produced using counter-rotating circularly polarized driving laser fields.
Although the resulting circularly polarized harmonics consist of relatively simple pairs of peaksin the spectral domain, in the time domain, the field is predicted to emerge as a complex series ofrotating linearly polarized bursts, varying rapidly in amplitude, frequency and polarization. Here,
we extend attosecond metrology techniques to circularly polarized light for the first time bysimultaneously irradiating a copper surface with circularly polarized high harmonic and linearlypolarized infrared laser fields. The resulting temporal modulation of the photoelectron spectra
carries essential phase information about the EUV field. Utilizing the polarization selectivity ofthe solid surface and by rotating the circularly polarized EUV field in space, we fully retrieve theamplitude and phase of the circularly polarized harmonics, allowing us to reconstruct one of themost complex coherent light fields produced to date.

5.2

Introduction

5.2.1

Circularly Polarized High Harmonic Generation
Tabletop EUV and soft X-ray radiation from high-harmonic generation (HHG) process isa unique light source for uncovering new fundamental understanding of dynamics in atoms[104,112], molecules [181] and materials [44,171]. Until very recently however, mostexperiments used linearly polarized harmonics which can easily be produced using single-color
driving lasers. In this case, HHG emerges when an electron that is ionized by the laser field isdriven back and recombines with its parent ion [63,64]. When implemented in a phase matchedgeometry, the dipole emission from each atom interferes constructively to generate a bright HHG

87beam [88]. However, until recently, it was not possible to generate bright, circularly polarized
HHG for probing magnetic materials [182] or chiral molecules [183]. This is because althoughatoms still undergo strong-field ionization in circularly polarized fields, the probability for anelectron to recombine with its parent ion to emit a HHG photon is greatly suppressed [63,184,185].
Fortunately, high-brightness circularly polarized harmonics can now be produced in the
EUV and soft X-ray regions by driving HHG with bi-chromatic circularly polarized counterrotating fields [68–71,73,186–189]. In this scheme, a unique HHG spectrum is generated,consisting of pairs of peaks, each with opposite circular polarization. Theory predicts that for
counter-rotating ω and 2ω laser fields, the circular harmonics are generated as a superposition of
three bursts of linearly polarized EUV light per optical cycle in the time domain, where thepolarization of each burst is rotated by 120o from its predecessor [71,187]. Experimentally, theelectron trajectories have been shown to predominantly move in a two-dimensional plane[71,190].

5.2.2

Direct Temporal Characterization of Circularly Polarized Harmonics
Despite of the promising applications of circular harmonic, no direct measurement of thetemporal characteristics of circularly polarized HHG exists, that could be used to inform andvalidate advanced theory, and to harness the enormous potential of extreme nonlinear optics togenerate arbitrary spectral-temporal-polarization-shaped light fields spanning the EUV and soft
X-ray regions. This is due to the novelty of high-brightness circularly polarized HHG, theircomplex temporal structure, and the fact that conventional methods [28,29,32,34,111,191] forpulse characterization are not directly applicable. Moreover, it was recently proposed that byusing crossed beams of circularly-polarized lasers, or taking into account the orbital angular88

momentum of the initial state, either isolated circularly polarized single attosecond pulses can beproduced [180,192] or else highly elliptical attosecond pulse trains [193,194]. Therefore, theability to directly characterize the temporal structure of circularly polarized harmonics would
benefit a whole new area of polarization-shaped x-ray attosecond science.
In this work, by using laser-dressed angle-resolved photoemission from solids, we extend
attosecond metrology techniques to circular polarization for the first time. By simultaneously
irradiating a copper surface with circularly polarized HHG and linearly polarized infrared laser
fields, the resulting modulation of photoelectron spectra caused by quantum-path interference
carries essential phase information about the EUV field. Utilizing the strong polarization
selectivity in photoemission from solid surfaces, as well as the ability to rotate the circularly
polarized EUV field in space, we retrieve the full waveform of circularly polarized high harmonic
fields. This represents the first direct measurement of circularly-polarized HHG in the time domain,
enabling us to fully characterize one of the most complex coherent light fields to date, where thespectral, temporal and polarization states are rapidly changing on attosecond timescales.

5.3

Experimental Setup
Figure 5.2 illustrates the circular high harmonic generation (HHG) beamline as well as thelinearly and circularly polarized laser fields for our experiment. We used a multi-pass Ti:Sapphirelaser system to generate 30 fs IR pulses of 780 nm (ħ1.60 eV) light, with pulse energy of 2.4
mJ, and at a repetition rate of 4 kHz. The laser output is then split: most of the energy (95%) isused to generate fundamental ( and second harmonic (2 circularly polarized and counterrotating driving beams for HHG, while a smaller portion (5%, linear polarization) is used to probe(dress) the photoelectrons generated by the circularly polarized harmonics. The 2 light is

89

generated using a 200 μm thick β-phase barium borate crystal, which produces 33 fs pulses at awavelength of 390 nm and with pulse energy of 0.7 mJ. The 2 light and the co-propagating
residual  light (pulse energy 1.2 mJ) are separated using a dichroic mirror, and sent into two
different arms of a Mach-Zehnder interferometer. The polarizations of the  and 2 beams arethen converted to right and left circular polarization using a combination of λ/2 and λ/4 waveplates.
The relative time delay between the 2 and  pulses (τRB) is adjusted using a closed-loop piezostage with sub-femtosecond scanning resolution.

Figure 5.2: Characterization of circularly-polarized HHG fields. (a) counter-rotating [780nm, right-circularly polarized (RCP)] and 2 [390nm, left-circularly polarized (LCP)] beams
from a Ti:Sapphire laser are focused into an Ar-filled hollow waveguide. The generated circular
HHG and a time-delayed linearly polarized 780nm dressing field are focused onto a clean Cu(111)
surface. Here, a specific geometry is plotted for illustration purpose, where the ŷ -pol componentof circular HHG field is aligned with the normal direction of sample ( n̂ ). In experiments, theorientation of the circular HHG field can be rotated by adjusting the temporal delay between the and 2 driving fields,RB.

90

Figure 5.3: Circularly-polarized HHG spectrum recorded by an EUV spectrometer. The 13thand 16th harmonic orders are right circularly polarized (RCP) and 14th and 17th orders are left
circularly polarized (LCP).

To generate circularly polarized high harmonics, the  and 2 beams are focused using
lenses of focal length 50 cm () and 75 cm (2), and recombined with a dichroic mirror, before
propagating into a 1-cm-long capillary waveguide. The waveguide has an inner diameter of 150
μm and is filled with Argon at a pressure of 30 Torr [69]. To measure the spectrum of circularly
polarized HHG, a toroidal Au mirror is inserted temporarily into the beamline at a grazing angle,
to reflect the harmonics into an EUV spectrometer. For angle-resolved photoemission experiments,
the HHG beam is then focused onto an atomically clean Cu (111) sample using a second toroidal
Au mirror, to a spot size of ≈100 µm FWHM. Any residual driving laser light is blocked by a 200
nm thick polycrystalline Al filter.
The linearly polarized IR probe (dressing) beam is recombined collinearly with the circular
HHG beam using an annular silver mirror. The FWHM of the IR beam on the sample is ≈250 µm,
which is larger than the harmonic beam to ensure efficient dressing of the photoelectrons. The
relative delay between the HHG and IR beams is controlled using an interferometrically stabilized

91

stepper-motor delay stage (with sub-femtosecond resolution). The intensity of the probe IR lighton the sample is adjusted to ≈3.75 µJ/pulse (peak intensity: 2.8×1011W/cm2) using a half wave
plate and a linear polarizer. The low intensity of the IR dressing light ensures that contributions
from higher-order sidebands (>1st order) are negligible. The IR dressing field is p-polarized, i.e.,
perpendicular to the sample surface for all the measurements.

5.4

Spectrum of Circularly Polarized Harmonics
The spectrum of circular harmonics consists of three pairs of harmonic orders [(13ω, 14ω),(16ω, 17ω), (19ω, 20ω) in Figure 5.3], each with right and left helicity, which is a consequenceof photon energy and spin-angular momentum conservation rules [68,69]. Consistent withexperiment and theory [68,69], the intensities of the 3mth orders (15ω and 18ω) are strongly
suppressed, because they do not conserve parity and spin angular momentum in the case of
perfectly circularly-polarized driving lasers. They are not completely suppressed however,
because a slight ellipticity was introduced into the 2ω driving field to serve the characterization
purposes.

5.5

Cu(111) Sample Preparation and Band Structure
To measure the phase of circularly polarized harmonics, a Cu(111) single crystal is usedas the photoemission target. An atomically clean Cu(111) surface is obtained using repeatedcycles of Argon ion sputtering (0.7 keV, incidence angle of 60o) at room temperature (300 K) in
UHV chamber ( base pressure <5×10-10 Torr), followed by annealing to 820 K for 10 minutes.
The sample is grounded during all static and dynamic measurements. The sample surface quality

92

is confirmed by monitoring the low energy electron diffraction (LEED) patterns and the Shockleysurface state in static photoemission spectra, as shown in Figure 5.4a.
The static photoemission spectra are measured using Helium Iα line radiation at 21.2 eVfrom a discharge lamp (Specs UVS300), and recorded by an angle-resolved photoelectron
spectrometer (ARPES, Specs Phoibos 100, acceptance angle is ±15o under wide angle mode),with an analyzer work function of 4.2 eV. The Cu(111) single crystal features a Shockley surface
state at 0.4 eV below the Fermi level, and two conduction bands: an upper band (at 2.8 eV below
the Fermi level) and a lower band (at 3.7 eV below the Fermi level). The upper band (d band)
exhibits Λ3 symmetry while the lower band (sp band) consists of a dominant state with Λ1
symmetry and a minor contribution from a Λ3 symmetry state.

Figure 5.4: Photoelectron spectrum of Cu(111). (a) ARPES spectrum along the M direction
taken using the He Iα (21.2 eV) line from a He discharge lamp. (b) ARPES spectrum of Cu(111)
excited by the circularly-polarized HHG. The spectrum with momentum span is recorded by ahemispherical electron analyzer. The analyzer work function is 4.2 eV, which correlates thephotoelectron energy with photon energy. The direct photoemission pathways from Cu d band
excited by the corresponding harmonic orders are labeled in the right panel.

93

5.6

Static ARPES Spectra of Cu(111) Excited by Circularly Polarized Harmonics
When excited by circularly polarized HHG, the photoemission spectrum from Cu(111) iscomprised of a manifold of modulated peaks ranging from 13 to 22 eV, as shown in Figure 5.4b.
Due to the complex spectra of circularly polarized HHG, consecutive ARPES peaks from the same
electronic state are separated by 1.60 eV (  L ). Two sets of strong photoelectron bands can beseen, corresponding to direct photoemission induced by circular harmonic pairs (13, 14) and(16, 17) from the sp and d bands. The photoelectron bands excited by harmonic pair (19, 20)are too weak to stand out from the secondary background, because of the low intensities of theseharmonics. The 3mth (m is an integer) order harmonics (15 and 18) are also strongly suppressed.
However, their presence can be detected from oscillations at a frequency of  in the photoelectroninterferogram at photoelectron energies 17 eV and 21.8 eV, which is discussed in section 5.7.3.
The photoemission cross-section of the sp band is very low when excited by 13 and 14,but increases for higher orders (16 and 17). At the same time, photoelectrons from the sp bands
overlaps with photoelectrons from the surface states excited by lower order harmonics because thebinding energy difference between the two states (~3.3 eV) is very close to twice of the energy
spacing between consecutive harmonics (3.2 eV). These parasite photoelectrons from sp andsurface states are spectrally well-separated from d band photoelectrons.

5.7

Measuring the Spectral Phase of Circularly Polarized Harmonics

5.7.1

Overview
Full characterization of the amplitude and phase of linearly polarized HHG fields has been

demonstrated using many approaches based on an electron-phase modulator [191], including attostreaking [28], RABBITT (resolution of attosecond beating by interference of two-photon

94transitions) [32,195], CRAB (complete reconstruction of attosecond bursts) [34], and PROOF(phase retrieval by omega oscillation filtering) which was successfully used to retrieve the phases
associated with isolated attosecond pulses necessarily with an ultrabroad spectrum [111]. In thesemeasurement schemes, the harmonic phases are imprinted as modulations on the photoelectronenergy-momentum spectrum in the presence of a dressing (or streaking) IR laser field [191].
In our work, photoelectrons ejected from the Cu(111) surface by the circularly-polarizedharmonics are simultaneously dressed by a linearly polarized IR field (780 nm, ћωL=1.6 eV). Theinterference between different quantum pathways induces a modulation of the photoelectron yield,
which was recorded as a function of temporal delay (τd) between the HHG and IR dressing fields
using a hemispherical electron analyzer. Because of a dramatic difference in the photoemission
cross-section of the Cu(111) surface for s- and p- polarized EUV light, photoelectrons ejected bya s-polarized field dominate the signal, i.e., HHG polarization parallel to the sample surface(section 5.7.2 (a)). This not only eliminates ambiguities resulting from the entanglement of
harmonic phases from two orthogonal polarizations ( x̂ and ŷ shown in Figure 5.2) in the electroninterferogram, but also eliminates the use of EUV multilayer polarizers, which have very low
efficiency and narrow bandwidth in this photon energy range. Moreover, a major advantage of
our scheme is that a simple rotation of the HHG field (achieved by varying the delay between theω and 2ω driving fields) allows us to record the perpendicular component of the HHG polarization(section 5.7.6).

95

5.7.2
(a)

PROOF Algorithm
Polarization dependent photoemission cross section

Figure 5.5: Photoelectron yields depend on the HHG polarization. (a) Experimental geometry.
The circular HHG and IR dressing fields arrive at Cu(111) surface collinearly with a 45o incidence
angle. n̂ is the vector normal to sample surface. The IR dressing field is linearly p-polarized, along
the surface normal n̂ . The x̂ and ŷ axes define two orthogonal linear components of thecircularly polarized HHG light, which rotate with the orientation of the HHG light. Here, asituation where the x̂ -pol HHG component aligns parallel to sample surface is plotted. (b)Photoelectron spectra of Cu(111) excited using linearly polarized equal-intensity HHG beams withs- and p- polarizations.

We will first derive the probability of strong-field ionization from solid surface excited bycircularly polarized EUV light in the presence of a linearly polarized dressing IR field. We use theatomic units except where stated otherwise. The photoelectron transition dipole is given by Ref[195] and Ref [138]

  

b d   i  dt  f r , t  r  E X t   d  i r , t  ,


(5.1)

where i  r , t  is the initial-state wavefunction of electrons inside the material, E X  t  the EUV
field,  d the time delay between the EUV and dressing IR fields and  f  r , t  the final-state

96

wavefunction of the photoelectrons. Polarization-dependent photoemission cross-section can be
determined by the selection rules. The d band of Cu(111), which is used for phase retrieval in our
experiments, is of Λ3 symmetry and can only be excited by EUV field parallel to the sample surface.
As a result, we can selectively study the spectrum and phase of x̂ - or ŷ -pol component byaligning it parallel to the sample surface (s-polarization). The experimental geometry is plotted in
Figure 5.5a. This conclusion is supported by the large difference in photoelectron yields excitedby HHG light of the same intensity, but linearly polarized harmonic fields either along s- and ppolarizations, as shown in Figure 5.5b. Since circularly polarized HHG were used to ejectphotoelectrons, we write down the EUV field E X  t  as

E X t   Ex t xˆ  E y t yˆ ,(5.2)

where Ex  t  and Ey  t  are two orthogonal linear components of the circularly polarized EUV
field. In our experiments, the orientation of the circularly polarized HHG field can be rotated byadjusting the relative delay between the fundamental IR (ω) and its second harmonic (2ω) driving
fields. The incidence-angle of the HHG light (45o) only affects the photoelectron yields, but does
not affect the measurement of the harmonic phases.

(b)

Amplitude of transition dipole under the strong-field approximation
After considering the geometrical projection of the dipole matrix element, we reduce

Equation 5.1 to a form only depending on Ex  t b d   i



2 
dt f r , t  s  Ex t   d  i r , t  .

2 

(5.3)

Here s represents an unit vector parallel to the sample surface. The initial-state wave function is

97



i r , t    i r eiEit ,(5.4)

with binding energy Ei . The final-state wavefunction  f  r , t  can be expressed in Volkov-wavef r , t    f r  k r , t  ,(5.5)

where  f  r  is the spatial wavefunction of the ionic state of the material, k is the final stateelectron wavevector and  k  r , t  describes the “dressed” photoelectron state (Volkov-wave
function) in the laser field [138] k r , t  1ei k  AL t r ei k ,t  ,
3/ 2
 





2 

(5.6)

 

where  k , t is the Volkov phase, defined as  k ,t  

1 2 1 k t   dt '  2k  AL  t '   AL2  t '   .
2
2 t

(5.7)

By plugging Equation 5.4-5.7 into Equation 5.3 and separating the spatial and temporal integration,
we can getb d   i1



8 3 / 2 e i k ,t e iEit E x t   d Dt dt .

(5.8)

Here, we define the canonical momentum p  t   k  AL  t  and the transition dipole  matrix element Dt    f r eip t r s  i r  . Strictly speaking, D  t  is time-dependent, imprinting

additional phases related to the material initial states and the photoemission time delay. In our
experiments, only the d band of Cu(111) is taken into account, which eliminates its dependence
on the initial states. The photoemission time delay in atomic systems can be calculated accuratelyby theory [32]. However, this is much more challenging for a solid-state system. RABBITT
measurements have been performed on Cu(111) surface using similar HHG photon energies. It

98

was found that the photoemission delay varies approximately linearly with a slope of ~15 as per
harmonic order, corresponding to ~0.36 rad in phase. As we will show later, this small correction
is within the uncertainty of our phase retrieval algorithm. So, for simplicity, we assume D(t) isconstant at Do.
Assuming the dressing laser field is EL t   E0 t  cos Lt  , where E0  t  is the envelope 

function, the term with the Volkov phase,  k , t , can be simplified following Ref [34]e  i k ,tkE  t 1
 i 02 cosL t 
i k 2t

i

L
2
1

e

e

e

,

(5.9)

where 1 is a static phase shift which does not contribute to temporal modulations [34].
Using the properties of Bessel function expansion [110]: e  J u   einn  

n

iu sin  

, we cansimplify Equation 5.9 toe  i k , te1
i k 2t i12e  kE0  kE0  iLt  iLt J

iJ
 0 2  1 2  e e
.


L
L











(5.10)

Here, only the terms with n = -1, 0, 1 are taken into consideration [ J 1 x    J 1 x  ], because
higher order sidebands can be neglected due to the relatively low peak intensity of the IR dressing
field (1.25 x1011 W/cm2). On the other hand, the EUV field can be described by Fourier transformE x t    Ax  e ix  e it d ,(5.11)



where Ax   and  x   are the amplitude and phase spectra of the x̂ -component of the circularly
polarized field, respectively. By plugging Equation 5.10 and 5.11 into Equation 5.8, we find that
only the terms with phase terms e i  , e i    and e i    contribute to the temporal
n d

n

L

d

n

L

d

99

modulations. Here we will ignore the pre-factors for simplicity and use the fact that n  L  n 1and n  L  n 1b d  
 kE0 kEJ 0  2  Ax  n e ix n e in d  iJ1  20   Ax  n1 e ix n 1 e i n L  d  Ax  n1 e ix n 1 e i n L  d L    L     

(iii)
(i )


(ii)
. (5.12)
The three components in Equation 5.12 represents the transition amplitudes of threequantum paths initiated from a same initial state and ending up with a same final state: (i) absorbing
an HHG and an IR photon (ћωn-1+ћωL); (ii) direct photoemission by a single HHG photon (ћωn);and (iii) absorbing an HHG photon and emitting an IR photon (ћωn-1-ћωL), as shown and labeledin Figure 5.6.

Figure 5.6: Schematic of the quantum paths that give rise to photoelectron interferences.

100

(c)

Quantum-path interference
The modulations on the photoelectron counts as a function of  d can be obtained by

calculating S  d   b d  . The interference between (i) and (iii) yields oscillation of frequency
22 L , while interference between (i) and (ii) or (ii) and (iii) can give oscillation with frequency  L :2  kE 
S 2L  2 J1  20  Ax n1 Ax n1 cos2 L d   x n1    x n1  ;   L 

(5.13)

 kE   kE SL  J 0  20  J 1  20  Ax  n Ax  n1 cos  x  n    x  n1    L d  2  L    L .
  Ax  n Ax  n1 cos  x  n1    x  n    L d   2 

(5.14)

 kE0 2  L 

With our experimental parameters, we estimate J 1 

 kE 
J 0  20   0.2 , which is in L 

good agreement with the observed amplitude ratio between 2ωL and ωL components (see Figure5.7c). At the energy corresponding to harmonics of (3m-1)th or (3m+1)th orders, the 2ωLoscillation diminishes as a result of the low intensity contributed by its suppressed neighboring
3mth harmonic order.
As shown in Equation 5.14, the phase differences of consecutive harmonics are imprinted
in the oscillation with ωL frequency. To further extract this relationship, we write down theoscillation with ωL frequency at the energy corresponding to the direct photoexcitation with nth
harmonic in a general form, in a way similar to the PROOF method [111].
Sn  d    n  sin L d   n  ,(5.15)

where  n  and  n  are the energy-dependent intensity and phase of the oscillation,
respectively, which are defined as

101  n  I    I 0  n1   2 I 0  n1 I 0  n1  cos  n1     n1 
kE0
I 0  n  0 n1,
2
I 0  n 
L(5.16)

andtan  n  

I 0  n1  sin   n     n1   I 0  n1  sin  n1     n 

I 0  n1  cos  n     n1   I 0  n1  cos  n1     n .

(5.17)

Here, I0(ωn) and ϕ(ωn) are the power and phase spectra of the harmonics, respectively.

5.7.3

Electron Interferogram
The x̂ -polarization ( x̂ -pol) of the circular harmonics was first aligned parallel to thesample surface. The temporal interferogram of photoelectrons, for τd in range of -5 to 5 fs, isplotted in Figure 5.7a. These data were obtained by integrating the photoelectron spectra over theentire momentum range and subtracting the background photoelectron spectrum obtained wellbefore time-zero. To eliminate photoelectrons originating from different states in the material[44,104], the angle-resolved spectra were filtered before integration (as illustrated in Figure 5.7b),so that photoelectrons only from the Cu d band contributed to the interferogram, which have analmost constant photoemission cross-section in the energy range of our experiments [196]. Any
residual contribution from sp band to the interferogram phases are not expected to influence the
HHG pulse train reconstruction, because they are only involved when retrieving phases in regions
between the major harmonic, where the spectral intensity of circularly polarized HHG is low.

102

Figure 5.7: Photoelectron interferogram as a function of time delay between the circular
HHG and linear IR field. (a) 2D map of photoelectron yields as a function of photoelectronenergy and pump-probe time delay. The selected lineouts on the right panel represent thephotoelectron-intensity modulation at the photoelectron energies corresponding to direct
photoemission from Cu(111) d band excited by labeled HHG orders. (b) ARPES spectrum along
the same direction of Figure 5.4(a) using circularly polarized HHG. The pink area represents the
spectral mask applied in the data analysis to remove the contributions from the sp band and thesurface state. The photoelectron bands corresponding to direction photoemission from Cu d band
are labeled by dashed lines. (c) Fourier analysis of the photoelectron-yield oscillations at energies
corresponding to direct photoemission by 14 and 15. The green area marks the frequency mask
used to extract the phases of the L oscillations in (a).

103

As can be seen in Figure 5.7a, the photoelectron yields as a function of τd oscillate with afrequency of ωL (periodicity of 2.6 fs), which can be understood through quantum-pathinterferences (Figure 5.6). Since the energy spacing between consecutive circularly polarizedharmonics is ћωL (Figure 5.3), there are three distinct quantum pathways for exciting electrons
from a given initial state to the same final state (path (i-iii) in Equation 5.12 and Figure 5.6). The
photoelectron yield oscillates with frequency ωL because of one-IR-photon-assisted interference
between quantum paths (i) and (ii), and (ii) and (iii). This cross section is much larger than that of
two-IR-photon-assisted interferences between (i) and (iii), which cause oscillations with frequency
2ωL, as used in RABBITT characterizations [32]. This conclusion is also supported by a Fourier
analysis of the photoelectron-yield oscillations (Figure 5.7c), where the amplitude of 2ωLcomponent is very weak for the 15th order, and is almost completely absent for the 14th order.

5.7.4

Extraction of the Interferogram Phase
To implement a phase retrieval algorithm to acquire the harmonic phases from thephotoelectron interferogram, we first need to extract the interferogram phases. To do this, we bin
the interferogram data into 100 meV wide electron energy regions and integrate the photoelectron
counts in each region to obtain a series of plots of the photoelectron yield versus relative timedelay between the HHG and IR dressing fields. Then, we apply a Fourier analysis to calculate therelative phases between the oscillations of different photoelectron energies.
In the Fourier analysis, we determine the relative phase between two energy-dependent
oscillation curves [ xE1 , t  and xE2 , t  ] by calculating their cross-correlation function

C E1 , E2 ,    ~x * E1 ,  ~x E2 ,   ,(5.18)

104

where E1 and E2 are the corresponding electron kinetic energies for these two oscillation curves,

  is the complex conjugate of ~x   . The peak*~x   is the Fourier transform of x t  and x frequency of CE1 , E2 ,   is the oscillation frequency shared by both curves, which equals to thefrequency of the IR dressing laser L, as shown in Figure 5.7c. The averaged phase ofC E1 , E2 ,   at this peak frequency corresponds to the relative phase difference between the two

oscillations [197]. To improve the signal-to-noise ratio, the measurement of interferogram isrepeated 30 times, which yields consistent phase relations. The averaged interferogram phases areused in the phase retrieval algorithm, which are shown as solid lines in Figure 5.8.

Figure 5.8: Reproducing photoelectron interferogram phase using retrieved HHG phase. The
solid lines represent the extracted interferogram phases from the experimental data. The symbols
are the reproduced interferogram phases using the retrieved harmonic phases. The error bars
represent the standard deviation from multiple trials using the genetic algorithm.

105

5.7.5

Retrieve Harmonic Phase from Interferogram Phase
To sum the derivation of equations in Section 5.7.2, the phases of the harmonics are

encoded in the ωL oscillations of the photoelectron-yield interferogram. At the electron kinetic
energy corresponding to direct photoemission by the nth order harmonics (ћωn), the oscillating
spectralintensityasafunction

of

τd

can

be

writteninthegeneralform:

S n  d    n sin  L d   n  , where   n  is the energy-dependent oscillation amplitude and n  the phase spectrum of the interferogram, which is related to the phase spectrum of HHG
 n  by tan  n  

I 0  n1  sin   n     n1   I 0  n1  sin   n1     n 

I 0  n1  cos  n     n1   I 0  n1  cos  n1     n .

(5.19)

I 0  n  is the power spectrum. Equation (5.19) bridges the interferogram phase [  n  ] andthe phase differences of corresponding neighboring harmonics [  n    n 1  ] and[  n 1    n  ], allowing us to retrieve the harmonic phases from the interferogram in an
iterative way. Here, the full interferogram phase spectra were fed into a genetic algorithm [198]toretrievetheharmonicRI         , where αcalc(ωn) and αexpt(ωn) are the interferogram phases


calc

n

by

minimizingtherankingfunction:

2

expt

n

phasesnnobtained from the retrieval algorithm and the experiment, respectively.
In the phase retrieval algorithm, the harmonic phases are represented by an array ofnumbers ranging from   to  . To minimize the ranking function R, a population of randomarrays are first assigned as the harmonic phases, covering harmonics of 12th-19th orders (on eitherside, a redundant harmonic order with zero intensity is included as the boundary conditions). In

106

the genetic algorithm, reproduction is carried out using Roulette-Wheel selection. Cloning,mutation and crossover operations are used to increase the diversity of population and to prevent
stagnation. To ensure uniqueness of the solution, the phase of 13th harmonic in x̂ - pol is set to bezero, including the phases in its adjacent area. To improve the convergence of the phase retrieval
for ŷ - component, proper helicities of harmonics are taken into consideration. The solution
searching for the phases of ŷ -pol component is confined in proper regions related to x̂ -polphases, i.e.,  y  x 
13

13



     
14  ,  , 14  ,  , etc. , where xn y  is the phase ofy  x 
2  4 42  4 4nth harmonic order in x̂ ( ŷ ) –pol, allowing the genetic algorithm to find the best fit to theexperimental data in these regions. In Figure 5.8, we plot the reproduced interferogram phases
compared with the experimental phases for x̂ - and ŷ - pol. The phases for major harmonics arelisted in Table 5.1. The robustness of the algorithm is tested by repeating the phase retrieval
processes for more than 50 times. The error bars are determined by the standard deviations ofretrieval results from multiple trials.

Phases of the harmonics (rad)Polarization 13th

14t

15th

16th

17th

x̂

0.0

0.57± 0.29

0.23 ± 0.58

0.97± 0.30

-0.19± 0.30

ŷ

-1.65± 0.58

1.88± 0.44

-2.32± 0.91

-0.37± 0.56

2.31± 0.50

Table 5.1: Phases of circular harmonics. The phases along x̂ - and ŷ -pol (see Figure 5.2)
are obtained using a phase retrieval algorithm. The uncertainty is determined as thestandard deviation of phase values retrieved from multiple trials.

107

5.7.6

Rotation of Circularly Polarized Harmonic Field
In our experiments, the two orthogonal linear components of the complex circularly

polarized HHG field are characterized by rotating the driving laser field with the three-foldsymmetry by 90o and aligning the component of interest parallel to the sample surface. The
successful reconstruction relies on the fact that, except their orientation, the x̂ - or ŷ -polcomponents of the EUV field are mostly unchanged throughout the rotation. In practice, therotation of the driving field is achieved by delaying the fundamental IR (ω) field relative to thesecond harmonic (2ω) field by 3/8 of the fundamental IR wavelength (292 nm in space or 0.973 fs
in time), which is controlled by a closed-loop piezo-stage (position error ±5 nm). As a result, theorientation of the combined driving field (with three-fold symmetry) [190], and hence thecircularly polarized HHG field, is rotated by 90o. Although this rotation has a negligible influenceon the HHG spectrum and flux, it has a significant impact on the photoelectron interferogram andthe corresponding harmonic phases, allowing us to reconstruct the temporal profile of the ŷ -pol
HHG field.
In Figure 5.9a, we plot the calculated 3D profile of the combined driving laser field, before
and after the ω field is delayed by 0.973 fs. A rotation of the combined laser field can be clearly
resolved. The corresponding 2D projections of the combined laser fields are shown in Figure 5.9b.
We note that when the temporal delay between ω and 2ω fields is changed, in addition to therotation of the driving field, a temporal shift (phase change) of the EUV field is also induced, as
illustrated by Figure 5.9c, which is taken into account in the phase retrieval algorithm. A 90orotation of the combined 2-color laser field can lead to a corresponding rotation of the circular
HHG by the same angle, which is confirmed by our numerical simulation (section 5.9) with bothdriving fields are perfectly circular (ε1=ε2=1.0), as shown in Figure 5.10(a-d).

108

Figure 5.9: Rotation of the clover-leaf-shaped driving field. (a) 3D plot of the combined electricfields of counter-rotating 780 nm (RCP) and 390 nm (LCP) fields. The red and green symbols
represent the combined field before and after delaying the IR field by 0.973 fs. (b) Ex-Ey 2D
projection of (a), where the red and green curves represent the field before and after the delay. The
colored area marks the same section of driving field so that a clear 90o rotation can be resolved.
(c) Electric field intensities along x̂ and ŷ before and after delay, plotted as a function of time.
The two fields are identical except for a delay of 0.33 fs.

109

Figure 5.10: Simulation of the EUV field rotation. (a) and (b) Projection views of the circular
HHG field before and after delaying the fundamental IR relative to second harmonic field by 0.973
fs. These results are obtained through our numerical simulation in Ar gas with both driving fields
perfectly circular. A clear rotation of the HHG field can be observed. (c) and (d)The x̂ and ŷcomponents of the EUV field before and after delaying  field with respect to 2 field by 0.973
fs. The two components are identical (comparing red solid and blue dashed lines), apart from atime delay of 0.33 fs. The ellipticity of both driving fields are 1.0 ( 1   2  1.0 , as definedpreviously). (e) and (f) The x̂ and ŷ components of the simulated EUV field with the same delayof  field, but with the ellipticity of 2 field to be 0.94 (  1  1.0,  2  0.92 ).

110

We note that the rotation of circularly polarized HHG field can be affected by the ellipticityof the driving fields. In order to test validity of our method when the driving field is slightly
elliptical, we simulated the circularly polarized HHG from Ar with the ellipticity of 2ω field (ε2)
to be 0.92. In Figure 5.10(e-f), the x̂ - and ŷ - components of the simulated HHG field are plotted
before and after the ω field is delayed. Similar to Figure 5.10(c-d) where the driving fields are
perfectly circular, E x ( E y ) after rotation is almost identical to E y ( E x ) before with a proper time
shift, which is a clear evidence of the 90o rotation of EUV field, even with a slight ellipticity ~0.92
of the driving laser fields.
Using the same procedure in Section 5.7, we characterized the perpendicular polarization
component) by aligning it parallel to the sample surface. Our approach therefore extends the
RABBITT [32] and PROOF [111] characterization approaches to be able to extract the 3D
temporal-polarization structure of an arbitrary polarized attosecond pulse train.

5.8

Reconstruction of Attosecond Pulse Train with Circular Polarization
With the knowledge of both the amplitude and phase, the temporal structure of thecircularly polarized harmonics can be uniquely determined. For the HHG pulse train
reconstruction, the harmonic phases are interpolated from the retrieved results to provide anenough temporal span. To properly account for the effects of harmonic phase errors from retrieval
process, the reconstruction calculation is repeated 100 times, in which a random number generator
is applied to vary the harmonic phases within the error bar region.
The 3D electric field of the experimentally reconstructed circular HHG pulse train isplotted in Figure 5.11. In different retrieval processes within experimental error bar, the threeburst structure of the pulse train similar to the one presented in Figure 5.11 can be consistently

111

reproduced, with some variations in the burst orientation and intensity. Three linearly polarized
bursts of the EUV fields with different orientations can be clearly distinguished in each IR optical
cycle (2.6 fs), in excellent agreement with the pulse-train structure generated by our numericalsimulations [199]. A typical duration of the EUV bursts is ~600 as, as shown in Figure 5.12.
The field orientation of each burst is obtained by averaging throughout each burst,
weighted by local field intensity. The deviations from different reconstruction trials are also taken
into account, providing the error bar. As shown in Figure 5.12, the field orientation of the EUV
bursts rotates approximately 120o from one to the next, which is a consequence of the three-foldsymmetry of the combined driving field, directly validating previous theory [71].

Figure 5.11: Experimental reconstruction of the circular HHG pulse train. Three-dimensional
field plot of the experimentally reconstructed pulse train of circular HHG, in direct comparison
with our numerical simulation. The pulse train from theory is offset by -3 units along the E x axis
for illustration purpose. Different colors (red, blue and green) highlight the three linear bursts withdifferent field orientations within each fundamental IR optical cycle (2.6 fs). The structure of threebursts and IR optical cycle are also labeled on the ŷ -projection of the electric field. The carrier
field of the experimental pulse train is reconstructed by assuming the phase of the x̂ component of
the 13th harmonic to be zero.

112

5.9

Numerical Simulation of Circular Harmonic Generation
The single source HHG has been computed in the dipole approximation using the Strong-

Field Approximation (SFA) method, without resorting to the saddle point approximation [199].
The harmonic spectrum is computed from the acceleration by integrating the momentum-spacecontributions of the Volkov waves. This method allows us to tackle the circular polarization
problem without the need of identification of the relevant semiclassical trajectories. Coulomb
corrections are introduced using the corresponding pre-factor [200]. The argon wave functions aredescribed as superpositions of Slater-type orbitals resulting from the Rothaan-Hartree-Fockmethod [201]. We take into account the electrons in the three valence orbitals, adding theircontributions to the HHG spectrum coherently.

Figure 5.12: Field orientation of circular HHG from Ar. (a) Field orientation and intensity
profile of the experimentally reconstructed circular HHG pulse train. The orientation of each EUV
burst, Φ, is defined as the angle of the electric field relative to the x̂ axis, as shown in the inset of(a). Φ is extracted by averaging the field orientation throughout each burst. (b) The overall fieldintensity profile of the pulse train between -15 and 15 fs. Three-burst structure per IR optical cycle(2.6 fs) can be distinguished throughout the entire pulse train. The red curve represents theenvelope of the pulse train with a FWHM of ~8 fs. A zoom-in view between -3 and 3 fs is presented
in a).

113

Figure 5.13: Normalized 3mth order harmonic intensity compared with numerical simulation.
(a) 2D map of harmonic intensity as a function of harmonic order and the ellipticity of the 2ω
driving field. The 15th and 18th orders are highlighted. (b) The normalized 3mth order intensity isplotted as a function of 2ω driving field ellipticity. The solid and dashed lines represent the resultsfrom simulation, with solid lines for 3m = 15 and dashed lines for 3m = 18. The symbols areexperimental results with the filled black square for 3m = 15 and the open red circle for 3m = 18.
The green area represented the estimation of the ellipticity of driving field in our experiment.

The driving field is described as a superposition of two counter-rotating elliptically
𝐴 (𝑡)𝐸𝑖

polarized laser pulses in the form of 𝑬(𝑡) = ∑𝑖=1,2 𝑖√1+𝜖𝑖2[𝑐𝑜𝑠(𝜔𝑖 𝑡)𝑥̂ + 𝜖𝑖 𝑠𝑖𝑛(𝜔𝑖 𝑡)𝑦̂] where 𝜖𝑖 isthe ellipticity. The driver frequencies, ω and 2ω, correspond to laser wavelengths of 𝜆1 =0.78 µm,
and 𝜆2 =0.39 µm respectively. The temporal envelope, 𝐴𝑖 (𝑡), has as a trapezoidal shape with 3
cycles of linear turn-on, 6 cycles of constant amplitude (16 fs) and 3 cycles of linear turn-off (interms of cycles of 𝜆1 ) . The electric field amplitudes ( 𝐸𝑖 ) have associated intensities of

I1  7.6  10 13 W/cm2 and I 2  3.8  10 13 W/cm2 which preserves the ratio between two driving
fields (2:1) used in our experiments. The field intensities are adjusted to match the cutoff energy
observed in the experiments. In the calculations present in this work, the relative carrier-phasebetween the pulses is zero at the peak of the envelope.

114

Figure 5.14: Experimental reconstruction of circularly polarized harmonics in comparison
with numerical simulation. (a) Experimentally reconstructed field intensity envelope of thecircular HHG pulse. The orientation of each EUV burst, Φ, is extracted by averaging the angle of
EUV field in each burst relative to the x-axis. (b) The results from our numerical simulation withthe ellipticity of 2ω driving field to be 0.94. A Gaussian envelop with 8 fs FWHM is applied on
the simulation result for comparison. (c-d) Comparison of the Ex and Ey components fromexperiments and simulations.

From Equation (5.19), we note that contributions from quantum pathways involving the
suppressed 3mth order harmonics are essential for retrieving the harmonic phases, because they
allow us to bridge the neighboring pairs of strong harmonics that are separated by 2ω L. The
appearance of 3mth order harmonics is a direct consequence of a slight ellipticity in the driving
laser fields, which was intentionally introduced into the 2ω driving beam.

115

In order to extract the ellipticity of the driving fields, we perform HHG simulations fordifferent ellipticities of 2ω driving field (ε2). As shown in Figure 5.13a, the intensities of 15th and18th harmonic increase monotonically as a function of the ellipticity of the 2ω driving field. Here
we define the normalized 3mth order intensity as

2 I 3m. By comparing with ourI3m 1  I3m 1

experimental results (symbols in Figure 5.13b) with the simulation results, we can determine thatthe ellipticity of 2ω driving field in our experiments is 0.92±0.04. In Figure 5.14, we plot theexperimentally reconstructed field-intensity envelop in direct comparison with the results from ournumerical simulations. The agreement between them is excellent.

5.10

Ellipticity of Circularly-Polarized High Harmonic Field
With the knowledge of intensity and phase of both x̂ and ŷ linear components, theellipticity of the circular harmonics can be uniquely determined. The ellipticity of a circular fieldis given by [68]E  E   4 E E cos     , E  E   E  E   4 E E cos    
2Ex  E y 
2

2

2

2

x

y

2 2

2

x

y

2 2

2

x

y

2

2

x

2

2

x

2

y

y

x

y

x

y

(5.20)

2

where E x ( E y ) and  x (  y ) are the amplitude and phase of x̂ ( ŷ ) component of the electricfield respectively. Here  =+1/-1 corresponds to full right/left circular polarizations, respectively.
Previously, the ellipticity of circular HHG from Ne was characterized using X-ray
magnetic circular dichroism (XMCD) [69]. Using our novel approach to access the harmonic
amplitudes, phases and polarization direction, we can analyze the chiral properties of HHG in any
spectral region, even when the photon energy does not overlap a characteristic absorption edge ofa magnetic material.

116

The ellipticity εX of the circular harmonics is plotted in Figure 5.15 and is also directlycompared with our numerical simulations. We find that the 3m+1th and 3m-1th orders are mostly
circularly polarized, while the 3mth (i.e., 15th) orders are strongly elliptical, consistent withprevious studies and theory [68,69]. Importantly, a reduction in the degree of HHG circularity
can be observed in the cutoff region for 3m+1th and 3m-1th orders, which is also quantitatively
confirmed by our numerical simulations.

Figure 5.15: Ellipticity of circular HHG from Ar. The experimentally measured andtheoretically calculated ellipticity of circular HHG from Ar. The open blue diamonds are thesimulation results with  1  1.0 and  2  0.92 , where  1 and  2 are the ellipticity of 780 nm and390 nm driving fields, respectively. The green dashed lines represent the results considering spinangular momentum conservation under a perturbative model in Ref. [68] and in Ref. [202]. The
HHG spectrum obtained from experiment is plotted in the lower panel.

117

Previously, the ellipticity of circularly polarized HHG has been described under aperturbative-optics models [68,202,203]. In Ref. [68], the spin of HHG photon is simply the sumof averaged spins of RCP and LCP photons involved in each harmonic channel [68]. However, as
pointed out in Ref. [202], this model has unphysical discontinuities, when describing the circularly
polarized HHG with some ellipticity. This problem was solved in Ref. [202] by considering an
ensemble of ω and 2ω photons with definite spin σ=±1, and their contributions to the HHG photon
spin are dealt with statistically, by considering different combinations of ω and 2ω photons ineach HHG channel when the light carries some ellipticity.
Here, ignoring the discontinuities from the first model [68], we note that both models[68,202] give same estimation on the HHG ellipticity as a function of harmonic order, as shownby the green dashed lines in Figure 5.15. This result is not surprising, since both models treat the
problem in a perturbative way. However, comparing the estimation from perturbative models with
our experimentally measured HHG ellipticity and numerical simulations, we find obvious
deviation at the cutoff region. For linear HHG, it is known that the high-energy harmonics in the
cutoff region are generated in a narrow temporal window, within one or two optical cycles at thepeak of the driving laser pulse [204,205]. For the same reason, we believe that the large deviationfrom circularity in the cutoff region can be attributed to the unequal intensity contributions fromthe linear EUV bursts and the breaking of the three-fold symmetry, due to the narrow temporalemission window and slight ellipticity in the driving fields.

5.11

Conclusion
In summary, we completely characterize the temporal, spectral and polarization states ofcircularly polarized harmonics generated by counter-rotating bi-chromatic driving fields. Using118

this novel generation scheme, we demonstrate that the circular harmonics consist of asuperposition of three linear EUV bursts with alternating polarization directions, repeating each
optical cycle, which validates theory in the time domain for the first time [71]. We note that
although some ellipticity has been intentionally introduced in the driving laser field to induce
weak 3mth order harmonics, which is necessary to bridge neighboring harmonic pairs in ourcurrent phase retrieval algorithm, this limitation can be removed by implementing a more
complicated algorithm such as FROG-CRAB [34]. With this implementation, our approach canbe applied to fully characterize the properties of both attosecond pulse trains and broadband
isolated attosecond pulses from arbitrary HHG fields, which is essential for investigations ofcoherent attosecond chiral and magnetic dynamics [206], where knowledge of the full temporal
structure of the EUV field is important.

Chapter 6
Influence of Microscopic and Macroscopic Effects on Attosecond Pulse Generation Using
Two-Color Laser Fields

6.1

Abstract
Attosecond pulses and pulse trains generated by high-order harmonic generation are

finding broad use for attosecond spectroscopies and imaging, enabling sub-femtosecond electron
dynamics to be probed in atomic, molecular and material systems. To date, isolated attosecond
pulses have been generated either by using very short few-cycle driving pulses, or by using
circularly polarized driving pules to enforce temporal gating, or by taking advantage of phasematching gating. Here I show that by driving the HHG with a two-color linearly polarized laser
field, the temporal window for phase matching is shorter than that for equivalent singe-color
driving laser. As a result, even when relatively long 25-femtosecond pulses are used to drive highharmonic generations, we can nevertheless generate quasi-isolated attosecond pulses with a peakwidth of ~450 as. Our theoretical simulations are in good agreement with our experimental
measurements, showing that the phase matching window decreases by a factor of 4 from four
optical cycles in the case of the single-color fundamental driver to one optical cycle in the case of
two-color (ω-2ω) laser drivers. Finally, I also show changing the relative delay between two-color
driving fields modulate the duration of attosecond bursts from 450 as to 1.2 fs.

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6.2

Introduction

6.2.1

Generation of Isolated Attosecond Pulses
Coherent attosecond pulses spanning the extreme ultraviolet (EUV) and soft X-ray spectral

range can be produced via high-order harmonic generation (HHG) of an intense driving laser field
in a gas medium [22,207]. Creation of such pulses have attracted extensive attentions, because
they open the possibilities to study the ultrafast dynamics in atoms and materials with
unprecedented time resolution. Indeed, coherent tabletop HHG sources were recently used tocapture chemical reactions in real time [181], to reveal correlated charge, spin and phonon
dynamics in materials with femtosecond time resolution [208], and to perform coherent imaging
on nanostructures beyond the wavelength limit [209]. Moreover, attosecond HHG pulses were also
implemented to track the fastest motion of electrons when they approach the atomic [59,124] ormaterial-final-band resonances [35], revealing the most intriguing observations on the attosecond
time scales.
The temporal structure of the HHG pulse trains is determined by the nontrivial electron
motion under the influence of the driving field. When driven by a monochromatic laser field, anelectron is tunnel ionized near the peak of the laser field and is driven away from its parent ion. A
EUV photon is emitted in the recombination step, when the laser field changes its direction anddrives the electron back to recollide with its parent ion [64,210]. When driven by a single-color
laser field, this process repeats twice every optical cycle at both the positive and negative peaks ofthe electric field. Therefore, the most straightforward way to achieve single attosecond pulse is by
reducing the number of electric field cycles to confine the recombination events to the most intense
half cycle of the driving field (i.e., intensity gating). In this scenario, the harmonic spectrum isusually centered at high energies (> 80 eV). The generation of isolated attosecond pulses with

121

ultrabroad-bandwidth or at low EUV photon energy regime require other gating techniques torestrict HHG process to one single half cycle of the driving field. The polarization gating techniquetakes advantage of the dependence of HHG efficiency on the circularity of the driving fields byusing two overlapping circular fields to create a temporal window with linear polarization. The
phase matching gating, on the other hand, utilize the macroscopic phase matching effect to balance
the negative dispersion of the ionized plasma and the positive dispersion and the neutral gas atoms
in the medium.

6.2.2

HHG under Two-Color Driving Fields
It has been long realized that the HHG emission can be controlled by introducing a second

driving field with different wavelength, which breaks the symmetry of the electron dynamics.
Two-color driven HHG has been shown to be useful in manipulating the temporal profile andpolarization state of the emitted attosecond pulses, as well as relaxing the conditions for single
attosecond pulse generation. For example, double optical gating technique introduces a secondharmonic pulse in addition to polarization gating, which allows to generate isolated attosecond
pulses [96]. Recently, circularly polarized attosecond pulses have been generated by mixing bichromatic circularly polarized driving fields with opposite helicities [30,69]. When driven by the
combination of an linearly polarized fundamental IR field and its second harmonic field with acomparable intensity, the harmonics were found to be stronger than those obtained in thefundamental field alone in both parallel and orthogonal configurations [211]. Moreover, when thepolarization of the two fields is the same the attosecond pulses are emitted once per IR optical
cycle and hence are carrier-envelop phase stabilized [72]. Nevertheless, there is no measurement
about the detailed temporal structure generated in this configuration so far, for which the direct

122

measurement of the HHG phases is required. At the same time, macroscopic phase matching under
the two-color scheme could also affect the intensity and temporal structure of the emitted
attosecond pulses by tailoring the ionization fraction of the gaseous medium through the changeof gas pressure and the confocal parameters [88,100]. Such information could bring insight to theway of coherently engineering the attosecond pulses using the two-color driving fields.
In this chapter, by using interferometric laser-assisted photoemission to measure the full
temporal profile of the HHG radiation, I show that quasi-isolated attosecond pulse with a peakwidth of ~ 450 as can be generated by mixing a multicycle 780 nm driving field with its second
harmonic field. In tandem with the numerical simulation, we demonstrate that it is caused by the
reduction of the temporal window of HHG phase matching in the two-color (ω-2ω) configuration
when compared to that obtained by the equivalent single-color driving laser. Our theoretical
calculations show that the phase matching window decreases by a factor of 4 from four optical
cycles in the case of the single-color fundamental driver to one optical cycle in the case of twocolor laser drivers. In the sub-cycle temporal regime, we also demonstrate the direct coherent
control of the attosecond pulses by changing the relative phase between the 390 nm and 780 nm
driving fields (  RB ). Our results show that when  RB gives the maximum HHG yield, the FWHM
(full width at half maximum) of the attosecond bursts in a pulse train reaches its minimum valueof ~450 as, which is close to the transform-limited width. In contrast, when  RB is adjusted awayfrom the optimum HHG yields, the peak FWHM is significantly increased due to the emergenceof additional temporal structures. By comparing the experimental results with the numericalsimulations and semiclassical calculations of the driving-field structure and electron trajectories, I
find this observation is a direct consequence of the broken symmetry introduced by the intense
second harmonic field, which strongly modifies the electron ionization, propagation and123

recombination dynamics during the HHG process. In the spectral domain, this corresponds to thedependence of the HHG phases on  RB .

6.3

Experimental Setup
The setup of our experiments is illustrated in Figure 6.1a. The linearly polarized HHG

beam is generated by mixing a near-infrared field (, 780 nm, IR) with its second harmonic field(2, 390 nm, blue) in a Mach-Zehnder interferometer. The pulse duration of the IR pulse is 26 fs.
The polarizations of the two driving fields are adjusted to be parallel to each other through the λ/2
waveplate in each interferometer arm, before they are collinearly focused into a 10-mm-longwaveguide filled with Argon gas at a pressure of ~30 torr. As a result, the HHG beam inherits thepolarization of the combined driving fields. In our experiments, the two-color driving fields canbe described as E t   A sin t   B sin 2t  0   RB  , where A and B are the amplitudes of the
IR and blue driving fields, respectively. φ0 is the relative phase corresponding to the highest HHG
yields. In the experiments, the relative phase φRB can be adjusted using a delay stage with subfemtosecond scanning precision as shown in Figure 6.1a. The peak intensity of the IR driving fieldis ~1.5x1014 W/cm2. To enhance the symmetry breaking between neighboring cycles in HHG, weapply a strong (non-perturbative) blue driving field in our experiments with a field intensity
approximately half of the IR.
The HHG spectra are measured directly using a EUV spectrometer after travelling througha 200 nm Aluminum filter to block the residual driving fields. The spectra plotted in Figure 6.1b
are corrected by considering the transmission of the Al filter reported in Ref. [212]. As shown in
Figure 6.1b, the spectra comprise both odd and even orders of harmonics with comparable intensity,
consistent with previous studies [72,213]. By adjusting φRB, we can control the total HHG yield124

with high precision, showing a modulation of the total yield with a periodicity corresponding tohalf optical cycle of the blue light field [72,213] (Figure 6.1b). This is a clear evidence that theharmonics are indeed generated by the two-color laser field. Notably, The HHG spectra show astrong continuum background, suggesting that the harmonic radiation is emitted in the form of an
isolated pulse [90,92,205,214] – though quasi-continuum radiation can be emitted throughadequate phase relationship between several pulses [215].

Figure 6.1: Characterizing the high-order harmonics using interferometric laser-assisted
photoemission method. (a) Linearly polarized ω (780nm) and 2ω (390nm) beams from a
Ti:Sapphire laser are focused into an Ar-filled hollow waveguide. The generated linearly polarized
HHG and a time-delayed p-polarized 780nm dressing field are focused onto a clean Cu(111)
surface. In experiments, the temporal structure of the attosecond bursts can be adjusted by the
relative phase between the ω and 2ω driving fields, φRB. (b) Upper panel: modulation of the total
HHG intensity as a function of φRB. Lower panel: The HHG spectra measured at the maximum (A)and the minimum (B) HHG intensities. The spectra were de-convoluted by considering thetransmission rate of 200 nm Al filter [212]. (c) Illustration of the quantum path interference of theinterferometric laser-assisted photoemission with HHG field generated by ω-2ω driving fields.

125

To fully reconstruct the temporal structure of the HHG pulse trains, I measure the spectralphases of the generated linearly-polarized HHG beam by attosecond metrology techniques. To
achieve this, I focus the HHG beam onto a clean Cu(111) surface to induce photoemission. The
photoelectrons are simultaneously modulated by a linearly-polarized IR field (780 nm, ћωL=1.6eV), which propagate collinearly with the HHG beam. The relative time delay between the HHG
pulse and IR dressing pulse (τd) can be adjusted by a second delay stage in our experiments. Boththe HHG and the dressing IR fields are adjusted to be p-polarized relative to the sample surface.
Due to the existence of both even and odd harmonics, the photoelectron yield at the kinetic energy
corresponding to the direct excitation by nth order harmonics (ћωn) is modulated due to theinterference (Figure 6.1c) between three quantum paths: (i) absorbing an HHG and an IR photon(ћωn-1+ћωL); (ii) direct photoemission by a single HHG photon (ћωn); and (iii) absorbing an HHG
photon and emitting an IR photon (ћωn-1-ћωL). The photoelectron yields as a function of τd arerecorded using a hemispherical photoelectron analyzer, which gives rise to the interferograms asshown in Figure 6.2a. This method was recently implemented to successfully reconstruct thecircularly polarized HHG beam [30].

6.4

Numerical Simulation
In order to understand the HHG driven by the two-color (ω-2ω) laser fields, numericalmacroscopic HHG simulations were performed using a combination of the strong fieldapproximation with the electromagnetic field propagator [216], where the harmonics are assumed
to propagate at the speed of light, and propagation effects in the driving fields such as thedispersion of neutrals and plasma, and group velocity mismatch [217] are taken into account. Notethat the time-dependent ionization population (computed via the ADK rates [218]) is taken into

126

account, thus including the nonlinear phase shifts in the driving field, whereas the nonlinear spatial
effects are not included. In the simulation, the driving pulses are modelled as a sin2 envelope with
the FWHM pulse duration of τR = 25 fs and τB = 35 fs for fields with wavelengths λ1=775 nm (ω)
and λ2=387.5 nm (2ω), respectively. The peak intensities are 1.17x1014 W/cm2 for the λ1 field andthe intensity of the λ2 field is half of that of the λ1 field, which are selected to be similar to theexperimental conditions.

6.5

Quasi-Isolated Attosecond Pulses
In Figure 6.2a, we plot the interferogram of photoelectrons excited by HHG beam at φRB ≈0, which gives the highest HHG yield. The yields of the photoelectrons excited by different
harmonic orders from the same d band of Cu(111) oscillate with a frequency of ωL as a functionof d. The time delay associated with the photoemission process is negligible as shown in recent
studies [30]. The information of the harmonic phases [ϕ(ωn)] is encoded in the phases of themodulations in the interferogram [α(ωn)], which is given by [30,111]tan  n  

I 0  n1  sin  n     n1   I 0  n1  sin  n1     n 

I 0  n1  cos  n     n1   I 0  n1  cos  n1     n 

,

(6.1)

where I0(ωn) is the power spectrum. Using the genetic algorithm (details are described in Ref. [30]),
I retrieve the harmonic phases that best fit the measured interferogram phases. The fitting results
are plotted in the right panel of Figure 6.2a for φRB ≈ 0. The measured HHG phases are plotted in
Figure 6.2b for both φRB ≈ 0 and φRB ≈ 0.5π. It was found that the HHG phases as a function ofharmonic orders gradually evolve from its originally linear shape to a more complicated “S” shape
as φRB changes from 0 to 0.5π. This trend can be successfully captured by the numerical simulationconsidering the single-atom response as shown in the inset of Figure 6.2b.

127

Figure 6.2: Measuring harmonic phases. (a) 2D map of photoelectron yields at φRB ≈ 0 as afunction of photoelectron energy and pump-probe time delay (τd). The experimentally measured
interferogram phase α(ω) as a function of photoelectron energy is plotted in the right panel, incomparison with the reconstructed phases obtained from phase retrieval program (black solid line).
The good agreement between the two indicates that the correct harmonic phases are retrieved fromthe experimental results. (b) Phases of major harmonic orders measured at φRB ≈ 0 and φRB ≈ 0.5π.
The uncertainty is determined as the standard deviation of phase values retrieved from multiple
trials. For the convenience comparison, the harmonic phases for both cases are offset so that thephase of the13th harmonic is zero. The solid lines highlight the variation of the harmonic phases
and serve the purpose of guiding the eyes. The difference of harmonic phases between φRB ≈ 0.5π
and φRB ≈ 0 is plotted in the inset. The phase difference obtained by the numerical simulation is
also plotted for comparison (dashed line).

With HHG spectra and phases measured, the time-domain structure of the HHG pulse train
can therefore be reconstructed. The temporal profile of HHG pulse train corresponding to φRB ≈ 0
is plotted in Figure 6.3a. The profile is observed to be a quasi-isolated attosecond pulse with apeak width of ~ 450 as (FWHM), accompanied by two much weaker attosecond bursts separated
by one optical cycle of the fundamental IR field. The peak width is very close to the transformlimited value (dashed line in Figure 6.3a), which is consistent with the linear dependence of HHG
phases to the harmonic orders as shown in Figure 6.2b. When φRB is changed to 0.5π, we observe
additional attosecond bursts in the temporal domain (Figure 6.3b). I note that the observation here

128

is distinctively different from several of previous time-domain studies of HHG driven by twocolor laser fields, where the attosecond pulses were found to be generated twice per IR-light cycle[219,220]. In those experiments, the blue-field intensity was <0.5% of the IR and therefore only
constitute a small perturbation onto the IR driving field [219,220].
Obviously, the generation of the quasi-isolated attosecond pulses driven by multicycle laser
fields cannot be explained by the single-atom response during the HHG process. In order tounderstand the results, I resort to the theory by considering the macroscopic response through thetime-gated phase matching [100,165,173,221] under the two-color configuration. In a hollow
waveguide fiber, the phase mismatch for qth order harmonics ( kq 2 ) is given by considering thephase mismatches contributed by 1) waveguide dispersion, 2) free-electron plasma dispersion and3) neutral atom dispersion. When including the second harmonic field (λ2), the phase mismatch is:k q  2 t   k qwaveguide  k qplasma  k qneutral  u2
 u2  
 e2
 e2
 n1 1 112  2n2 2 112   n1 P 1 2  t   2n2 P 2 2  t 
4a  
me c
me c
 4a


,

(6.2)



4 2  0
4 2  0
1   t   2n2 P
1   t  n1 P12
where η(t) is the time-dependent ionization rate,  0 is the linear susceptibility of the gas due tothe neutrals and P is the gas pressure inside the waveguide. The harmonic generation process n1(n2)photons were absorbed from the λ1(λ2) field, so it gives q=n1+2n2. In Equation (6.2), I havecalculated the waveguide dispersion term where a is the radius and u11 = 2.405 is the first zero of
the Bessel function J0 [100,165]. One can also define the coherence length asLcoh2 ,q t    kq2 t  . The perfect phase matching occurs when the three terms in Equation (6.2)
are compensated ( kq 2 t   0 ), which happens at a critical ionization rate. This condition is129

achieved during a finite temporal window in which the harmonic signal builds up constructively,
giving rise to a selection of the number of attosecond pulses that are efficiently emitted.

Figure 6.3: Quasi-isolated attosecond pulses generated through macroscopic phase matching.
(a) The quasi-isolated attosecond pulses reconstructed using experimentally measured spectra andphases for φRB≈0. (b) same as (a) for φRB≈0.5π. (c) Time-dependent coherence length for a onecolor (λ1=775 nm, dashed line) and two-color (λ1=775 nm, λ2=387.5 nm, solid lines) HHG driven
in a 30 torr Ar-filled waveguide. In the one-color case I present the coherence length of the 21st
harmonic (33.6 eV, dashed black), whereas in the two-color case we have selected the 15th
harmonic (24 eV, red solid, corresponding to n1=n2=5), the 18th harmonic (28.8 eV, green solid,
n1=n2=6), and the 21st harmonic (33.6 eV, blue solid, n1=n2=7). The dot-dashed line represents
5Labs, where the absorption length is Labs =1 mm for 33.6 eV harmonics [222]. (d) The attosecond
pulse trains obtained from numerical simulation with two-color driving fields at φRB=0. The results
are calculated by considering the 1D propagation in an Ar-filled waveguide with different
interaction length (L= 1mm, 5mm and 10mm). It is clear that the number of attosecond bursts
decreases as L increases, as a result of the reduction of phase matching window, consistent withthe analytical representation in (c). When L=10mm, only three attosecond pulses are isolated for
HHG emission, which is consistent with the experimental results shown in (b).

130

In Figure 6.3c, I compare the time-dependent coherence length for a one-color (λ1=775 nm,
dashed line) and two-color (λ1=775 nm, λ2=387.5 nm, solid lines) driving schemes in a 30 torr Arfilled waveguide (a=75 μm). The peak intensity of the one-color driving field is 1.3x1014 W/cm2.
The peak intensities for the two-color configuration are the same as those used in the numericalsimulation as previously described. As shown in Figure 6.3c, I plot the time-dependent coherence, H 21
length of the 21st harmonic ( Lcoh, dashed black) in the one-color configuration, whereas in the
2 , H 15
two-color case I have selected the 15th harmonic with n1=n2=5 ( Lcoh, solid red), the 18th
2 , H 18
2 , H 21
harmonic with n1=n2=6 ( Lcoh, solid green) and the 21st harmonic with n1=n2=7 ( Lcoh,solid blue). Note that there are many different photon combinations, but for simplicity I haveplotted those where n1=n2. The appearance of different absorption channels in phase-matching
would be similar to that occurring in non-collinear [117,180] or vortex-combination [223] HHG
schemes, but its analysis lies beyond the scope of this paper. The optimal phase-matchingconditions are achieved if Lcoh  5 Labs (pink dot-dashed line) where Labs is the absorption length[163]. One can clearly observe that introducing the second-harmonic driver significantly reduces
the phase-matching window from ~4 optical cycles of the fundamental IR field in the one-color
configuration to less than one optical cycle in the two-color configuration, allowing for thegeneration of few attosecond bursts driven by multicycle laser fields. Numerical macroscopic
HHG simulations by considering different propagation lengths (L) in the gas-filled waveguide areplotted in Figure 6.3d. Indeed, the harmonic emission is narrowed to few attosecond bursts as themedium length is increased to L=10 mm, in very good agreement with the analytical representation
presented in Figure 6.3c and the experimental results.

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6.6

Controlling the Sub-Cycle Temporal Structure of Attosecond Pulses
By varying the relative delay between two drivers φRB, I directly control the temporal

structure of the emitted HHG pulse train. This is clearly shown by the variation of the pulse train
structures (Figure 6.3a-b) and the HHG phases (Figure 6.2b) for different values of φRB. In Figure
6.4a, I plot the evolution of the FWHM duration of attosecond bursts as a function of φRB, while
the results obtained from numerical simulation are plotted in Figure 6.4b. Because of thecomplicated pulse train structure in the time domain, especially when φRB = 0.5π, I calculate theduration of the attosecond bursts (σ) obtained in experiments and simulations in a statistical way
by defining the mean square weighted deviation (MSWD) within one IR-light cycle (TIR): wt t  t  dt , where w(t) is the pulse-train envelope intensity and t   wt  t dt is  wt dt
 wt dt
TIR

2

TIR

2

0

0

TIR

0

0

0

TIR

0

the center-of-mass (COM) of the attosecond pulse. For an ideal Gaussian pulse,  is the RMS
width. Figure 6.4a shows that the attosecond burst reaches its shortest duration (Δt≈450 as) when
the HHG efficiency is highest at φRB ≈ 0, while the peak width gradually increases when the
spectral intensity approaches its minimum at φRB ≈ 0.5π. This result can be consistently reproducedby the numerical simulation regardless of the macroscopic propagation distances (Figure 6.4b),
indicating that it originates from the single-atom response to the two-color driving fields. To mybest knowledge, this represents the first direct measurement of the correlation between theattosecond pulse duration and HHG efficiency.
To understand this dependence of the attosecond pulse duration on φRB, the semiclassicalcalculation of electron trajectories driven by two-color laser fields is performed. The calculation
extracts the electron ionization and recombination times within the IR optical cycle (dotted lines
in Figure 6.4d-e) and the corresponding field strength at the moment of ionization and132

recombination for different φRB (Figure 6.4c). In the three-step model [64,210], an electron istunnel-ionized in the first step as the electric field passes through its maximum value. In the quasistatic limit, the ionization rate is given by the ADK model [218], which is proportional to 4 2m E 3 e a , where Ea=15.8 eV is the ionization potential of argon, Eion is the field strengthexp 
 3e Eion at the moment of ionization, and me and e are the mass and charge of an electron, respectively. The
values of Eion are directly obtained from the semiclassical calculations and can be controlled byadjusting φRB. Indeed, by comparing Eion (red dashed line in Figure 6.4b) and the HHG yields
obtained from our numerical simulations (red circles in Figure 6.4b), the comparison reveals a
direct correlation between the HHG yield and the driving field strength at the moment of ionization.
Therefore, within the three-step model of HHG, the efficiency of harmonic emission is dominantly
controlled by the total ionization rate at the first step for the two-color driving configuration.
On the other hand, the ionization and recombination of electrons are affected by thesymmetry of the two-color fields in the time domain. As shown in Figure 6.4c, when φRB = 0, thelaser field exhibits a single maximum in each IR optical cycle (  1ion ). The field strength of thesecond maximum is much lower (~60% of the peak field strength) so that the corresponding
electron ionization probability is negligible. As a result, the attosecond EUV burst is only emitted
in the corresponding narrow recombination window around  1emit as shown by the time-frequency
analysis [224] of our simulation results in Figure 6.4d, resulting in the short pulse duration in thissituation. In contrast, when φRB = 0.5π, there exist two maxima with comparable field strength
within each IR optical cycle (  2ion and  3ion ) as shown in Figure 6.4c. Both maxima in this situation
can induce ionization and the subsequent recombination events (  2emit and  3emit ), which give rise133

to HHG emission shown in Figure 6.4e, resulting in the additional attosecond EUV bursts in thetime domain (Figure 6.3b).

Figure 6.4: Coherent control of attosecond pulses by two-color driving fields. (a)Experimentally measured HHG intensity and FWHM peak-width of the attosecond bursts as afunction of φRB. (b) Same as (a) obtained from the numerical simulation. The yellow-coloredsection highlights the range of φRB probed in the experiments as shown in (a). The normalized fieldstrength at the moment of ionization Eion is plotted for comparison (red dashed line). The values
Eion are normalized to the maximum value, while the amplitude of the values (ΔEion) is multipliedby 3 for the convenience of plotting. (c) The time-dependent electric field of the two-color driving
field at φRB=0 and φRB=0.5π. (d) The time-frequency analysis for the pulse train generated at φRB=0
obtained from the numerical simulation. The electron ionization (magenta open circles) andrecombination times (green open diamonds) are plotted and overlaid on the figure. Clearly, at
φRB=0 electrons recombine and harmonics efficiently generate only in a single time window (  1emit )within every IR optical cycle, corresponding to a single ionization time  1ion (also see (c)). (e) Sameas the (d) for φRB=0.5π. Different from the situation when φRB=0, the two-color fields drive
electrons to recombine and emit HHG photons in two time windows (  2emit and  3emit ) every IR
optical cycle, which corresponds to ionization times  2ion and  3ion , respectively. This additional
ionization and recombination time window leads to additional attosecond temporal structure in the
HHG pulse trains driven by two-color laser field as shown in Figure 6.3b.

134

6.7

Conclusion
In summary, I experimentally investigated the HHG pulse train generated by two-color

linearly polarized laser fields. By measuring the HHG phases with interferometric laser-assisted
photoelectron spectroscopy, I demonstrated that the temporal structure of the attosecond pulse
trains can be coherently controlled in the two-color scheme by adjusting the symmetry of the
combined field. It was also found that the phase-matching window for efficient harmonic emission
can be significantly reduced when the two-color driving field was used, allowing generation of thequasi-isolated attosecond pulses from the multicycle driving fields. I expect the ability to generateand coherently control short attosecond pulses, as I have demonstrated in the experiments, canmake the two-color driven HHG a powerful tool for various attosecond studies.

Chapter 7
Direct Time-Domain Measurement of Attosecond Final-State Lifetimes in Photoemission
from Solids

This chapter is adapted, with permission, from:Z. Tao†, C. Chen†, T. Szilvási, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, Direct
Time-Domain Observation of Attosecond Final-State Lifetimes in Photoemission from Solids.
Science 353, 62 (2016) (†These authors contributed equally to this work).

Figure 7.1: Illustration of photoelectron lifetime measurement. (a) High harmonics of a
femtosecond laser kick electrons out of a nickel crystal, and the electrons then travel towards thesurface. The interaction of the electrons with an infrared laser probe allows us to determine howlong electrons linger in excited states before they escape the crystal. Art by Steve Burrows, JILA.

136

7.1

Abstract
The photoelectric effect represents a fundamental light-matter interaction that has been

harnessed to map the band structure of materials with stunning detail. Recently, attosecond
spectroscopies have made it possible to measure differences in transport times for photoelectrons
from localized core levels and delocalized valence bands. In this work, we use attosecond pulse
trains to directly and unambiguously measure the difference in lifetimes for photoelectrons thatare born into free-electron-like states compared with those that are excited into unoccupied excitedstates in the band structure of a material (Ni(111)). A significant increase in lifetime of 212±30 as
occurs when the final state coincides with a short-lived excited state. Moreover, a strong
dependence of this lifetime on emission angle is directly related to the final-state band dispersion
as a function of electron transverse momentum. This finding emphasizes the importance of thematerial band structure on photoemission response times and corresponding electron escape depths,
and represents the shortest lifetime of any state measured to date.

7.2

Introduction

7.2.1

Combing Angle-resolved Photoemission with Tabletop High Harmonic Light
The electronic band structure of materials consists of occupied and unoccupied bands thatemerge as the electron wave functions of adjacent atoms in the lattice overlap with each other. In
general, valence bands will have more wave function overlap and will therefore be wider than, forexample, the core levels. The periodicity of a crystal lattice imposes an energy momentum relationthat is described as the dispersion relationship for electrons in each band, making the electronic
structure of solids quite complex. Fortunately, angle-resolved photoemission spectroscopy(ARPES) can resolve both the energy and momentum of photoelectrons by probing photoemission

137

from materials at well-defined angles. This information is directly related to the energy andmomentum of electrons in various bands in the material, providing powerful and detailed
information about band dispersion and the Fermi surface [148]. Indeed, ARPES is one of the mostversatile and powerful tools for studying the electronic structure of materials. When combinedwith ultrafast lasers, time-resolved ARPES makes it possible to capture fast changes in the band
structure of materials near the center of the Brillouin zone on picosecond (ps) and femtosecond (fs)time scales [225]. More recently, advances in tabletop high-harmonic generation (HHG) [88,207]
have resulted in coherent extreme ultraviolet (EUV) and soft X-ray beams that are ideally suited
for ARPES, opening up time-resolved studies over the entire Brillouin zone [171,208]. HHG isemitted as a series of attosecond pulse trains with unique characteristics of good energy resolution(≈100-300 meV), combined with sub-fs time resolution. These new capabilities have made it
possible to observe and control the fastest electron dynamics in molecular and material systems[226,227]. Recent work probed how fast a material can change its magnetic or insulating-tometallic states [208,228], uncovering which microscopic mechanisms were responsible for driving
fast phase or state transitions in materials. Other work probed the ≈ 7 fs lifetime of core-excitedstates of adsorbates on surfaces [112].

7.2.2

Direct Time-domain Measurement of Photoelectron Lifetime
The time delay associated with the photoemission process itself has been probed in isolated

atoms and solid-state materials using HHG [42,44,54,56,104,142,229], by taking advantage oflaser-assisted photoemission [55]. Although there still exists some discrepancy between
experiment and theory [42,136], it is generally agreed that the photoemission time delay fromisolated atoms consists of a Wigner time delay due to scattering of photoelectrons in the atomic

138

potential, convolved with a measurement-induced delay due to propagation of the photoelectrons
in the laser field [230]. Compared to the case of isolated atoms, interpreting photoemission timedelays from solids is more complicated, because of the complex band structure of materials andvarious many-body interactions that photoelectrons can experience as they approach the surface.
Previous attosecond-streaking studies of a transition-metal W(110) surface revealed a considerable
time delay (~110 as) between photoemission from core-level and valence-band states [44]. In
contrast, no delay was measured for the free-electron metal Mg, although a similar time delay to
W was expected [142]. Various theoretical models have been proposed to explain thephotoemission time delay from solids in terms of transport time [138,140], degrees of initial-state
localization [138,141], and band-structure effects [143,231], highlighting the complex physics thatis not yet well understood. Moreover, the use of isolated attosecond pulses in these experiments isnecessarily accompanied by a broad EUV bandwidth, that then gives rise to an integrated broadband photoemission feature comprised of multiple valence bands of a material [44,56,142]. Other
recent work used synchrotron sources to measure the final-state linewidths of photoemission fromCu, and found pronounced variations in the ARPES spectra with small changes in photon energythat could not be assigned to a free-electron final state. This work identified the direct transitions
and measured very broad linewidths of >3 eV for final states in the 20 to 150 eV photon energy
range [144].
In this work, we directly and unambiguously measure the influence of the band structureof a material on the lifetime of photoelectrons. To accomplish this, we use attosecond pulse trainsof well-defined harmonics to measure the photoemission time delays for both free-electron final
states as well as final states corresponding to the unoccupied bulk bands of the transition metalNi(111). We distinguish photoelectron lifetimes from individual valence bands and final states in139

Ni(111), with additional help from selection rules for photoemission for EUV fields with different
polarizations [148,157]. Our results show that photoelectrons experience an abrupt increase inlifetime by ≈ 212 ± 30 attoseconds when the final-state corresponds to an unoccupied excited statein the Ni band structure. This increase in lifetime can be interpreted physically as a variation of
the attenuation length of the final-state wavefunction inside the crystal. Therefore, through a direct
time-domain measurement of attosecond photoelectron dynamics, our results provide new insights
into the fundamental concepts of photoelectron lifetime, inelastic mean free path and also groupvelocity. Moreover, we observe large angle- (momentum-) dependent variations in photoemission
time delay, which are directly related to the final-state band dispersion. This new capability toprobe band-structure effects on photoemission significantly enhances the use of HHG forexperimentally accessing quasiparticle lifetimes, electron mean free paths, electron-electron
scattering and dynamical screening, all of which represent grand challenges for condensed mattertheory.

7.3

Ni(111) Surface Preparation
The Ni(111) single crystal films are prepared by sputtering 200 nm of nickel onto α-

Al2O3(0001) substrates as described in reference [232]. The sample cleaning is performed in thesame UHV chamber for photoemission measurements, with the base pressure <5×10-10 torr. The
atomically clean Ni(111) surface is obtained using repeated cycles of Argon ion sputtering (0.5keV, incidence angle of 60o) at room temperature (300K) in UHV chamber, followed by annealing
to 900 K for 15 minutes. The sample is grounded during all static and dynamic measurements.

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7.4

Experimental Setup for Attosecond Photoelectron Lifetime Measurement

Figure 7.2: Experimental Setup of photoelectron lifetime measurement.

Figure 7.2 illustrates the setup of our experiments. We used a multi-pass Ti:sapphire laser
system to generate 26 fs pulses in the near infrared (IR) at a wavelength of ~780 nm (ħωL≈1.60eV), with pulse energy of 2.4 mJ, and at a repetition rate of 4 kHz. Most of the laser energy (95%)
is used for high-harmonic generation, while a small portion (5%, linear polarization) is used toprobe (dress) the photoelectrons generated by the attosecond EUV pulse trains. To drive highorder harmonic generation, 95% of the IR beam is focused using a 50 cm focal length lens into a
1-cm-long capillary waveguide, which has an inner diameter of 150 μm. Different gas targets (Xe,Kr, Ar, Ne) are used in order to cover a broad range of harmonic orders or photon energies (11th– 41st spanning from 17 to 66 eV). The corresponding EUV pulse train is comprised of ~200 as[full width at half maximum (FWHM)] bursts with a ~15 fs (FWHM) envelope [32] in the time
domain. The EUV light inherits the polarization of the driving laser, which is manipulated by a
λ/2 wave plate. The control over EUV photon energy and polarization allows us to selectively

141

excite photoelectrons from different initial states into different final states (free-electron-like orunoccupied bulk band) and study their individual lifetime (Figure 7.3a). The comb of linearlypolarized high-order harmonics are then focused onto an atomically clean Ni(111) surface using a
toroidal mirror (coated with B4C or gold depending on HHG orders used) at grazing incidence, toa spot size of ~100 µm FWHM. Any residual laser light is blocked by a 200 nm thick Al filter.
To measure the photoelectron lifetime, the linearly polarized IR probe is recombined
collinearly with the high-order harmonics onto an atomically Ni(111) surface using an annularsilver mirror (Figure 7.2 and Figure 7.3a). In the presence of both two fields, there are two distinct
quantum paths exciting the electrons from an initial state to the same final photoelectron energy(sideband), namely (i) absorption+  L and (ii) emission-  L of an IR photon,

as shown in Figure 7.3b. The attosecond beating due to the interference between two quantum
pathways modulates the photoelectron spectra as a function of relative delay between the EUV
pump and IR probe fields (τd). This allows us to extract photoelectron dynamics on attosecond
time scales and Å length scales (in a technique called RABBITT) [32,104]. RABBITT andattosecond streaking have been shown to yield the same temporal information about thephotoemission process [136,230]. In the experiment, the relative time delay between the HHG
pump and IR probe is controlled using an interferometrically-stabilized stepper-motor delay stage(with sub-fs resolution). The FWHM of the IR beam on the sample is ~250 µm, which is larger
than the HHG beam to ensure efficient dressing of the photoelectrons. The intensity of the probe
IR light is adjusted to be ~3.75 µJ/pulse (peak intensity ~3×1011W/cm2) using a half wave plate
and a linear polarizer. The low intensity of the IR dressing light ensures that the contributions fromhigher-order sidebands (>1st order) are negligible. The IR probe is fixed to be p-polarized, i.e.,
perpendicular to the sample surface for all measurements.

142

Figure 7.3: Photoemission time delay on and off resonance in the band structure. (a) Using
high-order harmonics, different photoelectron final states can be accessed, corresponding to freeelectron-like states or excited states in the band structure. The damping length of the final-state
wavefunction inside the crystal is significantly increased when the transition coincides with a finalstate resonance. (b) Schematic of quantum path interferences from a single initial state using a
combined EUV and infrared laser fields.
Band Structure of Ni(111) along Г-L Direction

7.5

We first probe the electronic band structure of Ni(111) along Г-L direction by studying thedependence of the static photoelectron spectra on the EUV photon energy and polarization. Under
the excitation of EUV harmonic light, photoelectrons with sufficiently large momentum along thesurface-normal direction escape the surface and are detected by a hemispherical electron analyzer.
In the angle-resolved photoelectron spectrum, the same initial Bloch wavepacket is excited tomultiple final states with different kinetic energies by multiple harmonic orders, manifesting itself
as a ladder of direct photoemission bands, each separated by twice the fundamental photon energy(  L ), as shown in Figure 7.4 (a-b), which are excited by s- and p-polarized EUV harmonics,
respectively.

143

Figure 7.4: Static ARPES spectra of Ni(111) excited by multiple orders of HHG with (a) sand (b) p- polarization. The energy resolution is ~0.3 eV, which is sufficient to distinguishphotoemissions from different initial bands ( 3 , 3 and 1 bands ). (c) EDC curves excited bys- (red) and p-polarized (blue) HHG in a normal emission geometry (integrated ~±2o around the Гpoint). The position of the Fermi level (black dashed line) is determined from the laser photon
energy (~1.6 eV) and analyzer work function (4.25 eV). The orange dashed line shows the shift ofthe high-energy peak with HHG photon energy, underlining the contribution of bulk-band
transitions. The intensity of the 3 band clearly shows a spectral resonance at ~24 eV.
To extract the band dispersion along the Г- L direction, We first obtain the photoelectronenergy distribution curves (EDCs) by integrating the angle-resolved spectra over an angle of ±2oaround the  point and deduct the secondary electron backgroud using the Shirley backgroundsubtraction [233], as shown in Figure 7.4c. The energy of Fermi level is determined with theknowledge of the photon energies and the analyzer work function. By utilizing the high energy
resolution of attosecnd pulse trains and the photoemission selection rule under polarized (s- andp-) HHG radiation [148,157], we can unambiguously indentify the peaks corresponding todifferent intial states. According to the DFT band-structure calculation for Ni(111) single crystal,
there are three bands along Γ-L contributing to the photoemission spectra: two valence bands with144

Λ3 symmetry ( 3 with lower binding energy of ~0.6 eV and 3 with higher binding energy of~1.25 eV) and one deeper valence band with Λ1 symmetry (~1.7 eV). Since Λ1 band can only be
excited with light field polarized perpendicular to the sample surface, we can assign the two
spectral peaks in the EDCs excited by s-polarized HHG light as photoelectrons from 3 and 3
bands. We then use double Voight function to fit the EDCs and extract their spectra intensity andbinding energy, as shown in Figure 7.5(a-b). The Gaussian linewidths of the Voight function are
fixed to the typical experimental resolution of 0.3 eV.

Figure 7.5: Static spectral analysis of Ni(111) exicted by HHG. (a) and (b) Experimentallymeasured EDCs excited by s-polarized HHG field (red open diamonds). The intensity and binding
energies of 3 and 3 bands are extracted by fitting the EDCs with double Voigt functions (solid
black lines). (c) and (d) The spectral intensity of 1 is extracted by taking the difference spectra
between EDCs excited by s- and p-polarized HHG fields (blue open circles). The intensity andbinding energies are determined by fitting the EDCs with a single Voigt function (solid black lines).

145

Figure 7.6: Band Mapping of Ni(111). (a) Band structure of Ni(111) along Г-L (normal tosurface). The band structure extracted from our data (open symbols) is compared to previous
experiments (solid lines) and DFT calculations (dashed lines). A free-electron final-state in aconstant inner potential is assumed and used to map the electron momentum normal to the sample
surface k  . The final state resonance is highlighted as direct transition from the 3 initial band tothe 1 final band. (b) Photoemission intensities of 3 and 1 bands relative to that of 3 band.
B

The red solid line represents the Lorentzian lineshape fitting to the intensity of 3 band as afunction of photon energy, yielding a linewidth of   3.68 eV. The blue solid line is a linear fit
to the intensity of 1 band.
To obtain the information about the 1 band, the EDCs excited by the same harmonicorders but with different polarizations (s- and p-) are normilized to equalize the intensity of the3 band. The additional spectral weight in photoelectron spectra excited by p-polarized HHG isassigned as photoemission from 1 band. We obtain its spectra intensity and binding energy byfitting it with a single Voight function, are shown in Figure 7.5(c-d).
We note that both s- and p-polarized photoemission exhibit peak shifts when excited bydifferent photon energies, which is clear evidence of contributions from bulk-band transitions.

146

Assuming direct transitions to a free-electron-like final state, we can map the electron momentum
in the surface-normal direction k  for different photon energies [148,196,234]. The extracted
band structure is plotted in Figure 7.6a. The good agreement between our photoelectron spectrawith previous studies using synchrotron light covering a similar energy range [234] (see Figure7.6a) underlines that high harmonics are ideal for capturing electron and band structure dynamics
with very high time resolution, good energy resolution, easy manipulation of the photon energy
and polarization [30], and perfect synchronization to the driving laser. Note that the calculated
valence bands using density functional theory (DFT) are ~0.7 eV deeper than experiment, due tostrong correlations present for Ni 3d electrons [234–236].

7.6

Spectral-Domain Measurement of Final-State Lifetime in Ni(111)The most pronounced feature of Figure 7.4 and 5.5 is the enhancement in the intensity of

the low-energy spectral peak (~-1.25 eV) when excited by ~24 eV HHG photons (15th order). This
resonant feature is clearly shown in the spectra excited by an s-polarized EUV field, indicatingthat the 3 band is the initial band. Considering the band structure of Ni(111) along Г-L, we canassign this spectral resonance to direct interband transitions from the 3 initial band to the highenergy 1 final band located at ~24 eV above the Fermi level, as indicated by the blue arrow in
B

Figure 7.6a.
In photoemission theory, the matrix element responsible for photoelectron spectra can be
written as M fi  f H int i , where i is the initial state,fis the final state andH int  AX  p  p  AX represents the interaction Hamiltonian between an electron and EUV
electromagnetic radiation with the vector potential AX. The final-state wavefunction can be

147

represented as a time-reversed low-energy electron diffraction (LEED) state, composed of a freeelectron wave arriving at the detector and a damped Bloch wave inside the crystal [148,237], witha characteristic damping length directly related to the inelastic mean free path and photoelectron
lifetime. The increase in photoemission intensity as a result of this final-state resonance can be
attributed to an increase in lifetime, with associated elongation of the characteristic damping lengthof the final-state wavefunction, which strongly enhances the spatial overlap between the initial andfinal-state wave functions - and hence the transition probability.
To determine the coupling of the intial bands to the unoccupied final bands and also thelinewidth of the final bands, we need to examine the dependence of indivual band-specific spectral
weight on photon energy. To take into account the intensity modulation of the HHG light, wenormalize the intensity of the 3 and 1 bands to the intensity of 3 band. The relative
intensities as a funtion of harmonic orders and photon energies are plotted in Figure 7.6b. The
resonant transition from the 3 band to the unoccupied bulk band 1B corresponds to an intensity
peak around 15th order harmonic. Lorentzian function fit gives the width γ=3.68 ± 0.88 eV
centered at 24.40 eV, which is consistent with previous studies [144]. Thus, the characteristic
lifetime can be extracted from  spec     179 ± 43 as. In contrast, intensity of 1 band varies
smoothly across the photon energy range used in our experiments.

7.7

Direct Time-domain Measurement of Final-State Lifetime in Ni(111)
To directly measure the photoelectron lifetime in the time domain, we use the RABBITT

method (see Figure 7.3b and section 7.4: experimental step) and measure the photoemission timedelay between valence bands of Ni(111) excited to bulk final bands or to free electron final states
located in the band-gap region [148,237] (Figure 7.3a). When the relative time delay d between

148

the HHG pump and the IR probe is changed, the intensity of side bands from a given initial bandis modulated as a result of quantum-path interferences [32,104],

S 2 n  d   A0 sin2 L  d   X   PE  ,where A0 is the intensity of modulation,  X (7.1)

2n1  2n1
represents a time delay contributed by
2L

the phase chirp between neighboring harmonics ( 2 n 1  2 n 1 ) and  PE is the photoemission delay.
Simultaneous measurement of two photoelectron wavepackets excited by the same harmonicorders allows us to cancel the influence of attochirp (τX). This method was first implemented andvalidated for isolated atoms [104]. In our experiments, we use the non-resonant photoemission
from 3 band as the timing reference, and determine the relative photoemission delay between
the resonant  PE 3   PE 3  and non-resonant  PE 1   PE 3  . This allows us to compare
time delays for comparable photoelectron energies, and cancel effects due to the attochirp or fieldinduced phase delays.

7.7.1

Time-Resolved Data Collection
To improve the signal-to-noise ratio and minimize the experimental systematic error, thesame pump-probe delay sequence from -3 fs to 3 fs in 0.2 fs step is repeated more than 200 times
with increasing and decreasing delays alternating between consecutive scans. Each individual
photoelectron spectrum in time-resolved measurements is integrated for 4~6 seconds to get
sufficient photoelectron counts. To minimize the space charge effects, the overall photoelectron
counts are adjusted well below the point where further reduction in HHG light intensity does notshift the spectra. The repetitive scans are analyzed individually and then averaged to obtain the149

results of photoemission delays. The error bars are determined by considering the variation of
measured time delays in individual scans.
We check the data quality by examining several factors: 1) We confirm that the peak
positions in each spectrum corresponding to different initial bands are not altered because of spacecharge effects, above threshold ionization (ATI) by IR field or the shift in high harmonic photon
energies. 2) Each sideband region of interest in every used scan shows a dominant 2ωL (ωL is thefundamental laser frequency) oscillation in Fourier analysis. The interferograms shown in Figure
7.7(b-d) and Figure 7.8c are obtained by summing the interferograms of individual scans satisfying
the two criteria above.

7.7.2

Photoemission Time Delay between 3 and 3 Bands
By utilizing the photoemission selection rules and using s-polarized HHG field in time-

resolved measurements [148,157], we can selectively excite 3 and 3 bands and

 

 

unambiguously determine their photoemission time delay [  PE 3   PE 3 ]. To cover as large
energy ranges as possible, different noble gases (Xe, Kr, Ar, Ne) are used for high-harmonic
generation (Figure 7.7a). The interferograms for d in range of -2 to 2 fs obtained using Xe, Kr andAr gas targets are plotted in Figure 7.7(b-d). The interferograms were obtained by integrating thephotoelectron spectra over ±2.5o around the Г point and subtracting the background spectrum
obtained well before time-zero. We intentionally selected a small angular region to avoid theambiguities resulting from angle-dependent photoemission time delay (see section 7.8). The side
bands around the spectral resonance (SB 14, 16 and 18) can be covered by various gases. The
extracted photoemission time delays are consistent between different gases, underlining therobustness of our results. This result indicates that the observed time delay origins from the

150

material properties and the atto-chirp contributed by HHG process is cancelled out using our
method. In the error analysis of experimental data, the deviation of results from different gases arealso taken into consideration.

Figure 7.7: Photoemission time delay measurements obtained using different noble gas
targets. (a) Photoemission time delays from laser-dressed harmonic sidebands for s- and ppolarized HHG for noble gas targets. A significant delay is introduced at harmonic order 15, due
to the > 200 attosecond lifetime of the excited-state in the material band structure. Theinterferograms covering the resonant energies are obtained by using (b) Xe, (c) Kr and (d) Ar as
gas targets for high-harmonic generation process. The photoemission is excited by s-polarized
HHG field. 11th to 17th HHG orders are covered by Xe, 13th to 19th by Kr and 15th to 25th by
Ar. By using Ne, the EUV probe can cover 23th to 41th HHG orders (not shown here). The
oscillation offsets at the side bands corresponding to the spectral resonance are highlighted bywhite dashed boxes, yielding consistent time delay using different gases.

151

To better visualize the dependence of photoemission time delay on photon energy, we
truncate the interferograms covering different energy regions [Figure 7.7(b-d)] and stitch them
together to generate the broad-range interferograms shown in Figure 7.8c. We observe thatphotoelectrons from the 3 band are significantly delayed for sideband (SB) 16 (corresponding
to a photon energy 25.6 eV), which manifests as a strong oscillation offset in the interferograms.
This time delay gradually vanishes at increasing and decreasing photon energies, showing a nonmonotonic change of  PE 3   PE 3  as a function of photon energies. To extract the 1D
lineouts corresponding to the 3 and 3 bands in the side-band region, we integrate thephotoelectron counts over a spectral window with an energy width of 300 meV and an angle widthof ±2.5o centered at  point of the sideband peak corresponding to the band of interest and plotthem as a function of the pump-probe delay (τd). 1D lineouts of 3 and 3 bands are plotted in
the panel on the right side of Figure 7.8c, making it possible to determine the precise values of PE 3   PE 3  as a function of photon energy.

To improve the efficiency in the data analysis on the huge dataset of time-resolved
measurements, we employed a cross-phase Fourier analysis to extract the relative phases between
the oscillations of two bands. The relative phase between two band-dependent 1D lineouts









[ x 3 , d and x 3 , d ] can be determined by calculating their cross-correlation function







 



C 3 , 3 ,   ~x * 3 ,  ~x 3 ,  ,(7.2)

x   is the Fourier transform of x  d  and ~x *   is the complex conjugate of ~x   . The
where ~



peak frequency of C 3 , 3 ,  is the oscillation frequency shared by both curves and equals to2ωL, as shown in Figure 7.8c. The averaged phase of C 3 , 3 ,  at this peak frequency

152

corresponds to the relative phase difference between the two oscillations. The extracted time delay
is consistent with the values directly extracted by fitting the 1D lineouts with sinusoidal functions.

Figure 7.8: Direct time-domain measurement of the final-state lifetime. (a) Photoemission
time delays  PE 3  PE 3 and  PE 1   PE 3 as a function of photon energy, clearly

 

 

 

showing an increase in lifetime by 212 ± 30 as when the final state corresponds to a short-lived
excited state in the band structure. (b) Spectral intensity of the 3 initial band as a function ofphoton energy. The blue point (14th order) is obtained from 390 nm driven HHG. The pink line
represents a Lorentzian fit, yielding a linewidth of   3.68 eV. (c) 2D map of photoelectron yields
as a function of photoelectron energy and pump-probe time delay  d , excited by s-polarized HHG.
To enhance the color contrast, 90% of the ground-state spectrum is subtracted to visualize theinterferogram. The relative delays between photoelectrons from the 3 and 3 initial bands are
manifested as a strong oscillation offset in the side bands (white dashed). 1D lineouts for 3 and3 initial bands in the corresponding regions are plotted in the right panel.

153

The results are summarized in Figure 7.8a. By comparing this relative photoemission delay
with the spectral resonance (Figure 7.8b), we find that the observed maximum in time delay
coincides with the spectral resonance at the same photon energy, strongly indicating that theobserved photoemission time delay originates from the unoccupied bulk final band Λ1B of Ni(111).
Most importantly, the time delay measured using laser-assisted photoemission,  chron = 212 ± 30
as, agrees with  spec = 179 ± 43 as from the spectral resonance, within error bars.

7.7.3

Photoemission Time Delay between 1 and 3 Bands
To extract the photoemission time delay  PE 1   PE 3  , we used p-polarized EUV

fields for photoemission. We note that in this situation, the low-energy side-band intensity of the
RABBITT oscillations has contributions from both 1 and 3 bands, which is labelled as3  1 . As a result, the relative time delay we extracted is  PE  3  1    PE  3  . Because
the quantum paths corresponding to these two oscillations start from different initial bands, we can
exclude the interferences between them. Rather, we expect a simple intensity addition of 1 and3 RABBITT signals. The results of  PE  3  1    PE  3  are plotted in comparison with thevalues of  PE  3    PE  3  as a function of harmonic order and photon energy in Figure 7.7a.
Due to non-negligible contributions of 3 band, large time delays can be observed around theresonant photon energy (~24 eV) when excited by p-polarized field.
In the strong field approximation, the modulation amplitude at SB 2n is proportional to thesquare-root of the spectral intensity product of the two neighboring one-HHG-photon-excited
direct photoemission bands which contribute to this side band: A2n  I 2n1I 2n1 [34]. The spectral

154weight of direct photoemission from the initial 3 and 1 bands have been determined by thestatic spectral analysis, as shown in Figure 7.6b. We note that for side band order lower than 14th and higher than 18th, the spectral weight of 3 band is so low that the time delay  PE 1    PE 3

can approximately equal to the time delay directly measured using p-polarized HHG field

 





 

 

[  PE 1   PE 3   PE 3  1  PE 3 ]. The values of  PE 1    PE 3 for SB 14, 16 and18 need to be extracted through a more detailed analysis, where the ratio of the modulation 

amplitudes between 1 and 3 bands is given by P  A2n 1  A2n 3 = 1.0, 0.64 and 0.76,

 

respectively. To extract  PE 1    PE 3 of these side bands, we employ the following fitting
algorithm:
1) We model the RABBITT oscillation lineout excited by p-polarized HHG field by considering
the relative contributions from both 3 and 1 bands:  .


Apmodelpol  d   P  sin2L  d   PE 1  sin 2L  d   PE 3

 

(7.3)

 

Here,  PE 3 is assumed to be zero for simplicity. The parameters P and  PE 3 aredirectly obtained from experimental results, with the latter from the time delays measured
using s-polarized HHG field. A modeled lineout with discrete delay points (τd) from -3.0 fs to
3.0 fs with 0.2 fs each step can be generated using Equation 7.3.
2) We fit this modeled lineout using a sinusoidal function to extract modeled time delay for ppolarized HHG field,  model  p  pol  .
3) By varying the value of  1  in range -200 as to 200 as with 2 as each step, we find the best
match of the modeled p-polarization time delay  model  p  pol  to experimentally measured 

one  expt  p  pol  (Figure 7.7a) and hence determine the value of  PE 1    PE 3 .

155

The fitting procedure can achieve convergence for the results of SB 14, 16 and 18, yielding 

the photoemission time delay  PE 1    PE 3 as shown in Figure 7.8a. The error bar isdetermined by considering the errors of the RABBITT measurements using both s- and p-polarized
HHG lights. In strong contrast to the 3 band, photoemission from the 1 band exhibits a verysmall time delay relative to that from the 3 band and varies smoothly across the photon energy
range.

7.7.4

Lifetime of the Unoccupied Bulk Final Band 1B
The fact that our measured lifetime agrees well with the lifetime extracted from the spectral

linewidth indicates that the lifetimes of photoelectrons originating from the 3 band are very small(within experimental error ~±53 as). At the same time, the photoelectron lifetimes corresponding
to the 1 band are ≈50 as. Since both bands exhibit a smooth variation in photoemission crosssection over the photon energy range used here, it is reasonable to assume their photoelectron
lifetimes are slowly varying as a function of photon energies. Thus, the results presented in Figure
7.8a can be interpreted as an abrupt increase of ~212 as in photoelectron lifetime from the 3 band
when the direct transition approaches a bulk final band ( 1 ), resulting in a final-state resonance.
B

156

Figure 7.9: Extraction of angle-dependent photoemission time delays. (a) Illustration of theangular regions for integration when extracting the angle-dependent photoemission time delay.
The energy region corresponding to SB16 is used as an example here. (b) Angle-dependent
photoemission time delays  PE  3 ,    PE  3 ,  at different side bands. The phases of the







RABBITT oscillations at different angles [  PE 3 ,  ] are used as the timing references for theother band.

7.8

Angle-Dependent Photoemission Time Delay Caused by Final-Band
Dispersion
Other important evidence that the band structure of the material strongly influences thephotoemission lifetime can be seen from the angle-dependence of the lifetime as a function ofelectron transverse momentum k // . To extract the angle-dependent photoemission time delay, we
first divide the angle-resolved photoemission spectra excited by s-polarized EUV light at each sideband into ten angular regions, with a ~3o angular span over the entire range of photoelectron
emission angle (~±15o), as shown in Figure 7.9a. The photoelectron counts in each region are
integrated and plotted as a function of pump-probe time delay τd to obtain the angle-dependent
RABBITT interferograms. Typical interferograms at the spectral resonance are plotted in Figure

7.10b. In this way, the angle-dependent photoemission time delays for photoelectrons from 3 and3 bands can be extracted. To cancel any time delays imposed by the incident and reflected laser
fields [60,118] and to extract the contribution due to band-structure effects, the RABBITT

157


oscillations from the 3 band were used as the timing reference at different emission angles (θ)













for each side band. The angle-dependent results of  PE 3 ,    PE 3 ,  are summarized in
Figure 7.9b for SB12 to SB24 (near the resonance). As illustrated in Figure 7.9b, pronounced
angle-dependence of photoemission time delays is only observed at the energies around thespectral resonance (SB16 and SB14), while it diminishes at the other sidebands, which strongly
supports the band-structure origin of these delays.

Figure 7.10: Angle-dependent photoemission time delays. (a) Angle-dependent photoemission













time delay  PE 3 ,    PE 3 ,  for SB 16 and 14 obtained using s-polarized HHG. The data
are points, while the solid lines are a fit to the final-state band structure obtained from our modeland DFT calculations. (b) Typical RABBITT interferograms for SB 16 with emission angles [(A)and (B)] labeled in (a). The chirp is highlighted with white dashed boxes. (c) Illustration of directtransitions in the direction   K for SB 14 and 16. Because different photon energies are used forthese two side bands, different k  along the Г- L direction are assigned according to the band-maps
in Figure 7.6a. The initial and final bands are highlighted by thick solid lines and the binding
energy of the initial band (pink) is corrected according to the binding energy obtained in our
experiments. The transitions are labeled as dashed arrows. Inset: Experimental geometry: IR andHHG beams are focused onto a Ni(111) surface at a 45o incident angle. θ is assigned to the emission
angle of photoelectrons relative to the sample normal direction (z).

158













Figure 7.10a shows the individual plots of  PE 3 ,    PE 3 ,  at SB 16 and SB14 tobetter visualize their angle dependent structure. As illustrated in Figure 7.10a, the time delay PE 3 ,    PE 3 ,  at SB 16 reaches its maximum value ~220 as at  point (θ=0), while it
reduces to ~30 as when the emission angle approaches θ=±15o. The overall trend is symmetric













around the  point. Most interestingly,  PE 3 ,    PE 3 ,  exhibits a double-peak shape at
SB 14 (Figure 7.10a), as the time delay increases to its maximum on-resonance value at θ≈±6o andthen decreases for larger emission angles.
To understand the band-structure effects on the angle-dependent photoemission timedelays shown in Figure 7.9b and Figure 7.10a , our first task is to determine the final band structure
along  - K direction. To do so, we employed DFT calculation to calculate the bulk electronic band
structure of Ni(111) up to ~40 eV above the Fermi level, which has shown good agreement to theexperimental results along Г-L momentum direction. The results exhibit that the final band(originally  1 band along Г-L) evolves into a downward dispersed Σ1 band, as shown in Figure
B

7.10c. This conclusion is also supported by the angle-resolved photoemission spectra in our
experiments, which exhibit an increment of the probed momentum range of k // when the EUV
photon energy approaches lower energies from the resonant photon energy (Figure 7.4a). We
believe that the final-band dispersion can lead to a mismatch between the EUV photon energies
and the resonant energies at different angles, hence affect the measured angle-dependent
photoemission time delays.
To quantify this, we model the photoelectron lifetime as a Lorentzian lineshape, centeredat the resonant energy [E0(θ)] with a linewidth assigned to the resonant spectral linewidth γ (Figure

1597.8b). Thus, the photoemission time delay as a function of photon energy  X and emission angle
θ is given by X ,  2M

 4X  E0  2   2

,

(7.4)

where M is the amplitude coefficient, which determine the maximum time delay at resonance[  X  E0   ]. The resonant energy E0   is directly extracted from the energy difference
between the initial and final bands at different transverse momenta k // along   K obtained from
DFT calculations. The emission angle θ can be directly related to k // by the well-known k //  , where Ekin is the photoelectron kinetic energy given byrelationship   arcsin
 2m E e kin 
Ekin   X  E B   , with EB and Φ the binding energy and the workfunction, respectively.
Because different values of the perpendicular momenta ( k  ) along Г-L direction are probed byusing different EUV photon energies (Figure 7.6a), the dispersion of the initial and final bands
along Г-L direction is also taken into account in this model. Compared to the final 1 band, theBdispersion of initial 3 band is smaller by one order of magnitude. As a result, we treat the initial3 band as a flat band along both   K and Г-L directions with a binding energy of 1.2 eV,
obtained from our experimental results. To determine the structure of final 1 band, we firstB

determine the values of k  from the band-mapping results in Figure 7.6a for photon energies
corresponding to different side bands (e.g., SB14: k   0.87 Å-1 and SB 16: k   0.61 Å-1),
respectively. Then the final Σ1 band along   K ( 1 in Г-L) is extracted from DFT calculation atB

the corresponding k  . We note that the effect of band dispersion along Г-L is mainly to shift the160energy of the final band, while the band dispersion along   K in the momentum range ofinterest are approximately the same at different k  .
With the values of E0   determined, we can fit the experimental results for SB 14 and SB
16 by adjusting the value of the amplitude coefficient M. We note that M is the only fitting
parameter used in this model, while the other parameters can be pre-determined by theexperimental parameters and DFT calculations, which underlines a very good constrain on thisproblem. The fitting results for SB 14 and SB 16 are plotted as solid lines in Figure 7.10a. To
obtained a good fitting to our experimental results, we find M≈1660 for SB 16 corresponding to a
maximum time delay of ~250 as at resonance, while M ≈ 950 for SB 14 corresponding to a
maximum time delay of ~140 as. The choice of different values of M at various photon energies
might indicate an influence of the band structure along Г-L on the photoelectron lifetime, which
requires further investigations.
Our results show that for SB 16, the HHG photon energy closely matches the resonant
excited state at the  point, yielding a maximum time delay for θ=0. Moreover, the time delay
monotonically decreases at larger emission angles, since final-state band dispersion causes thetransition to be off resonance. On the other hand, for SB 14 where the photon energy is ~3.2 eVlower, the resonant transition is not accessible at the  point, but is on resonance and yields a
maximum time delay at a ~ 6o emission angle (Figure 7.10c). The key factor for this agreement isthe dispersion of the Σ1 final band (~ -14 eVÅ /  ), which determines the slope of the time delay
as a function of θ.

161

7.9

Classical Considerations on Photoemission Time Delay and Screening of IR
Field
To understand the experimentally measured time delay, it is helpful to first consider the

problem in a classical picture. In a classical model, the drift velocity v(t) of an electron under theeffects of IR dressing field with a vector potential A(t) can be determined if we know the velocity(vi) of this electron at a specific time (τi), v t ,  i , vi   eeAt   vi A i  . If we measure
me
me



the energy of this particle at infinitely far away, the first time-dependent term can be ignored att   , hence we have its final drift velocity v f  i , vi   vi eA i  .
me

Considering the situation when the IR dressing field is applied onto a metal surface and theelectron escapes from inside the metal, two consequences are expected: 1) the electron moves
faster inside metal than in vacuum, since the workfunction (Φ) is deducted when the electron
escapes across the interface; 2) the intensity of the IR dressing field is strongly screened inside
metal [60]. So, it is more convenient to consider the electron dynamics inside metal separately.
We can take the metal-vacuum interface as the boundary, after which the electron moves as a freeparticle in the IR dressing field. By knowing the time when the electron arrives right above themetal surface (τs) and the escaping velocity (vs), the final drift velocity is given byv f vs , s   vs 

E fin eA s  .
me

As

a

result,themeasured

photoelectronenergyise11A s   v s which is valid for mostme v 2f  me v s2  v s  eA s  , if we assume2
2
me

attosecond-streaking or RABBITT experiments. Here we assume the IR dressing field arrives at162

t=0, while the EUV photon arrives and excites photoelectrons at t=τ0, hence the relative pumpprobe delay is τd = - τ0.

Figure 7.11: Classical model for photoemission time delay. (a) Streaking curves for surfaceborn photoelectrons (black line) and bulk-emitted electrons at initial depth of 10 Å. Two skindepths of IR fields are considered: l0~1 Å (red line) and l0   (blue line). The IR field strengthis E L 0  2x1013 V/m with a FWHM pulse duration of 30 fs. (b) Time delay between streaking

curves of bulk-emitted and surface-born electrons [  bulk    surface  ] as a function of the initial
depth of the bulk-emitted photoelectrons. The results for skin-depth of l0~1 Å (red open circle),l0   (blue filled diamond) and the IR field distribution calculated using Fresnel equations (pinkopen square) are plotted. The black solid line represents the transport time delay. (c) Results of 1D
semiclassical simulations. Photoelectrons emitted 10 Å below the surface are delayed by 267 ascompared to those emitted from 2 Å. Inset: profile of the electric field strength normal to surface( E z ) across the interface.

163

For an electron born at depth λ inside the metal, the time it arrives the surface is s   0   vtr   IR , where vtr is the velocity of this electron in the metal (only considering thedirection normal to the surface) and ΔIR is attributed to any time delay induced by the IR dressing
field inside metal. If we plot Efin as a function of τd, typical attosecond-streaking curves can be
reproduced and a time delay ( t   vtr   IR ) can be observed for this bulk-emitted electroncompared to an electron born right above the surface at t=τ0. In addition, the escape velocity vs of
the bulk-emitted photoelectron can also be affected by the IR dressing field inside the metal,
yielding an additional time-dependent distortion to the streaking curve.
It is key to understand the effects of IR dressing field on ΔIR and vs when the electron isescaping from the metal. To evaluate these effects, we employed a 1D classical model to calculate
the electron dynamics inside the metal and to yield the (τs , vs) for the bulk-emitted electrons.
Another surface-emitted electron is used as a reference to compare the streaking curves. The vector
potential of the IR dressing field in the free space above the metal surface is described by an cosine
 t2 
function modulated by an Gaussian envelop with a width of τL: At   A0 exp  2  cos Lt  .
 2 L 

The electric field inside the crystal as a function of time and depth (z) is given byE t , z    zAt 
 exp   with additional field damping is accounted for by a skin-depth l0. Wet l0 

assume both electrons (bulk-emitted and surface-born) own the same final energy (Efin) without
the IR dressing field. As a result, the initial kinetic energy of the bulk-emitted inside metal is Efin+Φ.
The deduction of the workfunction Φ is performed at interface (z=0). For simplicity, we only
consider photoelectrons with final energy Efin = 20.0 eV and metal workfunction W= 5.0 eV, which
is close to our experimental conditions.

164

In Figure 7.11a, we plot a comparison of streaking curves for electrons emitted from 10 Å
below the surface and a surface-emitted electron. As shown in the figure, when the skin depth l0 ~1 Å, significant time delay can be observed, while this time delay diminishes when l0   ,leading to a slight advancement in time. In both situations, the measured time delay does notchange for IR field strength varying several orders of amplitude. Photoelectrons emitted fromdifferent depths are examined using our classical model, as shown in Figure 7.11b. Under thisclassical model, the measured streaking time delay [(bulk) – (surface)] is generally in agreementwith the transport time of the bulk-emitted electrons, when the skin depth l0 ~ 1 Å. However, themeasured time delay can be significantly different from the electron transport time when l0   .
From a detailed analysis, it is interesting to see that the effects of the IR dressing field inside thecrystal does not contribute significantly to ΔIR, which is typically several attoseconds even whenl0   . On the other hand, it is the difference in escape velocity vs that causes the strong deviationof the measured time delay from the transport time.
The analysis above underlines the importance of the skin depth of the IR field to theinterpretation of the experimentally measured time delays, which is also noted in recent
experimental and theoretical studies [56,60,238]. When the skin depth is long, the measurementinduced time delay cannot be simply canceled by comparing photoelectrons under the influenceof the same dressing field. This is different from measurements on isolated atoms, where themeasurement-induced delay was shown to be universal. The IR field distribution and phase acrossthe metal-vacuum interface of a Cu(111) surface have been studied in details in recent angledependent RABBITT measurements, where the Fresnel equations are found to give a correct
description of the IR field at the interface and can account for the observed time delay when the165

IR fields is incident at normal and grazing angles. According to the Fresnel equations, the IR fieldin the direction normal to the surface is given by [239]sin  1 2in expik z z   r  exp ik z z exp iLt  ( z  0)0Ez z, t    1 2,
  0 sin in  1  r   expi L    0 sin 2 in 1 2 z  exp iLt  z  0
 c








(7.5)

where θin is the incident angle with respect to the surface normal, kz the wave vector withkz 

L 1 2
 0 cosin , r the reflaction coefficient withccos  /       sin    ,rcos  /       sin   
in

in

12
0
12
0

12

2

0

in

12

2

0

(7.6)

in

and ε represents the complex dielectric constant of metal. The metal-vacuum interface is definedat z=0, with z  0 for positions inside the metal and z  0 outside the metal. Consider the electricfield at the interface (z=0), the ratio of the electric field inside and outside the metal is given by
Ezin z, t  0 1  r  . Given the dielectric constant of nickel to be   20  30i [240], theoutE z  z , t  z 0E zin z , t 
ratio is out
 0.04 , representing a strong damping of the evanescent wave of the IR
E z  z , t  z 0

dressing field inside metal. The calculated IR field intensity inside nickel is shown in the inset of
Figure 7.11c. When this field profile is taken into consideration, the results of the classical model
for bulk-emitted electrons are in good agreement with the transport time as shown in Figure 7.11b,
because the effects of the IR dressing field are negligible on both ΔIR and vs.
We note that the discontinuity of Ez at the interface could lead to an unrealistic surface
charge distribution, which has been noted in previous studies [239,241]. However, with selfconsistent calculations where the dynamical charge distributions are taken into considerations

166[241], the screening length of Ez has been shown to be ~0.5 Å in metals, consistent with recent
experimental results [56]. On the other hand, for dielectric materials with smaller reflection
coefficients, the measured time delay will be more suspectible to IR induced phase change, rather
than to the electron transport time.

7.10

Semiclassical Model for RABBITT Measurements
To correctly take into account the interference effects in RABBITT measurements, we

calculate the accumulated quantum phases of the photoelectrons through scattering on thepotentials of IR dressing field and crystal lattice along their 1D trajectories. The transition
amplitude for a final state v f

as a function of  d is hence given by (atomic units) [34]a v f ,  d   i 





dt e

d p E X t   d ei IR t  i C t e1
i  v 2f    EB  t
2


,

(7.7)

where dp is the dipole-matrix element from the ground state to the continuum final state v f , EX(t)describes the EUV field for photoionization, EB the binding energy of the initial state and thespectral intensity is given by a v f , d  . The phase induced by IR laser field φIR(t) is given by2

 IR t    dt ' v  Az , t '  A2 z , t ' / 2 ,


t

(7.8)

where v is the drift velocity of the electron inside and outside the metal, z(t’) describes thephotoelectron trajectory and A(z, t’) is the IR vector potential acting on the electron along its
classical trajectory. The field intensity of the IR field across the metal-vacuum interface iscalculated following the Fresnel equations. On the other hand, the phase induced by scattering on
the crystal lattice is given by the integration of the crystal potential along the electron trajectoryz(t’):

167


 C t    dt 'U  z , t ' .
t

(7.9)

The crystal potential U(z) is defined as Chulkov potential [242] with the parameters for nickel
given in Ref. [243]. The Chulkov potential of nickel is plotted in Figure 7.12a with the first atomic
layer moved to z=0, defining the metal-vacuum interface. To account for the energy conservation
at the interface, the work function Φ is directly deducted from the kinetic energy of photoelectrons
when calculating its classical trajectory.
 t2 
The temporal structure of EX(t) is given by E X t   E X 0 exp  2  e iit , where E X 0
 2 X  iis the field strength, τX the width of the Gaussian envelop and ωi the frequency of ith order of
harmonics. By defining EX(t) in a form of attosecond pulse trains comprised of multiple HHG
orders (13th, 15th, 17th and 19th orders in this example) and x ≈ 6.38 fs, typical RABBITT
interferograms can be generated as shown in Figure 7.12b. The interferogram is obtained bysubtracting the ground-state photoelectron spectrum well before time-zero. Similar toexperimental data, the time delays can be determined by extracting 1D lineouts at side bands asshown in Figure 7.8c. In this case, we focus on the SB 16 with photoelectron energy ~20.6 eV. On
the other hand, when the EX(t) is modeled as isolated attosecond pulses with a single HHG
continuum and x ≈ 0.43 fs, the interferogram turns into attosecond-streaking spectrogram asshown in Figure 7.12c. The time delay can be obtained by directly fitting the center of energy(COE) as a function of time delay τd, as shown in the figure. For direct comparison with the
RABBITT results, we assign the center energy of the isolated EUV pulse at 16th HHG order. In
both cases, the phase of IR dressing field is used as timing references.

168

Figure 7.12: Semiclassical simulation results. (a) Chulkov potential of nickel. (b) Typical
RABBITT interferogram calculated by semiclassical model when the EUV field EX(t) is given as
an attosecond pulse train. Four harmonic orders (13th, 15th, 17th and 19th) are involved in thissimulation and x ≈ 6.38 fs. The side band region of 16th order is labeled by dashed white box. (c)Typical atto-streaking spectrogram calculated by the same semiclassical model as (b) with EX(t)modeled as an isolated EUV pulse. The pulse duration is x ≈ 0.43 fs and its center energy is
assigned at 16th harmonics for direct comparison with the RABBITT results in (b). A direct fittingof the center of energy (COE) profile is presented as the white dashed line. The field strength of
the IR dressing field is 2x1014 V/m for both calculations.
Both the RABBITT and attosecond-streaking spectrograms are calculated forphotoelectrons emitted from different depths inside metal using our 1D semiclassical model [118].
The results are summarized in Figure 7.13a. As shown in the figure, the time delay extracted fromthe RABBITT and attosecond-streaking spectrograms are identical for photoelectrons started fromdifferent depths. This conclusion is consistent with previous theoretical studies [56,60,238]. More
importantly, the extracted time delays from both methods are in good agreement with the transport
time of photoelectrons inside the metal, consistent with conclusion obtained from the classicalmodel. This represents direct measurements of photoelectron lifetime of metals using bothmethods.

169

Figure 7.13: Results of the semiclassical model. (a) Time delay extracted from RABBITT andatto-streaking spectrograms for photoelectrons emitted from different initial depths. The solid
black line represents the time delay calculated direction from transport model. We note thephotoelectron kinetic energy inside metal is 25.6 eV, because the workfunction used in thecalculation is 5 eV. (b) The quantum phases of photoelectrons accumulated by scattering on IR
dressing field or on Chulkov potential of nickel. The phase from Chulkov potential is shifted by astatic phase of ~7 rad for comparison with the phase from laser field. The photoelectron consideredin this plot starts from 10 Å below the metal surface with 20.6 eV as its final energy. The pumpprobe delay τd is fixed to be zero. The horizontal axis represents the variable of integration in
Equation 7.6.

We note that the effects of the phase accumulated by scattering on the Chulkov potential(φC) on the extracted photoemission time delay is negligible, which is typically ~10 as. This result
is consistent with previous studies. However, it is interesting to see that this negligible contribution
is not because the absolute value of φC(t) is small. Instead, it is because the time variation of thisphase is much smaller than that from IR laser field [φIR(t)]. To illustrate this, we plot φC(t) andφIR(t) in Figure 7.13b. We note that the φC(t) is shifted by a static phase ~7 rad for direct comparison
with φIR(t). From Equation (5.7), we can see that any static (time-independent) phase shift does
not change the calculated photoelectron spectrum I v f ,  d   av f ,  d  , whereas it is the time2

varying components of both phases that contribute to the variation of spectrograms. For φC(t), the170time-varying component is induced by the change of photoelectron trajectories driven by the IR
laser field, which is much smaller than φIR(t) as shown in Figure 7.13b. As a result φC(t) has very
limited effects on extracted time delay in solids.

7.11

Inelastic Mean Free Path and Photoelectron Lifetime at High Energies
In photoemission theory, the photoelectron lifetime can be understood as the time taken for

the wave function to evolve from a Bloch wave inside thematerial into a free-electron wave
function outside the solid. Semiclassically, this is also the time the photoelectron spends moving
a distance corresponding to an inelastic mean free paths (IMFP) [237,244]. Figure 7.11c showsthat photoelectrons emitted 10 Å below the surface are delayed by 267 as compared withphotoelectrons that originate 2 Å below the surface. This allows us to directly relate our timedomain measurements to the lifetimes and (IMFP) of photoelectrons, which determines the surface
and bulk contributions [148,237]. The inelastic mean free path is the distance over which a particle
travels within its lifetime, but a remaining question is what photoelectron velocity should be used
to calculate the inelastic mean free path? In a periodic crystal lattice, the elastic interaction withatoms strongly modifies the electron energy-momentum relationship, giving rise to the electronic
band structure. It is generally believed that the velocity of photoelectrons can be represented by
the group velocity of the corresponding final bands [44,56,142], which is given by the derivativeof the energy with respect to the momentum wave-vector k  . To determine the final-state groupvelocity, we employed an ab-initio calculation for the bulk band-structure of Ni along Г-L,
including the high-energy valence bands (~24 eV). We find the final band involved in the resonanttransition (highlighted in Figure 7.6a) has a narrow bandwidth (~4 eV), consistent with ourexperimental spectra. The corresponding group velocity is given by E f k   3.0 eVÅ /  ,171

and is significantly smaller than the velocity of a free electron with the same kinetic energy (~19.1
eVÅ /  ). Using the calculated group velocity, we obtain an inelastic mean free path of at most 1
Å, which is much smaller than values reported in other studies [245,246]. We believe thisdiscrepancy is because the group velocity is not the transport velocity for high-energy
photoelectrons. As pointed out in previous studies, high energy (>20 eV) photoelectrons leave thecrystal before they feel the influence of the crystal lattice, so they behave more like plane waves
with a free-electron dispersion. By using the corresponding free-electron velocity in our analysis,
we extract a IMFP of ~6 Å for photoelectrons that are emitted on resonance, consistent withprevious studies [196]. In contrast, the inelastic mean free paths of photoelectrons from the 3and 1 bands, as well as those from the 3 band away from the spectral resonance, are estimatedto be approximately 3 Å, manifesting their surface-emission nature.
The short IMFP of the 3 and 1 band photoelectrons is not surprising: in the strongdamping region (electron energy from ≈ 20 to 100 eV), many materials have IMFP comparable tothe interatomic distances, and the photoelectric current is dominated by the top-most layer.
Moreover, the IMFP of Ni was reported to be shorter than in other metals. In a one-step model
description, this can be attributed to the fact that the propagation of the final-state wavefunction isstrongly damped inside the material in this energy region, because of inelastic electron-electroninteractions.
We note that the photon energies used in our experiments are lower compared to previous
attosecond-streaking experiments. Indeed for higher photon energies, the density of final bands
increases for direct transitions from initial bands in a reduced-zone scheme, and as a result, bandstructure (matrix-element) effects become less significant. However, it has been shown in previous
photoemission studies that energy- and angle-dependent band-structure effects can still be172observed from noble metals [247–249] and semiconductors [250] even when photon energy is ashigh as 300 eV. For a final band ~100 eV above the Fermi surface, the typical linewidth is ~ 7 eV,corresponding to a lifetime of ~90 as, which could still introduce significant contributions to thephotoemission time delay measured.

7.12

Conclusion and Outlook
In conclusion, we directly and unambiguously measure the influence of the final-state band

structure on photoelectron lifetime by probing the energy-, angle- and polarization-dependent
photoemission time delays from multiple valence bands of Ni(111). We measure a significant
increase of photoelectron lifetime when the direct transition coincides with a final-state resonancein the band structure. In the case of angle-dependent measurements, we show that the large
variation in the measured photoemission time delay can be explained by the dispersion of the final
band. Our results highlight the importance of the material band structure on photoemission timedelays, which must also be taken into account even at higher photon energies. In future, thisapproach can be used to experimentally access quasiparticle lifetimes, electron mean free paths,
electron-electron scattering and dynamical screening, all of which represent grand challenges forcondensed matter theory. Moreover, other effects, including Cooper minima [251] and Fano
resonances [252], could also lead to significant delays, making attosecond studies of metal valencebands a challenging, but also rich and interesting, problem.

Chapter 8
Attosecond Electron-Electron Scattering and Screening in Transition Metals

Figure 8.1: Illustration of the photoemission process from Cu(111) and Ni(111) surfaces. (a)Using HHG pulse trains, photoelectrons are excited either from a Cu(111) or Ni(111) surface. Dueto the different band structure in these two materials, photoelectrons from Ni(111) experience more
electron-electron scattering, which reduces the lifetime of photoelectrons by 100 as compared with
Cu(111) as they escape from the material surface. The enhanced scattering also reduces theinelastic mean free path.

174

8.1

Abstract
Electron-electron interactions are the fastest processes in materials, occurring on

femtosecond to attosecond time scales, depending on the electronic band structure of the material
and the excitation energy. Such interactions can play a dominant role in light-induced processes
such as nano-enhanced plasmonics and catalysis, light harvesting or phase transitions. However,to date it has not been possible to experimentally distinguish fundamental electron interactions
such as scattering and screening. Here, we use sequences of attosecond pulses to directly measureelectron-electron interactions in different bands of different materials with both simple andcomplex Fermi surfaces. By extracting the time delays associated with photoemission, we showthat the lifetime of photoelectrons from the d band of Cu are longer by ~100 attoseconds comparedwith those from the same band of Ni, because of enhanced electron-electron scattering in theunfilled d band of Ni. Moreover, we also show that screening influences high-energy
photoelectrons (≈20 eV) significantly less than low-energy photoelectrons. As a result, highenergy photoelectrons can serve as a direct probe of spin-dependent electron-electron scattering,yielding information that is directly applicable to quantifying the contribution of screening to lowenergy excitations near the Fermi level, and providing valuable information for a host of magnetic
materials.

8.2

Introduction

8.2.1

Probing Excited-State Electron Dynamics in Materials
Excited state electron dynamics in materials play a critical role in light-induced phasetransitions in magnetic and charge density wave materials, in superdiffusive spin flow, in catalytic
processes, and in many nano-enhanced processes. However, to date exploring such dynamics is175challenging both experimentally and theoretically. Using femtosecond lasers in combination withadvanced spectroscopies, it is possible to measure the lifetime of excited charges and spins directlyin the time domain [253]. To date, such studies have been applied to a wide variety of materials,
including noble metals and semiconductors [253–256], ferromagnetic metals [227,257–259],
strongly correlated materials [208] and high-Tc superconductors [260,261]. These studies havesignificantly improved our understanding of the fastest coupled interactions and relaxation
mechanisms in matter. However, to date experimental investigations of electron dynamics havebeen limited to femtosecond time-scale processes in materials with low charge densities [208,260–
262] or to Fermi-liquid metals with low excitation energies (<3.0 eV above EF, where EF is the
Fermi energy) [255–257], due to the visible-to-UV-wavelength photon energies used in theseexperiments. In this region, two fundamental electron interactions – electron-electron scatteringand charge screening due to a rearrangement of adjacent charges - contribute to the signal, making
it challenging to independently probe these dynamics. On the theory side, initial studies in the late
1950s were enabled by the seminal Fermi-liquid theory of Landau [263–266]. In exciting recent
developments using the self-energy formalism of many-body theory and the random phaseapproximation, calculations of electron-electron interaction in materials - that include the material
band structure - have now become possible [267–272]. Thus, experimental approaches that candistinguish between different electron-electron interactions, particular with band specificity, arevery important and timely.

8.2.2

Distinguishing Attosecond Electron-Electron Scattering and Screening
High harmonic generation (HHG) provides attosecond pulses and pulse trains that are

perfectly synchronized to the driving laser, and which are ideal for probing the fastest coupledchargeandspindynamics

inatoms,

molecules,

andmaterials

176[32,35,42,54,55,88,104,136,229,273,274]. To date, two approaches have been used to probe
attosecond electron dynamics in matter through photoemission, taking advantage of laser-assisted
photoemission (LAPE) sidebands [55,273]. For atoms, since the energy separation between
different states is large, attosecond streaking using isolated attosecond pulses (with an energy
resolution of several eV) has been applied very successfully [42,229]. The same approach has also
been used to measure the transit time for a photoelectron to be emitted from the surface of amaterial. The RABBITT method [32,54,104] (reconstruction of attosecond beating by interferenceof two-photon transitions) has also been very successfully applied to atomic and material samples,
where quantum interferences between neighboring two-photon transition pathways can modulatethese sidebands as a function of the relative time delay between the HHG pump and infrared (IR)probe pulses: any time delay in photoemission from different initial or final states will lead to aphase delay in the interferograms [104,136].
In chapter 7, by combining attosecond HHG pulse trains with time- and angle-resolved
photoemission (ARPES), I demonstrated the ability to resolve attosecond electron dynamics indifferent individual final states in materials for the first time, with ≈20 attosecond time resolution.
I used attosecond-ARPES to measure a photoelectron lifetime of ~210 as, which was measured
for a final-state that coincides with an unoccupied excited state in the band structure of Ni[35,274]. I also showed that the photoelectron lifetime sensitively depends on the band dispersionof the material i.e., the photoelectron emission angle. That work demonstrated that atto-ARPES
can probe intrinsic properties of materials. A great advantage of atto-ARPES is that it achieves
good energy resolution (<0.3 eV), to enable band-selectivity as well as angle-resolved studies,combined with the ability to change the HHG polarization, which are all critical for harnessing
photoemission selection rules. This makes it possible, in principle, to selectively capture electron

177

dynamics in different initial or final bands in many materials, since the typical separation between
neighboring valence bands is <1 eV.
Here, we use sequences of attosecond pulses coupled with time-, energy-, polarization-,and angle-resolved photoelectron spectroscopy (atto-ARPES) to distinguish electron-electroninteractions for electrons excited from different initial bands during the photoemission process.
The high photoelectron energies (E-EF >20 eV), combined with attosecond time resolution, allows
us to independently measure electron-electron scattering in metals with simple and complex Fermi
surfaces, without the influence of screening, for the first time. To achieve this, we extract the timedelays associated with photoemission from individual valence bands in Ni(111) and Cu(111). We
find that the lifetime of photoelectrons from a d band of Cu is longer by ~100 attoseconds
compared to the lifetime of those from the same band of Ni. We attribute this difference to the factthat the d band in Ni is not fully occupied, resulting in enhanced electron-electron scattering andhence a shorter photoelectron lifetime (see Figure 8.1a). Then, using a spin-dependent scatteringmodel to compare electron-electron interactions in Cu and Ni, we show that the photoexcitedelectron lifetime in Ni involves enhanced electron-electron scattering throughout the energy range
from 0.5 eV to 40 eV. Moreover, because screening influences high-energy photoelectrons (≈ 20eV) significantly less than low-energy photoelectrons [270,272], they can serve as a direct probeof spin-dependent electron-electron scattering. The resulting Coulomb interaction information we
extract is applicable across a broad energy range - from the Fermi energy on up – and can separate
and quantify the contribution of screening to low-energy excitations, where both screening andscattering contribute to the signal. Our atto-ARPES approach thus makes it possible toindependently distinguish the fastest electron-electron scattering and screening dynamics in metals
on attosecond time scales, providing valuable information for a host of magnetic materials.

178

8.3

Sample Surface Preparation
The Cu(111) and Cu(001) samples used in our experiment are commercially available

single crystals (Princeton Scientific Corp, diameter 10 mm × thickness 2 mm) with surface polishroughness of <10 nm and orientation accuracy <0.1 deg. Sample cleaning is performed in the same
UHV chamber used for the photoemission measurements, with a base pressure <5×10-10 torr. The
cleaning procedure for both crystal surfaces follows the same sequence of repeated cycles of Ar
ion sputtering (beam energy 0.7 keV) at room temperature and subsequent annealing to 820 K.
The sample surface quality is confirmed by monitoring the low energy electron diffraction (LEED)
patterns and the sharpness of the static photoemission spectra. For the Cu(111) single crystal, theShockley surface state can be clearly distinguished for an atomically clean surface. The samples
are electrically grounded during all static and dynamic measurements. The preparation of Ni(111)
surface is discussed in Section 7.3.

8.4

Static ARPES Spectra of Cu(111) and Cu(001)The band structure of Cu is plotted in Figure 8.2, which is quite similar to Ni band structure

discussed in chapter 7 [35]. According to the theoretical density-functional theory (DFT)
calculations, there are three bands along Γ-L that contribute to the photoemission spectra: twovalence bands with Λ3 symmetry ( 3 with lower binding energy of ~2.8 eV and 3 with higher
binding energy of ~3.5 eV) and one deeper valence band with Λ 1 symmetry (~ 3.8 eV). In our
experiments, photoemission from the Cu(100) surface is also measured, whose surface normal
corresponds to a band structure along  - X direction (Figure 8.2).

179

Figure 8.2: Band structure of Cu. (a) Band structure of Cu along the Γ-L direction for Cu(111),
and Γ-X for Cu(100), showing the evolution of the  bands to  bands across the Γ point. Dueto the photoemission selection rules [157], transitions from  2 bands are forbidden in the normal
emission geometry from Cu(100). The colored areas indicate the perpendicular momentum regions
measured in our experiments. Blue arrows indicate the direction in which the HHG photon energy(  X ) increases.

Figure 8.3: Photoelectron spectrum of Cu(111) and Cu(001). (a) ARPES spectrum ofCu(111)along the  - K direction taken using the He Iα (21.2 eV) line from a He discharge lamp.
(b) ARPES spectrum of Cu(001) along the  - X direction taken using the He II line (40.8 eV) froma He discharge lamp.

180

We measure the static spectra of the sample at the room temperature using a helium
discharge lamp (Specs UVS300, unpolarized radiation). Cu(111) is measured using the He Iα line
at 21.2 eV, which can clearly visualize the surface state. The static spectra of Cu(001) surface isobtained using the He II line at 40.8 eV, which is near the HHG photon energy range used in timeresolved experiment on Cu(001). The spectra are recorded with a hemispherical angle-resolved
electron analyzer (Specs Phoibos 100, acceptance angle is ±15o under wide angle mode) and taken
along the  - K direction of the surface Brillouin zone for the Cu(111) surface, and along the  X direction for the Cu(001) surface. During the measurement, the sample is mounted on a XYZ

manipulator with azimuthal angle adjustment so that the crystal orientation and position of thepump and probe beams can be adjusted.
Figure 8.3a shows the band structure of Cu(111) along  - K measured using the He Iα
source. The Cu(111) single crystal features a Shockey surface state at ~0.4 eV below the Fermi
surface. In terms of the bulk bands, all the original Λ bands along the Г-L direction evolve into Σbands along the  - K direction. The 3 band with a binding energy ~2.8 eV splits into 1 andΣ4 bands. The lower 3 and Λ1 bands with binding energies around 3.6 eV evolve into 1 , Σ2and Σ3 bands. Due to the wave-function symmetry of these bands, they can be selectively excitedwith polarized HHG light [157].
The band structure of Cu(001) along  - X measured using a He II line is plotted in Figure
8.3b. Since emission from the Г-point of the bulk band structure in Cu(001) occurs at photon
energy around 41 eV, which is close to the He II 40.8eV photon energy, Figure 8.3b manifests thetransition along the Δ axis of the bulk Brillouin zone [144,275]. The two degenerate upper bands
at the Г-point with binding energy ~2.9 eV are Δ1 and Δ2 bands. The lower two bands crossing at
the Г-point with binding energy ~3.7 eV are Δ2’ and Δ5 bands. These two lower bands both exhibit

181

double components due to spin-orbit splitting. In the normal emission geometry, photoemission
from bands with a Δ2 geometry is forbidden because of selection rules, allowing us tounambiguously extract the photoemission delay between the Δ1 and Δ5 bands.

8.5

Experimental Setup
The experimental setup (Figure 8.4) is similar to the one described in section 7.4. Most of

the output of a near infrared (IR) femtosecond laser is used to generate high harmonics in various
noble gases (Xe, Kr and Ar), which are then focused onto single crystal Cu(111) and Ni(111)
surfaces. In the spectral domain, these harmonics span ~15-45 eV (corresponding to 11 - ),each with a linewidth of ~0.3 eV, and separated by 2ωL≈3.2 eV, where L is the frequency of thedriving infrared (IR) laser. The residual phase-locked laser field is used to simultaneously irradiate
the material together with a high harmonic field, which induces sidebands of the photoelectron
peaks corresponding to the absorption or emission of an IR laser photon. The photoelectron
spectrum is then collected using a hemispherical photoelectron analyzer (Specs Phoibos 100). Notethat it has already been shown that RABBITT and attosecond-streaking yield the same temporal
information about the photoemission process [136], while ARPES adds significant advantages ofband-specificity [35]. Moreover, by simultaneously measuring two photoelectron wavepackets
from different initial states excited by the same harmonic orders, we can eliminate the influenceof the HHG phase [104].

182

Figure 8.4: Experimental setup.

8.6

Static HHG Photoelectron Spectra Analysis of Cu(111)
Experimentally, the band dispersions of Cu(111) along the Γ-L direction are extracted fromthe HHG-excited photoelectron spectra in a normal emission geometry. As shown in Figure 8.5(ab), photoemission dipole transitions couple different initial and final states (bands) of Cu(111) thatcan be selected using different harmonic orders and polarizations. By utilizing the good energy
resolution of attosecond pulse trains and photoemission selection rules for s- and p-polarized HHG
beams, we can unambiguously identify the peaks corresponding to different initial states [35,38].
Since the Λ1 band can only be excited by light fields polarized perpendicular to the sample surface[148,157], we can assign the two spectral peaks excited by s-polarized HHG light as
photoelectrons from 3 and 3 bands. We first obtain the photoelectron energy distribution
curves (EDCs) corresponding to surface normal (Γ-L) emission by integrating the angle-resolved

183

spectra over an angle of ±2o around the  point and deduct the secondary electron backgroundusing the Shirley background subtraction [250]. The energy of the Fermi level of each EDC isdetermined with the knowledge of the photon energies and the analyzer work function. We thenuse a double Voigt function to fit the EDCs and extract their spectra intensity and binding energy,
as shown in Figure 8.6(a-b). The Gaussian linewidths of the Voigt function are set to theexperimental resolution of ≈0.3 eV.

Figure 8.5: HHG-excited photoelectron spectra of Cu(111). Static ARPES spectra excited bys-polarized (a) and p-polarized (b) HHG field, generated using different noble gases (Xe, Kr and
Ar).

184

Figure 8.6: Static spectral analysis of photoemission from Cu(111) surface. (a) and (b)Experimentally measured EDCs excited by s-polarized HHG field (magenta open circles). The
intensity and binding energies of 3 and 3 bands are extracted by fitting the EDCs with double
Voigt functions (solid black lines). (c) and (d) The spectral intensity of 1 is extracted by taking
the difference spectra between EDCs excited by s- and p-polarized HHG fields (cyan open squares).
The intensity and binding energies are determined by fitting the EDCs with a single Voigt function(solid red lines).
To extract information about the 1 band, the EDCs excited by the same harmonic orders
but with different polarizations (s- and p-) are normalized to equalize the intensity of the 3 band.
The additional spectral weight in the photoelectron spectra excited by p-polarized HHG is assigned
as photoemission from the 1 band. We obtain its spectral intensity and binding energy by fitting
it with a single Voigt function, as shown in Figure 8.6(c-d).

185

Figure 8.7: Band mapping of Cu(111). Band structure of Cu(111) along the Γ-L direction from
DFT calculation (dashed lines), compared with experimental results of band mapping. Theinterband transition 3  B1 is highlighted by the blue arrow, which corresponds to the spectral
enhancement of the photoelectron spectrum at harmonic orders 15 and 17 as shown in Figure 8.5.

The band-mapping results using s- and p-polarized HHG light are plotted in Figure 8.7,
showing good agreement with the band structure obtained from DFT calculations. The innerpotential of Cu(111) is 8.6 eV according to previous photoemission studies [276]. We observe astrong dispersion of the photoemission peaks as a function of photon energy, indicating thatphotoemission from bulk states contributes to the signal.

186

Figure 8.8: Comparison of photoelectron lifetimes in Cu(111) and Ni(111). The comparison
includes photoelectron lifetimes for photoemission either into excited-states (on resonance, ≈ 25eV) or into free-electron states (off resonance). (a) Normalized spectral intensity of the Cu(111)3 band as a function of photon energy. The filled symbol (14th order) is obtained from HHG
driven by 390-nm laser field. The red solid line represents the Lorentzian fitting to the spectral
intensity, which yields a linewidth of 2.13±0.65 eV. (b) Photoemission time delay PE 3   PE 3 as a function of photon energy for both Cu(111) and Ni(111) surfaces. The time

 

 

delay  PE  5    PE 1  measured for a Cu(100) surface is also plotted for comparison. The open
triangle represents the lifetime derived from the linewidth in (a). (c) Illustration of the quantumpath interference in RABBITT measurements. Electrons from two initial states ( 3 and 3 ) are
excited by multiple harmonic orders into different final states. By absorbing and emitting one
additional IR photon (  L ), quantum-path interference causes spectral modulation at the side bands(SB) in between neighboring harmonic orders. (d) Two-dimensional map of photoelectron spectral
intensity as a function of photoelectron energy and HHG-laser field time delay  d . The relative
time delay between photoelectrons from 3 and 3 initial bands are highlighted as large offsets
in oscillations in the sideband region by white dashed boxes. Right panel: 1D lineouts for the
spectral modulations with angular integration of ±2.5o around the Γ point of photoelectrons from3 and 3 initial bands in the selected side-band region.

187

8.7

Final-State Resonance in Cu(111)
To distinguish the influence of wavefunction localization (excited bulk states vs. free-electron final states) as well as the influence of the fundamental electron interactions on thephotoelectron lifetimes, we first identify where the final-state resonances occur in Cu(111). The
spectral intensity of 3 band photoelectrons excited by the s-polarized HHG field is plotted in
Figure 8.8a. A strong enhancement of the spectral intensity peaked around the 15th order at theresonant photon energy of ~26 eV can be observed, which can be attributed to the interband
transition from the 3 initial band to the excited B1 final band (Figure 8.7). The Lorentzian
linewidth is  spec =2.13±0.65 eV (Figure 8.8a), which is consistent with the linewidth recently
obtained using high-resolution photoemission at a synchrotron radiation source [144]. The
predicted final-state lifetime is therefore given by  spec    spec = 309±94 as.

Figure 8.9: Spectral intensity of 3 and 1 bands. Photoemission intensities of the 3 (redopen circle) and 1 (blue open diamond) bands relative to that of the 3 band. The red solid line
represents the Lorentzian lineshape fit to the intensity of the 3 band as a function of photon
energy, which yields a linewidth of 2.13±0.65eV (Figure 8.8a).

188

 

 

The photoemission time delay  PE 3   PE 3 at the Γ point can be extracted from the
RABBITT interferogram (Figure 8.8d) and is summarized in Figure 8.8b. The quantum paths
involved in RABBITT interference are illustrated in Figure 8.8c. These interferograms (Figure8.8d) were obtained by integrating the photoelectron spectra over ±2.5o around the Г point

 

 

(normal-emission direction). Comparing Figures 8.8a and 8.8b, we find that  PE 3   PE 3 atthe Γ point reaches its maximum value  chron =291±48 as at the resonant photon energy for theinterband transition 3  B1 (Figure 8.7), which is in good agreement with  spec ≈ 309±94 as.
This agreement indicates that the lifetime of photoelectrons emitted from the initial 3 band 

(  PE 3 ) is short at the Γ point. Considering the fact that the spectral intensity of 3 is a smoothfunction of photon energy (Figure 8.9), it allows us to directly assign the measured time delay tothe lifetime of photoelectrons from the 3 band. Compared to Ni(111) results in chapter 7 [35],the resonant linewidth from the initial 3 band in Cu is narrower, which is consistent with thelonger lifetime measured in the time domain (291±48 as for Cu vs. 212±30 as for Ni).

8.8

Momentum Dispersion of Photoelectron Lifetime
To extract the angle-dependent photoemission time delay, we first divide the angle-

resolved photoemission spectra into 16 angular regions at each side band, with a ~1.8o angularspan over the entire range of photoelectron emission angle (-15o< θ <15o), as shown in Figure
8.10a. The photoelectron counts in each region are integrated and plotted as a function of pumpprobe time delay τd to obtain the angle-dependent RABBITT interferograms. Typicalinterferograms at the spectral resonance are plotted in Figure 8.10(b-c). In this way, the angle-

189

dependent photoemission time delays for photoelectrons originating from different bands[  PE  ,  ] can be extracted.

Figure 8.10: Extraction of angle-dependent photoemission time delay. (a) Illustration of theangular regions for integration when extracting the angle-dependent photoemission time delay.
The energy region corresponding to SB16 is used as an example here. (b-c) Typical RABBITT
interferograms for SB16 with emission angles [(A) and (B)] labeled in (a). The offset is highlighted
with white dashed boxes.

Figure 8.11: Angle-dependent photoemission time delay. (a) Angle-resolved photoemission
time delay at the resonant photon energy (SB16) of photoelectrons from both the 3 and 3
initial bands (s-polarized HHG fields) are plotted as blue and magenta symbolled lines. The
photoemission delay of 3 band photoelectrons at θ=0 (  PE  3 ,0  ) is used as the timingreference. The red dashed line indicates the angle dependent delay imposed by the IR dressing
field. (b) Experimental geometry of laser excitation and photoelectron detection.

190

The angle-dependent time delay due to band-structure effect alone should be symmetricwith respect to the Γ point. However, as shown in Figure 8.11a, when we use the phase

corresponding to the 3 band at the Γ point [  PE 3 ,0 ] as the timing reference, the time delays
extracted from the two equivalent regions on the two sides of the Γ point are different. This is also
observed in other side-band regions (both resonant and non-resonant). The photoemission timedelay induced by IR probing field can be simulated using the semiclassical model (details
presented in Ref. [60,118]). The IR field distribution is calculated using the Fresnel equations withthe optical parameters of Cu extracted from Ref. [240]. The simulation results are plotted in Figure
8.11a as the red dashed line. The results show that the photoelectrons emitted from IR incident
side is relatively delayed to those emitted from the IR exit side. The delay depends linearly on theelectron emission angle θ (Figure 8.11b) with a slope of ~2.15 as/deg. This slope is quantitatively
consistent with the background delay of the experimentally measured angle-dependent
photoemission time delay (Figure 8.11a) and is also qualitatively consistent with the field effects

reported before [60,118]. We note that because the binding energies of Cu(111) 3 and 3 bandare very similar (ΔE≈0.7 eV), the IR-laser induced angular dependence of time delay are very
similar for photoemission from both bands. This allows us to cancel the laser-induced time delay









by subtracting the photoemission time delay of one band from the other (  PE 3 ,  PE 3 , ).

8.9

Lifetime of Photoelectrons from Λ1 Band 

The relative photoemission time delays between L1 and 3 bands [  PE  1    PE 3 ]can be extracted from experimental results of both s- and p-polarized HHG fields. The time delay PE 3  1   PE 3  is the result obtained with p-polarized HHG field, where the low-energy

191

photoemission peak has the convoluted contributions from both  1 and 3 bands. The retrieval
algorithm was described in detail in Chapter 7. The retrieved photoemission time delay PE  1    PE  3  is plotted in Figure 8.12. The error bar is determined by considering the errorsof the RABBITT measurements using both s- and p-polarized HHG lights. We note that for Cu(111) 

surface,  PE  1    PE 3 exhibits a discontinuity and abrupt change in sign at SB 18. This isdue to the contribution of surface state excited by p-polarized HHG field. We also note that it does
not affect our conclusion, which was based on the results measured using s-polarized HHG field.

Figure 8.12: Photoemission time delay of 3 and 1 bands relative to the 3 band. Timedelays  PE  3    PE  3 

and  PE  3  1    PE  3  as a function of photon energy are

obtained by exciting photoemission using p-polarized HHG fields, respectively.
 PE  1    PE  3  is extracted by combining the s- and p-polarized delay and the relative
photoemission intensity of  and  bands in Figure 8.9. The ellipse marks the contribution of
surface state to the RABBITT interferogram at sideband 18.
1

3

192

8.10

Photoelectron Lifetime Difference between Cu and Ni
For photoemission through free-electron final states (away from any final-state

resonances, >25 eV in Figure 8.13a), we find that the photoelectron lifetime from the 3 band ofCu(111) is ~100 as in the normal-emission geometry. Moreover, this lifetime is a smooth functionof the final-state energy (Figure 8.13a). The associated time delay can be clearly seen in theexperimentally measured interferograms of Cu(111) as an obvious phase shift in the oscillationsof the RABBITT quantum interferences (Figure 8.13b), which interestingly is absent in Ni(111)for free-electron final states [35].
We note that we can exclude the possibility that the finite photoelectron lifetime in thisenergy range in Cu(111) is caused by another final-state resonance because we did not observe
any photoelectron yield enhancement in this energy range (Figure 8.8a), and because that itexhibits little momentum (angle) dispersion - unlike the lifetime on resonance (Figure 8.13c). To
further reinforce this conclusion, we also measured the photoemission time delay between the  5and  1 bands along the Γ-X direction for Cu(100) as the two  3 bands cross the Γ point of theBrillouin zone (Figure 8.4). As shown in Figure 8.8b, a similar lifetime difference between  5and  1 band photoelectrons was observed on the Cu(100) surface when there is no spectral
resonance. Excluding final-state effects, the ~100 as lifetime difference of photoelectrons from the3 band for Cu(111) and Ni(111) must be attributed to differences in the fundamental electronelectron interactions experienced by the high-energy photoexcited electrons during photoemission
from these two materials.

193

Figure 8.13. Origin of different photoelectron lifetimes for photoemission into free-electron
final states for Cu and Ni. (a) Photoelectron lifetime emitted from 3 band measured using attoARPES as a function of the final-state energy (E-EF) for both Cu(111) and Ni(111), in comparison
with the FEG model (black) [268], Goldmann et al. (cyan) [277] and Eberhardt and coworkers(green dashed) [144] models. The lifetime corresponding to free-electron final states arehighlighted in the colored region. (b) Interferograms for sidebands 20-24 (off-resonance, awayfrom final-band resonance) measured from Cu(111) and Ni(111) [35] surfaces. Large offsets inthe side band regions can be observed in the Cu(111) data, as highlighted by the white dashedboxes. Such offsets are absent for Ni(111). (c) Angle-dependent photoemission time delay of PE 3 ,  PE 3 , measured on Cu(111), which clearly shows the difference between a









resonant transition to a bulk final band (side band 16) and those to free-electron final states (sideband 20: square; sideband 22: triangle; sideband 24: circle; and sideband 26: diamond).

8.11

Dynamic Screening and Inelastic Electron-Electron Scattering
The photoexcited electron lifetime for free-electron final states is mainly determined by

the competition between dynamic screening and inelastic electron-electron scattering during thephotoemission process [268]. This is because for highly excited electronic states (>20 eV), other194

decay channels including scattering with photoholes, phonons and impurities are expected to havenegligible contributions.

8.11.1

Absence of Electron Screening
In Figure 8.13a, we compare our measured photoelectron lifetime from the 3 band of Cu

to two empirical models (Goldmann et al. [277] and Eberhardt and coworkers [144]) that are basedon bulk excited-state linewidths, as well as a free-electron gas (FEG) model with rs=2.67 for Cu[268], where rs is the electron-density parameter defined for an electron density n by n  3 4rs3 .
As shown in Figure 8.13a, both empirical models agree well with the photoelectron lifetime onresonance (SB16) as expected; however, they overestimate the photoelectron lifetime offresonance. This is not surprising – the Goldmann and Eberhardt models are derived from a
linewidth analysis of photoemission and inverse photoemission experiments [144,277], that are
mainly based on contributions from resonant (bulk final state) excitations. Compared to steadystate photoemission and inverse photoemission studies, our time-domain approach has uniqueadvantages that allows us to measure the intrinsic high-energy photoelectron lifetime at arbitraryenergies (including transitions both on and off final-state resonance). Note that the FEG modelcannot reproduce the photoelectron lifetime on resonance, which is also not surprising since it does
not account for the real band structure of the material. Most interestingly, the FEG model matches
the off-resonance (i.e., free-electron final-state) lifetime measured on Cu(111) very well, but not
for Ni(111), which is ~100 as shorter. Note that this trend is very different from the hot-electron
lifetimes measured at low energy (<3 eV above EF), which exhibit a strong deviation from the
FEG model due to the added presence of screening of d-band electrons [267,269].

195

The absence of electron screening effects in high-energy photoemission can be understoodby considering two different aspects. First, screening of d band electrons can be estimated byconsidering an effective dielectric constant  d  1   d that is induced by the polarizable
background of d electrons. At low energies, the corrected lifetime is larger than the value predictedby the FEG model by d [278–280]. As pointed out by Quinn [281,282],  d reduces as afunction of excitation photon energy. As a result, the variation of the photoelectron lifetime due tod-electron screening is estimated to be only a few percent of the FEG lifetime at energies >20 eV.
Second, on ~100 attosecond time scales, we also need to consider the dynamics of electron
screening in metals. As shown in previous theoretical studies, the buildup of charge screening in
metals is not instantaneous, but takes approximately half of a plasma period to fully develop [283],
which corresponds to ~200 as in both Cu and Ni. Because the off-resonance photoelectron
lifetimes are much less than this, it appears that photoelectrons from Cu (and Ni) escape before
dynamic screening can influence the photoelectron lifetime in this energy range. As a result,dynamic electron screening has negligible influence on the photoelectron lifetimes at energies >20 eV.

8.11.2

Influence of Inelastic Electron-Electron Scattering
On the other hand, the ~100 as lifetime difference between photoelectrons from the d bandsof Cu and Ni can be attributed to the differences in the band structure of these materials, which
results in different electron-electron scattering rates between photoelectrons and other electrons in
the conduction bands during photoemission. The scattering rate between photoelectrons andunexcited bulk electrons is calculated using Fermi’s Golden rule, following the formulas presented
in Refs. [258,284,285]. Because the unoccupied states above EF in Ni are electronic states with

196

minority spins (  ), spin-dependent electron-electron scattering needs to be taken into account.
Here, we consider a high-energy photoexcited electron with energy E above EF. This electron
decays into a lower energy state E’ by exciting one of the other electrons in the band (a scattering
partner) from its original state ε into an unoccupied state above EF, ε+Δ, where Δ=E-E’ is theenergy transfer (see Figure 8.14a). We note that the scattering process illustrated in Figure 8.14a
is responsible for removing photoelectron signal from our measurement. Thus, the experimentallymeasured photoelectrons are those that escape without scattering; nevertheless, the lifetimes ofthese electrons are influenced by scattering within the occupied bands, and experience different
phase shifts in our RABBITT measurement.
The two-electron scattering processes presented in Figure 8.14a can be either a spinconserving or a spin-flipping process for the photoexcited electron. In order to calculate the total
probability of inelastic scattering between two electrons as shown in Figure 8.14a, the full quantum
states of the two-particle initial and final states need to be taken into consideration. The Coulomb
matrix element is given by M EEk'k','k,1'k12 2  Ek , k11 V E' k ' ' ,  ' k2 2 , where (E, k, σ) and (E’,k’, σ’) are the initial and final states of the photoexcited electron and (ε, k1, σ1) and (ε’, k2, σ2) are
the initial and final states of the scattering partner. V is the screened electron-electron interaction.
The spin indices σ1, σ’ and σ2 either equal to σ or are opposite to σ (  ). Specifically, when  '  
the scattering is a spin-conserving process for the photoelectron, while if  '   , it is a spinflipping one. The scattering process conserves the total spin-angular momentum of the twoelectron system.
In principle, the matrix element M EEk'k',',k1'k12 2 can be calculated by considering the
momentums of different states if the wave functions and selection rules are known. However, ingeneral this information is not available allowing accurate calculations in materials. Here, we

197

assume the matrix element is independent to the momenta (k) of both electrons involved in the
scattering process (random-k approximation) in order to investigate the effects of electron-electron
scattering on the photoelectron lifetime. Indeed, as shown in Figure 8.13c, we verified
experimentally that the assumption of a momentum-averaged Coulomb matrix element M is valid.
The random-k approximation was first used by Berglund and Spicer to calculate the photoemission
from Cu and Ag [149] and was later used to calculate electron-electron scattering probability indifferent materials [258,286–288]. It has been shown that with a proper choice of the momentumindependent matrix element, the results obtained from the random-k approximation can be in good
agreement with more sophisticated calculations [289,290]. By considering the random-k
approximation we can evaluate the electron-electron scattering rate by including the appropriate
densities of states (DOS) and Fermi functions. As a result, we can write an averaged matrix element, 1
as M EE' ', ' 2  E , 1 V E '  ' ,  '2 , where the energy (E), spin (σ) and orbital character(α, β) of the initial and final states are taken into consideration.
The spin-conserving processes scatter the photoexcited electron from an original state (E, ) into an unoccupied state (E’,  ), and can be formulated as 1) E , E '  ,E ,d   ,    ,      M EE' , '  M  ' , E ' s, p, d
  dE '  E '  Es, p, d

EF

,

EF

2

EF 

I

,

(8.1)


2
 ,
 ,
E ,         M E ' , '   
II


where

 ,  E   1  f E  E 

 ,  E   f E  E  ,

(8.2)

198

where f E  is the Fermi-Dirac distribution function, and  E  is the spin-dependent density ofstates (DOS) of the  orbital that is obtained from a DFT calculation (see the inset of Figure 8.14c).
  E  E' is the energy transfer between the two electrons and σ1 can be either σ (same to) or 

(opposite to photoexcited electron). The first term (I) in the integrand describes the scattering
between a photoexcited electron with spin  and an unexcited electron with the same spin. The
fact that electrons are indistinguishable particles is taken into account. The second term (II)describes scattering between the photoexcited electron and an electron with opposite spin (  ). We
first neglect the interference term after expanding the modulus square of the matrix elements [287].
Then, following Ref. [258], we do not further distinguish between s, p and d states for the matrix
element and assume M is energy-independent. As a result, we only have matrix elements for
scattering between electrons with the same spins ( M  ) and opposite spins ( M   ).We howevernote that energy-dependent matrix element (M) is obviously observed in current work and otherstudies, which we will illustrate by using different (but constant throughout energy) values of M
to make agreement with low-energy and high-energy results obtained in experiments. Beyond theapproach in Ref. [258], we assumed that the wavefunction character (s, p or d) of the initial andfinal states of an electron are the same during the scattering process. This was suggested by a recent
all-electron ab initio calculations showing that the scattering rate between two electrons will bevanishingly small if the wave-function characters of the initial and final states are different [269].
For simplicity, we only distinguish between different wave functions for the scattering partner inthe DOS assignment, but do not further distinguish it for the photoexcited electrons. As a result,the DOS used for the final state of photoexcited electrons is the total DOS, E  .With the, s , p ,dconsiderations above, the Equation 8.1 is reduced to -1991) E , E '  22d   ,    ,     2 M    ,    ,      M    .
EF  
s, p, ddE '   E '  



E

,

EF

 s , p ,d

EF

(8.3)
Similarly, the scattering rate for the spin-flip process is given by -   dE '   2) E , E ' EEF

  s , p ,d

,


2
E
E'  E  d  ,    ,     M   .
F s, p, d

(8.4)

F

Note that in this process, the photoexcited electron must scatter with an unexcited electron with
opposite spin in order to conserve spin-angular momentum in the process. The total scattering rate
(which will correspond to the inverse of the lifetime   ) is hence given by1  E , E ' '  1) E , E '    2) E , E ' ,  E   ' ,

(8.5)

which yields
2
EF

2
1
 E
  dE '    ,  E '  d   ,    ,     2 M    ,    ,      M   
E
E


F
F
  E  .
 s, p, d  s , p , d  E '  ,s, p, ds, p, dEFEF  

(8.6)

2d  ,    ,      M   We note that because the unoccupied states above the Fermi energy in Ni are dominated byelectronic states with minority-spin polarization (  ), spin-dependent scattering needs to be taken
into account, which results in spin-dependent excited-electron lifetime in the ferromagnetic
materials such as Ni [257,258,285]. Assuming        and M   M   M , Equation (8.6)
is simply reduced to1
2 3 2 M E  EF 2 , which is the well-known ( E  EF ) 2 scaling of E  

hot electron lifetimes excited close to the EF [149].

200

Figure 8.14: Spin-dependent model of electron-electron scattering. (a) Illustration of theelectron-electron scattering process described by Equation (8.1). The photoexcited electron (redcircle) can decay by exciting another unexcited electron (blue circle) to a state above EF.
M  E  E ' is the Coulomb matrix element, which we find is mostly constant for Cu (at 1.4)
across a broad energy range, but varies for Ni due to stronger screening at low energies. (b)Comparison between the spin-dependent scattering model (red (Cu) and blue (Ni)) and theexperimentally measured lifetime of photoexcited electrons in Cu and Ni. The low-energy data
(0.5-3 eV) are measured using Tr-2PPE method, extracted from Ref. [255] for Cu and from Ref.
[258] for Ni. The high-energy data (15 - 40 eV) are directly measured in our experiment using
atto-ARPES. The data that overlap final-state resonances in both materials are represented bycrosses to distinguish them from the off-resonant results of interest here. The yellow-area estimatesd-electron screening effects by considering the optical constants of Cu [291]. The experimentallymeasured low-energy electron lifetime approaches the bare electron-electron scattering limit (solid
blue line, M=1.8 for Ni) at an energy Ecrit ≈ 3eV. Inset: the phase space of the two materials
calculated from Equation (8.1), assuming M   M   1.0 . The blue dashed line (Ni, ΔDOS) is
the results with the DOS of Ni downshifted by 1.8 eV. (c) Lifetime ratio     as a function of
excited electron energy (E-EF) for Ni obtained using Equation (8.1). Inset: spin and orbitaldependent DOS of Ni and Cu obtained from DFT calculations.

201

We first evaluate the influence of the DOS on the available phase space for scattering byassuming M   M   1.0 in Equation (8.6). As shown in the inset of Figure 8.14b, the phasespace increases monotonically as a function of the photoexcited electron energy above EF for both
Ni and Cu, and indeed, the phase space of Ni is larger than Cu in the energy range of ourexperiments, indicating that a higher scattering rate and a shorter photoelectron lifetime would
indeed be expected. The additional phase space of Ni is dominated by the unoccupied DOS above
EF, as evidenced by the fact that the available phase space of Ni moves closer to Cu as its DOS isdown-shifted by 1.8 eV to artificially remove the peaked unoccupied DOS (dashed line in the inset
of Figure 8.14b). Figure 8.14b plots the experimentally measured and theoretically calculated
lifetimes of photoexcited electrons from 0.5 to 40 eV. Although the focus of this work is on
photoelectron lifetimes in the high energy >20 eV region, a comparison with Tr-2PPE data allows
us to gain valuable physical insights. In general, electron-electron Coulomb interactions areenergy-dependent due to different screening properties at different energies in a material [149].
Here, for convenience, we assume the Coulomb matrix element M is a constant, and select values
by fitting to the experimental data. For example, to further determine the Coulomb matrix element,
we compare the photoexcited electron lifetimes measured using spin-integrated Tr-2PPE on Cu[255] and Ni [258] with our atto-ARPES results and models. For Cu, we have M   M   M ,because the DOS for electrons with majority (  ) and minority (  ) spins are the same. For Ni, as
shown in the inset of Figure 8.14c, there are spin-dependent DOS differences, so we assume
2

M  M   0.5 and M 

M   M 22

in order to get agreement with the spin-dependent

electron lifetime measured at low energies [258]. The spin-averaged excited electron lifetime

202



2  is plotted in Figure 8.14b as solid lines for Cu and Ni, while Figure 8.14c plots the ratio 

between spin-up and spin-down electron lifetimes (     ). Most interestingly, we find that thephotoexcited-electron lifetime in Cu can be explained by a mostly energy-independent Coulomb
matrix element (M=1.4) throughout the entire energy range from 0.5 to 40 eV. The presence of thed-band screening in the low energy range (<3 eV) is well known for Cu, which increases the
lifetime by approximately a factor of 2.5 [267,291]. In stark contrast, our atto-ARPES
measurements suggest a stronger energy dependence of the Coulomb matrix element in Ni: M=1.8
is best for high energy photoelectrons and is close to that observed in Cu, while M=0.6 is best for
low energy photoelectrons - that are influenced by both screening and scattering.
Considering that electron screening does not have a strong influence on the photoelectron
lifetime in the high energy range [270,272], we can extract the influence of the bare electronelectron Coulomb interactions (no screening) at high energies and extend the corresponding matrix
element to the low energy range (the solid blue line in Figure 8.14b). The measured lifetime at low
energy in Ni is more than one order of magnitude longer than the bare electron-electron scattering
limit, as shown in Figure 8.14b. Most interestingly, we find that the experimentally measured lowenergy excited electron lifetime in Ni gradually approaches the bare electron-electron scattering
limit (solid blue line in Figure 8.14b) defined by our atto-ARPES measurement at an energy Ecrit~ 3 eV above EF. This further corroborates our findings, since screening is expected to diminish
at these higher energies [272]. Comparing Cu and Ni, our results strongly suggest the presence ofenhanced electron screening in Ni at low energies, which can be attributed to the high DOS at the
Fermi energy based on our DFT calculations (see Figure 8.14c inset) [258]. From the above, we
conclude that atto-ARPES can extend measurements of photoexcited-electron lifetimes to higher
energies (>20 eV) to distinguish and quantify fundamental electron interactions such as electron

203

scattering and screening, as well as the influence of resonant interband transitions. Compared toother approaches, atto-ARPES also has the unique ability to distinguish band-specific electronelectron scattering for direct comparison with theory, and can exclude other contributions such as
hot electrons and intermediate-state refilling [227,258,292].
8.11.3Spin-Dependent Electron-Electron Scattering
Finally, we note that spin-dependent electron-electron scattering in ferromagnetic materials

is responsible for many interesting phenomena, including laser-induced demagnetization [227],superdiffusive spin transport, and giant magnetoresistance [293]. Low-energy spin-dependent
electron lifetimes have been studied, providing much valuable information [257,258]. However,to date it is not possible to experimentally isolate electron-electron scattering, because of strong
contributions to the measured lifetimes from electron screening from localized d and f band
electrons, as well as contributions from other interactions e.g., phonons and impurities. By probing
high-energy photoelectron lifetimes, where electron screening becomes negligible, spin-resolved
atto-ARPES could probe spin-dependent electron-electron scattering, which could help uncoverfundamental magnetic properties.

8.12

Atto-ARPES as a Probe of High-Energy Final States and Matrix Element
Effect in Photoemission
We note that when photoexcitation is on-resonance with an excited bulk band final state,

the photoemission time delay strongly varies as a function of the electron transverse momentum,i.e., photoelectron emission angle θ, which is a consequence of the final-band dispersion [35]. This
allows us to clearly understand what transitions are involved, and when we are observing lifetimesof bulk final bands. To extract the final (excited) state time delay, we simulated and subtracted the204

angular dependence of the time delay imposed by the incident and reflected laser fields [60,118] .

















The angle-dependent photoemission time delays (  PE 3 ,   PE 3 ,0 and  PE 3 ,  PE 3 ,0 )excited with an s-polarized HHG field are plotted in Figure 8.15a. Most interestingly, we find thetime delay disperses oppositely around Γ point as a function of θ for 3 and 3 band
photoelectrons. Using the model presented in Ref. [35], and the band structure along the  - 
direction obtained from density-functional theory (DFT) calculations (Figure 8.15b), we model the
lifetime dispersion of 3 band photoelectrons, as shown by the red-solid line in Figure 8.15a. The
maximum time delays at θ≈5o (A) and θ≈16o (B) corresponds well to the features on 1 final bandB( 1 band along Γ-L direction) as shown in Figure 8.15b, which corroborates the contribution ofBthe 1 final band in this resonant transition.
B









In contrast, we find that the dispersion of the measured time delay  PE 3 ,  PE 3 ,0

is in good agreement with the shape of the 1 band ( 1 band along the Γ-L direction), as shownAA

by our model results (pink curve) in Figure 8.15a. The maximum delay (C) at θ≈8o corresponds tothe peak in the 1 band shown in Figure 8.15b. More interestingly, by adjusting the polarizationA

of the HHG field from s to p, we can tune the transition from 3 initial band to out of the resonancewith the 1 final band and directly into a free-electron-like final state as illustrated by the nonA













dispersive feature of the time delay  PE 3 ,  PE 3 ,0 in Figure 8.15a. We believe this result
is due to photoemission selection rules, which determine the allowed transitions for different
polarizations. Indeed, we find different initial bands are excited with s- and p-polarized HHG fields
( 1 by s-pol and  4 by p-pol), as clearly evidenced by the different dispersion in the photoelectron
spectra (Figure 8.15c).

205

Figure 8.15: Extracting final-state effects in photoemission. (a) Angle-resolved photoemission
time delay at the resonant photon energy (side band 16) of photoelectrons from both the 3 and3 initial bands (both s- and p-polarized HHG fields, for a 3 initial band). The photoemission

delay of 3 band photoelectrons at θ=0 (  PE 3 ,0 ) is used as the timing reference. The solid lines
plot the theoretical lifetime dispersion considering two interband transitions ( 3  1B and3  1A ). The peaks of the lifetime dispersion curves are labeled (A, B and C). The verticaldashed line at θ=13o represents the angle limit of our hemispherical analyzer. To measure the timedelays at larger angles, the sample was rotated by 12o. (b) Band structure of Cu(111) along the   transverse momentum ( k // ) direction obtained by DFT calculations. The perpendicular
momentum k  is determined by the band-mapping results of Figure 8.7b. The labeled band
structure features (A, B and C) correspond to the labels in (a). (c) ARPES spectra obtained using
the 17th order harmonic for both s- and p-polarizations. The dispersion of the upper band isobviously opposite in both cases, as highlighted by the yellow dashed lines, indicating that
different initial states ( 1 and  4 ) are selectively excited by HHG fields with different
polarizations.

206

High-energy final states are very important and represent one of the major difficulties ininterpreting photoemission spectra [144,148,294], which has been extensively studied by twophoton photoemission [295,296], inverse photoemission [148,297] and very-low-energy electron
diffraction (VLEED) techniques [298]. Here, through angle- and polarization-dependent
photoemission time delay measurements at the resonant energy, we show that our method can bea very sensitive tool to measure the selectivity of dipole transitions between different initial andfinal states.

8.13

Conclusion and Outlook
In conclusion, we show that attosecond electron interactions in metals can be studied using

energy-, polarization-, and angle-resolved atto-ARPES, allowing us to distinguish thecontributions of occupied and unoccupied bands to the photoelectron lifetimes. Strong electronelectron scattering in the unfilled d band of Ni decreases the lifetime of photoelectrons by ~100 as
relative to the photoelectrons emitted from the same band of Cu. Most interestingly, we find thatdynamical screening influences high-energy photoelectrons much less than low-energy
photoelectrons, and is different for Cu and Ni due to the difference in material band structure. As
a result, spin-dependent atto-ARPES with high-energy excitation is a unique tool to exclusively
study the fundamental processes of spin-dependent electron-electron scattering in magnetic
materials, and also quantify the contributions of scattering and screening for low-energy
excitations. In the future, atto-ARPES can also be used to extract valuable information about
fundamental electron-electron interactions in a host of materials including strongly correlatedmaterials and modern quantum materials.

Chapter 9
Conclusion and Outlook

9.1

Conclusion
The work presented in this thesis demonstrated interferometric laser-assisted

photoemission () as a quantitative technique to study both the attosecond pulses and theattosecond electron dynamics in solids with unprecedented temporal and spatial resolution. Within
the pursued metrology approach, photoelectron wave packets are launched inside the solid throughexcitation by extreme ultraviolet (EUV) attosecond pulse trains. The generated photoelectrons aresubsequently probed by their interactions with an infrared laser pulse perfectly phase-locked to theattosecond excitation. The presence of IR field induces the absorption and stimulated emission of
IR photons for photoelectrons, forming sidebands around the direct photoemission peak. Quantum
path interference modulates the sideband intensities at twice the frequency of the IR field, with theoscillation phase related to the arrival time of photoelectrons into the laser field. There are two
factors determining the instant of photoelectron release: 1) the timing (phase) of the EUV photons
and 2) the photoemission time. Therefore, the I-LAPE technique (i.e., RABBITT) can serve as aprobe of the attosecond pulse structure as well as the photoelectron lifetime. To summarize theprimary results of the thesis:Tomographic reconstruction of the circularly polarized harmonic field have revealed the

3D structure of the circular attosecond pulse train produced by bi-chromatic counter-rotating

208

driving fields. It was demonstrated experimentally that the circular harmonics generated throughthis method was composed of three linear burst within one optical cycle of the IR field, with eachburst rotated 120o from the previous one, collaborating with the theoretical calculations. This
represent the first ex-situ spatial-temporal (4D) reconstruction of attosecond pulses. It was enabledby the capability to rotate the circular attosecond pulse train and the polarization sensitivity insurface photoemission. The full temporal characterization also provides a universal way tomeasure the ellipticity of circular harmonics.
Linearly polarized high harmonic radiation generated by two-color field consisting of thefundamental IR laser (ω) and its second harmonic (2ω) is an attractive attosecond light source
because it has broader spectral coverage and the pulse train structure can be controlled by tuning
the relative delay between two drivers. We fully characterized the evolution of the attosecond pulse
structure as a function of the ω-2ω delay. We found that the delay corresponding to the highest
overall harmonic yield also results in the shortest attosecond pulse train.


Attosecond RABBITT experiments performed on Ni(111) surface confirm the influence ofmaterial bandstructure on the photoelectron lifetime. In particular, there is an abrupt increase inphotoelectron lifetime when the photoexcitation resonantly couples the bound electron with anunoccupied final band, resulting in a photoemission time delay for the resonant excitationcompared to non-resonant excitations. This lifetime elongation peaks at the resonant photon energy
and decreases at higher or lower photon energies. A strong emission-angle-dependent
photoelectron lifetime was observed, which perfectly tracks the dispersion of the final band. This
serves as a strong proof for the final band effect on the photoemission time delay.
After identifying the unoccupied final band effect, we move forward to investigate how theoccupied initial band influence the photoelectron lifetime. The pinpoint of the final bands allows

209

us the select the photon energies to excite the electron wave packets into free electron final states
instead of coupling to a final bulk band. We found that for the same free electron final states,
photoelectrons in Cu(111) has a longer lifetime than Ni(111). Unlike the resonance case, this delay
is non-dispersive, exempting the final state as an origin. Combined with a numerical simulation,
we show that the longer photoelectron lifetime in Cu(111) is due to lower scattering cross section
as a result of its fully occupied d band. We also found that distinct from the lower excited statesprobed by two-photon photoemission, the high energy photoelectrons escape so fast that the
dynamical screening has a negligible influence on it.


As a bonus of the final state effect on the angular dispersion of the photoelectron lifetime,

attosecond metrology can act as a sensitive probe of the coupling between different initial and final
states. This is a peek into the mysterious “black box” of transition dipole matrix in photoemission.
The results will inform the theoretical effects to shed light on this problem.
The outlook is very bright for using attosecond metrology in combination with angleresolved photoemission spectroscopy to study the material systems. Since only electronic systems
are evolving on the attosecond time scale while the lattice is essentially frozen, attosecond
spectroscopy provides a powerful tool to study the electron-electron many-body interactions
adiabatically. This avenue holds great potential in solving the central mysteries in condensed
matter physics such as superconductivity, nano-plasma, ultrafast demagnetization andphotoinduced metal-insulator phase transition.

9.2

Ongoing Experiment
We have investigated two transition metal surfaces Ni and Cu in the thesis. In theimmediate future, we will expand our study to more metallic systems to answer several important210

questions: 1) how the localization of the valence band impacts the photoemission time? 2) Is it
possible to find a system with photoelectron lifetime comparable to the time scale of dynamical
screening to investigate the interaction between two processes? 3) It is reported that the surface
state of metal has a varying mean free path depending on photon energy. Can we directly measure
it in the time-domain? Is surface state emission really from “surface”? Besides the valence bandsof meal single crystals, we are also investigating the following dynamics.

9.2.1

Time-Domain Study of Fano Resonance in Solid-State Materials
Spectral phase measurement of Fano resonance has been performed on gas phase argon

atoms using RABBITT technique. In the proximity of the 3𝑠 2 3𝑝6 → 3𝑠1 3𝑝6 4𝑝 autoionizing
resonance, the oscillation phase of the nearest sideband to this resonance strongly depends on theharmonic photon energy relative to the resonant energy. This phase variation reflects theinteraction between the continuum [3𝑝−1 𝜖𝑠, 𝑑] and quasi-bound [3𝑠 −1 4𝑝] states.
Same Fano type resonance was also observed in photoemission from nickel [252,299,300].
It was first reported by Guillot et al. in 1977 [252], followed by a detailed analysis by Barth et al.
in 1979 [299]. They observed that M2,3VV Auger electrons peaked around 6 eV started to emerge
and increased dramatically when the photon energy passed through the 3p threshold at about 66
eV. The dependence of Auger spectral intensity on photon energy showed a typical Fano typeasymmetric line shape with a minimum at 66 eV and a maximum at 71 eV. More interestingly, it
was demonstrated that not only the Auger peak but also the Ni 3d band photoemission was coupled
to the 3p excitation. Furthermore, this 3p-3d interaction differs from the top to the bottom part ofthe d band.

211

The investigation of photoelectron lifetime in Ni(111) are mainly focused on valence bands
at photon energies < 60 eV in thesis. Since bright high harmonic source can reach wavelength all
the way to 13 nm using the current laser system, time-domain study of Fano resonances in Ni(111)
can be achieved. It could shed light on the influence of core-level excitation to valence band, whichis scarcely addressed before.

9.2.2

Doped Metal and Surface-Adsorbate Systems
The RABBITT experiments so far have been mainly performed on metallic samples. In

these experiments, the dressing IR field can be assumed to be substantially screened in the interiorof the solid, which is a result of the optical properties of metals. This assumption is proved to bevalid in multiple publications [56,60]. This assumption simplifies the interpretation of streaking
and RABBITT measurements because the liberation of photoelectrons inside the metal can beseparated from their subsequent interaction with the dressing field in vacuum.
Conversely, the screening effect is less pronounced in semiconductors and dielectrics. This
gives rise to the complication in determining when the photoelectrons are “streaked” or “dressed”
by the IR field. For dielectrics, the refractive index is close to unity. The IR field can efficiently
penetrate into the material. It was proposed that the photoelectrons from dielectric wereimmediately accelerated by the dressing field and could serve as the reference to measure the“absolute” photoemission time in metals [301]. The approach to implement this scheme is toabsorb certain dielectric layers on top of metal surface. Nepple et al. studied the solid stateXe/W(110) system and concluded that the absolute timing measurement relied on an improved
understanding of the photoemission time from isolated atoms, which is not yet achieved [301].

212

Another interesting problem is how the orbital hybridization changes the excited-state
lifetime. In chemisorbed system like O2/metal and CO/metal, the electrons are transferred between
the molecular HOMO/LUMO and the conduction bands of metal [302,303]. Even in physisorbed
noble atoms on metal surface, the electrons that are resonantly excited to the autoionization state
can transfer into the metal surfaces [304]. This charge transfer can influence the Auger electron
and autoionization state lifetime and has been a hot topic for static and time-resolved
photoemission. Chemisorbed molecules can also induce changes on the final states of metals andimpact the photoelectron lifetime.
It has been observed that the asymmetric molecular potential will lead to the dependenceof Wigner time delay on photoelectron emission angle. For chemisorbed systems like CO/Ni(111)[121], the molecular potential is perturbed by the surface and photoemission can only occur on
one side. Attosecond photoemission can be a sensitive probe to map the potential change.
Furthermore, collective nuclear motion such as the vibrational, rotational and stretching mode of
the molecule can be excited by IR laser. The impact of these dynamics on photoionization time
will shed light on the surface-adsorbate interactions.
We have successfully fabricated single-layer and double-layer graphene on Ni(111) surface
in our setup using chemical vapor deposition (CVD) growth method [151,305]. Their band
structures have been mapped using linearly polarized harmonics. We also added alkali Na atoms
to modify the electronic structure of Graphene/Ni(111) system. The ARPES spectra reveal distinct
spectral change during the course of atomic Na absorption onto and intercalation into
Graphene/Ni(111). Subsequent oxygen exposure of adsorbed/intercalated surface shows that the
on-top Na atoms that are not “protected” under the graphene layer can be oxidized and form
sodium oxide layer.

213This manipulation of oxygen-sodium-graphene-nickel layered system gives us the accessto a wide range of photoionization dynamics. 1) For double-layer graphene grown on nickel, thefirst layer is strongly hybridized with nickel but the second layer is almost free-standing [306]. It
is therefore a candidate system to study the photoelectron lifetime of hybridized states. 2) The
study of transport time difference for Na on top of graphene and intercalated under graphene will
give information on electron transport through a single layer graphene. 3) During the oxygen
exposure, metallic sodium on top of graphene will be oxidized into dielectric sodium oxide. This
metal to dielectric transitions might lead to different interactions between photoelectrons with the
dressing field.

9.3

Future Opportunities
Within the past three years, time-resolve photoemission spectroscopy benefit from twodevelopments in high harmonic generation. 1) Narrow bandwidth harmonics driven by 2ω light
substantially improved the energy-resolution and photon flux for high harmonic-based
photoemission experiments [171]. 2) Different methods of circularly polarized high harmonic
generation were demonstrated [30,68–70]. In terms of the detection of photoelectrons, spinresolved photoemission can add unique information especially for itinerant magnetic systems.
Recently spin detector based on very low-energy electron scattering from an oxygen-passivatedepitaxial Fe film improved the efficiency of spin-detection by a factor 20 compared to previous
Mott- or spin polarized lower energy electron diffraction (SPLEED-) based detectors. It has been
demonstrated that the combination of spin-resolved detector with high-flux 2ω–driven highharmonic sources can be used to track the band stricture evolution during ultrafast ferromagnetic
to paramagnetic phase transitions in cobalt. Our results show that the phase transition in cobalt

214

cannot be explained by a collapse of exchange splitting in spin-polarized bands, but instead shows
rapid band mirroring after excitation.
Similar experimental setup can be used to study spin-dependent photoemission time insolids. In Chapter 8 of the thesis, we have shown through simulation that the photoelectron lifetimein Ni(111) is spin-dependent due to electron-electron scattering. The experimental measurementof such dynamics will shed light on influence of spin on electron correlations.

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