Attosecond Probing of Electron Dynamics in Atoms and Molecules Using Tunable Mid-Infrared Drivers

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Attosecond Probing of Electron Dynamics in Atoms and Molecules using Tunable Mid-Infrared Drivers

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

Timothy Thomas Gorman, B.S., M.S.

Graduate Program in Physics

The Ohio State University

2018

Dissertation Committee:

Professor Louis F. DiMauro, Advisor

Professor Jay A. Gupta

Professor Daniel J. Gauthier

Professor Douglass Schumacher © Copyright by

Timothy Thomas Gorman

2018 Abstract

The spectral intensity and phase of the complex photoionization and photorecombination dipole matrix elements of gas phase atoms and molecules were studied with sub- femtosecond time resolution and sub-eV energy resolution using attosecond spectroscopy and high-harmonic spectroscopy. Using attosecond spectroscopy, the phases of two-photon transitions near autoionizing resonances were studied in Ar and He with harmonic combs of odd harmonics only as well as combs of even and odd harmonics. The Ar 3s3p64p and the He 2s2p resonances were explored using the Reconstruction of Attosecond Beat- ing by Interference of Two-Photon Transitions (RABBITT) method and by probing omega oscillations that occur with pulse trains comprised of even and odd harmonics. The RAB-

BITT measurements revealed that the 3s3p64p resonance produced phase excursions of

¤ 0.2 rad and that the 2s2p resonance exhibited phase excursions of ¤ 0.5 rad. These results are in excellent agreement with ab initio calculations solving the time-dependent

Schrödinger equation. The RABBITT measurements also revealed a small phase structure

(¤0.05 rad) in Ar, which is attributed to the tentatively-labeled 3s23p44s4p resonance. The complexity of this resonance is beyond current modeling capabilities. Comparatively, the omega-oscillation experiments revealed resonant phase excursions for the Ar 3s3p64p and

He 2s2p resonances that starkly deviate from theoretical predictions using the same theoretical framework. These measurements emphasize that two-photon ionization involving electron correlations is incomplete and in need of further investigation.

Using high-harmonic spectroscopy, the ﬁrst measurements fully characterizing the spectral intensity and phase of the methyl chloride molecular Cooper minimum (CM) were performed. These experiments revealed that the CM of methyl chloride is located at ¤ 43 eV and is accompanied by a ¤ 120 as group delay minimum. The key to accurately identifying this resonance was the development of an in-house algorithm that was able to accurately calculate attochirp contributions present in the measurements. Also using high- harmonic spectroscopy, the spectral intensity and phase of two-center interferences in CO2,

i OCS, and N2O were studied in the molecular frame. These measurements determined that the phase jump associated with CO2 two-center interference is negative in sign, settling an existing discrepancy in the literature. This result is also in agreement with theory calculations performed using a time-dependent density functional theory framework. Additionally, the complex interplay of the OCS Cooper-like minimum (¤ 43 eV, ¤ 140as) and the molec-

ular structure of OCS in the recombination dipole matrix element of the highest occupied

molecular orbital was explored and characterized. The careful understanding of such com-

plex interferences is expected to serve as the foundation for future measurements studying

attosecond electron correlations in molecules.

ii I would like to dedicate this dissertation to my wife, Dr. Emily Gorman, and my mother

and father, Denise and Thomas Gorman, for always believing in me.

iii Acknowledgments

First and foremost, I would like to thank my two advisors, Professors Louis DiMauro and Pierre Agostini. They have provided me with a world-class research laboratory, a friendly and productive working environment, and priceless guidance throughout my entire graduate student career. I would also like to thank my undergraduate research advisors,

Dr. Perry Yaney and the late Dr. Peter Powers, for ﬁrst introducing me to lasers and all of their excitement at the University of Dayton.

In no particular order, I would also like to thank the many other people who have positively impacted my research career:

• Drs. Chantal Sudbrack and Timothy Gabb at the NASA Glenn Research Center for

expanding my research knowledge into metals processing.

• Drs. Stephen Schoun and Antoine Camper for ﬁrst introducing me to attosecond

science and molecular alignment.

• Drs. Cosmin Blaga and Yu Hang “Marco” Lai for working with me to better under-

stand strong ﬁeld ionization, laser pulse characterization, and laser operation.

• Drs. Hyunwook Park and Zhou Wang for teaching me about laser-cluster interac-

tions, and particularly Dr. Wang for excellent debates concerning challenging physics

questions.

• Dietrich Kiesewetter for great acquisition and analysis software, which he developed

for the group, for participating in physics discussions that helped expand my under-

standing of relevant physics topics, and for many hours spent together learning how

to build complex interferometers.

• Dr. Timothy Scarborough, with whom all of the work in this dissertation was done

in collaboration. Tim has an excellent work ethic, is a critical thinker, and is an all

iv around good person to work with. I could not have asked for a better colleague with

whom to perform my dissertation research.

• Greg Smith and Stephen Hageman for many long, detailed physics discussions. I

gained much of my physics knowledge through conversations with them.

• Andrew Piper and Daniel Tuthill for productive discussions concerning autoionizing

resonances.

• Our research collaborators at Louisiana State University, including Dr. Ken Schafer,

Dr. Mette Gaarde, Dr. François Mauger, and Dr. Paul Abanador for many in-depth

discussions of high-harmonic spectroscopy.

• Our research collaborators at the University of Virginia including Dr. Robert Jones,

Dr. Péter Sándor, and Sanjay Khatri for teaching me about molecular strong-ﬁeld

ionization and surface harmonic generation.

• Our research collaborators at the University of Central Florida, University of Auburn,

and the Vienna University of Technology including Dr. Luca Argenti, Dr. Joachim

Burgdörfer, Stefan Donsa, Saad Mehmood, Dr. Nicolas Douguet, Cariker Coleman,

Dr. Guillaume Laurent, and John Vaughan for excellent discussions on autoionizing

resonances.

• Our research collaborators at CEA Saclay including Dr. Thierry Ruchon, Dr. Pascal

Salieres, Dr. Romain Géneaux, Dr. Lou Barreau, and Celine Chappuis for originally

teaching me about autoionizing resonances and orbital angular momentum of light.

• All of the following people for useful and productive conversations: Kent Talbert, Eric

Moore, Bryan Smith, Dr. Kaikai Zhang, Dr. Junliang Xu, Dr. Sha “Lisa” Li, Dr.

Abraham Camacho, Dr. Douglass Schumacher, and Dr. Enam Chowdhury.

• All of the following for being such great friends as I made my way through graduate

school: Luke Gorman, Andrew Gorman, Julie Buchanan, Elizabeth Lannan, Solani

v Harawa, Dr. Janet Harawa, Mike DeBrosse, Kevan Kramb, Dr. Christopher Jaymes,

Dr. Steven Tjung, Thuc Mai, Dr. Matt Sheﬃeld, Dr. Alex Dyhdalo, Michael Chilcote,

and David Maguire.

Lastly, I would like to thank all of the people I mentioned for always being willing to discuss my numerous questions, physics or otherwise.

vi Vita

2012 ...... B.S. Physics. University of Dayton, Dayton,

OH.

2014 ...... M.S. Physics. The Ohio State University,

Columbus, OH.

2014 to present ...... Graduate Research Associate. The Ohio

State University, Columbus, OH.

Publications

Zhang, K., Lai, Y. H., Diesen, E., Schmidt, B. E., Blaga, C. I., Xu, J., Gorman, T. T.,

Légaré, F., Saalmann, U., Agostini, P., Rost, J. M., and DiMauro, L. F. (2016). Universal pulse dependence of the low-energy structure in strong-ﬁeld ionization. Physical Review A,

93(2):021403.

Géneaux, R., Chappuis, C., Auguste, T., Beaulieu, S., Gorman, T. T., Lepetit, F., Di-

Mauro, L. F., and Ruchon, T. (2017). Radial index of Laguerre-Gaussian modes in high- order-harmonic generation. Physical Review A, 95(5):051801.

Scarborough, T. D., Gorman, T. T., Mauger, F., Sándor, P., Khatri, S., Gaarde, M. B.,

Schafer, K. J., Agostini, P., DiMauro, L. F. (2018). Full Characterization of a Molecular

Cooper Minimum Using High-Harmonic Spectroscopy. Applied Sciences, 8(7):1129.

Sándor, P., Sissay, A., Mauger, F., Abanador, P. M., Gorman, T. T., Scarborough, T. D.,

Gaarde, M. B., Lopata, K., Schafer K. J., Jones, R. R. (2018). Angle dependence of strong-

ﬁeld single and double ionization of carbonyl sulﬁde. Physical Review A 98(4):043425.

vii Fields of Study

Major Field: Physics

viii Table of Contents

Page

Abstract...... i

Dedication ...... iii

Acknowledgements ...... iv

Vita...... vii

Table of Contents ...... xii

List of Figures ...... xxxii

List of Tables ...... xxxiii

1 Introduction ...... 1

1.1 Attosecond Science...... 1

1.1.1 High-Harmonic Generation ...... 1

1.1.2 Attosecond Metrology as a Spectroscopic Tool ...... 3

1.2 Attosecond Spectroscopy: Two-Photon Direct Ionization as a Scheme for

Attosecond Science...... 6

1.3 High-Harmonic Spectroscopy: A Self-Probing Scheme for Attosecond Science 8

1.4 Outlook ...... 9

2 Theory of High-Harmonic Generation...... 11

2.1 A Strong-ﬁeld Mechanism for Harmonic Generation ...... 11

2.2 Microscopic Theory of High-Harmonic Generation: Recollision ...... 11

2.2.1 Semiclassical Model: A Model for a Simple Person ...... 13

2.2.2 HHG Cutoﬀ and Attochirp ...... 15

2.2.3 Scaling of HHG ...... 17

ix 2.2.4 Attosecond Pulse Trains and High Harmonics ...... 19

2.2.5 Quantum Mechanical Model: The Lewenstein Model ...... 20

2.2.6 Strong Field Ionization ...... 24

2.2.7 Propagation ...... 26

2.2.8 Photoionization and Photorecombination ...... 29

2.2.9 Quantitative Rescattering Model ...... 32

2.2.10 Femtochirp of the Dipole Phase ...... 33

2.3 Macroscopic Considerations ...... 36

2.4 Conclusions ...... 40

3 Two-Photon Ionization Theory ...... 41

3.1 XUV Metrology and a Direct Photoionization Scheme for Attosecond Spec-

troscopy...... 41

3.2 Odd Harmonic Orders: A Two-Omega Oscillation ...... 44

3.3 Even and Odd Harmonic Orders: An Omega Oscillation ...... 47

3.4 Conclusions ...... 51

4 Impulsive Molecular Alignment ...... 52

4.1 Introduction to Alignment ...... 52

4.2 Theory of Impulsive Molecular Alignment...... 54

4.3 Dependence on Laser Parameters ...... 57

4.4 Conclusions ...... 59

5 Experimental Methods and Apparatus ...... 60

5.1 Statement of Contributions...... 60

5.2 Methods ...... 60

5.2.1 Molecular Alignment in an HHS Experiment ...... 61

5.2.2 Wavelength Tuning for Resonance Spectroscopy ...... 63

5.3 Laser Systems ...... 64

5.3.1 Home-built Laser as a Source for Near-Infrared Femtosecond Light . 66

x 5.3.2 Optical Parametric Ampliﬁer ...... 69

5.4 Interferometers ...... 72

5.4.1 HHG-Kick Interferometer ...... 72

5.4.2 Attobeamline for Two-Omega and Omega Oscillation Measurements 73

5.5 Magnetic Bottle Electron Spectrometer...... 78

5.5.1 Principles of Operation ...... 79

5.5.2 Resolution ...... 81

5.5.3 Detection Gas ...... 82

5.5.4 Data Acquisition Hardware and Software ...... 84

5.6 Omega and Two Omega Acquisition and Analysis Procedures...... 85

5.6.1 Data Acquisition Procedures ...... 86

5.6.2 Data Analysis: Extracting Oscillation Phases ...... 87

5.6.3 Data Analysis: Separation of Delay Contributions in RABBITT . . . 92

6 High-Harmonic Spectroscopy of the CH3Cl Cooper Minimum ...... 97 6.1 Atomic Probes of Molecular Electron Dynamics...... 97

6.2 Molecular Cooper Minima in High-Harmonic Spectroscopy ...... 98

6.2.1 Statement of Contributions ...... 100

6.3 Theoretical Background ...... 100

6.3.1 Origin of the Cooper Minimum ...... 100

6.3.2 Cooper Minima in a Coherent HHS Measurement ...... 103

6.4 HHS of a Cooper Minimum in CH3Cl : The Experiment...... 104 6.4.1 Experimental Details ...... 104

6.4.2 High-Harmonic Spectral Intensity and GD Analysis ...... 105

6.4.3 Results ...... 106

6.5 Conclusions ...... 114

7 Attosecond Spectroscopy of Autoionizing Resonances in Ar and He ...... 115

7.1 Introduction to Autoionizing Resonances ...... 115

xi 7.1.1 Statement of Contributions ...... 119

7.2 Theoretical Description of Autoionizing Resonances in Two-Photon Ionization 119

7.3 Results ...... 124

7.3.1 Ab Initio Calculations of Autoionizing Resonance Properties . . . . . 124

7.3.2 Experimental Details ...... 127

7.3.3 Ar and He Photoionization Spectra ...... 128

7.3.4 Ar and He RABBITT: Two-Omega Oscillation Results ...... 133

7.3.5 Ar and He with Even and Odd Harmonics: Omega Oscillation Results 138

7.4 Conclusions ...... 144

8 High-Harmonic Spectroscopy of Two-Center Interference in CO2,N2O, and OCS...... 145

8.1 Two-Center Interference Introduction...... 145

8.1.1 Statement of Contributions ...... 147

8.2 Theoretical Description of Two-Center Interference...... 148

8.3 Time-dependent Density Functional Theory Calculations ...... 150

8.4 Results ...... 154

8.4.1 Experimental Details ...... 154

8.4.2 CO2 Interferences at 785 nm ...... 156

8.4.3 Two-Center Interference in CO2,N2O, and OCS in the MIR . . . . . 160 8.5 Model Calculations ...... 170

8.6 Conclusions ...... 174

9 Conclusions...... 175

9.1 Summary...... 175

9.2 Outlook ...... 176

Bibliography ...... 192

xii List of Figures

Page

1 A typical CO2 spectrum recorded with a magnetic bottle electron spectrometer that was generated at 1300 nm, spectrally ﬁltered through Al, and de-

tected with Ne. As shown in the ﬁgure, this comb is only comprised of odd

harmonics. Of note, the cutoﬀ around ¤72 eV is due to the Al ﬁlter used in

the experiment. The HHG cutoﬀ extends past this point...... 2

2 Experimental layout for the ﬁrst measurement of attosecond pulse trains in

2001. In this experiment, 800 nm light was used in a collinear interferometer

where the XUV and IR copropagated. The inset shows a cartoon depicting

the measurement process. Adapted from [1]...... 4

3 Cartoon depiction of the two attosecond probing schemes. In (a), we see

that the AS scheme uses a well-characterized attosecond pulse from an Ar

generator to interrogate a N2 detection gas. In (b), the HHS scheme uses a well-known detection gas, in this case Ar, to interrogate the XUV generation

gas, N2. In truth, neither the generator nor the detector is well-characterized in group delay. To this end, both generator and detector are studied in this

dissertation. Adapted from [28]...... 6

4 Example results from an experiment where harmonics were generated in Ar

and then were used to photoionize a separate Ar detector gas. Centered

around ¤26.7 eV and surrounded by a red circle, one can see the imprint of

the 3s3p64p resonance on the group delay. Additionally, circled in green at

¤34 eV, the 3s23p44s4p imprints itself on the group delay...... 7

5 Comparison of methane (CH4) and methyl chloride (CH3Cl) GD after removal of attochirp and other contaminating delays. The dip in group delay

in CH3Cl centered around 45 eV corresponds to a Cooper minimum. . . . . 9

xiii 6 A cartoon depiction of a typical harmonic spectrum generated in a gas sam-

ple. The spectrum is a comb of odd harmonic “spikes” separated by two

times the fundamental laser frequency 2ω0. At the lowest energies the harmonic intensities drop oﬀ rapidly, followed by a relatively ﬂat plateau region

and then cutoﬀ. The existence of the plateau region indicates a diﬀerent

mechanism for HHG: recollision...... 12

7 The three step model. The Coulomb potential is in green, the laser potential

is the dashed black line, and the combined potential is shown in red. In the

ﬁrst step (a) a single electron is freed from the parent ion through tunneling

ionization. In the second step (b) the electron is propagated in the laser

ﬁeld, where it gains energy from the ﬁeld. Finally in the third step (c) the

electron recombines with the ground state of the parent ion, releasing an

XUV photon. Adapted from [28]...... 13

8 Electron trajectories for birth times between t 0 and t 0.35T0, where T0

is the laser period. The trajectories that return tox 0 (0 B t B 0.25T0) are

shown in blue, and the trajectories that do not return (0.25T0 @ t B 0.35T0) are shown in red...... 14

9 Numerical calculations of the semiclassical three step model. In (a) we see

that there are two sets of ionization or “birth” times corresponding to two

diﬀerent sets of recombination times. These are called the “long” and “short”

trajectories with corresponding long excursion times and short excursion

times, respectively. In (b) the recombination times, in laser cycles (L.C.),

for the long and short trajectories are plotted versus the electron return en-

ergy...... 17

xiv 10 Scaling examples of HHG. In (a) we can see that the harmonic cutoﬀ position

scales linearly with the laser intensity at a ﬁxed wavelength, and in (b) we

can see the that the cutoﬀ scales quadratically with the laser wavelength at

a ﬁxed intensity. Because of the quadratic scaling, the laser wavelength can

be increased as a more eﬀective way to extend the HHG cutoﬀ at a given

intensity...... 19

11 Pulse train attributes in HHG. The laser ﬁeld attributes are shown in red

whereas the XUV ﬁeld attributes are shown in purple. In (a) we see that

every intense half cycle leads to bursts of XUV light. Then in (b), under

Fourier transform, we see that the harmonic spectrum for the same pulse

train is comprised of discrete odd harmonics separated by 2ω0. Adapted from [51]...... 20

12 Ratios of experimental single ionization yields to theoretical ionization yields

for linear polarization at various values of γ from Lai et al. [56]. In these

experiments, diﬀerent values of γ were achieved using diﬀerent combinations

of driving wavelengths, interaction intensities, and atomic targets. Above

γ ¤ 0.7, the ADK model signiﬁcantly deviates from the experimental results. 27

13 Comparing semiclassical and SFA GD calculations. The SFA calculation

shown by the dotted lines produces a similar attochirp to the semiclassical.

In the cutoﬀ, however, the SFA predicts notably diﬀerent behavior, namely

a diﬀerent dependence on IP...... 29

14 Charge migration calculation from Kuleﬀ et al. [18]. In this diagram we can

see the remaining hole density as a function of time in the singly-ionized C-

terminally methylamidated dipeptide Gly–Gly–NH–CH3. The hole density

migrates from one side of the molecule to the other simply through electron

correlations...... 33

xv 15 Figures depicting the intensity dependence of the high-harmonic dipole phase.

In (a), adapted from [65], we see the that the intensity dependence of Φ is

nearly linear for the 19th harmonic of 800 nm, generated in an Ar gas jet.

The solid circle is an experimental data point extracted from [68]. In (b)

N we see from [66] that αj is larger for the long trajectories and smaller for the short trajectories away from the cutoﬀ. One may expect this, due to the

extended excursion amplitudes of the long trajectories compared to the short. 35

16 2D calculations of the coherence length across the laser focus for the 21st

(a-b), the 35th (c-d), and the 45th (e-f) harmonics of a 750 nm driving

ﬁeld in Ne at 4 1014W ~cm2. The confocal parameter here is 7 mm. Longer

coherence lengths, indicated by brighter (whiter) colors indicate better phase-

matching. (a), (c), and (e) represent phase-matching of the short trajectories

while (b),(d), and (f) represent phase-matching of the long trajectories. Red

arrows represent the direction of harmonics generated at their respective

positions. It can be easily seen that the phase-matching conditions for the

long trajectories are less regular than for the short trajectories, due to the

increased sensitivity of the long trajectories to the laser intensity. Adapted

from [69]...... 38

17 Far-ﬁeld image of two spatially-separated, delay-variable harmonic sources.

In this image we can see the spatial separation of long and short trajectory

HHG contributions. The bright patch of XUV in the middle corresponds

to the short trajectories and the dimmer outer ring corresponds to the long

trajectories. The vertical fringes are a consequence of the temporal overlap

of the two HHG sources. Adapted from Bellini et al. [44]...... 40

18 Depiction of angular momentum channels available depending on the parity

of the initial ground state. For the cases of Ar and Ne, there are six total

pathways to the same ﬁnal energy, while for the case of He, there are only

two pathways to the same ﬁnal energy...... 42

xvi 19 Diagram of allowed XUV + IR pathways in two-photon transitions from a

ground state (g) to a ﬁnal state at the energy of the qth harmonic above

the ionization potential (IP). In pathways (1) and (3) an XUV photon is

absorbed ﬁrst followed by the emission or absorption of an IR photon. In

pathways (2) and (4), an IR photon is ﬁrst absorbed or emitted followed by

XUV absorption. All pathways contribute to the photoelectron spectrum,

but only pathways (1) and (3) contribute to the phase...... 43

20 Example of the two-photon process that is fundamental to RABBITT mea-

surements. In this type of process, two separate two-photon pathways in-

terfere in a sideband at the energy of an even harmonic. One pathway is

comprised of the (q-1)th harmonic plus the absorption one IR photon, and

the other is the (q+1)th harmonic plus the emission of an IR photon. This

type of interference would result in an oscillation at two times the fundamen-

tal laser frequency...... 45

21 A depiction of the interaction that is responsible for omega oscillations. Just

like in Figure 20, two, two-photon pathways interfere at the energy of the qth

harmonic. The primary diﬀerence here is that the qth harmonic is present

and contributes to the interference signal at this energy – the source of the

omega oscillation...... 49

22 Models describing molecular alignment. In (a) we see the diﬀerence between

molecular alignment and molecular orientation. Molecular alignment does

not specify head-to-tail alignment whereas “orientation” does. In (b) a “kick”

aligning pulse is incident on a randomly aligned sample of gas molecules,

coherently populating a wave packet of J-states. The wave packet then

promptly aligns, dephases to random alignment, and then realigns at a later

time, called a revival. These revivals occur at delays that are fractions of the

rotational period or at the full rotational period T0...... 53

xvii 23 Revival Structure of O2. The solid black line represents the thermally-

2 averaged expectation value cos θ for the O2 rotational wave packet, and the dotted red line is the envelope of that expectation value. It is clear that

the wave packet reaches high degrees of alignment multiple times per rota-

tional period. The exact structure of the fractional revivals strongly depends

on the nuclear spin statistics of the molecule. Adapted from [92]...... 58

24 Cartoon evolution of RABBITT scans over many diﬀerent driving laser wave-

lengths. The blue bars represent harmonics and the red bars represent side-

bands. The phase diﬀerences between neighboring harmonics are imprinted

on the sidebands separating the two, represented here by stars. Starting

from the top left and ending in the bottom right, the generating wavelength

is scanned so that one harmonic, H3, thoroughly samples the “zig-zag” phase

feature. When the scan is over, the zig-zag is imprinted on the neighboring

sidebands SB2 and SB4 denoted by the red lines. The diﬀerence in sign be-

tween SB2 and SB4 is due to the phase diﬀerence in emission or absorption

of an infrared (IR) photon...... 65

25 Block diagram of the laser system. Adapted from [28]...... 66

26 SHG FROG results for the output of the laser compressor before entering

the OPA. (a) The originally-recorded FROG spectrogram. (b) The recon-

structed spectrogram. (c) Reconstructed ﬁeld intensity and temporal phase.

(d) Reconstructed spectral intensity and spectral phase...... 70

xviii 27 Schematic of the HE-TOPAS OPA. The beam enters in the top right with

approximately 6 mJ of energy at a pulse duration of 55 fs. The beam is

then split three times. The smallest energy arm is used to generate white-

light (WL) in a sapphire plate. That white light is then used to seed DFG

in a BBO creating the signal and idler wavelengths. This output is then

used to seed two more ampliﬁcation stages. The three delays and the three

BBO rotation angles in the OPA have been motorized for simple wavelength

tuning. Figure adapted from [28] ...... 71

28 OPA output (a) pulse energies and (b) pulse durations. The most power at a

single wavelength is at 1300 nm, but the shortest pulse duration is centered

around 1500 nm...... 72

29 A schematic of the compressor, OPA, and HHG-kick interferometer. The

output of the compressor pumps the commercial HE-TOPAS OPA. The out-

put of the OPA consists of the collinear signal and idler (separated by a

not-pictured wavelength separator) and a depleted pump. The signal or idler

is used for HHG and the depleted pump is controlled by a variable delay and

half waveplate for use as a “kick” in molecular alignment experiments. . . . 73

30 The attobeamline and an example RABBITT scan. The attobeamline di-

rects the majority of the driving pulse (dark purple) for harmonic generation

and saves a small amount of the pulse called the “dressing” (light purple) for

recombining with the XUV in the MBES. The alignment pulse (red) avoids

the CaF2 BS and is focused using the same optic as the HHG driving pulse.

After generation, the alignment pulse is clipped by an aperture and the HHG

laser is ﬁltered by a metallic ﬁlter. The XUV and dressing are then recom-

bined at HM and refocused by the TM into the MBES for RABBIT scans

like the one pictured on the right. Key: BS = beamsplitter; Filter = Al or

Zr metallic ﬁlter; HM = hole mirror; TM = toroidal mirror ...... 75

xix 31 HHG focal spot imaging. (a) The 2f-2f setup for 1:1 reimaging of the ﬂuo-

rescence from the HHG source. (b) An image taken of the ﬂuorescence from

the HHG source. When both the HHG pulse and the kick pulse are present,

two ﬁlaments can be seen. That makes this setup indispensable for spatially

overlapping the two pulses...... 77

32 Al and Zr Filter Properties in the XUV spectral region. (a) Filter Transmis-

sion. (b) Imparted Group Delay. The transmission curves and the Zr GD

were calculated using refractive indices calculated from [109], which are also

available through the Center for X-Ray Optics (CXRO) “X-Ray Interactions

with Matter” database. The Al GD was calculated using an Al refractive

index that was calculated by Rakic [110]...... 78

33 A schematic of the MBES. An inert gas is emitted from the gas needle where

XUV light pointed out of the page is incident. An electron is then ionized at

an initial angle θi relative to the TOF axis. From there the electron is guided along the magnetic ﬁeld lines of the permanent magnet onto the ﬁeld lines of

the solenoid. The electron travels along these until it is incident on an MCP

detector at a ﬁnal angle θf . The vertical slit is used for increasing the TOF resolution of the MBES. It does this by restricting the detected electrons to

a small solid angle, thereby restricting the detected ionization volume to a

small region near the laser focus...... 79

34 Calculated cross sections for typical inert gas detectors. Ar easily has the

largest cross-section at low energies, but it drops dramatically around 50 eV

due to the presence of the Cooper minimum. He and Ne both have smoother-

varying cross sections, but He has a much smaller cross section than Ne across

the entire energy range shown. This makes Ne the most desirable detection

gas for HHS experiments...... 83

xx 35 Information pathway of voltage data from the TOF detector into averaged

and CFD data sets. Both CFD and averaged data sets are saved for each

measurement...... 85

36 A diagram showing the allowed polarization orientations for omega and two-

omega oscillation experiments. Two-omega oscillations will be detected with

either vertical or horizontal XUV+IR polarizations due to the even parity

of the RABBITT interaction. Omega oscillation measurements, however,

require a laser polarization along the TOF axis due to the odd parity of the

interference responsible for it...... 86

37 (a) RABBITT spectrogram and (b) the same spectrogram after an FFT

along the delay axis. In (a) two-omega oscillations are clearly visible in the

lower-intensity sidebands though are washed out in the harmonics due to the

color scale. After an FFT, we can see that both sidebands and harmonics are

oscillating. Additionally, we can see that there is no sign of higher frequency

oscillations...... 88

38 Example of windowing for RABBITT data in unaligned OCS @ 10% in He

for 1250 nm. In the top plot, the FFT two-omega intensity is shown versus

photon energy. In the bottom plot is the corresponding oscillation phase.

Red windows correspond to sidebands and green windows correspond to odd

harmonics. The phases are extracted from the shaded regions, which are the

FWHM of each oscillation peak...... 90

xxi 39 800 nm atomic delay calculations. In (a) we see the absolute value of τat. The original calculation is only available over a restricted energy range, which

requires extrapolation to higher and lower energies. The ﬁnal atomic delay

after extrapolation is also interpolated to a ﬁner grid. In (b) the signed values

of the atomic and continuum-continuum delays are shown. τcc at 800 nm also requires some extrapolation at lower energies for ease of application to MIR

data sets where electrons can have kinetic energies smaller than the energy

of an 800 nm photon...... 95

40 Diagram of the angular momentum channels in single photon ionization of

Ne and Ar. According to the dipole selection rules, both ground states have

two channels to the continuum: p s and p d. Only the 3p d channel

for Ar, though, exhibits a CM. There is no such minimum in Ne...... 101

41 Model calculations of atomic CMs. In (a), the ground state wavefunctions

for Ne, Ar, and Kr are depicted along with their corresponding 0 d-waves.

The ground state of Ne does not contain a radial node, whereas the ground

states of Ar and Kr do. Thus Ar and Kr exhibit CMs. In (b), Rl1 are shown for all three atoms. The sign changes occur at ¤ 2 Ry in Ar and ¤ 2.5

Ry in Kr. Adapted from [114]...... 102

42 SFA ﬁtting program used to calibrate I0 and the corresponding τSFA. In (a), the in-house GUI is shown, and in (b) various frames are shown from the

ﬁtting process. In (b), the data are shown by the black curves. The τSFA

calculations are shown as dashed lines from the lowest-intensity τSFA (blue)

to the highest-intensity τSFA (red). The algorithm calculates the lowest-

intensity τSFA ﬁrst (left), then iteratively increases the intensity and recal-

culates τSFA (middle) until the preset limit is reached (right). In this case,

13 2 the data is from CH4 and the final fitted intensity is ¤ 5.57 10 W ~cm . The vertical dashed lines indicate the fitting region selected in this example. 107

xxii 43 Laser intensity scaling of the HHG group delay Xe. In (a), The measured

group delays (open circles) were ﬁtted using using the intensities indicated by

the legend. The lowest-intensity curve was ﬁtted for intensity using Equation

110, and the intensity was scaled to the measured laser input power for each

additional scan (solid lines). Each data set and ﬁt combination was separated

by 0.2 fs for visual clarity. In (b), the SFA ﬁt errors at each intensity are

shown. Quantitative agreement, within the experimental error, is found at

all intensities...... 109

44 Laser intensity scaling of the HHG group delay CH4. In (a), The measured group delays (open circles) were ﬁt using using the intensities indicated by

the legend. As with the Xe case, the lowest-intensity curve was ﬁtted for

intensity using Equation 110, and the intensity was scaled to the measured

laser input power for each additional scan (solid lines). Each data set and

ﬁt combination was separated by 0.2 fs for visual clarity. In (b), the SFA ﬁt

errors at each intensity are shown. Within the experimental error, 6 of the 7

experimental intensities showed no deviation from the SFA calculations. The

only outlier at 8.51 1013 W ~cm2 sits 1.25 standard deviations from zero. . 110

45 Inﬂuence of the CM on the spectral intensity of CH3Cl relative to CH4. In (a),

the normalized spectra of CH4 (blue) and CH3Cl (green) were taken back- to-back in identical conditions. The envelopes of both spectra were retrieved

through a second order Savitzky-Golay smoothing. Panel (b) shows the ratio

of the CH3Cl envelope to the CH4 envelope. The minimum at ¤ 43eV is

attributed to the presence of the CH3Cl CM...... 112

xxiii 46 Comparison of the GD and phase features between CH4 (blue circles) and

CH3Cl (green circles). In (a) τSFA (solid red curve) is shown along with

the high-harmonic delays (τHHG) from the experiment. The laser intensity

13 2 was ﬁt using the CH4 GD, resulting in an intensity of 5.56 10 W ~cm ,

and then, this same intensity was used to calculate τSFA for the CH3Cl GD.

The CH3Cl data is shifted by -0.2 fs for visual clarity. (b) Removal of τSFA

isolates τtarget. By doing so, the CM in CH3Cl is revealed as a ¤ 120 as “dip” in the GD. (c) Group delays were then integrated to show a phase shift of

¤ 2.6 rad across the CM. Additionally shown are the results for Ar at 1.3 µm

and 2.0 µm from Schoun et al. [10]...... 113

47 XUV photoionization of an Ar atom in the presence of an autoionizing res-

onance. In this case, there are two pathways to the continuum: a direct

transition to the continuum and a transition to a bound state which decays

into the continuum. Because there are two pathways to the same ﬁnal state,

an interference in the photoionization spectrum occurs...... 116

48 Example calculations of autoionizing resonance parameters for a few values

of the asymmetry parameter q. Panel (a) shows that the cross section can

change from asymmetric to symmetric depending on q. Panel (b) shows that

the position of the phase jump associated with the resonance depends on q

and does not necessarily need to occur at 0...... 117

xxiv 49 Depictions of the RABBITT interaction in the presence of an autoionizing

resonance for the case of the Ar 3s3p64p resonance. In (a), the phase of the

electron ionized by the (q+1)th harmonic is imprinted onto the neighboring

qth and (q+2)th sidebands. In (b), the speciﬁc notation used in the derivation

is shown. Sge symbolizes the ground state, Ω is the XUV frequency, Sφe

represents the resonant bound state with energy Eφ, SψαEe at energy E with T f angular momentum α and ψγEf at energy Ef with angular momentum γ represent the intermediate and ﬁnal states, respectively, connected by an

infrared photon at frequency ω...... 120

50 Illustrations of the two-photon interactions used to probe autoionizing reso-

nances in this chapter. (a) and (b) show the “two-omega oscillation” RAB-

BITT interaction and the “omega-oscillation” interaction, respectively. The

phase in the RABBITT interaction is probed through the sidebands neigh-

boring the resonant harmonic, and the phase in the “omega-oscillation” ex-

periment is probed through the resonant harmonic and its two neighboring

harmonics...... 124

51 Diagram of the collinear frequency-doubling design used to generate high-

harmonic spectra comprised of even and odd harmonics. The fundamental

frequency ω is ﬁrst doubled using Type I second harmonic generation in a

Beta Barium Borate crystal. The two colors are then copropagated through

a rotatable calcite crystal for tunable delay pre-compensation between the

two colors. Lastly, the polarization of the fundamental beam is rotated 90X

to match that of the second harmonic polarization...... 128

xxv 52 Spectral intensity structures of the autoionizing resonances in Ar and He.

Panel (a) shows the photoionization cross section of the Ar 3s3p64p reso-

nance measured by Berrah et al. [175]. In panel (b) is plotted an Ar ab-

sorption measurement recorded on a photographic plate measured by Baig et

al. [176]. Decreased plate density implies increased (decreased) absorption

(transmission). Panel (c) shows the He 2s2p cross section calculated by Mor-

gan et al. [178] using parameters extracted from a ﬁt of XUV transmission

measurements...... 130

53 Photoelectron spectra of Ar and He using XUV generated in Ar and CO, re-

spectively, with diﬀerent HHG wavelengths. In (a), Ar photoelectron spectra

recorded with HHG driving wavelengths between 1400 nm (purple) and 1520

nm (yellow) in 2 nm steps are shown. A slight decrease in signal occurs in

the region around the 3s3p64p resonance (green vertical line, E =26.6 eV).

No clear feature is seen around the 3s23p44s4p resonance (blue vertical line,

E = 33.8 eV). In (b), He photoelectron spectra recorded with driving wave-

lengths between 1460 nm (purple) and 1510 nm (yellow) are shown. The

signal increases just below the 2s2p resonance and then decreases quickly at

the resonant position. Normalized spectra of Ar and He are shown in panels

the signals were normalized. After normalization, the Ar 3s3p64p resonance

and the He 2s2p resonant structures are clearer and qualitatively similar to

the corresponding cross sections in Figure 52. No obvious features can be see

in the vicinity of the 3s23p44s4p resonance...... 132

xxvi 54 An example of how resonances appear in group delay through our HHG

wavelength-scanning technique. In the top-left panel, no obvious resonant

phase features can be discerned with only one driving wavelength. As more

phase measurements are recorded at diﬀerent driving wavelengths, mirrored

resonant phase structures are revealed around the Ar 3s3p64p and 3s23p44s4p

resonances. The mirrored structure is due to the fact that the resonant phase

is imprinted into the two neighboring sidebands, but with opposite phases

due to the diﬀerence in IR absorption and emission phases...... 135

55 (a) Ar experimental RABBITT results with diﬀerent HHG generation gases,

CH4 (blue) and Ar (purple). Also shown are Ne photoionization results with an Ar HHG generator (green). The resonant features survive changes in

generation gases, but are no longer present with a Ne target; therefore, the

resonant features belong to Ar. (b) Comparison of the Ar photoionization

phase results to the ab initio theoretical calculations. The experiment has

good agreement between the resonant shapes predicted by theory...... 137

56 (a) He 2s2p experimental RABBITT results. The sideband just below the

resonant energy sees an increase in phase where as the sideband just above

see a decrease. This is qualitatively similar to the results recorded using

the so-called “Rainbow RABBIT”, which spectrally resolves the resonance

within one sideband [153]. (b) Comparison of the He experimental results

to the theoretical calculations. Here there is excellent agreement between

experiment and theory, verifying the model’s ability to reproduce two-photon

ionization phases of RABBITT...... 138

xxvii 57 A photoelectron spectrum resulting from photoionization with an XUV comb

comprised of even and odd harmonics. The even harmonics in the region

around the 3s3p64p resonance are kept at 25% of the intensity of the neigh-

boring odd harmonics. The XUV light was generated by a 1515 nm driver

and its spatiotemporally-overlapped second harmonic. The second harmonic

power is ¤ 1% that of the fundamental...... 139

58 Experimental and theoretical results for the omega-oscillation phases around

the Ar 3s3p64p resonance, where an odd harmonic is resonant. Panel (a)

shows that the even and odd harmonic phases are roughly separated by π.

In the resonant odd harmonic (green dashed box), there is not much struc-

ture. In contrast, the neighboring even harmonics (blue dashed box) exhibit

pronounced structures much larger than the measurement error bars. In the

lower even harmonic, the phase increases, then goes through a sharp decrease.

In the upper even harmonic, the phase decreases sharply and then recovers

within a few hundred meV. The theoretical results shown in (b) (odd) and (c)

(even) are dramatically diﬀerent from the experimentally retrieved phases.

At the time this dissertation was written, these diﬀerences were not yet resolved.141

59 Experimental and theoretical results for the omega oscillation phases around

the He 2s2p resonance. An odd harmonic is resonant in this case. In panel

(a) the odd harmonic phases do not exhibit any signiﬁcant structures larger

than the measurement error. In panel (b), however, the neighboring even

harmonics exhibit structure: an increase in phase for the lower even harmonic

and a decrease in phase for the upper even harmonic. Panels (c) and (d) show

the theoretical results for He, which again do not agree with our experimental

results. The origin of this discrepancy was not understood at the time this

dissertation was written...... 143

xxviii 60 A cartoon depiction of two-center interference. Two center interference oc-

curs when an electron with de Broglie wavelength λde Broglie is incident on a molecule with lobe separation R and, for the case of destructive interference,

satisﬁes the criterion of Equation 132. Isosurfaces of the HOMO orbitals

for each molecule are represented with red and blue, which indicate opposite

phases. The interference can be aﬀected by the recombination dipole phase

of the respective centers, for example if one center is of 2p character and the

other is of 3p...... 147 X X 61 Gabor Analysis of aligned CO2 TDDFT calculations for (a) 15 and (b) 45 . The forward tilting spectra (dashed purple line shown in ﬁrst half-cycle),

are the short trajectory responses in CO2. These are clearly more dominant than the backwards tilting (dashed white line shown in ﬁrst half-cycle) long

trajectories. A minimum clearly appears around 45 eV at 15X ...... 151

62 Alignment-dependent harmonic spectral yields in CO2 at 1500 nm with an intensity of 0.6 1014 W ~cm2. The color bar indicates log scale. These

spectra are extracted from the time signal during the ﬁrst half-cycle after the

ﬁeld ramp up. Each angle is normalized to the angle-averaged signal. This

is done for a more direct comparison to the experimental results. Two-center

minimum positions are indicated by the white diamonds, and the dashed line

indicates 1~ cos2 θ dependence to guide the eye...... 153

xxix X 63 Calculated HHG spectral intensity and phase for CO2 at θ 5 . In (a), the total spectral intensity (black dashed) and its x-component (red) and y-

component (blue) are plotted. A deep minimum appears in the total spectral

intensity and it’s x-component. In (b), the corresponding spectral phases are

plotted. A negative phase jump occurs at the energy of the spectral minimum

in both the total spectral intensity and in the x-component. The negative

phase jump would correspond to a minimum in the GD. In order to help

guide the eye, the green dashed line indicates the expected position of the

minimum based on Equation 132 ...... 154

64 CO2 molecular-frame HHG GD and intensity, generated with a 785 nm driver. In panel (a), the angularly-resolved GD results are shown for 0X, 45X, and 90X.

At 0X, the GD bumps up compared to 90X, opposite that predicted by the

TDDFT calculations. In (b) the delay-dependent spectrogram (corrected for

Al transmission and Ne cross-section) is shown, and the same data normalized

to the quasi-unaligned region is shown in (c). Both plots exhibit a minimum

at 0X(dashed green) that is gone by the time the molecule reaches 90X (dashed

blue). Panel (d) shows the laser intensity-dependence of the 0X minimum.

The interference minimum moves from high energies to low energies as the

laser intensity is decreased...... 157

65 Shown are HHS results of unaligned CO2,N2O, and OCS at. In (a) are the spectral intensities at 1300 nm (each normalized to unity) and in (b)

are the GD wavelength scans centered around 1300 nm. The spectra and

GDs have been corrected for the eﬀects of the Al ﬁlter and Ne detection gas.

Additionally, the attochirp has been removed from the GD measurements.

These results demonstrate the unique aspects of OCS, including a sharp drop

in signal after ¤ 43 eV and a GD dip centered at the same position...... 159

xxx 66 Molecular-frame spectral data for all three molecules at 1300 nm, extracted

from the HHG-alignment delay-dependent and alignment polarization-dependent

signals. All plots have been corrected for the Ne cross-section and the Al ﬁlter

transmission, and the spectrograms (a, c, e) have been normalized to unity

in order to compare the diﬀerent spectral proﬁles on the same color scale. In

panel (a) are the results for CO2 versus HHG-alignment delay and in (b) are shown the spectra obtained by rotating the alignment polarization at a ﬁxed

delay. The same is done for N2O in (c) and (d) and for OCS in (e) and (f). 162 67 Molecular-frame spectral intensity enhancements relative to unaligned sam-

ples for all three molecules at 1300 nm, extracted from the HHG-alignment

delay-dependent and alignment polarization-dependent signals. In panel (a)

are the enhancements for CO2 versus HHG-alignment delay and in (b) are shown the enhancements obtained by rotating the alignment polarization at

a ﬁxed delay. The same data sets are shown for N2O in (c) and (d) and for

OCS in (e) and (f). CO2 and N2O both demonstrate similar enhancement behaviors: deepest minimum at 0X with large enhancements at higher ener-

gies for large angles. For OCS, however, the deepest minimum is between

22.5X and 45X with positive enhancements localized to an energetic region

around 43 eV...... 164

68 Enhancement curves at 0X for all three molecules as the laser intensity is

varied. All of the intensity scans demonstrate the intensity-independence of

the minima positions. This supports the interpretation that the minima are

due to the electronic structure of the respective molecules...... 166

xxxi 69 Molecular-frame GD measurements for (a) CO2, (b) N2O, and (c) OCS. For each measurement, the alignment pulse polarization was rotated and

RABBITTs were then recorded. The attochirp, ﬁlter delay, and Ne atomic

delay have been removed from all measurements. The most striking features X are at 0 , where OCS bumps up in GD (a positive phase jump), and CO2

and N2O dip down (a negative phase jump)...... 168 70 0X GD wavelength scans for all three molecules. The above results show

that OCS goes through a smooth “bump up” in GD, where as the other

two molecules “dip” down. The results for CO2 are in agreement with the TDDFT predictions. The OCS GD has been shifted by 300 as for visual

clarity...... 169

71 Comparison of the theoretical reference spectrum to the theoretical unaligned

spectrum for the case of CO2. The unaligned spectrum was formed by coherently averaging angular spectra calculated with Equation 137. The unaligned

spectrum exhibits a drop in signal for energies above the 0X two-center inter-

ference, due to the superposition of interferences from all angles...... 172

72 Toy model comparison against results for (a) CO2 and (b) OCS. The dotted lines represent the experimental data and the solid lines represent the model

calculations. Qualitative agreement is found for both molecules. The results

indicate that the two-center interference of OCS must be positive, opposite

that of the other two molecules...... 173

xxxii List of Tables

Page

1 Relevant molecular parameters for impulsive molecular alignment...... 55

2 Ionization potential values for various inert gas detectors. The ﬁrst values are

for ﬁnal ion states with total angular momentum of J=3/2, and the values in

parentheses are for ﬁnal ion states with angular momentum J=1/2. It is clear

that for Kr and Xe, the splitting is much larger than our spectral resolution

(¤ 100 meV). This results in two separated electron peaks for every XUV

photon. Additionally, this has the added complication of harmonic peaks

overlapping with sidebands for certain driving wavelengths...... 82

3 SFA parameters used in toy model...... 171

xxxiii Chapter 1: Introduction

“The rabbit-hole went straight on like a tunnel for some way, and then dipped suddenly down, so suddenly that Alice had not a moment to think about stopping herself before she found herself falling down a very deep well.” —Lewis Carroll, Alice in Wonderland

1.1 Attosecond Science

With the ﬁrst measurements of sub-femtosecond pulses in 2001 [1][2] came a wave of experiments attempting to create shorter pulses for studying electron dynamics on their natural timescale, the attosecond (10-18 s). Indeed, over the last decade and a half, pulses as short as 43 attoseconds [3][4] have been reported, and many experiments have been conducted to explore attosecond electron dynamics with the primary motive of making “movies” of the electrons [5][6]. Even with these advancements, measuring true attosecond electron dynamics in an attosecond-pump-attosecond-probe format has remained out of reach of current experimental capabilities [7]. In order to circumvent the need for attosecond-duration pump and probe pulses, various methods utilizing recollision dynamics [8][9], high-harmonic spectroscopy (HHS)[10][11], or methods involving attosecond pulses combined with femtosecond infrared (IR) pulses have been developed [12]. Though the interpretation of these experiments can be cumbersome, particularly using HHS, researchers have been able to extract information on laser-induced electron dynamics, electron correlations, and electron-nuclei dynamics using such methods [13][14][15][16]. Until recent years, these types of experiments have been largely limited to atoms, but molecular attosecond dynamics have been predicted to be of great relevance to chemistry, photovoltaics, and biology [17][18][19][20].

1.1.1 High-Harmonic Generation

When attosecond physics is discussed, it is usually spoken about hand-in-hand with high-harmonic generation (HHG). This is because HHG is the primary, but not sole source

1 100 CO2

10-1

10-2

10-3

10-4

10-5

10-6

-7 Normalized Amplitude (arb. units) 10

10-8 20 30 40 50 60 70 80 90 Harmonic Energy (eV)

Figure 1: A typical CO2 spectrum recorded with a magnetic bottle electron spectrometer that was generated at 1300 nm, spectrally filtered through Al, and detected with Ne. As shown in the figure, this comb is only comprised of odd harmonics. Of note, the cutoff around ¤72 eV is due to the Al filter used in the experiment. The HHG cutoff extends past this point.

[21], of attosecond pulses. Brieﬂy, HHG is a phenomenon that occurs when an intense laser

(¤1014W/cm2) interacts with an inert gas and generates a plateau-like spectrum containing odd-harmonic orders, and sometimes even orders, of the fundamental driving laser wavelength. This spectrum typically covers the extreme ultraviolet (XUV) region and extends into the soft X-rays, possibly containing hundreds of harmonic orders. For example, in

Figure 1, a high-harmonic spectrum from CO2 demonstrates a harmonic plateau where the intensity of successive harmonics decreases more slowly than a power law that you would expect from a perturbative interaction.

Other sources do exist for creating XUV and X-ray pulses, such as X-ray Free Electron

Lasers (XFELs) and synchrotrons, but XFELs can only typically produce pulses as short as tens of femtoseconds [22] and synchrotrons are limited to the hundreds of femtoseconds

2 range [23]. Though neither of theses sources are currently able to produce attosecond X-ray pulses, XFELs show promise as future attosecond sources [24]. Other than the obvious advantage of having the shortest pulses, how does high-harmonic generation compare to these other sources? One often-claimed advantage of HHG is that it typically requires a much smaller footprint than XFELs and synchrotrons; laboratory-sized experiments compared to building-sized experiments. This is generally true, but even this advantage is becoming less clear with the advent of large user facilities like the Extreme Light Infrastructure (ELI) [25].

When it comes to the yield of XUV light, HHG pales in comparison to that of XFEL pulse energies and synchrotron average powers. A typical energy conversion from the infrared laser to XUV light is on the order of 10-6. Since usual infrared laser pulses are on the order of 1 mJ, this means that each attosecond pulse has ¤1 nJ, whereas XFELs can output ¤1 mJ per pulse [26]. This small yield of XUV increases the diﬃculty of HHG-based attosecond experiments and makes the production of pulses with even ¤1uJ very challenging [27].

Nevertheless, HHG is the best tool to date for attosecond experiments, so it is the one that was used to investigate attosecond dynamics in this dissertation.

1.1.2 Attosecond Metrology as a Spectroscopic Tool

In 2001, as mentioned above, the ﬁrst two experiments were conducted to measure the pulse durations of XUV light generated from HHG. Both of those experiments consisted of a high-harmonic source and also a nonlinear process (two-color photoionization in a detector gas) to make the pulse duration measurement. Paul et al. [1] measured the group delay

(GD) of an attosecond pulse train through XUV-IR delay-dependent sideband modulations at the energies of even harmonics in a photoelectron wave packet generated in a detector gas, depicted in Figure 2. The modulation at the energy of the qth harmonic is given by

¢ XUV Sq τXUV IR cos 2ω0τXUV IR ∆φq , (1)

where Sq is the modulation intensity of the wave packet, ω0 is the driving laser frequency,

XUV τXUV IR is the delay between the XUV and IR pulses, and ∆φq is the phase diﬀerence

3 Figure 2: Experimental layout for the ﬁrst measurement of attosecond pulse trains in 2001. In this experiment, 800 nm light was used in a collinear interferometer where the XUV and IR copropagated. The inset shows a cartoon depicting the measurement process. Adapted from [1]. between neighboring (q-1)th and (q+1)th harmonics. These phase diﬀerences can be recast into delays using the discrete derivative approximation

XUV ∆φXUV XUV dφ ¤ q τq , (2) dω 2ω0

XUV where τq is the group delay measured at harmonic q. Here 2ω0 is the spacing between the two neighboring odd harmonics. Together with a measurement of the spectral intensity, one can then reconstruct the XUV pulse duration.

Where Paul et al. measured an attosecond pulse train, Hentschel et al. [2] measured the pulse duration of an isolated attosecond pulse. The method used by Paul et al. is unable to retrieve the GD of isolated pulses, so another method called “attosecond streaking” was used instead. In that method, the duration of the isolated attosecond pulse is extracted from the delay-dependent modulation of photoelectrons ionized by the attosecond pulse in the presence of a strong, few-femtosecond infrared ﬁeld.

4 At the time of these measurements, the experiments were only considered from a metrological standpoint. Upon careful considerations, however, it can be shown that the measurements also contain information about the electronic structure of the generation and detection gases. Speciﬁcally, the HHG process imprints electronic structure of the genera-

XUV tion gas onto the GD of the XUV light τq , and additionally, the nonlinear measurement at acquires a delay oﬀset τq , known as the atomic delay at harmonic q, which describes the electronic structure of the detection gas.

This leads us to how pulse duration measurements can be used from a spectroscopic point

at of view. First, because τq tell us about the timing associated with the ionization process, we can study it to probe attosecond dynamics in the context of a direct photoionization experiment. Secondly, because electronic information from the generation gas is imprinted

XUV onto τq , we can also study electron dynamics in a sort of self-probing scheme of the XUV generation gas. To summarize, we have the following types of spectroscopic experiments from the measurement of an attosecond pulse duration:

1. a direct photoionization experiment of the detection gas, typically called “attosecond

spectroscopy” (AS),

2. and a self-probing experiment of the HHG gas called “high-harmonic spectroscopy”

(HHS).

In Figure 3(a), we can see an AS (direct photoionization) experiment that assumes a well- characterized XUV attosecond pulse which is used to characterize dynamics in the detection gas. In contrast, Figure 3(b) shows an HHS (self-probing) experiment that assumes a well- known detection gas which is used to characterize the XUV pulse, shedding light on the electron dynamics involved in the HHG process. Of course, in a real experiment, neither the source nor the detector is perfectly well-known.

5 Figure 3: Cartoon depiction of the two attosecond probing schemes. In (a), we see that the AS scheme uses a well-characterized attosecond pulse from an Ar generator to interrogate a N2 detection gas. In (b), the HHS scheme uses a well-known detection gas, in this case Ar, to interrogate the XUV generation gas, N2. In truth, neither the generator nor the detector is well-characterized in group delay. To this end, both generator and detector are studied in this dissertation. Adapted from [28].

1.2 Attosecond Spectroscopy: Two-Photon Direct Ionization as a Scheme for Attosecond Science

As mentioned previously, the AS scheme focuses on studying atomic ionization delays through direct photoionization. Measuring and studying these delays gives access to precise timing information about electron correlations involved in ionization. For example, this access has been exploited to study ionization delays in the presence of autoionizing resonances

[29], ionization delays aﬀected by electron-nuclei correlations [30], ionization delays between diﬀerent vibrational states [11], and ionization delays due to electron-electron correlations

[31]. In this dissertation, I will describe two types of experiments in the AS scheme that study ionization delays in the presence of autoionizing resonances in Ar and He. In the ﬁrst experiment we use a high-harmonic comb comprised of only odd harmonics to precisely measure ionization delays associated with resonances in Ar and He: the Ar 3s3p64p and

3s23p44s4p resonances and the He 2s2p resonance. A GD measurement of the resonant Ar

6 Figure 4: Example results from an experiment where harmonics were generated in Ar and then were used to photoionize a separate Ar detector gas. Centered around ¤26.7 eV and surrounded by a red circle, one can see the imprint of the 3s3p64p resonance on the group delay. Additionally, circled in green at ¤34 eV, the 3s23p44s4p imprints itself on the group delay. ionization delays is pictured in Figure 4. Our measurements of the Ar 3s3p64p resonance and the He 2s2p resonance are in excellent agreement with theoretical predictions and previous measurements, and our measurement of the 3s23p44s4p resonance is the ﬁrst reported measurement in the relevant energetic region of Ar. These results additionally serve as a benchmark for cutting-edge ab initio theories of two-photon ionization. In the second experiment, we use harmonic combs comprised of even and odd harmonics to study the Ar

3s3p64p resonance and the He 2s2p resonance. We do this for two reasons: to study the continuum-continuum delays around the resonance relative to the odd-harmonic only measurement, and, on a more fundamental level, to study spectral phase measurements with combs of even and odd harmonics, which to date, are largely unexplored. These experiments will serve as benchmarks for extending the AS scheme to more complicated atomic and molecular systems.

7 1.3 High-Harmonic Spectroscopy: A Self-Probing Scheme for Attosecond Science

The HHS scheme focuses on studying the XUV intensity and group delay in order to

ﬁnd deviations from an otherwise featureless harmonic generator. This idea is based on the principle that the high-harmonic generation dipole can be written as

Dω W ESdωS2, (3) where D is the HHG dipole, W is the returning electronic wave packet, E is the returning electron’s kinetic energy, d is the photorecombination transition dipole, and ω is the XUV photon energy. It has been theoretically shown by Le et al. [32] that the photorecombination dipole dω contains gas-dependent attosecond electron dynamics, and that the returning wave packet W E is largely independent of the generation gas. The goal is then to extract information about the photorecombination dipole in order to study attosecond dynamics. Because the photorecombination dipole can be thought of as the inverse of the photoionization dipole, this scheme has been used to study ionization delays in the presence of Cooper Minima [10][33], shape resonances [34], sub-femtosecond nuclear motion [35][36], and attosecond charge migration [37].

In this dissertation, I will describe two experiments performed using HHS. In the ﬁrst experiment, I will describe my eﬀorts to extend standard atomic HHS techniques to small molecules, particularly methane and methyl chloride. With this study I will show that is possible to fully characterize, through spectral intensity and group delay, a molecular

Cooper minimum in methyl chloride. The main results of the methyl chloride GD measurement, to be explained later, are shown in Figure 5. In the second experiment, we explore the eﬀect of nuclear substitution on two-center interferences in carbon dioxide (CO2), carbonyl sulﬁde (OCS), and nitrous oxide (N2O). These experiments demonstrate that HHS of two-center interferences is sensitive to the interplay between geometric (nuclear positions)

and electronic features. Additionally, these measurements show that the sign of the phase

8 Figure 5: Comparison of methane (CH4) and methyl chloride (CH3Cl) GD after removal of attochirp and other contaminating delays. The dip in group delay in CH3Cl centered around 45 eV corresponds to a Cooper minimum.

jump associated with two-center interference in CO2 is opposite that of previously reported results. These experiments serve as foundational studies for future experiments studying

attosecond charge migration through HHS.

1.4 Outlook

There has been signiﬁcant progress in developing attosecond techniques, but there is still

much work to be done to achieve true attosecond-pump-attosecond-probe experiments. I

believe these types of experiments will become attainable in the not-too-distant future, and

as the ﬁeld progresses, I believe that we will continue to see shorter and brighter attosecond

pulses, the continuing spread of attosecond science to other ﬁelds, and the discovery of

new processes occurring on the shortest time-scales. Because the experiments described in

this dissertation have thoroughly established new attosecond techniques in atoms and small

molecules, I believe that we have paved the way for studying attosecond dynamics in larger

9 molecules. These experiments will serve as a strong foundation for extending attosecond techniques into larger molecules important for chemistry and biology.

10 Chapter 2: Theory of High-Harmonic Generation

“I seem to be looking through the surface of atomic phenomena into a strangely beautiful interior world.” —Michael Frayn, Copenhagen, Act Two

2.1 A Strong-ﬁeld Mechanism for Harmonic Generation

Figure 6 depicts a cartoon of a typical harmonic spectrum from a strong laser (>1013 W ~cm2) interacting with a gas sample, where each “spike” represents a harmonic order such that each harmonic is separated by two times the fundamental driving frequency. At the lowest energies, the harmonic yield decreases with increasing harmonic order according to a power law until a certain point when the spectrum begins to plateau. The plateau extends for many harmonic orders until it sharply drops in the cutoﬀ region. When the spectrum changes from power-law to plateau, the mechanism for harmonic generation changes from a perturbative regime to a nonperturbative regime.

The theory of low-order harmonic generation is well-understood [38], and we will not focus on its mechanism. Instead we will focus on the nonperturbative mechanism that gives rise to the harmonic plateau that is followed by a sharp cutoff. Our discussion will start with describing the single-atom recollision model of high-harmonic generation: first semiclassical then quantum mechanical. Following the single-atom picture we will discuss the macroscopic effects of high-harmonic generation and then the theory underpinning high- harmonic spectroscopy.

2.2 Microscopic Theory of High-Harmonic Generation: Recollision

The ﬁrst high-harmonic spectra were recorded by McPherson et al. [39] and Li et al.

[40] in 1987 and 1989, respectively. It was not until 1993 that Schafer et al. [41] explained the mechanism of high-harmonic generation in the context of electron recollision. Shortly after, Corkum’s paper [42] broke down the recollision process into the now famous “three-

11 Figure 6: A cartoon depiction of a typical harmonic spectrum generated in a gas sample. The spectrum is a comb of odd harmonic “spikes” separated by two times the fundamental laser frequency 2ω0. At the lowest energies the harmonic intensities drop off rapidly, followed by a relatively flat plateau region and then cutoff. The existence of the plateau region indicates a different mechanism for HHG: recollision. step model” (TSM) of ionization, propagation, and recombination, as seen below in Figure

7. The beauty of these explanations was that they were the first to explain the position of the harmonic cutoff and other important strong-field phenomena in the context of a singular process, recollision. In the following year, a quantum-mechanical explanation of high-harmonic generation (HHG) was given by Lewenstein [43]. Until the paper by Bellini et al. [44] in 1998 though, the temporal coherence of this light was not yet understood.

They showed that there were two dominant, spatially-separated coherence times. And of course, the later measurements followed in 2001 to measure the true pulse durations created by HHG.

In the following subsections, we will describe a semiclassical model that will roughly explain the timing and energetic cutoﬀ associated with HHG. We will then describe and break down Lewenstein’s model into the three steps, ending with a description of photorecombination.

12 Figure 7: The three step model. The Coulomb potential is in green, the laser potential is the dashed black line, and the combined potential is shown in red. In the first step (a) a single electron is freed from the parent ion through tunneling ionization. In the second step (b) the electron is propagated in the laser field, where it gains energy from the field. Finally in the third step (c) the electron recombines with the ground state of the parent ion, releasing an XUV photon. Adapted from [28].

2.2.1 Semiclassical Model: A Model for a Simple Person

In the semiclassical picture of high-harmonic generation, we will ignore the purely quantum-mechanical features, particularly those associated with ionization and recombination, which we will return to in the following sections. We instead first focus on the second step, propagation, which alone is able to explain the dominant timing of HHG and the HHG cutoff. We begin by following Zenghu Chang’s textbook, Fundamentals of At- tosecond Optics [45], writing down the equation of motion of an electron released into a laser field from its parent ion in one dimension (SI units)

d2x e E cosω t, (4) dt2 m L 0 where x is the electron’s displacement from its parent ion, t is time, e is the electron charge, m is the electron mass, EL is the electric field of the laser, and ω0 is the central radial frequency of the laser field. We are ignoring the effect of the Coulomb potential, because the electron’s average excursion, which will be shown momentarily, is much larger than the radius of a typical rare gas atom used in HHG. This equation can then be integrated once to get the electron’s velocity at time t

13 Figure 8: Electron trajectories for birth times between t 0 and t 0.35T0, where T0 is the laser period. The trajectories that return tox 0 (0 B t B 0.25T0) are shown in blue, and the trajectories that do not return (0.25T0 @ t B 0.35T0) are shown in red.

eEL vt sinω0t sinω0t ¥, (5) mω0

where t is the time that the electron entered the electric ﬁeld or “birth time.” After another

integration of vt we get the electron’s displacement

eEL ¥ ¥ x t 2 cos ω0t cos ω0t ω0 sin ω0t t t . (6) mω0

eEL If we normalize x t by the prefactor x0 2 , also known as the quiver amplitude, and mω0 plot xt~x0 for a few birth times shown in Figure 8, we can see that depending on when the electron enters the electric field it may or may not recollide with the parent ion at a later time. For a typical HHG laser intensity, the quiver amplitude is on the order of a few nanometers to 10s of nanometers, much longer than the radius of a typical atom which is on the order of an angstrom. This justifies our previous assumption that the propagation is largely unaffected by the Coulomb potential of the parent ion.

Let us now consider the electron’s recollision with its parent ion. If the electron recol- lides, it can do one of three things [42][46]:

1. The electron can scatter inelastically oﬀ the ion, potentially ejecting a second electron

14 or more.

2. The electron can scatter elastically oﬀ the ion, creating higher energy electrons, not

possible from directly ionized electrons.

3. The electron can recombine with the parent ion to emit a single photon, responsible

for the HHG process.

Though the ﬁrst two processes described are very useful features to study in strong-ﬁeld physics, they do not contribute to the HHG process. Therefore, we will focus our attention on the third process, recombination.

The electron returns with kinetic energy

1 KEt mv2t 2U sinω t sinω t¥2 (7) 2 p 0 0

where Up, the ponderomotive energy, is the time-averaged energy of an electron in an oscillating electric ﬁeld

2 eEL Up 2 . (8) 4mω0 It now should be easy to take the logical leap that after the electron recombines with the ground state, a photon will be emitted with energy that is equal to the kinetic energy of the returning electron plus the ionization potential

Ò 2 hωt, t IP 2Up sinω0t sinω0t ¥ , (9) where ω is the XUV frequency. With this, we have a simple explanation of the origin of the high-harmonic plateau.

2.2.2 HHG Cutoﬀ and Attochirp

Equation 9 explains the origin of the high-harmonic spectrum, but we still have not explained the benchmark features of the spectrum: the harmonic cutoﬀ and the timing

15 (“chirp”) of the attosecond pulses. In order to answer both questions, let us calculate the recombination times within a cycle and the electron kinetic energies associated with those recombination times. The electron recombines when it returns to x 0, so the recombination time can be found by numerically solving

xt 0 cosω0t cosω0t ¥ ω0 sinω0t t t (10) for t given t and plugging t and t back into Equation 7. The results of this calculation can be seen in Figure 9(a). In it we can see that the highest electron return energy is approximately 3.17Up such that the harmonic cutoﬀ is

Ò hωcutoff IP KEcutoff IP 3.17Up. (11)

Ò Additionally, it is clear from Figure 9(a) that energies less than hωcutoff are degenerate; i.e. there are two separate sets of birth and recombination times that have the same return kinetic energy. Electrons released into the ﬁeld at times t @ 0.05T0 return at a later time than electrons born at times t A 0.05T0, where T0 is the period of the driving laser. The former are aptly referred to as the “long trajectories” and the latter are referred to as the

“short trajectories.” Other trajectories with longer excursions leading to recombination are

possible, but due to electron wave packet spreading these trajectories suﬀer from a reduced

recombination cross section [47][43][48] so that they do not signiﬁcantly contribute to HHG

and will be ignored henceforth. Also, from Figure 9(a), we can deduce that these long and

short trajectories will recombine with the parent ion every half cycle.

It is now obvious from the calculation in Figure 9(a) that electrons with diﬀerent return

energies recombine at diﬀerent times. This feature is the origin of the intrinsic group delay, dφ , in the emitted photons. In Figure 9(b), this is shown more clearly where the return time dω is plotted versus electron return kinetic energy. In the energetic region away from the cutoﬀ, the slopes of the group delays (GDs) for the short and long trajectories, also called the group d2φ delay dispersions (GDDs) , have opposite signs such that short trajectories have normal dω2

16 (a) (b)

Figure 9: Numerical calculations of the semiclassical three step model. In (a) we see that there are two sets of ionization or “birth” times corresponding to two diﬀerent sets of recombination times. These are called the “long” and “short” trajectories with corresponding long excursion times and short excursion times, respectively. In (b) the recombination times, in laser cycles (L.C.), for the long and short trajectories are plotted versus the electron return energy. dispersion (“reds before blues”) and the long trajectories have anomalous dispersion (“blues before reds”). These intrinsic GDDs are referred to as the “attochirp” of the trajectories.

In a real experiment, if we had both trajectories signiﬁcantly contributing to HHG, then we would have both normal and anomalous dispersion contributing to the GDD of the spectrum. This would make the spectrum appear as coming from an eﬀectively random, incoherent source: i.e. no attosecond pulses. Luckily, a macroscopic tool called “phase- matching” allows us to select only the short trajectories for use in experiments. This will be discussed later, after the quantum-mechanical description of HHG.

2.2.3 Scaling of HHG

The above numerical calculations were done with an overall normalization to the ponderomotive energy, which has masked some useful scaling features that we will now explore.

As written above in Equation 8, the ponderomotive energy scales with the laser parameters

EL and ω0 as S S2 EL 2 Up 2 ILλ , (12) ω0

17 where IL is the laser pulse intensity and λ is the central wavelength of the driving laser. From Equations 11 and 12, we can see that the cutoﬀ energy of the harmonics will scale linearly with the laser intensity and quadratically with the wavelength. This is very useful because the ionization potential (IP) of each generation gas provides a natural limit to IL by way of ground state depletion through ionization. This will also be explored further in

the quantum-mechanical description. Additionally, increasing the laser wavelength leads to

favorable scaling of the attochirp [49][47]. This can be seen by calculating Equation 9 for

varying laser intensities and laser wavelengths. Both the cutoﬀ scaling and attochirp scaling

versus IL and λ can be seen in Figure 10. From these calculations we can see that increasing

IL and/or increasing λ decreases the attochirp of the emitted attosecond pulses, producing closer to Fourier-limited pulses. For our experiments, extension of the cutoﬀ is critical

because we are exploring relatively low IP molecules ( ¤12 eV) compared to typical HHG

gases. The decreased attochirp is not critical for us because our goal is to simply measure

deviations from a typical attochirp, but for experiments aimed at generating ultra-short

attosecond pulses, this scaling is critical [3][4]. It has also been shown by two studies that

longer driving wavelengths suﬀer decreased HHG eﬃciency that scales as ¤ λ56[47] and

as ¤ λ6.4[50]. Neither of these is very favorable for HHG in the mid-infrared (MIR), but

the reduced yield can be somewhat compensated with increased gas density in generation.

18 (a) (b)

Figure 10: Scaling examples of HHG. In (a) we can see that the harmonic cutoff position scales linearly with the laser intensity at a fixed wavelength, and in (b) we can see the that the cutoff scales quadratically with the laser wavelength at a fixed intensity. Because of the quadratic scaling, the laser wavelength can be increased as a more effective way to extend the HHG cutoff at a given intensity.

2.2.4 Attosecond Pulse Trains and High Harmonics

Though the process is called high-harmonic generation, a keen reader would notice that the semiclassical calculations up until this point have produced continuous spectra rather than spectra comprised of only odd harmonics. This is because these calculations have been for birth times within only one half cycle of the driving laser and therefore for only one set of recollision times. If we consider a multi-cycle laser pulse, then we could imagine that this process happens twice per cycle for many laser cycles. As depicted in Figure 11, this leads to bursts of attosecond pulses, or an attosecond pulse train (APT), where each pulse in the train is separated by T0~2 in time. By the nature of the Fourier Transform, we can then see that this leads to a comb of odd harmonics spaced by 2ω0, two times the fundamental driving frequency. Comb spacing becomes more favorable at MIR wavelengths, because as

ω0 decreases so does the separation between odd harmonics, eﬀectively providing a ﬁner energetic sampling for both the HHS and the AS methods.

19 (a) (b)

Figure 11: Pulse train attributes in HHG. The laser ﬁeld attributes are shown in red whereas the XUV ﬁeld attributes are shown in purple. In (a) we see that every intense half cycle leads to bursts of XUV light. Then in (b), under Fourier transform, we see that the harmonic spectrum for the same pulse train is comprised of discrete odd harmonics separated by 2ω0. Adapted from [51].

2.2.5 Quantum Mechanical Model: The Lewenstein Model

Though the semiclassical model describes the most prominent features of HHG, it falls short of a full quantum-mechanical description for multiple reasons. The two most obvious reasons are that it cannot describe the ionization process or the recombination process.

Ionization is important to understand because it largely governs the yield of HHG, and photorecombination is important because it is the nonlinearity responsible for generating high harmonics. Additionally, the semiclassical model does not properly describe the GD in the presence of the harmonic cutoﬀ. Because we probe the recombination step to understand electron dynamics in our HHS experiments, it is imperative to detail the quantum- mechanical model, or Lewenstein Model [43], that can describe the aspects beyond the reach of the semiclassical model. To describe the atomic and molecular quantum-mechanical responses that are sources of HHG, we will follow Smirnova et al. [52].

Recall from nonlinear optics that the source of electromagnetic radiation from an atom or molecule, the polarization P r, t nDt, is proportional to the number density, n, and the induced dipole, Dt. The induced dipole of an atom or molecule in a laser ﬁeld is

20 given as

Dt `ΨtS dˆ SΨte , (13) and it is the origin of the HHG process, where Ψt is the wavefunction of the atomic or

molecular system and dˆ is the dipole operator. In atomic units (hÒ m e 11, the time

evolution of Dt is governed by the time-dependent Schrödinger Equation (TDSE)

∂ SΨte i Hˆ t SΨte , (14) ∂t where Hˆ t is the Hamiltonian. Solving this equation directly for the true wavefunction is computationally challenging, so we will continue our derivation in the single active electron

(SAE) approximation where we assume that only a single ionized electron is affected by the laser field and all other electrons are frozen in the field. The Hamiltonian in the SAE can be written as 2 kˆ Hˆ t V rˆ Vˆ t, (15) 2 C L ˆ where k i©r is the kinetic momentum operator, VC rˆ contains the Coulomb potential ˆ due to the ionic core, and VLt d ELt rˆ ELt is the potential due to the laser field in the dipole approximation and length gauge. The solution to Equation 14 can be written as2 t SΨte i S dt Uˆt, t VˆLt Uˆ0t , t0 Sge Uˆ0t, t0 Sge , (16)

t R Hˆ τdτ where Uˆt, t e t is the propagator for the full Hamiltonian Hˆ t, Uˆ0t , t0 2 t ˆ R Hˆ0τdτ k e t0 is the propagator for the ﬁeld-free Hamiltonian Hˆ V rˆ, and Sge repre- 0 2 C 3 sents the ground state wavefunction at the initial time t t0. As a reminder, the propagator time-evolves a state from some initial time to some ﬁnal time. For example, when Uˆ0t, t0 iIP t t0 acts on Sge, we get SΨgt e ¡ Uˆ0t , t0 Sge e Sge, where IP is again the ionization

1For the remainder of the quantum-mechanical description of HHG, we will remain in atomic units. 2For a derivation of Equation 16 please refer to [53]. If this derivation is not clear, the wavefunction can be tested as a solution by inserting it into the left-hand side of Equation 14 and using the Leibniz Integral Rule to get the right-hand side of Equation 14. 3Note that the integrals of the Hamiltonians in the propagator equations are time-ordered.

21 potential. The beneﬁt of using Equation 16 rather than 14 is that approximations like the following are more transparent.

The second approximation that we will apply is to assume that the electron’s excursion is large compared to the Coulomb potential, just as in the semiclassical model. This is called the strong ﬁeld approximation (SFA). The SFA approximation leads to a full Hamiltonian of the form 2 SFA kˆ Hˆ t ÐÐ Hˆ t Vˆ t, (17) V 2 L

t t R Hˆ τdτ SFA R Hˆ τdτ called the Volkov Hamiltonian, so that Uˆt, t e t ÐÐ UˆV t, t e t V in Equation 16. The eigenstates of this Hamiltonian are known as Volkov states which are plane waves with kinetic momentum kt p At, where p is the canonical momentum and At is the vector potential of the laser ﬁeld at time t. In the position basis they take the form 1 arTp Atf ei pAt ¥ r. (18) 2π3~2

The Volkov states form a complete basis set at each time t such that 1ˆ R dp Sp Ate `p AtS, a fact which will be used later in the derivation. If UˆV t, t acts on a Volkov state Sp At e then we get

iSV p,t,t UˆV t, t Tp At f e Sp Ate , (19) where t 2 SV p, t, t S p Aτ¥ dτ. (20) t So, in plain words, the Volkov propagator takes a plane wave with kinetic momentum p At and turns it into a state with kinetic momentum p At and an accumulated phase SV p, t, t . Within the SFA, Equation 16 becomes

t SΨSFAte i S dt UˆV t, t VˆLt Uˆ0t , t0 Sge Uˆ0t, t0 Sge . (21)

Already we can see the beginnings of the three step model. The integrand of the ﬁrst term

22 represents the evolution of the ground state electron before the arrival of the laser ﬁeld, followed by ionization at time t, and then the evolution of the ionized electron in the laser

ﬁeld until some later time t. In the SFA, the dipole of Equation 13 becomes

t ˆ DSFAt i aUˆ0t, t0gT d S dt UˆV t, t VˆLt Uˆ0t , t0 Sge c.c., (22)

where aUˆ0t, t0gT represents the ground state time-evolved under the ﬁeld-free Hamiltonian. In writing Equation 22 we have assumed that the ground state is negligibly depleted, that

the continuum-continuum transitions are negligible, and that there is no permanent dipole

in the ground state. If we now resolve the Volkov Propagator on the Volkov basis set and ˆ use the deﬁnition of VLt d ELt, Equation 22 becomes

t iSp,t,t DSFAt i S dt S dp d p Ate ELt dp At c.c., (23)

where dp At ¡ `p AtS dˆ Sge and

t 2 p Aτ¥ Sp, t, t ¡ S IP dτ. (24) 2 t

Equation 24 is typically referred to as the “action” of the electron in the laser ﬁeld, which is actually the energetic term of the classical action.

Now we can fully see the three-step picture through DSFAt:

1. strong-ﬁeld ionization: ELt dp At ,

2. propagation: eiSp,t,t ,

3. and photorecombination: dp At.

If we Fourier Transform DSFAt, we can get the spectral response

iNω0t DSFANω0 S dte DSFAt. (25)

23 Here we have inserted N, the harmonic order, to reﬂect the multi-cycle nature of HHG. The

2 far-ﬁeld HHG spectrum SINω0S due to the dipole radiation is then

2 4 2 SINω0S Nω0 SDSFANω0S . (26)

In Equations 23, 25, and 26, we have a manageable framework that we can use to eﬀectively

describe the HHG process. In the following sections, we will brieﬂy describe each step of

the TSM in the SFA picture, and we consider each step as it applies to our experiments.

2.2.6 Strong Field Ionization

The ﬁrst step is of course strong-ﬁeld ionization, wherein a multiphoton process from

a strong laser ﬁeld removes an electron from a parent ion. In the HHG process, ionization

largely determines the ﬁnal HHG yield through the ionization rate. The cycle-averaged

ionization rate from a non-resonant strong ﬁeld is given by the so-called “PPT ionization

rate” [54]4

2 w E , ω c2 fl, mIP 2nSmS11 γ2SmS~23~4 PPT L 0 nl 3 ELn ~ (27) 22IP 3 2 Amω0, γexp ¤ gγ© , 3EL where n* is the eﬀective principle quantum number, l and m are the orbital and magnetic quantum numbers, respectively, and γ is referred to as the “Keldysh Parameter”, named for

L.V. Keldysh who ﬁrst explored strong-ﬁeld ionization [55]. For a more detailed description of the functions and variables in Equation 27, refer to the appendix of [56]. γ is given by ¿ Á IP γ ÀÁ . (28) 2Up

4The abbreviation “PPT” stands for names of the those who ﬁrst detailed the PPT ionization rate equation: A. M. Perelomov, V. S. Popov, and M. V. Terent’ev.

24 If we consider wPPT in the situation where γ Q 1, then we will retrieve the “multiphoton regime” ionization rate 2K0 EL wMPI , (29) ω0

5 where K0 IP ~ω is the requisite number of photons required to ionize the atom. If we instead consider the situation γ P 1, we will retrieve the “ADK”6 or “tunneling regime”

ionization rate for an alternating electric ﬁeld [57]

2nSmS1 2 3~2~ 3~2~ w E c2 fl, mIP e 2 2IP 3EL e 2 2IP 3EL . (30) ADK L nl 3 ELn

We can see from Equation 29 that the multi-photon ionization rate scales like a power law that is dependent on the laser intensity and frequency, as one might expect. In the tunneling regime, however, Equation 30 tells us that the ionization rate scales exponentially with the laser intensity but has no dependence on the laser frequency.

One of the limits of the tunneling regime occurs when the driving wavelength is kept

ﬁxed, but the intensity is continually increased. At some point, the Stark potential, which is comprised of both the Coulomb and laser potentials, will dip below the energy of the initial state. In that case, the electron can leak out over the potential barrier in a process called over-the-barrier ionization (OTBI) or barrier-suppression ionization (BSI) [58]. The laser ﬁeld strength for which this process occurs can be given by (in atomic units)

IP 2 F , (31) OTBI 4Z where Z is the charge of the atomic core. In the OTBI regime, the ionization probability quickly approaches unity so that the ground state is rapidly depleted. Because of this,

FOTBI can serve as a rough upper bound for allowable laser ﬁeld strengths in HHG from atoms and molecules. It is important to note two limitations of this estimate though.

5“ ” denotes a floor function. 6In an analogous naming method to “PPT”, the abbreviation “ADK” stands for the last names of the authors who first described tunneling ionization in an alternating electric field: M. V. Ammosov, N. B. Delone, and V. P. Krainov.

25 First, it is known that Equation 31 typically underestimates the OTBI intensity because its derivation neglects the AC Stark shift [58]. Secondly, for the case of molecules, strong-ﬁeld ionization is considerably more complex, so the OTBI intensity should only be used as a rule of thumb.

Figure 12 shows a detailed experimental comparison of these two models recently carried out by Lai et al. [56]. From it we can see that below γ ¤ 0.7, the ADK and PPT ionization rates are able to accurately predict the total yield from experimental ionization studies.

On the other hand, above γ ¤ 0.7, ADK strongly deviates from the measured ionization yield. For some of the experiments in this dissertation, we will be generating harmonics in a gas with IP ¤ 12 eV and a laser wavelength centered around 1.3 µm with a focal spot intensity of ¤ 1.0 0.2 1014 W ~cm2. This gives us a Keldysh parameter of γ ¤ 0.6. For such an IP, the OTBI intensity would be closer to ¤ 8.3 1013 W ~cm2, so our experiments are somewhat into the tunneling regime and at or above the OTBI intensity. It is still reasonable to assume, though, that our HHS experiments reside in or near the tunneling regime.

Another important feature of ionization in the tunneling regime is that the laser ﬁeld

ﬁxes the quantization axis for the ionization dipole matrix element to be along the laser polarization. Because of this, the m 0 channel dominates the m 1 channels in Equation

30. This results in the electron being ejected primarily along the laser polarization, which

is therefore the polarization of the emitted high-harmonic light. Additionally, this result

simpliﬁes the interpretation the photorecombination step, which will be shown in Section

2.2.8.

2.2.7 Propagation

Analogous to the semiclassical model, the propagation of the electron in the laser ﬁeld

largely determines the GD and the harmonic cutoﬀ of the emitted XUV. Here we will again

follow Smirnova et al. [52] in order to derive the cutoﬀ and return times that contribute to

HHG in the SFA. By the end of the derivation, the deviations from the semiclassical model

26 Figure 12: Ratios of experimental single ionization yields to theoretical ionization yields for linear polarization at various values of γ from Lai et al. [56]. In these experiments, different values of γ were achieved using different combinations of driving wavelengths, interaction intensities, and atomic targets. Above γ ¤ 0.7, the ADK model significantly deviates from the experimental results. and their origins should become clear.

Equations 23 and 25, rewritten below as Equations 32 and 33 for convenience, are responsible for the high-harmonic spectrum INω0.

t iSp,t,t DSFAt i S dt S dpd p Ate ELt dp At c.c. (32)

iNω0t DSFANω0 S dte DSFAt (33)

iSp,t,t In order to calculate INω0, we have to deal with the term e that describes

1 t 2 the electron’s propagation in the field, where Sp, t, t ¡ R p Aτ¥ IP dτ is the 2 t action. The action is formed by summing all electron trajectories with canonical momentum p initialized at the ionization time t ti and propagated in the laser field until a return time t tr. So in order to calculate INω0, we need to figure out how S contributes to the three integrals over over p, t, and t, and therefore which trajectories most contribute to

the three integrals. This problem is typically handled using the saddle point approximation

27 to weed out the most important trajectories. The saddle point approximation says that the greatest contributions of S to the three integrals will be for the p, ti, and tr where S varies slowly

©p,t,tS 0. (34)

Using the saddle point approximation on Equation 24, we ﬁnd the following:

p At¥2 s i IP 0, (35) 2

tr ¥ S ps A t dt 0, (36) ti

p At¥2 s r IP Nω , (37) 2 0

where ti ionization time, tr recombination time, and ps are the solutions to the saddle point approximation. It is important to note that ti , tr , and ps are all inherently complex numbers. Equation 35 can be interpreted as a statement of conservation of energy during ionization, and Equation 37 can be interpreted as a statement of conservation of energy during photorecombination, wherein a photon is emitted upon recombination. One can also note that the semiclassical result would be obtained if IP was set to zero in Equation 35.

Equation 36 says that the electron must return to the parent ion, therefore requiring that the electron’s displacement is zero in the HHG process.

With Equations 35, 36, and 37, we can now calculate the energetic properties of the HHG ¡ ¥ 7 dipole. Plotted in Figure 13 are the GDs, tr Re tr , imparted to the attosecond pulses from either trajectory versus photon energy using the semiclassical model and the Lewen- stein model for the case of linear polarization. The Lewenstein model produces a modified result compared to the semiclassical model. Where the cutoff in the semiclassical model scales like IP+3.17Up, the cutoff from the Lewenstein model scales like 1.32IP+3.17Up.

7 Im ti ¥ and Im tr ¥ have been interpreted as the time it takes for the electron to tunnel ionize and the time it takes to recombine, respectively. The exact interpretation of these quantities is hotly debated and beyond the scope of this dissertation. For further reading, please refer to [52], [59], and [60].

28 Figure 13: Comparing semiclassical and SFA GD calculations. The SFA calculation shown by the dotted lines produces a similar attochirp to the semiclassical. In the cutoff, however, the SFA predicts notably different behavior, namely a different dependence on IP.

Additionally, we can clearly see that the GDD behavior in the Lewenstein cutoff strongly deviates from the semiclassical picture. This cutoff deviation is important for our HHS experiments, because the molecules we are interrogating have low IPs. As explained above in Section 2.2.6, low IPs limit the maximum intensity allowed for HHG, which brings the cutoff to lower energies and consequently closer to experimentally relevant features. Lastly, and not shown here, the scaling of the HHG GD with the laser intensity and wavelength are the same in the Lewenstein model as in the semiclassical model.

2.2.8 Photoionization and Photorecombination

The last step of the TSM is photorecombination. In this step, the electron wave packet recombines with the parent ion, ﬁnally releasing an attosecond pulse. Photorecombination can be thought of as the inverse process of single-photon ionization. In order to relate our HHS self-probing experiments to AS-based direct photoionization experiments, it is important to clarify this relationship. We will follow the derivation of the important aspects of photorecombination given by Schoun [28], Starace [61], and Le et al. [62].

29 In photoionization experiments done with an angle-averaging detector, the measured photoelectron yield is proportional to the photoionization cross section given by8

2 d σion σion SS dΩkdΩr, (38) dΩrdΩk where Ωr θr, φr are the angular variables relating to the parent ion, and Ωk θk, φk are the angular variables for the scattering electron. The integrand, also known as the diﬀerential cross section (DCS), is given by (in SI units)

2 2 d σion 4π ωk 2 SdionS , (39) dΩrdΩk c where dion is the ionization dipole matrix element, ω is the XUV photon’s angular frequency, and k is the magnitude of the wavevector for the scattering electron. Similarly, the DCS for photorecombination is given by

2 2 3 d σrec 4π ω 2 SdrecS . (40) dΩrdΩk ck

These two diﬀerential cross sections are related to each other by the law of detailed balance for direct and time-reversed processes by [63]

1 d2σ d2σ 1 ion rec . (41) 2 2 k dΩrdΩk dΩrdΩk ω

The photoionization and photorecombination dipoles can be given in the length gauge by:

` S S e dion Ψk r n Ψg , (42) and ` S S e drec Ψg r n Ψk , (43)

8In the following equations “d” is used for derivative and dipole. When used for a dipole, it is accompanied by a subscript.

30 where

`rSΨge Ψgr RnlrYlmΩr (44) is the ground state wavefunction, and

1 ` S e º l i σL ω δL ω r Ψk Ψk r Q i e REL r YLM Ωr YLM Ωk (45) k LM

are the continuum wavefunctions. In Equations 42-45, n is the XUV polarization direction;

YlmΩ Ylmθ, φ are the spherical harmonics; n, l, and m are the principle, orbital, and magnetic quantum numbers of the ground state bound electron; E, L, and M are the

analogous quantum numbers for the continuum states; and “+” (“-”) indicates outgoing

9 (incoming) electrons. σL is the Coulomb phase shift given by σL arg ΓL 1 iγ¥,

th where γ Z~k with the asymptotic nuclear charge Z=1, and δL is the L partial wave

phase shift due to the short range potential. For hydrogen, δL 0, and for all other atomic

cases, δL is function of ω that is not necessarily equal to zero. In HHS though, the HHG process is our detector and it is not angle-integrated. From

Section 2.2.6, we know that the laser-ﬁeld deﬁnes the quantization axis, preferentially ion-

izing electrons with k þ n, along the laser polarization, and similarly in the recombination

process the electron is restricted to return with k þ n. Because of this, our considerations

are reduced to the z-component of the recombination dipole dz,rec where Ωk θk, φk 0. It can be shown (see Appendix D of [28]) that

¢ ¨ ¨dz,ion, if l is odd d ¦ , (46) z,rec ¨ ¨ ¤¨ dz,ion, if l is even where the phase of dz,rec is given by

¢ ¨ ¨arg dz,ion¥ , if l is odd arg d ¥ ¦ , (47) z,rec ¨ ¨ ¥ ¤¨π arg dz,ion , if l is even

9This is not a typo.

31 and the group delay of dz,rec is given by

∂arg d ¥ ∂arg d ¥ z,rec z,ion , for all l. (48) ∂ω ∂ω

Equation 48 is often referred to as the Wigner delay [64], which is the target-specific delay associated with photoionization. In our HHS experiments we use the HHG process to specifically probe Sdz,recS and ∂arg dz,rec¥ ~∂ω, which are related to the ionization counter- parts through Equations 46-48. Because the first two steps also affect the amplitude and phase of HHG spectra, it is important to remove their contributions in order to study the photorecombination step. The theory for doing so will be described in the following section.

Now that we have thoroughly covered the TSM, we can expand on why the third step is the step that we want to probe. The argument goes as follows. In a pump-probe way of thinking, tunneling ionization of the HHG process can be thought of as a “pump” that ionizes an electron, leaving a corresponding “hole” in the parent ion. While the ionized electron is propagating in the laser ﬁeld, the remaining hole may undergo oscillations between localized states or even electron-correlation-driven charge migration as seen in Figure 14 on femtosecond to sub-femtosecond timescales. These oscillations would necessarily aﬀect drec upon the electron’s return, and because there is an electron-return-energy-to-time-delay mapping like the one shown in Figure 13, we can think of the returning electron as a probe that has the ability to map out hole dynamics as a function of time through drect. In truth, the details of this time-to-energy mapping are not always clear, especially in molecules, and exploring the feasibility of this interpretation is one of the goals of this dissertation.

2.2.9 Quantitative Rescattering Model

A couple assumptions that we have made for the above derivation in Section 2.2.5 are

(1) that a plane wave is suﬃcient for describing the propagating electron and (2) that the eﬀect of the Coulomb potential on the propagation can be ignored. These assumptions are not true in general, so extensions to the SFA model such as the quantitative rescattering

(QRS) model [32] have been created. This extension is also known as the scattering wave

32 Figure 14: Charge migration calculation from Kuleﬀ et al. [18]. In this diagram we can see the remaining hole density as a function of time in the singly-ionized C-terminally methylamidated dipeptide Gly–Gly–NH–CH3. The hole density migrates from one side of the molecule to the other simply through electron correlations.

SFA (SW-SFA) model. The main assumption of QRS is the ansatz that the spectral dipole can be decomposed as

2 DNω0 W ESdz,recNω0S , (49) where D is the HHG dipole, W is the returning electronic wave packet at the return time tr (in a sense constructed from the first two steps), and E is the returning electron’s kinetic energy. It has been shown through the work of Le et al. [32] that the wave packet W is largely independent of the generation atom or small molecule, such that if the wave packet is accurately calculated, it can be removed from experimental HHG data in order access dz,rec specifically. We use this ansatz of separability in order to interpret our experiments by first calculating the GD due to the SFA propagation and then removing it from our measured

GD. Although we are explicitly not including the Coulomb potential in our calculations of the SFA GD, I will show in later chapters that we are still able to extract interesting and sensitive features from the HHG spectra.

2.2.10 Femtochirp of the Dipole Phase

Before we move on to the macroscopic picture, there is one last factor that contributes to the phase of a harmonic pulse train that needs to be considered. Up until this point, we have ignored the temporal envelope of the oscillating electric ﬁeld. Doing so has conveniently

33 hidden an important chirp contributor called the “femtochirp.” To better understand the femtochirp, let us rewrite the HHG dipole from Equation 33 in a discrete fashion, following

Varjú et al. [65] and Gaarde et al. [66],

N ¡ N ¥¦ DN Q Aj exp iΦj rj ti, trp , (50) j

N where N is the harmonic order, j denotes a particular trajectory (long or short), Aj is the N amplitude of trajectory j for harmonic N, and the phase Φj along path rj is given by

t r p Aτ¥2 ΦN r t , t , p¥ Nω t S IP dτ. (51) j j i r 0 r 2 ti

Let us assume that the temporal envelope of our laser ﬁeld takes the following form

¡ 2~ 2¦ I t I0 exp 4ln 2 t τ0 , (52)

where 2τ0 is the full-width at half max of the temporal profile. As seen in Figure 15(a), the dipole phase varies nearly linearly with intensity when far from the cutoff with a slope defined by

∂ΦN αN j αN A 0 , (53) j ∂I j

N where αj is diﬀerent for the long and the short trajectories. This linear dependence is only true in the “long-pulse” limit: i.e. many cycles [66]. A more general approximation that is good down to three optical cycles is given by [67]

N N ~ Φj αj Up ω0, (54)

which will be used in the following section. It can be seen in Figure 15(b) that for a

N wide range of return energies in the plateau, αj varies nearly linearly, and therefore varies N linearly with Up. Additionally, αj is large for the long trajectories and small for the

34 (a) (b)

Figure 15: Figures depicting the intensity dependence of the high-harmonic dipole phase. In (a), adapted from [65], we see the that the intensity dependence of Φ is nearly linear for the 19th harmonic of 800 nm, generated in an Ar gas jet. The solid circle is an experimental N data point extracted from [68]. In (b) we see from [66] that αj is larger for the long trajectories and smaller for the short trajectories away from the cutoﬀ. One may expect this, due to the extended excursion amplitudes of the long trajectories compared to the short.

short trajectories until the cutoﬀ where the intensity-dependence of the two trajectories

merge. This makes physical sense, because we expect the intensity-dependence of the long

trajectories to be more severe due to their extended time spent in the continuum.

If we now consider the fast intensity variation of the laser pulse envelope and its eﬀect

on the chirp of the dipole phase at the peak of the laser pulse for a harmonic in the plateau,

we will ﬁnd that

∂2ΦN ∂ΦN ∂2I ∂2ΦN ∂I 2 I ∂ΦN I bN j j j ¤ 8 ln 2 0 j 8 ln 2 0 αN , (55) j 2 2 2 2 2 j ∂t ∂I ∂t ∂I ∂t τ0 ∂I τ0

N where we have used Equations 52 and 53 to get to the ﬁnal form on the right hand side. bj is called the “femtochirp” for harmonic N from trajectory j. This intensity-dependent chirp plays a fundamental role in the macroscopic selection of the short trajectories discussed in the next section.

35 2.3 Macroscopic Considerations

As mentioned above in Section 2.2.2, macroscopic effects are used to select the short trajectories over the long trajectories. Here we will follow [66] to develop our understanding of the macroscopic phase-matching process.10 Let us consider a Gaussian laser beam focused into a thin gas jet such that the intensity is low enough to ignore ionization effects and the gas jet is thin enough to ignore dispersive effects. We will assume cylindrical coordinates such that z is along the laser propagation direction and r is the perpendicular polar coordinate.

Phase-matching is the matching of phase fronts of the nonlinear polarization generated by the laser to the phase fronts of the generated harmonic. This is equivalent to matching the wavevectors of the source polarization and harmonic

ksource kω, (56)

where ksource ©φsourcer, z and kω correspond to the wavevectors of the source and harmonic frequency ω, respectively. The two largest contributors to ksource in this case are © N kdip,j r, z Φj r, z , discussed in Section 2.2.10, which is now a function of position across the laser focus, and kfocusr, z ©φfocusr, z. Equation 56 can then be written as

N ω kdip,j r, z kfocus r, z k0 kω, (57) ω0

where k0 and ω0 are the wavevector and radial frequency of the laser ﬁeld. We can instead rewrite Equation 57 as a diﬀerence called the “phase mismatch” given by

ω ω ∆kωr, z Vkdip,jr, z kfocusr, z k0V . (58) c ω0

If we now use the approximation from Equation 54 and the following equation for the phase

10The following derivation is done using SI units.

36 accrued from a focusing Gaussian beam

2z 2k r2z φ r, z tan1 0 , (59) focus b b2 4z2 where b is the confocal parameter, which is twice the Rayleigh Range, and insert them into

Equation 58, we get the following on-axis r 0 mismatch at harmonic energy ω

dU ~ω ω 2 ∆k z αN p 0 . (60) ω j 2 dz ω0 b1 2z~b

Upon a close inspection of the preceding equation, we can see that the second term is

always positive, so we must adjust the ﬁrst term to be negative in order to get perfect

phase-matching. This can be accomplished by placing the gas jet “downstream” of the

laser focus, because in those positions, dUp~ω0~dz will be negative. If we want to calculate oﬀ-axis phase-matching, which is beyond the scope of this dis-

sertation, the calculations quickly become complex. We can gather some understanding,

however, of oﬀ-axis phase-matching for the long and short trajectories from the calcula-

tions of Chipperﬁeld et al. [69], seen in Figure 16. Plotted are the coherence lengths Lcoh

for harmonics 21, 35, and 45, where Lcoh is deﬁned as Lcoh r, z 2π~∆kωr, z. Longer coherence lengths, indicated by the color bar, correspond to smaller phase mismatch, and

therefore reﬂect better phase-matching conditions. If the gas jet is placed approximately

2 mm downstream of the laser focus, then the short trajectories for all three harmonics

plotted in (a), (c), and (d) can be phase-matched well for nearly all radial distances. On

the other hand, the long trajectories have a much more complicated phase-matching proﬁle

shown in (b), (d), and (f), making it diﬃcult to properly phase-match.

It is also clear from Figure 16 that the harmonic divergences depicted with red arrows

are larger for the long trajectories than the short trajectories. This fact can also be used to

help select the short trajectories over the long trajectories. A 2D proﬁle of a harmonic beam

can be seen in Figure 17. The bright patch of XUV in the middle corresponds to the short

trajectories and the dimmer outer ring corresponds to the long trajectories. Experimentally,

37 Figure 16: 2D calculations of the coherence length across the laser focus for the 21st (a-b), the 35th (c-d), and the 45th (e-f) harmonics of a 750 nm driving ﬁeld in Ne at 41014W ~cm2. The confocal parameter here is 7 mm. Longer coherence lengths, indicated by brighter (whiter) colors indicate better phase-matching. (a), (c), and (e) represent phase-matching of the short trajectories while (b),(d), and (f) represent phase-matching of the long trajectories. Red arrows represent the direction of harmonics generated at their respective positions. It can be easily seen that the phase-matching conditions for the long trajectories are less regular than for the short trajectories, due to the increased sensitivity of the long trajectories to the laser intensity. Adapted from [69].

38 this separation is accomplished by propagating the XUV through a hard aperture that can remove the long trajectory outer ring.

In reality, phase-matching is not as simple as Equation 60 indicates. We also have to consider (1) dispersion from the gas medium of nonzero length and (2) dispersion from free electrons, which are created by ionization of the gas medium by the laser ﬁeld. Consideration of these two sources of dispersion leads us to a modiﬁed on-axis phase mismatch equation

ω ω dU ~ω ω 2 ∆k z ∆n ω ∆n ω αN p 0 , (61) ω at el j 2 c c dz ω0 b1 2z~b

where we have now included the eﬀects of the neutral atomic dispersion ∆natω natω

natω0 and similarly the free electron dispersion ∆nelω nelω nelω0. It is easy

to show that ∆natω is generally a negative number and ∆nelω is generally a positive number.

The aforementioned dispersive eﬀects have the following consequences. First, free elec-

trons contribute an overall negative factor to the refractive index that scales proportionally

to the free electron density ρ and quadratically with wavelength λ. If we consider the phase-

matching process on axis where the laser intensity is highest, we can deduce that the “blue”

wavelengths in the bandwidth of the laser pulse will advance faster than the “reds,” due to

the λ2 scaling of the free electron refractive index. This leads to a “blue-shifting” of the

central driving frequency of the laser pulse, consequently shifting the harmonic spectrum

to higher energies [70]. Secondly, at high laser intensities, where ρ is relatively large (¤ 1%

of neutral density [71][72][73]), the free electron phase mismatch begins to overwhelm the

neutral atomic dispersion leading to increased overall phase mismatch. This is acts as an-

other limit, in addition to the OTBI intensity, on the maximum allowable laser intensity for

HHG. Thirdly, the phase mismatch scales linearly with harmonic frequency ω, which means

that phase-matching at the highest harmonics becomes increasingly diﬃcult. This makes

it challenging to phase-match well across an entire harmonic spectrum, and therefore it is

inherently diﬃcult to get high production across an entire harmonic spectrum.

39 Figure 17: Far-ﬁeld image of two spatially-separated, delay-variable harmonic sources. In this image we can see the spatial separation of long and short trajectory HHG contributions. The bright patch of XUV in the middle corresponds to the short trajectories and the dimmer outer ring corresponds to the long trajectories. The vertical fringes are a consequence of the temporal overlap of the two HHG sources. Adapted from Bellini et al. [44].

2.4 Conclusions

In this chapter, we have described the classical and quantum-mechanical microscopic theories underlying high harmonic generation and high harmonic spectroscopy. Addition- ally, we have carefully detailed that macroscopic considerations of high harmonic generation necessary to select short trajectories over long trajectories. These theoretical considerations will serve as the framework for the experiments in Chapters 6 and 8. In the next chapter, we will expand upon two-photon perturbation theory, which is the foundation of our attosecond spectroscopy-based experiments.

40 Chapter 3: Two-Photon Ionization Theory

“In my painful experience, the truth may be simple, but it is rarely easy.”

—Brandon Sanderson, Oathbringer

3.1 XUV Metrology and a Direct Photoionization Scheme for Attosecond Spectroscopy

Chapter 2 covered the underlying theory of high-harmonic generation (HHG) in order to clarify the self-probing scheme of high harmonic spectroscopy. In this chapter, we aim to lay the theoretical foundation for two topics: (1) characterization of the group delay

(GD) of our attosecond pulse trains through a nonlinear process in a detection gas, and (2) using the same process to instead characterize ionization of the detection gas in the direct ionization scheme of attosecond spectroscopy. In this scheme, our harmonic comb is incident on the detection gas while spatio-temporally overlapped with a delay-variable IR ﬁeld.

By measuring the extreme ultraviolet-infrared (XUV-IR) delay-dependent photoionization yield, we are able to extract timing information about the XUV and simultaneously timing information about the ionization process itself.

In these experiments both the XUV intensity and IR intensity are kept weak enough to be properly described by two-photon perturbation theory, so we will start from the time-dependent perturbative picture to describe the two-photon ionization process following the articles by Laurent [74], Dählstrom [75], and Toma [76]. For the remainder of the chapter then, we will lay the foundation for extracting attosecond dynamics by detailing the underlying perturbation theory. The precise experimental methods used to extract the relevant physics will be detailed in Chapter 5.

In the two-photon picture, the probability to end in a state with a ﬁnal momentum k and energy at a time delay τ between the XUV and IR (in atomic units) is given by

2 P k, , τ SMdk, Mdak, , τ Mdek, , τS , (62)

41 Figure 18: Depiction of angular momentum channels available depending on the parity of the initial ground state. For the cases of Ar and Ne, there are six total pathways to the same ﬁnal energy, while for the case of He, there are only two pathways to the same ﬁnal energy.

where Md, Mda, and Mde are the matrix elements for direct XUV single-photon ionization, XUV ionization plus an absorbed IR photon, and XUV ionization plus an emitted IR photon. Expanding P k, , τ gives us

S S2 S S2 S S2 P k, , τ Md Mda Mde MdMda MdMde MdaMde c.c. , (63) which is the starting equation for the two diﬀerent types of XUV pulse trains studied in this dissertation: (1) odd harmonics only or (2) both even and odd harmonics. We will see that when only odd harmonics are present, the dominant interference oscillates at two times the IR laser frequency called the “two-omega oscillation,” and when even and odd are harmonics are present, there is an additional oscillation at the laser frequency called the

“omega oscillation.”

Before deriving the two different types of oscillations, it is important to discuss the various ionization pathways leading to the final ionized electronic state. Primarily, the ionization pathways are dictated by the available angular momentum channels. In a two- photon process, the number of angular momentum channels to the final continuum state can easily be deduced by consecutive applications of the dipole selection rules. For all of our experiments, we use linearly polarized XUV and IR. In that case, the absorption or

42 Figure 19: Diagram of allowed XUV + IR pathways in two-photon transitions from a ground state (g) to a final state at the energy of the qth harmonic above the ionization potential (IP). In pathways (1) and (3) an XUV photon is absorbed first followed by the emission or absorption of an IR photon. In pathways (2) and (4), an IR photon is first absorbed or emitted followed by XUV absorption. All pathways contribute to the photoelectron spectrum, but only pathways (1) and (3) contribute to the phase.

emission of a photon is accompanied by ∆l 1 and ∆m 0. Additionally, our detectors are angle-averaging such that all possible m states contribute to the ﬁnal photoelectron distribution. For the case of single-photon ionization (Md), where the ground state is l 1, such as in Ar and Ne, the ﬁnal electronic state is a coherent superposition of s (l 0) and d

(l 2) channels. For an l 0 ground state, such as in He, the final state is comprised of only the p channel (l 1). For the case of two-photon ionization (Mda and Mde), one must sum over all intermediate angular momentum channels to fully describe the ionization process as seen in Figure 18. For the case of l 1 and considering all m, there are six possible contributions to the final state of p and f character. If l 0, the final state has only two contributions, one of s character and the other d.

Finally, for the case of two-photon absorption with a single XUV photon and a single IR photon, there are two diﬀerent pathways that lead to the same ﬁnal state: 1) XUV absorbed

ﬁrst or 2) IR absorbed or emitted ﬁrst. Both of these cases can be seen in Figure 19. It has been shown that both pathways contribute to the total photoelectron yield, but only XUV-

ﬁrst pathways contribute to the phase of the photoelectron spectrum [76]. Furthermore,

43 XUV-ﬁrst pathways dominate the photoelectron yield due to the small polarizability of the ground state. Because of this, only XUV-ﬁrst pathways will be considered for the remainder of this dissertation.

3.2 Odd Harmonic Orders: A Two-Omega Oscillation

The ﬁrst measurement of group delay of a pulse train comprised solely of odd harmonics was done by Paul et al. [1] in 2001. The measurement method was later called Recon- struction of Attosecond Beating by Interference of Two-photon Transitions (RABBITT) by Muller ([77]). As mentioned in the introduction, this method has not only been used for metrological purposes but also attosecond spectroscopy in the direct photoionization scheme. See [29], [30], [11], and [31] for examples. RABBITT relies on the fact that there is no electron population at the energy of an even harmonic q; i.e. Md 0. Applying this

th to Equation 62 at an energy q corresponding to the q even harmonic, we see that

S S2 S S2 P2ω k, q, τ Mda Mde MdaMde MdeMda. (64)

We are only left with the two-photon transitions from odd harmonics q 1 and q 1

to a “sideband” state at the energy of an even harmonic q. Because the two pathways

are of course degenerate, they interfere, and from this interference we are able to extract

the interesting timing features. It is also important to note that because both pathways

(XUV+IR and XUV-IR) involve an even number of photons, the interference terms in

Equation 64 are of even parity. Therefore, electrons emitted in either direction along the

XUV polarization oscillate in phase with each other. A cartoon example of this type of

measurement can be seen in Figure 20.

2 2 In order to see how the GD is extracted, let us deﬁne C0 ¡ SMdaS SMdeS and expand

44 Figure 20: Example of the two-photon process that is fundamental to RABBITT measurements. In this type of process, two separate two-photon pathways interfere in a sideband at the energy of an even harmonic. One pathway is comprised of the (q-1)th harmonic plus the absorption one IR photon, and the other is the (q+1)th harmonic plus the emission of an IR photon. This type of interference would result in an oscillation at two times the fundamental laser frequency.

11 Mda and Mde as ˜ Mda MdaEq1E0, and (65) ˜ Mde MdeEq1E0 ,

where ˜ S S ¡ at ¦ Mda Mda exp iφq1 and (66) ˜ S S ¡ at ¦ Mde Mde exp iφq1

at at are complex amplitudes with phase contributions φq1 and φq1 called the “atomic phase” resulting from the ionization process. E0, Eq1, and Eq1 are the complex electric ﬁelds given in atomic units by

E0 exp iω0 t τ¥ , ¡ XUV ¦ Eq1 exp i q 1 ω0t φq1 , and (67) ¡ XUV ¦ Eq1 exp i q 1 ω0t φq1 , where ω0 is the IR laser frequency, and we have included the electric ﬁeld amplitudes in the

11By writing Equation 65, we are implicitly assuming that the two-photon ionization is dominated by a single angular momentum channel. A more general form would be given as Mda,de Mda,de Mda,de corresponding to the two diﬀerent intermediate angular momenta λ l1, where l is the angular momentum of the ground state.

45 S S S S XUV XUV amplitudes Mda and Mde . φq1 and φq1 are the phase contributions to the ionization

process that are inherent in the harmonics. If we now include the deﬁnition for C0 and expand Equation 65 with Equations 66 and 67 we get

XUV at P2ω k, q, τ C0 SMdaSSMdaS exp ¡i 2ω0τ ∆φ ∆φ ¦ c.c. , (68)

XUV XUV XUV at at at where ∆φ φq1 φq1 and ∆φ φq1 φq1. Simplifying Equation 68 and deﬁning

B0 ¡ 2 SMdaSSMdaS, we get the ﬁnal form

XUV at P2ω k, q, τ C0 B0 cos ¡2ω0τ ∆φ ∆φ ¦ . (69)

Equation 69 is the working equation for all spectroscopic measurements performed with

RABBITT. In an experiment, C0 and B0 are treated like ﬁt parameters in order to extract the phase oﬀsets in the cosine. From here we can see the path towards the two types of experiments laid out in Chapter 1. If we want to perform an HHS experiment, then we would study ∆φXUV , because it carries information about the Dω detailed in Chapter 2.

In that case we would calculate ∆φat and remove it from the measured data in order to study

∆φXUV . If instead we would like to measure attosecond physics in the direct ionization scheme, we would calculate and remove ∆φXUV from the measurement in order to study

∆φat. When calculating ∆φat in later chapters, we make a fundamental assumption that the at cc cc cc cc atomic phase can be split into ∆φ ∆φ ∆η, where ∆φ φq1 φq1 is the continuum-

continuum phase diﬀerence and ∆η ηq1 ηq1 is the “Wigner phase” diﬀerence, where the Wigner phase is simply the phase acquired through single-photon ionization. Note that this

Wigner phase diﬀerence is angle-integrated in comparison to the z-component measured in

HHS, which can be seen in Equation 47 in Chapter 2. Exactly how these contributions are calculated and removed, respectively, will be detailed in Chapter 5. In our experiments, we typically recast these phase diﬀerences as delays using the energy diﬀerence between

46 neighboring harmonics and the discrete derivative approximation such that

∆φXUV τ XUV ¡ and (70) 2ω0

∆φat τ at ¡ , (71) 2ω0 which are referred to as the XUV delay and atomic delay, respectively. One can then reconstruct the entire GD of the harmonic spectrum by concatenating each τ XUV from

each sideband. The RABBITT method is then in essence a measurement of the discrete

derivative of the spectral phase as a function of photon energy.

3.3 Even and Odd Harmonic Orders: An Omega Oscillation

Although the RABBITT method for studying attosecond physics is beautiful and sim-

ple, it is limited to the speciﬁc case where an attosecond pulse train is comprised of only

odd harmonics.12 If we would like to measure the phases of isolated attosecond pulses,

which have continuous spectra, or pulse trains comprised of even and odd harmonics, we

need another method to access the phases. Indeed, multiple methods have been developed

up until this point, including Frequency-Resolved Optical Gating for Complete Recon-

struction of Attosecond Bursts (FROG-CRAB) [78], an updated Volkov projections-based

FROG algorithm [3], Phase Reconstruction by Omega Oscillation Filtering (PROOF) [79],

and Improved PROOF (iPROOF) [74]. The various versions of FROG-CRAB rely on a

computationally intensive and somewhat opaque reconstruction method that is not very

convenient for rapid data analysis. PROOF, ﬁrst developed by Zenghu Chang’s group, re-

lies on computationally intensive genetic algorithms that do not provide unique solutions

because the algorithms only consider the phase of the relevant omega oscillation. The other

method, iPROOF, can be used to reconstruct attosecond pulse trains of arbitrary shape,

and it is able to provide unique solutions because it incorporates the amplitudes of the

omega oscillation into the reconstruction procedure.

12More generally, it is constrained to trains with a period of half a laser cycle.

47 All of the above methods have been used to varying degrees, and what is clear is that there is no consensus on the best method for reconstructing the phases of attosecond XUV light of continuous spectra or spectra with even and odd harmonics. In our experiments studying two-photon interferences that include even and odd harmonics, depicted in Figure

21, we decided to explore iPROOF for both its ease of implementation and its ability to provide unique solutions. We found, however, that it was too-greatly affected by transmission functions to reproduce the spectral phases of fake data sets. Particularly, it was found through simulations that general types of amplitude modulations, due to a detector transmission function, that are on the order of 10 % of the signal will lead the algorithm to find incorrect solutions for most spectral profiles [Daniel Tuthill, personal communication, 2018].

These types of modulations are very possible and common in our detector, so any retrieved spectral phases would be rampant with error. In reality, this aspect of pulse reconstruction has not been explored by the community when reconstructing even and odd pulse trains like ours or for isolated attosecond pulses. Understanding these types of systematic errors for the aforementioned reconstruction methods is active research in our group, so instead of focusing on retrieving the spectral phase, we will focus on the origin of the omega oscillation phase. Thus, in Chapter 7, when we discuss our even-and-odd harmonic photoionization results, we will compare to theoretical calculations of the omega oscillation.

In the following, we will derive the origin of an omega oscillation that can be found by analyzing the terms that were dropped from Equation 62 in order to perform the RABBITT analysis. If we instead keep the terms involving the interference of a direct transition with a two-photon transition along the z-direction (the laser polarization direction) only for the energy of the qth harmonic and drop the interference terms relevant to RABBITT,13 then we have the ﬁnal state probability

S S2 S S2 S S2 Pω0 kz,ˆ q, τ Md Mda Mde MdMda Md Mda MdMde Md Mde. (72)

13The two-omega oscillation terms of course exist, but their interpretation is opaque and irrelevant for the current derivation so they have been dropped.

48 Figure 21: A depiction of the interaction that is responsible for omega oscillations. Just like in Figure 20, two, two-photon pathways interfere at the energy of the qth harmonic. The primary diﬀerence here is that the qth harmonic is present and contributes to the interference signal at this energy – the source of the omega oscillation.

We have chosen to only look along the z-axis, because, when integrated over all angles, the interference terms MdMda Md Mda MdMde Md Mde 0. This can be understood through

parity considerations. The single-photon matrix element Md is of odd parity, and the two-

photon matrix elements Mda,e are of even parity. The products MdMda,e are then of odd parity such that a total angular integration will erase the contributions of the interference

terms. Furthermore, this means that these interference oscillations do not aﬀect the total

ionization probability. In order to resolve these oscillations, we must then restrict our

observation to a fraction of the total angular distribution.

2 2 2 For simplicity, let us drop the constant terms SMdS SMdaS SMdeS and rewrite the interference terms as

Pω kz,ˆ q, τ Md Mda Md Mda Md Mde Md Mde, (73)

which can further be written as

Pωkz,ˆ q, τ TM T SMdaS Re ¡exp ¡i arg ¡M Mda¦¦¦ d d , (74) T T S S ¡ ¡ ¡ ¦¦¦ Md Mde Re exp i arg Md Mde

49 leading us to the form

S S S S ¥ S S ¥ Pω kz,ˆ q, τ 2 Md Mda cos arg Md Mda Mde cos arg Md Mde . (75)

Using the matrix element and electric ﬁeld deﬁnitions from Equation 67 and Equation 25 of [75] it can be shown that

¥ XUV at arg Md Mda ω0τ ∆φa ∆φa , and (76) ¥ XUV at arg Md Mde ω0τ ∆φe ∆φe . (77)

With the above deﬁnitions it is then possible to ﬁnally show

S S S S XUV at S S XUV at Pω kz,ˆ q, τ 2 Md Mda cos ω0τ ∆φa ∆φa Mde cos ω0τ ∆φe ∆φe , (78) XUV XUV at cc at where ∆φa φq1 φq, ∆φe φq φq1, ∆φa φq1 ηq1 ηq, and ∆φe cc φq1 ηq1 ηq. Just as for the two-omega oscillation described above, the XUV phase diﬀerences between neighboring harmonics and the atomic phases accumulated through

diﬀerent ionization pathways contribute to the total phase of the omega oscillation. With

a little more algebra, one can show that

Pωkz,ˆ q, τ A cos ωτ Ψ , (79)

where the amplitude and phase are given by:

~ S S S S2 S S2 S SS S XUV XUV at at1 2 A 2 Md Mda Mde 2 Mda Mde cos ∆φa ∆φe ∆φa ∆φe ,

(80)

XUV at XUV at SMdaS sin ∆φ ∆φ SMdeS sin ∆φ ∆φ Ψ arctan a a e e . (81) S S XUV at S S XUV at Mda cos ∆φa ∆φa Mde cos ∆φe ∆φe

As is quite obvious from the above equations, extracting and separating the spectral phase

50 and atomic phase is nontrivial. This inherent complexity in the omega-oscillation is one of the reasons that the physics community has not reached a consensus on a standard reconstruction technique. As we will show in Chapter 7, one can use the omega-oscillation

“as is” to probe atomic ionization delays without having to reconstruct the attosecond pulses through a spectral phase retrieval.

3.4 Conclusions

We have now established the two-photon theory necessary for understanding our experiments. First we showed the origin a two-omega oscillation in the RABBITT interaction, which can easily provide a structure for measuring the GD of harmonics and also the GD associated with two-photon ionization. We have also described the origin of an omega oscillation that occurs with pulse trains comprised of even and odd harmonics. Both of these interactions will be explored in the presence of resonant ionization features in Chapter 7.

51 Chapter 4: Impulsive Molecular Alignment

“ ‘The whole business of [lasers] is one of these things you just can’t aﬀord to let go,’ John Pierce told an interviewer around that time.” —Jon Gertner, The Idea Factory: Bell Labs and the Great Age of American Innovation.

4.1 Introduction to Alignment

So far, we have laid the theoretical ground work for the experiments to be described later in this dissertation. Before we move on to those experiments, we should discuss one more tool that is critical for our high-harmonic spectroscopy (HHS) experiments: molecular alignment. For some HHS experiments, rich dynamics such as charge migration [37] or two-center interference [80] only manifest themselves with access to the molecular frame.

This is because the recombination dipole matrix element drec,z may vary as a function of the molecular angle relative to the high-harmonic generation (HHG) laser polarization, θ.

If the molecular sample is instead left unaligned, this often “washes out” the interesting physics. Thus, it is desirable to have control of the molecular angle θ.

Molecular “alignment” is the process of creating a molecular ensemble that has a particular nuclear axis that is anti-parallel and parallel to a lab-frame axis. This is in contrast to molecular “orientation” which creates an ensemble of molecules that have a particular nuclear axis that is anti-parallel or parallel to a lab-frame axis, but not both. This diﬀer- ence is only important for molecules with permanent dipoles such as polar heteronuclear molecules like CO because one cannot orient a homonuclear diatomic molecule like N2. The diﬀerence between the two methods is shown in Figure 22(a). Due to the technical demands of molecular orientation [81], we use molecular alignment in order to understand certain molecular ionization processes. Some physics may be lost by not orienting polar molecules, but theory has shown that certain phenomena only expected in an oriented sample can

“survive” alignment [82], extending the usefulness of molecular alignment.

52 (a) (b)

Figure 22: Models describing molecular alignment. In (a) we see the diﬀerence between molecular alignment and molecular orientation. Molecular alignment does not specify head- to-tail alignment whereas “orientation” does. In (b) a “kick” aligning pulse is incident on a randomly aligned sample of gas molecules, coherently populating a wave packet of J-states. The wave packet then promptly aligns, dephases to random alignment, and then realigns at a later time, called a revival. These revivals occur at delays that are fractions of the rotational period or at the full rotational period T0.

There are predominantly two mechanisms for aligning a molecular ensemble in the lab

frame with a laser pulse: adiabatic alignment and impulsive alignment. Adiabatic align-

ment is a method that relies on the laser pulse duration τ being on the order of or longer

than the rotational time period Trot of the molecule of interest. In this case, the initially unaligned molecules slowly and smoothly align to the laser polarization, but then return to an unaligned sample in the same way when the laser pulse exists the interaction region.

Impulsive alignment requires a laser pulse duration such that τ P Trot. In this case, the laser pulse “kicks” the molecules and coherently populates a rotational wave packet through

two-photon Raman transitions. Just after the initial kick, the molecules will align along the

laser polarization in the lab frame. After the laser pulse has left, the rotational wave packet

will evolve until a later time called a “revival.” At this time, the molecules re-align in the

lab frame as seen in Figure 22(b). For our experiments, we chose impulsive alignment so

that we do not have strong external ﬁelds perturbing our laser-molecule interactions.14

14Advanced methods of molecular orientation and alignment also use laser pulses with a slow turn-on and sharp turn-oﬀ, e.g. [83][84].

53 4.2 Theory of Impulsive Molecular Alignment

In this section, we will brieﬂy establish the mechanism responsible for molecular alignment, and then we will describe some of the relevant laser conditions necessary for alignment.

In the following derivation, we will again assume that our “kicking” laser polarization is along the lab frame z-axis where the lab-frame coordinates are denoted by (x, y, z) and the molecular-frame coordinates are denoted by (X, Y, Z). We follow Ortigoso et al. [85] and Hamilton et al. [86] and restrict our discussion to rigid rotors, which are a class of molecules that do not experience centrifugal distortion. Additionally, we will also assume that the Born-Oppenheimer Approximation is valid, which implies that the rotational degrees of freedom are decoupled from the vibrational and electronic degrees of freedom. Rigid ¢ rotors have an inertial tensor I such that its principle moments of inertia are given by

IXX IYY x IZZ 0, and all oﬀ-diagonal terms are zero. Similar to the inertial tensor, the ¢ components of the polarizability tensor α are such that all oﬀ-diagonal elements are zero

and αXX αYY ¡ αÙ x αZZ ¡ αÕ, where αÙ represents the polarizability perpendicular to

the nuclear axis and αÕ represents the polarizability parallel to the nuclear axis.

Assuming a linearly-polarized driving ﬁeld of the form Et 1~2 ¡εteiωt εteiωt¦,

the Hamiltonian of a rigid rotor in the presence of a laser ﬁeld is given by ﬁeld-free Hamil-

tonian plus the induced Hamiltonian

Ht Hmol Hindt. (82)

Hmol is given in atomic units by

J 2 H BJ 2, (83) mol 2 2µRe where J is the total angular momentum operator, µ is the reduced mass of the molecule, Re ~ 2 is the length of the nuclear axis, and B 1 2µRe is referred to as the rotational constant.

The eigenstates of Hmol are of course the spherical harmonics YJM θ, φ ¢ SJMe, and J and M are the orbital angular momentum and magnetic quantum numbers associated with

54 1 Molecule B(cm ) ∆α αÕ αÙ(a. u.) αÙ (a. u.) Tr (ps)

CO2 0.389 [87] 15.4 [87] 14.7 [87] 42.7 [87] N2O 0.41 [87] 18.85 [87] 13.16 [87] 40.5 [87] OCS 0.20286 [88] ¤27.15 [89] ¤26.08 [89] ¤81-82 [90][91] N2 2.01 [87] 6.276 [87] 9.785 [87] 8.26 [87] CO 1.93 [87] 3.536 [87] 11.83 [87] 8.6 [87]

Table 1: Relevant molecular parameters for impulsive molecular alignment.

J. The eigenvalues of Hmol are given by

Emol BJJ 1. (84)

With Equation 84, we can deﬁne a rotational time period for the lowest nonzero rotational

state J 1 such that Tr 1~ 2B. As will be shown later, Tr corresponds to the time at

which the ﬁrst full revival occurs. Hind for a rigid rotor is given by

1 2 2 H t ε t ¡∆α cos θ αÙ¦ , (85) ind 4

where θ is the angle of the molecular nuclear axis relative to the laser polarization in the

lab frame, and ∆α is called the polarizability anisotropy given by ∆α αÕ αÙ. The ﬁrst

term on the right hand side of Equation 85 is the term responsible for molecular alignment,

and the second term serves as an overall shift to the energy, but is otherwise not related to

alignment. Because of this, the polarizability anisotropy is obviously a critical property of

rigid rotors that determines their ability to align; larger anisotropies lead to higher degrees

of molecular alignment. Some examples of molecular parameters relevant to the alignment

interaction are shown in Table 1. It is important to note that molecular alignment does not

depend on the permanent dipole.

It can be shown that Y10 cos θ, which means that interactions involving cos θ will exchange one unit of angular momentum and not alter M. One can then deduce that

interactions involving cos2 θ will exchange two units of angular momentum. Therefore,

55 when Hind acts on a state SJMe, one can show that

∆J 0, 2 and (86)

∆M 0, (87) which are of course the dipole selection rules for two-photon absorption of linearly polarized light. These types of transitions are known as Raman transitions, which involve the absorption or emission of one photon to an intermediate state and then absorption or emission of another photon back to the initial state or to a state two steps higher (Stokes process) or lower (Anti-Stokes Process).

Before the laser pulse arrives, the initial J-state population NJ of the ensemble of rigid rotor states is given by a Boltzmann distribution

NJ gJ exp Emol~kBT ¥ , (88)

where gJ is the multiplicity of a particular J-state that depends on the nuclear spin statistics, kB is the Boltzmann constant, and T is the rotational temperature of the ensemble. The Stokes and Anti-Stokes processes are out of phase with each other and happen with equal probability unless they run into the ground rotational state, where only the Stokes process is possible [92]. Thus, if T is large enough so that transitions do not signiﬁcantly

reach the ground state barrier, Stokes and Anti-Stokes processes will interfere with each

other resulting in a weakly-populated rotational wave packet. So in order to have the Stokes

dominate and create a coherent rotational wave packet, the distribution of rotational states

needs to be as close to J=0 as possible, i.e. as cold as possible. Another beneﬁt of us-

ing a cold gaseous sample is the reduction of centrifugal distortion eﬀects [93], which can

contribute to dephasing and therefore a less-coherent rotational wave packet.

The wavefunction describing the rotational wave packet excited by the total Hamiltonian

56 Ht is given by

ΨtM Q aJ tYJM , (89) J where aJ t are complex amplitudes such that aJ t 0) are determined by Equation 88.

After propagating ΨtM through the total Hamiltonian for the duration of laser pulse,

ΨtM is allowed to evolve under Hmol. If we want to know the degree of alignment at some time t after the excitation pulse, we can calculate the expectation value acos2 θf t.

In order to compare to the experiment, we must ﬁnd aacos2 θff t, which is the thermal

average over the initial distribution of J states. An example of aacos2 θff t calculated for

O2 is shown in Figure 23, where we can see the revival structures of O2. At times 0, 0.25π~B, 0.50π~B, 0.75π~B, and 1π~B are the prompt alignment in the presence of the laser pulse, the

quarter revival, the half revival, the three-quarters revival, and the full revival, respectively.

At each of these revivals, the molecular ensemble deviates from near random alignment

aacos2 θff 0.33. If we focus on the half revival, we can see that aacos2 θff goes through

a minimum corresponding to “anti-alignment,” a molecular distribution centered around

θ 90X, followed by a maximum corresponding to “alignment,” a molecular distribution

centered around θ 0X. The same happens at the full revival, but the full revival is π out

of phase with the half revival. Between the maxima and minima within the half and full

revivals, we can think of molecular distribution as smoothly varying from 0X to 90X and 90X

to 0X, respectively.

4.3 Dependence on Laser Parameters

We have only brieﬂy mentioned how molecular alignment depends on the laser pulse

parameters, and so we have left out some important details. Molecular alignment is sen-

sitive to the temporal duration, central wavelength, and peak intensity of the laser pulse.

As mentioned in Section 4.1, the laser pulse duration τ must be much smaller than the

rotational period Trot. It has additionally been shown that the optimal pulse duration can range anywhere from 1-5% of the rotational period [94], which is species-dependent in that

it depends on the polarizability anisotropy and the rotational constant B in a nontrivial

57 Figure 23: Revival Structure of O2. The solid black line represents the thermally-averaged 2 expectation value cos θ for the O2 rotational wave packet, and the dotted red line is the envelope of that expectation value. It is clear that the wave packet reaches high degrees of alignment multiple times per rotational period. The exact structure of the fractional revivals strongly depends on the nuclear spin statistics of the molecule. Adapted from [92].

fashion. As an example, for impulsive alignment of N2 with an 800 nm laser pulse, it has been found that the optimal pulse duration is approximately 120 fs [95].

The central wavelength of the aligning laser pulse eﬀects molecular alignment through

the wavelength-dependent polarizability anisotropy. To this author’s best knowledge, there

has not been any investigation of the wavelength-dependence of molecular alignment, but

a study was performed by Herring et al. [96] that explored the wavelength-dependence

of the polarizability anisotropy for the case of N2. In this study, they found that the anisotropy increases as the driving frequency increases towards the ionization threshold.

This indicates that shorter driving wavelengths would be better for molecular alignment, but

shorter wavelengths at a ﬁxed intensity would also result in increased ionization probability,

which brings us to our last point.

Typical laser intensities used in molecular alignment are on the order of 1013 W ~cm2.

As a rule of thumb, the kicking pulse intensity should usually be kept weaker than the HHG laser intensity. This is done in order to decrease the total amount of molecular ionization or molecular dissociation prior to HHG so that the high harmonic process is probing the ground state of the neutral molecule. Techniques have been developed utilizing multiple

58 kicks so that the laser intensity can be kept low [97]. Using multiple kicking pulses is much like pushing a child on a swing: low force at the correct frequency so that the child’s swing amplitude becomes large. These techniques have been used for HHS before [98], but experimental implementations are complex and are highly sensitive to laser pointing drifts.

Because of this, we do not utilize the multi-kick scheme is this dissertation.

4.4 Conclusions

In this chapter, the theoretical foundations of impulsive molecular alignment were established. We began by detailing a quantum-mechanical model of impulsive molecular alignment. Next we discussed the molecular parameters that inﬂuence alignment, namely the polarizability anisotropy, and then established the importance of rotationally-cold molecular gas samples for high degrees of molecular alignment. Lastly, we described how molecular alignment depends on the aligning laser pulse parameters: pulse duration, wavelength, and intensity. This method will be used extensively in Chapter 8 to study molecular-frame physics in CO2,N2O, and OCS.

59 Chapter 5: Experimental Methods and Apparatus

“If a drop of water enters the soil at a particular angle, with a particular pitch, what’s to say a man can’t ride one note into the earth like a ﬁreman’s pole?” —Sarah Ruhl, Eurydice, Second Movement, Scene 15

5.1 Statement of Contributions

In this Chapter, I describe the methods and apparatus used to perform our experiments.

Before describing those methods and apparatus, I would like to directly state my most sig- niﬁcant contributions. I designed and built the interferometer that allowed us to perform molecular alignment with assistance from Dr. Timothy Scarborough. The imaging apparatus used to spatiotemporally overlap the alignment pulse and the HHG pulse was designed by me and constructed by me with assistance from Dr. Scarborough. The wavelength tuning method that was used to carefully measure the phase or group delay of resonances was developed by Dr. Scarborough and me. The laser system compressor upgrades described in this chapter were conceived and implemented by me. All of the pulse duration measurements using the Frequency-Resolved Optical Gating method were performed by me using an interferometer that I designed and built. The XUV-IR interferometer or “attobeamline” existed when I joined the group, but, as described below, I greatly increased the achromaticity of the interferometer by implementing a thin, uncoated CaF2 beamsplitter instead of typical wavelength-speciﬁc beamsplitters. Doing so made implementation of the wavelength scanning technique possible.

5.2 Methods

Up until this point, we have established the theory for high-harmonic spectroscopy

(HHS), attosecond spectroscopy (AS), and molecular alignment. In this section, we will describe how we apply molecular alignment to HHS experiments, and we will also describe

60 a unique implementation of wavelength scanning for both omega oscillation-based experiments and two-omega oscillation-based experiments, where the two-omega oscillations are measured using the Reconstruction of Attosecond Beating by Interference of Two-photon

Transitions (RABBITT) method.

5.2.1 Molecular Alignment in an HHS Experiment

As discussed in Chapter 4, the degree of molecular alignment depends on the polarizability anisotropy, the gas temperature, and the laser intensity, wavelength, and pulse duration. The polarizability is an intrinsic property tied to the choice of molecule, whereas we have external control over the gas temperature and the laser parameters. In order to increase molecular alignment it was explained in Chapter 4 that the gas sample should be rotationally cold. Though it is true that lower temperatures will provide higher degrees of alignment, it is not strictly necessary. Varying degrees of molecular alignment have been observed in gases like N2,O2, CO, CO2, CS2, and C2H4 at room temperature [99] and at

temperatures as high as 400 K for I2 [100]. Even so, we would like to rotationally cool our gases as much as possible. To this end, we use the inherent collisions of a supersonic gas jet for our high-harmonic generation (HHG) experiments (as opposed to a cell-based HHG source) to rotationally cool our molecules. These collisions transfer energy preferentially from the rotational degrees of freedom to the translational degrees of freedom [101]. Mov- ing further along the molecular gas jet away from the nozzle allows for more collisions and consequently more rotational cooling with maximal cooling occurring at a distance of ¤ 20 nozzle diameters away from the nozzle. Increasing the gas pressure and nozzle diameter also lead to more rotational cooling. We are limited in distance from the nozzle due to the rapidly decreasing gas jet centerline density and we are limited on the maximal backing pressure and nozzle diameter by the pumping speed of our vacuum apparatus. In practice, we adjust the laser focus position and gas jet pressure to maximize rotational cooling (measured through alignment-dependent modulations in the harmonic yield) without sacriﬁcing phase-matching of the harmonic spectrum and without increasing the operating background

61 chamber pressure above ¤ 10 mtorr. This balancing act typically restricts our laser focus to be approximately 1 millimeter (¤ 5 nozzle diameters) from the exit of the nozzle, and nozzle

backing pressures for a continuous nozzle of approximately half an atmosphere absolute. As

an example, we can estimate the terminal rotational temperature of a neat CO2 gas sample, one of the molecular samples used for the HHS experiment described in Chapter 8. From

[101] we can calculate the terminal rotational temperature with ¤ 8 PSI of backing pressure and a nozzle diameter of 200 µm. In that case, the rotational temperature is approximately

45 K, 15% of the source temperature of 300 K. In our experiments, we are are not fully 20 nozzle diameters away from the nozzle so our rotational temperature is likely above 45 K.

An additional limit on rotational cooling is the eﬀect of clustering [102], where collisions between molecules in a supersonic gas jet can lead to clustering of molecules into large

“clumps” that can be as large as tens of thousands of molecules in size. Clusters are known to affect the HHG yield, but have not yet been shown to affect the group delay (GD) of high harmonic spectra [103]. Even so, we wanted to minimize or expose any potential clustering effects in our highly polarizable molecular samples. To this end we seeded our most polarizable gas, OCS, at 5 and 10% in He for all experiments. The low polarizability of helium makes it notoriously difficult to cluster, so it acts as a barrier to cluster formation.

For our molecular alignment experiments, our aligning laser pulse, or “kick” pulse, interaction intensities are in the range of a few 1013 W ~cm2 and typically no higher than

1014 W ~cm2. We control the power through an adjustable aperture or “iris” as our group refers to it. Using an iris to adjust the power is advantageous because it shrinks the beam size and lowers the power at the same time. Shrinking the unfocused kick beam increases the size of the focal spot, guaranteeing that our HHG spectra are only generated from an aligned sample. Typically we adjust the total kick pulse power and beam size so that at the unfocused kick beam is approximately half the diameter of the HHG beam. As an example, our kick pulse is typically 800 nm and our HHG pulse is usually 1300 nm. If the beam size of the unfocused kick beam is half the size of the unfocused HHG beam, then the focal spot size of the kick is approximately 120% of the HHG spot size. As for the central frequency

62 of the kick pulse, it is ﬁxed at ¤ 800 nm by our experimental setup detailed in the following sections.

Lastly, for all of our HHS experiments employing molecular alignment, we probe the aligned sample around the half revival Trot/2. Around this delay time, and if the HHG and kick pulses have the same polarization, the rotational wave packet will smoothly vary from 0X to 90X relative to the HHG laser polarization as a function of delay. We choose the half revival over the full revival for two reasons. The ﬁrst is that at long delays between the HHG and aligning pulses, we can expect more dephasing of the excited rotational wave packet due to centrifugal distortion. The second reason is that the molecules studied in

Chapter 8 have relatively large rotation periods where Trot can be as large as 80 ps; this means that our delay stage (seen in Figure 29 for reference) needs to be carefully calibrated to retroreflect over approximately half an inch to an inch. This is of course not beyond our capability, but smaller delays will lead to more repeatable experimental settings. So in practice we choose to probe around the half revival to study the HHG spectra as a function of molecular angle, which we also do in two different ways. We both scan the HHG-kick delay around the half revival using the same polarization for each beam, and we rotate the polarization of the kick pulse at the fixed delay corresponding to 0X molecular alignment for the same polarization. Using the two methods provides for a check of our molecular angle-dependent results.

5.2.2 Wavelength Tuning for Resonance Spectroscopy

As Section 5.4 will show, we have gone to great lengths to make our experimental apparatus as achromatic as possible. Our primary reasons for doing this are two-fold.

First, it allows us to investigate wavelength-dependent HHG phenomenon, and second, wavelength tuning can be used in resonance spectroscopy in HHS or in AS. As described in

Chapter 3, the RABBITT method is a measurement of the GD in the discrete derivative approximation such that the phase diﬀerence between the (q-1)th and (q+1)th harmonics is imprinted on the sideband at the energy of the qth harmonic. In order to get a more

63 continuous sampling of the GD versus energy, we can tune the driving wavelength to ﬁll in the energetic gaps as one may want in the case of a sharp resonance.

A critical parameter for wavelength tuning in resonance spectroscopy is the harmonic separation 2ω0 relative to the resonance width ∆ER. If 2ω0 Q ∆ER such that only a single harmonic can be incident on the resonance in each RABBITT scan, then the shape of the

reconstructed group delay will reﬂect the shape of the phase of the resonant structure. In the

other extreme where 2ω0 P ∆ER, multiple harmonics can sample the resonance at once, such that the discrete derivative approximation to the GD is accurate, meaning that the

shape of the reconstructed group delay will reﬂect the shape of the group delay of the resonant

structure. In the regime where 2ω0 ¤ ∆ER, the interpretation of the wavelength scan

becomes vague and diﬃcult to interpret. A cartoon example of the case 2ω0 Q ∆ER is shown in Figure 24. Here we have a “zig-zag” phase feature that is due to some ﬁctional resonance.

If we want to map out the resonance phase shape, we simply tune a single harmonic over the

resonance from one side to the other in small steps by tuning the HHG driving wavelength.

This serves to imprint the phase diﬀerence, symbolized by black stars, from the resonant

harmonic H3 and the non-resonant harmonics H1 and H5 on sidebands SB2 and SB4 for

each scan, respectively. After the wavelength scan is complete, the reconstructed GD feature

is the solid red line. Notice that the phase diﬀerences imparted to SB2 and SB4 are inverted

due to the sign change in absorption or emission of an IR photon. This type of measurement

is the basis for much of the results shown in Chapters 7 and 8.

5.3 Laser Systems

For all experiments described in this dissertation two lasers were used: a home-built

system and a commercial Spectra Physics Spitﬁre Ace. Both lasers are Ti:sapphire-based

emitting at central wavelengths in the range 784-800 nm with pulse durations between 50

and 60 fs and rely on the principle of chirped-pulse ampliﬁcation, wherein a femtosecond-

scale pulse is stretched to hundreds of picoseconds, ampliﬁed and then compressed to tens

of femtoseconds [104]. From there, both systems are predominantly used to pump optical

64 H1 SB2 H3 SB4 H5 H1 SB2 H3 SB4 H5 H1 SB2 H3 SB4 H5

Increasing # of Wavelength Scans

H1 SB2 H3 SB4 H5 H1 SB2 H3 SB4 H5 H1 SB2 H3 SB4 H5

Increasing # of Wavelength Scans

Figure 24: Cartoon evolution of RABBITT scans over many different driving laser wavelengths. The blue bars represent harmonics and the red bars represent sidebands. The phase differences between neighboring harmonics are imprinted on the sidebands separating the two, represented here by stars. Starting from the top left and ending in the bottom right, the generating wavelength is scanned so that one harmonic, H3, thoroughly samples the “zig-zag” phase feature. When the scan is over, the zig-zag is imprinted on the neighboring sidebands SB2 and SB4 denoted by the red lines. The difference in sign between SB2 and SB4 is due to the phase difference in emission or absorption of an infrared (IR) photon.

65 Figure 25: Block diagram of the laser system. Adapted from [28]. parametric amplifiers (OPAs) that are able to generate wavelengths from 1100 - 2200 nm, extending into the mid-infrared (MIR). The main difference is that the Spitfire and its coupled OPA output twice as much energy as the home-built system and its corresponding

OPA. Because these systems are so similar, and the vast majority of the data in this dissertation was collected with the home-built system, we will focus on describing the operation and upgardes of the home-built system only.

5.3.1 Home-built Laser as a Source for Near-Infrared Femtosecond Light

Here we will give a brief overview of the home-built laser system and its upgrades. For a more detailed description of the laser system please refer to Stephen Schoun’s dissertation

[28]. As seen in Figure 25, our home-built laser system can be broken down into 5 main stages leading into the OPA: oscillator, stretcher, regenerative ampliﬁer (referred to as regen hereafter), multipass ampliﬁer and compressor. The oscillator is a Venteon from Venteon

Technologies GmbH. It produces ¤ 8 fs pulses centered around 785 nm with approximately

1 nJ per pulse at a repetition rate of 80 MHz through Kerr-lens modelocking [105]. The oscillator is pumped at 532 nm by an upgraded diode-pumped solid-state (DPSS) Millennia

66 Pro from Spectra-Physics.15 Additionally, a trigger signal is taken out of the oscillator by picking oﬀ a small portion of light from a reﬂection of an internal wedge. This trigger will be used to control the Pockels cell in the regen for pulse-picking.

After a pulse leaves the oscillator it is sent to an all-reﬂective stretcher. The stretcher uses a 1200 grooves/mm gold grating from Spectrogon combined with a focal mirror with f=150 cm radius and a 20.3 cm diameter, and two ﬂat mirrors to stretch the pulse duration out to ¤ 230 ps. The stretcher stretches the pulse by adding a large amount of negative group delay dispersion (GDD) and clipping the spectrum from the oscillator on either side of its bandwidth down to about 100 nm. By propagating through the stretcher, the pulse energy is approximately halved.

After leaving the stretcher, the pulse enters the regen. The regen is a laser cavity that uses another Ti:sapphire crystal that is pumped at 527 nm by a ¤ 8 mJ Q-Switched,

intracavity-doubled, DPSS DM20 laser produced by Photonics. First the 80 MHz pulse

train is picked down to 1 kHz by the Potassium Dideuterium Phosphate (KD*P ) Pockels

cell (model 700-KDP by Medox Electro-Optics, Inc.) receiving the trigger signal from the

oscillator. The picked pulses are the kept in the regen cavity for approximately 20 passes

through the crystal. After the regen, the pulse train has been cut to 1 kHz and each pulse

has ¤ 1.5 mJ of energy and 27 nm bandwidth, reduced through gain narrowing, centered

around 785 nm.

Following the regen is the second stage of ampliﬁcation: the multipass ampliﬁer. Unlike

the regen, the multipass is not a cavity. It is a bow-tie optical layout that allows the seed

from the regen to twice pass through another Ti:sapphire crystal that is pumped at 527 nm

by a DPSS DM40 laser produced by Photonics. The DM40 pumps the multipass crystal

with approximately 3 times as much energy as the DM20 pumps the regen crystal. At this

higher average power, there is potential to overheat and/or burn the crystal. To mitigate

these risks and to reduce thermal eﬀects in the ampliﬁcation processes, the crystal is kept

in a vacuum chamber pumped by a StarCell ion pump from Varian, and the chamber allows

15This is a newer version of the same Millennia oscillator pump used at the time Stephen Schoun’s dissertation [28] was written.

67 the crystal to thermally contact an attached liquid nitrogen trap for cooling. The vacuum is able to thermally insulate the crystal so that during operation it is kept at approximately

150 XC and is free of dirt and dust. Upon exiting the multipass the pulse energy has been boosted from 1.5 mJ to approximately 7.5 mJ with a spectrum still centered around 785 nm.

The last stage of the home-built laser system is the compressor, which is located in a separate room from the rest of the laser. This is done in order to reduce the distance between the compressor and the OPA. In previous builds of the laser system, it was found that a longer propagation distance between the compressor and the OPA allowed for too much self-phase modulation in the compressed pulse. In order to reduce laser pointing slow-drift

(on the order of to minutes to hours) leading to the compressor, we have implemented a laser-pointing locking program that was created by Dietrich Kiesewetter. This program uses a feedback algorithm to move a single motorized mirror while tracking the weak transmission of the beam that leaks through a turning mirror onto a camera just in front of the compressor. The feedback algorithm operates by recording a spatial image of the beam for

30 seconds, calculating the mean position of the beam proﬁle, and then shifting the beam by the diﬀerence in pixels between the new mean and the locking position.

The home-built compressor is comprised of four flat turning mirrors, two 1500 grooves/mm gold-coated gratings from Spectrogon and a roof-top retro-reflector. The compressor adds positive GDD in order to compensate for the large amount of phase added by the stretcher and also to compensate for any additional phase acquired through the amplification processes. The first iteration of the compressor design employed two gold-coated gratings, one small and one large, that had ZKN7 substrates with a thermal conductivity of 1.0 WK1m1.

It was found that these gratings continually swelled over the span of many hours causing the beam exiting the compressor to signiﬁcantly drift in pointing. The last place that the beam contacts the compressor, where the beam is most intense and mostly likely to be subject to localized thermal swelling, is on the small grating. Because of this the small grating was swapped out for an identical one with a ZERODUR substrate, which has a

68 higher thermal conductivity 1.46 WK1m1, circa 2009. This solved some of the drifting, allowing the compressor to fully warm after approximately 8 hours of warm-up time. In

2017 it was decided to purchase a ZERODUR-based grating for the large grating as well.

This replacement further reduced the necessary compressor warm up time to approximately

2 hours.

Another problem that we experienced with these gratings is that the small grating slowly burns over months at the last place that laser beam contacts it. It was found that the best solution to this problem was to wash the gratings in-place with distilled water on a monthly basis. This cleaning process and the new gratings have allowed us to maintain a compressor with high total transmission ¤ 81% and a clean Gaussian spatial mode. The output of the compressor is a ¤ 6 mJ pulse centered around 785 nm with a pulse duration of 50 fs. The results from a Frequency-Resolved Optical Optical Gating (FROG) reconstruction taken with a home-built second harmonic generation FROG apparatus can be seen in Figure 26.

5.3.2 Optical Parametric Ampliﬁer

Once the laser pulse exits the compressor, it is typically used to pump an HE-TOPAS

OPA by Light Conversion. The OPA is based on a series of diﬀerence frequency generation (DFG) processes between the 785 nm “pump” (horizontal polarization) and a “signal” wavelength in the range 1100-1600 nm (vertical polarization) that generates an “idler” wavelength in the range 1600-2500 nm (horizontal polarization). The DFG process obeys momentum conservation such that

kI kP kS, (90)

where kI , kP , kS, are the wavevectors of the idler, pump, and signal, respectively. A diagram of the OPA’s optical layout can be seen in Figure 27. The process begins

with white-light generation in a Ti:sapphire plate where the ﬁrst signal wavelengths are

generated. From there the white light is recombined with a small portion of the pump in a

beta barium borate (BBO) crystal where diﬀerence frequency generation (DFG) ampliﬁes

69 (a) (b)

Figure 26: SHG FROG results for the output of the laser compressor before entering the OPA. (a) The originally-recorded FROG spectrogram. (b) The reconstructed spectrogram. (c) Reconstructed ﬁeld intensity and temporal phase. (d) Reconstructed spectral intensity and spectral phase.

70 Figure 27: Schematic of the HE-TOPAS OPA. The beam enters in the top right with approximately 6 mJ of energy at a pulse duration of 55 fs. The beam is then split three times. The smallest energy arm is used to generate white-light (WL) in a sapphire plate. That white light is then used to seed DFG in a BBO creating the signal and idler wavelengths. This output is then used to seed two more amplification stages. The three delays and the three BBO rotation angles in the OPA have been motorized for simple wavelength tuning. Figure adapted from [28] the signal and creates an idler pulse. The signal from this is taken to seed two more DFG processes that each have successively higher pump powers. After the third and final stage of amplification, the total output energy of the signal plus idler is approximately 2.4 mJ. The energy per pulse across the OPA spectrum can be seen in Figure 28(a). The pulse durations recorded by a SHG FROG and a TIPA single-shot autocorrelator from Light Conversion for a few select wavelengths can be seen in Figure 28(b). It has been consistently found by our group that the pulse durations across the OPA spectrum can vary widely with a consistent shortening of the signal and idler pulse durations near the degeneracy point, 1600 nm. The depleted pump, which has approximately 2 mJ of energy, is also available for use, though

its spectrum and spatial proﬁle are greatly modulated due to self-phase modulation and the

DFG process. The two processes have the eﬀect of leaving the pump with a “doughnut-like”

spatial mode and depleting the spectral intensity around the central frequency.

71 (a) (b)

Figure 28: OPA output (a) pulse energies and (b) pulse durations. The most power at a single wavelength is at 1300 nm, but the shortest pulse duration is centered around 1500 nm.

5.4 Interferometers

In order to perform our experiments, we utilize two separate interferometers. We will

ﬁrst describe the HHG and molecular alignment interferometer that allows us to generate harmonics from aligned molecules. We will then move on to describe the “attobeamline” which contains the interferometer used to measure the phases of extreme ultraviolet (XUV) spectra.

5.4.1 HHG-Kick Interferometer

Figure 29 shows the optical table that leads to the experimental apparatus. It holds the compressor, OPA, and interferometer used to spatiotemporally overlap the HHG and molecular-alignment kick pulses. For all of our experiments, the HHG pulse is comprised of either the signal or the idler and the depleted pump is used for molecular alignment when applicable. In order to control the HHG-kick relative delay, we use a motorized delay stage. Two turning mirrors after the delay stage allow for spatiotemporal overlap without detuning the retroreﬂection of the delay stage, and a zero-order half waveplate is used to control the polarization of the kick relative to the HHG. At the exit of the interferometer, the two beams are horizontally separated and coparallel. This type of interferometer allows for

72 Figure 29: A schematic of the compressor, OPA, and HHG-kick interferometer. The output of the compressor pumps the commercial HE-TOPAS OPA. The output of the OPA consists of the collinear signal and idler (separated by a not-pictured wavelength separator) and a depleted pump. The signal or idler is used for HHG and the depleted pump is controlled by a variable delay and half waveplate for use as a “kick” in molecular alignment experiments.

simple wavelength tuning of the signal or idler with little change to the depleted pump, which provides consistent molecular alignment conditions across a range of HHG wavelengths.

The most important update of this interferometer compared to previous iterations is that the HHG and kick pulses share all of the same optics after leaving this table, aside from the beamsplitter seen in Figure 30. In previous iterations, the HHG-kick interferometer was spread out over two optical tables and the pulses were recombined in vacuum. This made for diﬃcult spatial overlap and unstable temporal overlap. Completing the interferometer on one table is of course more stable, and because these beams share the same focusing optic, simple coparallelization is nearly suﬃcient to have the beams spatially overlapped in the HHG interaction region.

5.4.2 Attobeamline for Two-Omega and Omega Oscillation Measurements

A schematic for the attobeamline, the primary experimental apparatus, can be seen in

Figure 30. Here we will give an overview of the apparatus, with focus on upgrades in design and operation. For a more detailed description of the apparatus, refer to Stephen Schoun’s

73 and Razvan Chirla’s dissertations [28][106]. The attobeamline is comprised of an XUV-IR

Mach-Zehnder interferometer and magnetic bottle electron spectrometer (MBES) that allows for measurement of XUV-IR delay-dependent, energy-resolved ionized electron yields.

Both the interferometer and the MBES are entirely contained in vacuum chambers symbolized by black boxes in Figure 30, which are necessary to reduce atmospheric absorption of the XUV light and scattering of the ionized electrons. The chamber containing the gas jet is referred to as the “generation chamber” and the other chamber containing the toroidal focusing mirror is referred to as the “mirror chamber.” The vacuums in the two chambers are maintained by a 1300 L/s turbomolecular pump on the generation chamber, a 360 L/s pump differential pump between the two chambers, and a 500 L/s pump on the mirror chamber. The generation and mirror chamber turbo pumps are backed by the same rough pump system, located in a different room, comprised of a high-pumping-speed blower backed by a high-throughput rotary piston pump. The differential pump is separately backed by a small rotary vane vacuum pump. Gas pressures in the generation chamber and mirror chamber during operation are typically in the few 10-3 torr and 10-6 torr, respectively.

The HHG and kick pulses enter the generation chamber through the same wedged window (not pictured), that is wedged in order to reduce dangerous back reﬂections returning to the OPA. From there the HHG pulse is incident on an uncoated 0.75 mm thick CaF2 beamsplitter where 93% of the beam is transmitted when the beam is s-polarized and

99.5% when the beam is p-polarized. The transmitted beam is used for HHG whereas the reﬂected beam, hereafter referred to as the “dressing beam,” is rerouted to be spatiotemporally overlapped with the XUV in the MBES. Previous iterations of the interferometer used wavelength-speciﬁc beamsplitters that made changing wavelengths cumbersome because each time the wavelength was changed the beamsplitter had to be changed. The thin

CaF2 beamsplitter imparts little GD and GDD for mid-infrared wavelengths, allowing us to keep temporal overlap and pulse durations as short as possible across our entire OPA spectrum. Additionally, CaF2 reﬂects nearly uniformly across the OPA spectrum allowing for similar generating and dressing conditions across diﬀerent wavelengths.

74 Figure 30: The attobeamline and an example RABBITT scan. The attobeamline directs the majority of the driving pulse (dark purple) for harmonic generation and saves a small amount of the pulse called the “dressing” (light purple) for recombining with the XUV in the MBES. The alignment pulse (red) avoids the CaF2 BS and is focused using the same optic as the HHG driving pulse. After generation, the alignment pulse is clipped by an aperture and the HHG laser is filtered by a metallic filter. The XUV and dressing are then recombined at HM and refocused by the TM into the MBES for RABBIT scans like the one pictured on the right. Key: BS = beamsplitter; Filter = Al or Zr metallic filter; HM = hole mirror; TM = toroidal mirror

First let us consider the HHG arm of the attobeamline. The HHG beam transmitted through the splitter and the kick beam are focused into the HHG gas jet by the same 2-inch diameter concave Ag mirror with varying focal lengths of 300, 400, and 500 mm depending on the experiment. The reflective focusing mirror was chosen over previous lens-based geometries in order to increase the achromaticity of the attobeamline. Specifically, using a reflective geometry provides for simpler, more repeatable HHG wavelength scanning, and it allows for the HHG and kick pulses to focus to the same position along the k-direction for molecular-alignment experiments.

We use two types of gas nozzles for our experiments: a 200 um continuous steel nozzle for neat samples at low pressures and a pulsed 200 um Even-Lavie (EL) valve [107] for samples diluted in carrier gases or other high pressure conditions. Using a pulsed jet relieves the pumping load on the turbo pumps by greatly reducing the total throughput of the nozzle.

We typically run the EL valve with opening times between 20-35 µs at a repetition rate of

1 kHz, to match that of our laser repetition rate. Doing so provides suﬃcient conditions

75 to properly phase-match HHG, while also keeping the generation chamber pressure at or below 10-3 torr. The gas jet position is motorized in all three dimensions in order to provide

ﬁne-tuning to the HHG phase-matching. Typically, the gas jet and focus are adjusted so that the beam waist is located 1-2 mm upstream of the gas jet, which is known to give the best phase-matching for the short trajectories [108] [69].

Overlapping the HHG and kick beams can be extremely difficult, sometimes taking hours to find spatiotemporal overlap, which is typically detected through a modulation of the high harmonic spectrum. In order to relieve some of the difficulty, we implemented a

2f-2f imaging apparatus, a schematic of which is shown in Figure 31(a). This allowed us to image the plasma ﬂuorescence induced by the two beams on a simple charge-coupled device (CCD) camera as seen in Figure 31(b). Because the imaging setup was horizontally displaced from the foci, the camera can only ﬁne-tune the vertical spatial overlap. The horizontal overlap is unable to be optimized with the camera except for when increased

ﬂuorescence is observed, indicating that the beams are both ionizing the same volume of the gas jet. It was discovered, though, that once the spatiotemporal overlap was found, it remained nearly overlapped day-to-day such that only minor adjustments were needed regain overlap. The adjustments now only take ﬁfteen minutes to a half hour to optimize.

After the gas jet, the infrared and XUV are incident upon a thin metallic ﬁlter of thickness typically between 100 and 200 nm where the infrared is totally reﬂected and the

XUV is transmitted to varying degrees. We typically use two diﬀerent types of ﬁlters, Al or

Zr, depending on what spectral range we are interested in studying. The transmissions of

Al and Zr filters and their corresponding imparted XUV GD can be seen in Figures 32(a) and 32(b), respectively. The Al filter typically transmits XUV well in the 20-72 eV range, with a slow turn off at low energies and a sharp cut off at 72 eV due to the Al L2,3-edge. The amplitude modulation due to the Al edge is useful for calibrating the energy scale of retrieved electron energy spectra. Zr on the other hand begins transmitting XUV around

60 eV and continues to until approximately 200 eV. For analysis of RABBITT experiments, it is necessary to calculate and remove the GD imparted by the ﬁlters as will be detailed

76 (a) (b)

Figure 31: HHG focal spot imaging. (a) The 2f-2f setup for 1:1 reimaging of the fluorescence from the HHG source. (b) An image taken of the fluorescence from the HHG source. When both the HHG pulse and the kick pulse are present, two filaments can be seen. That makes this setup indispensable for spatially overlapping the two pulses. later in this chapter.

The diverging XUV is then incident on a 5 mm or 10 mm hole mirror and then a f=750 mm toroidal focusing mirror that focuses the XUV into the MBES. The hole mirror serves as a beam recombiner for the XUV and dressing and also filters any long trajectory contributions as they are more divergent than the short trajectory contributions [44]. The toroidal mirror is used in a 2f-2f configuration, meaning that the HHG focus is 2 focal lengths upstream of the toroidal mirror and the reimaging focus in the magnetic bottle is two focal lengths downstream. This is done in order to have 1:1 imaging of the HHG source X in the MBES. Additionally, the toroid is used at a glancing angle (θi ¤ 75 ) for increased reflection of the XUV light.

Now considering the dressing arm, we propagate the dressing beam through two matched

4X Infrasil (fused quartz) wedges. These wedges are doubly motorized with a 25-mm-travel

Thorlabs DC servo actuator (Z625BV by Thorlabs) and a 500-µm-travel piezo-actuated

stage (P-625.1CL from Physic Instrumente) to control coarse and ﬁne temporal overlap

between the XUV and IR in the MBES, respectively. From there, the dressing beam is

77 (a) (b)

Figure 32: Al and Zr Filter Properties in the XUV spectral region. (a) Filter Transmission. (b) Imparted Group Delay. The transmission curves and the Zr GD were calculated using refractive indices calculated from [109], which are also available through the Center for X- Ray Optics (CXRO) “X-Ray Interactions with Matter” database. The Al GD was calculated using an Al refractive index that was calculated by Rakic [110].

focused by a plano-convex f=500 mm CaF2 lens so that the dressing is also imaged 1:1 in the MBES using the 2f-2f conﬁguration of the toroidal mirror. The dressing beam then recombines with the XUV on the hole mirror so that the two copropagate to the MBES.

Before describing the MBES, it is important to note that the dressing intensity in the

MBES must be kept weak, ¤ 1011 1012 W ~cm2, in order to restrict the nonlinear detection process to the perturbative regime. To this end, Dietrich Kiesewetter created a motorized variable aperture that can control the dressing intensity. This aperture is placed after the imaging lens but before the toroidal mirror as seen in Figure 30.

5.5 Magnetic Bottle Electron Spectrometer

The detection apparatus used for all of our experiments is the MBES, a schematic of which can be seen in Figure 33. In such a detection apparatus, the XUV and IR are focused by the toroidal mirror just above a thin capillary gas source referred to as the “needle.”

Electrons are ionized from the gas by the XUV or combined XUV + IR field and released into a magnetic field created by a 1-inch diameter “flat” cylindrical Neodymium magnet.

From there the electrons are guided along helical trajectories around the magnetic ﬁeld lines

78 Figure 33: A schematic of the MBES. An inert gas is emitted from the gas needle where XUV light pointed out of the page is incident. An electron is then ionized at an initial angle θi relative to the TOF axis. From there the electron is guided along the magnetic field lines of the permanent magnet onto the field lines of the solenoid. The electron travels along these until it is incident on an MCP detector at a final angle θf . The vertical slit is used for increasing the TOF resolution of the MBES. It does this by restricting the detected electrons to a small solid angle, thereby restricting the detected ionization volume to a small region near the laser focus. of the Nd magnet and smoothly transferred onto the magnetic field lines of a 1 meter-long, 1 wire-loop/mm, solenoid that enshrouds the flight tube all the way to the MCP detector. The

flight tube solenoid is additionally surrounded by a Faraday cage and mu metal shielding in order to reduce the effects of external electric and magnetic fields, respectively. The key benefit of using a MBES is that it has high detection efficiency compared to other field-free time-of-flight (TOF) detection apparatus, and it preserves the energy of the electrons for electron spectra calibration. We will proceed by breaking down the critical aspects of the

MBES. For a more complete description of the design and operation of this particular MBES, please refer to Stephen Schoun’s dissertation [28] and Christoph Roedig’s dissertation [111].

For a general description of MBES operation, please refer to the paper by Kruit [112].

5.5.1 Principles of Operation

As mentioned above, the MBES relies on guiding ionized electrons along magnetic ﬁeld lines to an MCP detector. The electrons are freed near the surface of the 5 T permanent magnet and then are quickly transferred over a short distance to the solenoid’s relatively

79 weak magnet ﬁeld of ¤ 0.5 mT without changing the electron’s kinetic energy.16 The electron’s initial kinetic energy is calibrated by the TOF of the electron down the solenoid- contained ﬂight tube to the detector using the equation

L2 1 tube Ecal m 2 , (91) 2 tflight where m is the electron mass, Ltube is the length of the flight tube from ionization to detection, and tflight is the TOF down the tube. The TOF is determined by the electron’s velocity projection along the primary axis of the time of flight vfz v cos θf . The closer vfz is to the total speed v, then the more accurate a calibration of the electron’s energy we can retrieve. How close vfz is to v can be determined through conservation of angular momentum considerations in the following way; since we know that angular momentum of an electron is conserved in the presence of magnetic fields we can say that

li lf , (92) and then deduce that m2v2 m2v2 sin2 θ sin2 θ . (93) 2 i 2 f q Bi q Bf

The ﬁnal speed along the TOF axis, z, is then given by

¼ 2 vfz v 1 Bf ~Bi sin θi. (94)

3 In our case Bf ~Bi ¤ 10 so that the TOF calibration of the electron’s energy is accurate to within a fraction of a percent.

Another requirement of the MBES is that the magnetic field of the magnet adiabatically transfers the electron to the magnetic field of the solenoid so that the electrons do not hop onto extraneous magnetic field lines. We had just argued, though, that the most accurate

TOF reconstruction of the electron energy needs the most disparate magnetic ﬁeld values

16This is because, of course, magnetic ﬁelds do no work.

80 possible between the magnet and solenoid. These contradictory requirements necessitated modeling calculations in order to design the optimal MBES geometry. As shown in Stephen

Schoun’s Dissertation [28], it was found that d 50 mm from the interaction region to the solenoid TOF tube was optimal for achieving adiabatic electron trajectory transfer while still allowing for eﬀective pumping of the interaction region in the needle chamber.

When running an experiment, we optimize the magnetic field in two ways. First, we adjust the position of the permanent magnet with an initial solenoid current. This allows us to maximize electron counts on the detector while maintaining a reasonable TOF signal, i.e. resembling an expected HHG spectral profile. After adjusting the permanent magnet position, we then adjust the solenoid current to maximize counts and optimize the spectrum in a similar way. Increasing the solenoid current most greatly affects the spectral intensity of the lowest energy electrons as these are the most easily gathered by the magnetic field.

Typical current values are between 0.1 A and 0.5 A.

5.5.2 Resolution

An important parameter of the MBES is the temporal resolution because it is related to the energetic resolution in the following way

∆E ∆T 2 constant. (95) E T

The temporal resolution of the MBES TOF is mostly dominated by the angular distribution of the collected electrons. Through modeling calculations completed by our group, it is known that the “ﬂat” magnet is able to gather trajectories with initial ionization angles

0X B θ B 106X. The resolution of the bottle is then determined by the MBES’s ability to parallelize the diﬀerent trajectories quickly over a short distance so that the diﬀerent trajectories travel with nearly the same TOF. It was found by Schoun that the TOF error ∆E can theoretically range anywhere from 0.6% to 3.6% depending on the asymmetry E parameter of the detector gas, but in practice is seemingly no larger than ¤ 1.3% based on

the widths of harmonics taken under diﬀerent HHG conditions.

81 An additional contributor to the MBES TOF resolution comes from the spread in ionization across the focal volume of the XUV. Referring back to Figure 33, the ionization volume is restricted by the focal spot size along the x and z axes to a few hundreds of microns at most, which is small compared to Ltube. The same is not true along the propagation direction of the XUV, in and out of the page, which can be multiple millimeters in length. This could lead to a signiﬁcant decrease in temporal resolution such that it was decided to add a vertical slit that is 1 mm wide and 10 mm tall approximately 3 cm away from the interaction region. This slit restricts the angular acceptance from electrons born along diﬀerent parts of the laser propagation direction, therefore increasing the MBES TOF resolution.

5.5.3 Detection Gas

The measurement as described above begins with the detection gas, so it is valuable to describe some of the key features of the different gases. We use inert gases as our detection gases, most typically Ar, Ne, and He. The ionization potentials (IPs) for these gases along with Kr and Xe are shown in Table 2. The two different values per atom correspond to the different IPs resulting from the spin-orbit interaction. For experiments with Ar, Ne, or He the spin-orbit splitting is small enough or nonexistent such that the difference in IPs of the two different final states is on the order of or smaller than our TOF resolution, so whether

Detection Gas Ionization Potential IP (eV) Xe 12.13 (13.44) Kr 14.0 (14.6) Ar 15.76 (15.94) Ne 21.56 (21.66) He N/A (24.59)

Table 2: Ionization potential values for various inert gas detectors. The first values are for final ion states with total angular momentum of J=3/2, and the values in parentheses are for final ion states with angular momentum J=1/2. It is clear that for Kr and Xe, the splitting is much larger than our spectral resolution (¤ 100 meV). This results in two separated electron peaks for every XUV photon. Additionally, this has the added complication of harmonic peaks overlapping with sidebands for certain driving wavelengths.

82 Figure 34: Calculated cross sections for typical inert gas detectors. Ar easily has the largest cross-section at low energies, but it drops dramatically around 50 eV due to the presence of the Cooper minimum. He and Ne both have smoother-varying cross sections, but He has a much smaller cross section than Ne across the entire energy range shown. This makes Ne the most desirable detection gas for HHS experiments. or not we use the J=3/2 or J=1/2 IP is negligible. This is not the case for Xe and Kr, where the spin-orbit splittings are on the order of MIR photon energies. Because these are so large, sidebands from one ﬁnal state can overlap with a harmonic of another other ﬁnal state, completely obscuring our phase retrieval. Due to this, we typically avoid using Kr and Xe as regular detection gases.

Another important aspect of the detection gas is the cross-section, examples of which are plotted in Figure 34. Ar has a very large cross-section at low energies, but sharply drops oﬀ in the presence of the Cooper minimum centered around 50 eV [113]. Ar also has autoionizing resonances (to be explored in Chapter 7) located very near the peak of its cross section. On the other hand, Ne and He have weak but smoothly varying cross sections. Because the features in Ar can greatly modify the spectral intensity and phase of the measured electrons and He’s cross section is so small, we have chosen to use Ne as the detection gas of choice for our experiments in Chapters 6 and 8. Ne and He also have autoionizing resonances, but they are much smaller modulations on the spectral intensity and phase when compared to the features in Ar. Ne and He both do not exhibit Cooper

83 minima, a well-known fact [114].

5.5.4 Data Acquisition Hardware and Software

After the electrons are incident on the MCP detector, a voltage signal is sent to and digitized by an Agilent Acqiris U1065A 10 bit, 8 GS/s analog-to-digital converter. The digitizer resolution is determined by its bin width of 0.125 ns, which is comparable to the 1 ns rise time of the MCP. The Acqiris is triggered at 1 kHz, the repetition rate of our home- built laser system, by a photodiode with a 1 ns rise time that is triggered oﬀ scattered 800 nm light after the compressor but before the OPA.

The Acqiris then sends a digitized voltage signal to our data acquisition computer. We utilize in-house acquisition software that was written in the C# programming language and can keep up with the 1 kHz repetition rate of our laser. The program preprocesses the data in two ways: averaging mode and constant fraction discrimination (CFD) mode. In averaging mode, every voltage signal above a threshold value, typically 0.15 V,17 is summed for a predetermined number of shots and then divided by the total number of shots. Every signal less than 0.15 V is set to zero to eliminate dark counts. In CFD mode, a threshold level at a constant value of the rise time, usually 50%, is used to count the number of electron hits per laser shot in each TOF trace. The hits are added up over the predetermined number of shots, creating a histogram representing the TOF trace. CFD mode also uses the 0.15

V threshold to eliminate dark counts. A block diagram summarizing the data pathways is shown in Figure 35.

When deciding which mode to use, it is important to know the beneﬁts and drawbacks of each method. Let us ﬁrst consider spectral resolution. CFD mode typically gives better spectral resolution at higher energies than averaging mode. This is because each MCP voltage spike has a 1 ns rise time, and a tail of multiple nanoseconds. In CFD, a count is binned at the 50% rise time, so the long tail is removed from the CFD histogram. On the other hand, the averaging mode incorporates this long tail so that high-energy harmonic

17In actuality, the voltage signals are negative, but the minus sign has been dropped for simplicity.

84 Figure 35: Information pathway of voltage data from the TOF detector into averaged and CFD data sets. Both CFD and averaged data sets are saved for each measurement. peaks and sidebands begin to overlap to the point of indistinguishability. For our apparatus, this is typically a problem for energies above 100 eV, but because our experiments are mostly focused at energies below 70 eV, this problem is not relevant to our experiments.

The other important difference between the two modes is how they handle high and low count rates. In the high-count-rate regime, multiple electrons are incident on the detector at the same time. This phenomenon is called “pulse pile up.” The CFD is unable to account for the increased number of counts because it only looks for a 50% rise time and does not detect the absolute amplitude increase due to multiple hits. Averaging mode is able to account for pulse pile up because it incorporates the increased voltage peak height. In the low-count- rate regime, a different problem arises for the CFD mode we call the “grass effect.” In this case, the binning of the CFD redistributes a continuous voltage signal across discrete peaks separated by bins filled with zero counts. These zero count bins produce extra noise that make it difficult to extract oscillation phases from scans comprised of these traces. Many of our data sets involve large dynamic ranges that have both high count spectral regions and low count spectral regions. Because of this we have decided to use the averaging mode for our primary data sets, and we use the CFD traces as a comparison when applicable.

5.6 Omega and Two Omega Acquisition and Analysis Procedures

In the following, we will break down the acquisition and analysis procedures for both the experiments studying omega oscillations and the experiments studying two-omega oscillations (RABBITT).

85 Figure 36: A diagram showing the allowed polarization orientations for omega and two- omega oscillation experiments. Two-omega oscillations will be detected with either vertical or horizontal XUV+IR polarizations due to the even parity of the RABBITT interaction. Omega oscillation measurements, however, require a laser polarization along the TOF axis due to the odd parity of the interference responsible for it.

5.6.1 Data Acquisition Procedures

As detailed back in Chapter 3, the RABBITT oscillation is of even parity and the omega oscillation of odd parity. This means that electrons ejected into the upper and the lower hemispheres along the XUV polarization oscillate in phase for the two-omega oscillation and out of phase for the omega oscillation. So if we collect equal portions of electrons from the upper and lower hemispheres, we will measure a two-omega oscillation, but the omega oscillation will be eliminated. We know from the previous section that the

MBES has ¤ 2π steradian collection volume. Because of this, the XUV can have vertical or horizontal polarizations in the MBES for a two-omega experiment, but can only have horizontal polarization for an omega oscillation experiment as seen in Figure 36. With horizontal polarization we only collect the upper or lower hemisphere of ionized electrons in the omega oscillation experiments, but not both. More carefully, any polarization that does not lead to collection of equal portions of electrons from the upper and lower hemispheres will work, but in practice we are restricted to vertical or horizontal polarization by our optical layout.

86 Before starting XUV-IR delay scans, we make sure to center the XUV temporal envelope onto the envelope of the IR dressing ﬁeld by using the coarse adjustment of the wedges to delay the IR dressing ﬁeld. We do this by centering our scan on the delay envelope of the spectral sidebands. Doing so helps us eliminate femtochirp contributions, which are dependent on the rate of change of the envelope intensity, from our retrieved harmonic phases. For two-omega scans, we record 10 cycles of two-omega oscillations with ¤7 points per oscillation. For omega scans, which are twice as long as two-omega scans, we sometimes run into the limit of our piezo stage depending on the driving wavelength so we typically record 7-8 cycles with 7 points per oscillation. Either situation allows us to clearly separate the relevant frequency from any neighboring frequencies.

Additionally, for every scan we carefully limit the intensity of our infrared dressing ﬁeld, controlled with the motorized aperture shown in Figure 30. Too much dressing intensity introduces higher order two-omega and omega interference processes that complicate the interpretation of our extracted phases. We gauge our dressing intensity by the appearance of higher frequency oscillations: four omega in the case of RABBITT and three omega in the case of an omega oscillation, which would be accompanied by extra two-omega or omega oscillations in the perturbative regime, respectively. Therefore, in every scan, we limit our dressing intensity to remove any signs of higher-order oscillation frequencies that can be seen through an energy-resolved fast Fourier transform (FFT) or an energy-integrated FFT. In experiments, this typically corresponds to limiting the delay-averaged sideband intensities to less than 50% of the neighboring harmonic intensities. An example of a retrieved RABBITT spectrogram and the corresponding FFT along the delay dimension for CO2 at 1300 nm can be seen in Figure 37.

5.6.2 Data Analysis: Extracting Oscillation Phases

Now that we have detailed our acquisition methods, we will move on to our analysis methods. At the end of each omega or two-omega delay scan, we retrieve a two-dimensional matrix with delay on one axis and TOF on the other. Our ﬁrst steps in analyzing a matrix

87 (a) (b)

Figure 37: (a) RABBITT spectrogram and (b) the same spectrogram after an FFT along the delay axis. In (a) two-omega oscillations are clearly visible in the lower-intensity sidebands though are washed out in the harmonics due to the color scale. After an FFT, we can see that both sidebands and harmonics are oscillating. Additionally, we can see that there is no sign of higher frequency oscillations. are the following: 1) turn a TOF signal vs. TOF into spectral intensity vs. photon energy;

2) choose the spectral regions for phase retrieval; and then 3) retrieve the phase with four diﬀerent methods. These steps of analysis are very common in our research group, so much so that Dietrich Kiesewetter wrote a general analysis software package in the Python programming language that can be applied to multiple experimental setups in the lab. The software package is comprised of two graphical user interface (GUI) programs, one used to convert experimental TOF signals to spectral intensity and the other for choosing the proper regions for phase retrieval, and a script that rapidly retrieves oscillation phases in multiple ways. Additionally, he has created a GUI for quick and simple viewing of GD data sets, allowing for rapid feedback. These pieces of software were critical for eﬃcient real-time analysis in these experiments.

In the ﬁrst GUI, the TOF data is converted by the Jacobian method to spectral intensity.

The energy axis is manually ﬁt by adapting Equation 91 in the following way:

½ m Ltube t t0 º , (96) 2 E IP E0

88 where t t0 is the TOF, t0 is the trigger delay, m is the electron mass, Ltube is the length of the flight tube, E is the kinetic energy of the electron, and IP is the ionization potential of the detection gas. E0 is an arbitrary energy offset that accounts for stray fields or other unknown shifts to the TOF-to-energy calibration. The fit parameters used are t0 and E0, whereas Ltube is fixed by the experimental geometry. Using Equation 96 then converts each TOF time bin to an energy bin. The Jacobian method creates many small energy bins at low energies and wide bins at high energies. This often creates highly pixelated spectral data at low energies, much like the “grass” effect described above. This is undesirable, so the spectral data is then rebinned to uniformly-sized energy bins typically 20 meV in size, effectively smoothing the lower energy regions.

After transforming with the Jacobian, our 2D matrix, hereafter referred to as “harmonic scan,” has one axis of delay and the other of photon energy. The next step is to ﬁnd the spectral regions over which to perform our phase analysis. This is done by performing an

FFT at each energy step of the harmonic scan and plotting the oscillation peak intensity and phases as seen in Figure 38. Red windows are placed on sidebands and green windows on odd harmonics through a peak-finding algorithm. Within each window we find the FWHM of each harmonic peak, represented by the shaded region; from this region we will extract the phase. If a harmonic or sideband is so heavily modulated that it is difficult to discern the FFT peak, then we use the delay-averaged scan as a reference to find harmonic and sideband peaks. If these are also difficult to discern (like in the case of weak signals), we try to restrict our phase extraction to a region of relatively flat, well-behaved phase. If these criteria cannot be met, then we exclude the sideband or harmonic from our analysis.

Before moving on, it is important to clarify that our delay wedge step size is not precisely calibrated to the XUV-IR temporal delay. Because of this, we do not exactly know the frequency at which we identify the oscillation peak in frequency space. However, reasonable estimates using the refractive indices of our wedges and physical considerations of our two- photon processes in the MBES allow us to deduce the frequency at which the peak appears.

This is how we identify all omega or two-omega peaks in the following analysis.

89 Figure 38: Example of windowing for RABBITT data in unaligned OCS @ 10% in He for 1250 nm. In the top plot, the FFT two-omega intensity is shown versus photon energy. In the bottom plot is the corresponding oscillation phase. Red windows correspond to sidebands and green windows correspond to odd harmonics. The phases are extracted from the shaded regions, which are the FWHM of each oscillation peak.

90 Once we have selected the regions over which to extract the phase, we retrieve the phase using the four following methods.18

1. FFT: In this method we apply a Blackman window along the delay dimension for

each energy bin in order to remove any high frequency content from the scan edges.

We also subtract the delay-averaged mean for each energy bin to remove the DC peak

in FFT space. After these steps we perform an FFT for each energy bin, ﬁnd the

relevant oscillation peak (omega or two-omega), and extract the phase. We then use

the phase values from each energy bin in the shaded region to calculate an intensity-

weighted average phase and standard deviation for the ﬁnal reported phase and error

bar.

2. Integrated FFT: We use the same Blackman windowing and DC subtraction as the

FFT method. The main diﬀerence here is that each harmonic is integrated within

the shaded region for each delay step before an FFT. After the FFT, we extract the

oscillation phase from each FFT frequency bin across the FWHM of the oscillation

peak. The ﬁnal reported phase and error are the intensity-weighted average phase

and standard deviation across the FFT oscillation peak.

3. Fit: Here we ﬁt the delay-dependent oscillation for each energy bin to the following

equation

S C0 B0 cos 2ω0τ ∆φ¥ . (97)

This takes into account DC contributions. The phase oﬀset contains the phase con-

tributions that we seek. The ﬁnal reported phase and error are the intensity-weighted

average phase and ﬁt error across the shaded region.

4. Integrated Fit: This method integrates the sideband or harmonic at each delay step

within the shaded region. The integrated peak is then ﬁtted versus delay using Equa-

tion 97. The final reported phase and error bar are the resultant fit phase and fit

18As a reminder, only the sidebands are analyzed in RABBITT, whereas the even and odd harmonics are analyzed when studying omega oscillation phases.

91 error.

For all RABBITT traces we compare the results from each method. If the four diﬀerent methods produce similar results, then we treat the data as high quality and proceed with analysis. If one or more of the methods disagrees for reasons that we cannot explain, we throw the data set out. When analyzing omega oscillations, we only use the FFT and the

Integrated FFT methods because the ﬁtting program equation did not allow for multiple oscillation frequencies.

As discussed in Chapter 3, correctly extracting spectral and atomic phase contributions from omega oscillations is still being researched. Instead we focus on comparing the omega oscillation to theory and do not further process the omega oscillation phases. On the other hand, separating the spectral and atomic contributions to the two-omega oscillation is possible, and it will be described in the following section.

5.6.3 Data Analysis: Separation of Delay Contributions in RABBITT

At this point in the analysis we have a phase value for each sideband, but need to separate the contributions to each sideband phase. We ﬁrst unwrap the harmonic and sideband phases together and then extract only the sideband phases. Next we apply the same discrete derivative approximation in Chapter 3 for each sideband q

∆φ τq ¡ . (98) 2ω0

We can then expand τq in terms of its diﬀerent delay contributions

τq τtarget τSFA τat τfilt, (99)

where τtarget is the molecule-speciﬁc delay from the recombination dipole matrix element dz,ion, τSFA is the HHG propagation phase contribution (Section 2.2.7), τat is the atomic delay due to the detection process, and τfilt is the delay acquired by the XUV after passing through the ﬁlter. In an HHS experiment we are interested in τtarget. It is important for us

92 to extract τtarget quickly in order to have real-time feedback. To this end, I have written a GUI in the Matlab programming language that allows for rapid calculation and removal of τat, τfilt, and τSFA from each τq. In the following we will detail how we calculate and remove the those contributions

We begin by calculating and removing the atomic delay from each experimentally- retrieved sideband delay. The atomic delay for 800 nm light has been accurately calculated for Ar, Ne, and He in Mauritsson et al. [115]. If we assume that the atomic delay can be separated into the contributions from the Wigner delay τη and continuum-continuum delay

τcc [116] then we can write

τat τη τcc. (100)

τη is given by ηl κA ηl κ@ τη , (101) 2ω0

where ηl is the Wigner-like phase for a particular angular momentum channel l, κA is the wavenumber of the intermediate state from the higher harmonic energy and κ@ is for the lower harmonic. τcc is given by

φcck, κA φcck, κ@ τcc , (102) 2ω0 where φcc is the continuum phase acquired from an IR transition from an intermediate state with wave number κ@,A to the final state with wave number k. This assumption of separability is valid for electron kinetic energies that are much larger than zero, because in these cases the atomic potential is less likely to affect the phase of the outgoing electronic wave packet. τη can be treated as wavelength independent, but τcc cannot. Because of this we cannot directly apply the atomic delay from Mauritsson’s calculation. Luckily, Dahlstrom et al. [75] have developed an analytical formula for τcc that includes the wavelength dependence. Using this we can do the following calculation to retrieve our wavelength-specific

93 atomic delay19

τat,exp τat,800nm τcc,800nm τcc,λ, (103)

where we have removed the 800 nm τcc,800nm and replaced it with the experiment’s wavelength- speciﬁc τcc,λ.

In Figure 39 we can see τat at 800 nm for Ne extracted from Mauritsson’s paper. Un- fortunately the calculated values by Mauritsson et al. only extend to 50 (Ar) or 80 eV (Ne and He) and only reach down to 2-3 eV above threshold. Often times our data sets extend beyond 50 eV and they sometimes reach under 2-3 eV. Because of this, τat has been ex- trapolated towards negative infinity at low energies and towards zero at high energies using power laws. These are reasonable because we expect the atomic delay to approach negative infinity near threshold and we expect the atomic delay to go to zero at high energies due to the diminishing effect of the atomic potential. Even though these extrapolations are indeed reasonable, they are assuredly not accurate at low energies. Any results at low energies that are affected by this extrapolation deserve extra scrutiny. For this dissertation, none of the main results rely on the lowest-energy data points.

After the atomic delay has been calculated and removed, we remove the delay contribution from the metallic ﬁlter. For essentially all of our experiments, we used a 200-nm-thick

Al foil to filter our XUV from our IR generating laser. In order to calculate the filter delay we used refractive indices calculated by Rakic [110]. The phase acquired by the harmonics passing through the filter is given by

Ò φfilt E nfilt E 1 E Lfilt~ ch , (104)

where nfilt E is the energy-dependent XUV refractive index, E is the XUV photon energy, Ò Lfilt is the thickness of the ﬁlter, c is the speed of light and h is the reduced Planck constant.

The true GD associated with this phase would be calculated by dφfilt~dE, but it is important to remember that we do not actually measure GD. We must calculate the discrete derivative

19This procedure for calculating our wavelength-speciﬁc atomic phase is the same as the one used by Stephen Schoun in [28]. For additional information, please refer to it.

94 (a) (b)

Figure 39: 800 nm atomic delay calculations. In (a) we see the absolute value of τat. The original calculation is only available over a restricted energy range, which requires extrapolation to higher and lower energies. The final atomic delay after extrapolation is also interpolated to a finer grid. In (b) the signed values of the atomic and continuum- continuum delays are shown. τcc at 800 nm also requires some extrapolation at lower energies for ease of application to MIR data sets where electrons can have kinetic energies smaller than the energy of an 800 nm photon. using the harmonic comb spacing just like we do in our RABBITT analysis. This is because the harmonics do not sample the GD profile of the filter, they sample the phase profile of the filter and the RABBITT measurement itself then acts as the discrete derivative of those discrete phase samplings. This type of difference only matters when the GD varies much more rapidly than the harmonic spacing, which is only the case near the Al L2,3-edge at 72 eV for our data sets.

An additional consideration when calculating φfilt is oxidation of the ﬁlter. According to a study by Powell et al. [117], one can expect approximately 15 nm of Al2O3 oxide capping layers on either side of our Al foils after long-term exposure to atmospheric conditions.

Though we take great care to limit atmospheric exposure, the ﬁlters see the atmosphere fairly regularly as we often have to vent the attobeamline to perform regular maintenance.

According to Powell, 15 nm of oxidation most decreases the overall transmission of the foil, most dramatically altering the spectral proﬁle below ¤ 18 eV. Overall, the XUV transmission with or without the oxidation has a similar shape above 18 eV, so using nfilt from Rakic

95 [110] is still reasonable.

Once the atomic delay and ﬁlter delay are removed, we are left with only τSFA to remove. This is done by an iterative ﬁtting processes. First we calculate the SFA GD for the short trajectory at a given laser intensity and wavelength. We shift our experimental GD to the

SFA GD through a least squares fitting, and then calculate the absolute difference between our data and the SFA GD. We repeat this for many intensities to form an error plot and then perform a minimization search for the intensity that gives the smallest absolute difference between our data and the SFA. This SFA-removal method relies on the SFA remaining an accurate description of the HHG process for small molecules. We will show that this process works well for small molecules like CH4 and CH3Cl in Chapter 6 and then we will apply it to the molecules studied in Chapter 8.

96 Chapter 6: High-Harmonic Spectroscopy of the CH3Cl Cooper Minimum

“All the proof of a pudding is in the eating.”

—William Camden, Remains Concerning Britain

6.1 Atomic Probes of Molecular Electron Dynamics

For much of its history, attosecond science and techniques for molecules have predominantly been calibrated and optimized in atomic targets [13]. Those same techniques are now being applied to more complex, biologically-relevant molecular systems [118] and solid targets [119]. As the complexity of the target systems increases, it is important to reassess the applicability of those techniques and methods to study electron dynamics. A particular phenomenon of recent interest in molecular attosecond science is the study of charge migration [17]. Charge migration is the electron-correlation-driven motion of a hole created after ionization from one spatial location of a molecule to another, and it is predicted to eﬀect phenomena relating to photovoltaics, biological processes, and chemical reactions

[17][18][19][20]. Kraus et al. [37] recently demonstrated high-harmonic spectroscopy (HHS) as a technique that is sensitive to attosecond charge migration. The interpretation of such results are complicated though, and requires a great amount of theoretical input. At this time in attosecond science, there is not a consensus on how such aﬀects should generally manifest in HHS measurements.

Given that attosecond science has a strong foundation in atomic targets, one way to reduce the complexity of such measurements is to study molecular targets that retain some atomic features. One such feature is the Cooper minimum (CM) [114]. A CM is a minimum in the photoionization cross section at a speciﬁc photon energy caused by a sign change in the bound-free transition dipole of a particular angular momentum channel. CMs occur for any atomic orbital that has a wavefunction with a radial node, except for the 2s orbital.

97 Furthermore, it has been shown that atoms with an atomic number Z A 10 have at least

one CM and can have as many as ﬁve CMs [120]. CMs in atoms were ﬁrst reported in alkali

atoms in 1948 [121], and CMs in molecules have been studied since the ﬁrst photoionization

experiments by Carlson et al. [122][123] in the 1980s.

Because CMs carry information about orbital nodal structure, it is easy to imagine how

such features could be used to probe attosecond charge migration. If charge were oscillating

across a molecule with a known CM, that oscillation would necessarily alter the electronic

structure, and therefore the CM. Furthermore, because CMs are ubiquitous across the

periodic table, they can in principle be used to probe dynamics in a multitude of atoms and

molecules.

6.2 Molecular Cooper Minima in High-Harmonic Spectroscopy

As described in Chapter 2, HHS probes the recombination dipole matrix element (RDME)

through the photorecombination step of high-harmonic generation (HHG), which can pro-

vide access to the spectral features of CMs. To date, there have been multiple experiments

studying the spectral intensity features of atomic CMs using HHS [124][125][126]. Prior to

the work of this dissertation, CMs were also explored in molecules by Wong et al. [127] [33]

and by Ren et al. [128]. In these studies, high-harmonic spectra were generated from N2

as well as multiple S- and Cl-containing molecules: CS2, CH2Cl2, trans–C2H2Cl2, CHCl3, and CCl4. These measurements identiﬁed and characterized minima in the high harmonic spectral intensities, which they attributed to CMs. The CMs did not always reveal themselves unambiguously, though. In the case of CS2, theoretical input was required to clearly identify the CM, and for the cases of CH2Cl2, CHCl3, and CCl4, other minima due to multi- center interferences populated the spectra in close proximity to the minima attributed to

CMs. For molecules such as these, where the HHG spectral intensities are complicated, the

measurements would have greatly beneﬁted from another dimension over which to study

the spectra. One such extra dimension is the group delay (GD), which is also available in

HHS measurements. However, a CM has only once been fully characterized in spectral

98 intensity and GD: measured in atomic Ar by Schoun et al. [10].

If we would like to extract the GD features of a molecular CM, we must ﬁrst verify that we can separate the GD of the electron propagation in HHG from the GD of the RDME.20

Rather than apply complex theories of HHG in molecules to do so, we have chosen to explore the applicability of extending atomic techniques to small molecules. In the paper by Schoun et al., the propagation GD was accounted for by calculating the classical GD due to the attochirp. Because the cutoff was far from the region of interest in that measurement, the classical GD was valid. In the case of low ionization potential (IP) atoms and molecules like the ones studied here, the cutoff is often near the spectral region of interest. This precludes the applicability of the classical calculation, due to the expected inaccuracy of it near the cutoff. Instead, a calculation of the propagation GD within the strong-field approximation

(SFA) [43] is required because it is expected to be more accurate in the cutoﬀ region.

Though GD propagation calculations are only strictly intended for use in atomic targets, they have been applied to molecules nonetheless. Their validity, however, has never been experimentally tested in molecules, and for that matter, molecules with CMs.

The applicability of the SFA GD to HHG in polyatomic molecules is nontrivial; along with distortion of the GD from CMs, interference between multiple ionization centers [129] or multiple orbitals [130][131] can aﬀect the GD. Furthermore, an implicit assumption in the SFA calculation is that the electron, following tunneling ionization, is unaﬀected by the ionic potential. The application of this assumption to species with a larger spatial extent is unclear, and it may break down in the case of molecules where the hole density can be comparable in size to the electron trajectory.

In this chapter, we study the applicability of the SFA approximation to the removal of propagation GD from HHS measurements in molecules. To this end, we have chosen to study CH4 (methane, IP 12.61 eV), a small molecule without a CM, and CH3Cl (methyl chloride, IP 11.26 eV), a CM-containing molecule. We will ﬁrst verify the applicability of the SFA GD in Xe (xenon, IP 12.13 eV), and then show that it is indeed possible to

20These contributions to the HHG GD were described in detail in Chapter 2.

99 extract target-speciﬁc spectral intensity and spectral GD information from small molecules, such as CH3Cl, in the SFA approximation.

CH3Cl serves as a good benchmark because it is a simple molecule known to contain a CM [123][132]. In fact, it has been calculated that the CH3Cl HOMO has ¤ 80% of its

population in the Cl p orbital [123]. The CH3Cl minimum, however, is not as pronounced

as the minima in the other CClnH4-n molecules, some of which were previously studied

with HHS in the studies by Wong et al. [33]. The reduced prominence of the CH3Cl CM makes it diﬃcult to unambiguously identify by spectral intensity alone: ideal for a ﬁrst test.

Additionally, its vapor pressure is high compared to that of CS2, which also has a weak

CM, making CH3Cl much more appealing from an experimental point of view.

6.2.1 Statement of Contributions

Before describing the theory of a Cooper minimum, I would like to state my direct con-

tributions to the experiments described in this chapter. All of the measurements detailed

in this chapter were performed by me with assistance from Dr. Timothy Scarborough. The

Cooper minimum spectral intensity analysis was performed by Dr. Scarborough with assis-

tance from me. The Cooper minimum group delay analysis was done by me with assistance

from Dr. Scarborough, and the algorithm used to calibrate the correct attochirp contri-

butions to our measurements was conceived and initially created by Dr. François Mauger.

The algorithm was then reﬁned and expanded by me for use in our group delay analysis.

The analysis of the intensity scans that were used to benchmark the ﬁtting algorithm was

carried out by Dr. Scarborough with assistance from me.

6.3 Theoretical Background

6.3.1 Origin of the Cooper Minimum

To gain a better understanding of molecular CMs, we will now discuss the basic at-

tributes of atomic CMs. The photoionization cross section (discussed in Section 2.2.8) of an

atom in the nonrelativistic dipole approximation is given by the incoherent sum of l l 1

100 (εs) (εd) (εs) (εd)

(2p) (3p) Ne Ar Figure 40: Diagram of the angular momentum channels in single photon ionization of Ne and Ar. According to the dipole selection rules, both ground states have two channels to the continuum: p s and p d. Only the 3p d channel for Ar, though, exhibits a CM. There is no such minimum in Ne. and l l 1 dipole matrix elements, where l is the orbital angular momentum quantum number [120]. It was understood by John W. Cooper that there is a general feature of atoms where minima occur in the l l 1 channel, depicted in Figure 40.21

The photoionization cross-section for a subshell with orbital angular momentum quantum number l, principle quantum number n, and energy nl to a continuum state with ﬁnal energy is given by [120]

2 2 4π a0α Nnl nl 2 2 σ l SR S l 1 SR S , (105) nl 3 2l 1 l 1 l 1

where a0 is the Bohr radius, α is the ﬁne-structure constant, Nnl is the subshell occupation number, and Rl1 are the dipole matrix elements of the two angular momentum channels.

The CM has its origin in the radial dipole matrix element Rl1. Both Rl1 are given by

Rl1 S Pnl r rP,l1rdr, (106) 0

where Pnl and P,l1 are the radial components of the bound and continuum wavefunctions,

respectively. For the cases of Ne, Ar, and Kr, Rl1 is several times larger than Rl1 for all

energies [134], so the cross-section is largely determined by Rl1. It is possible for Rl1 to go through a zero at a particular if the ground state wavefunction has a radial node. Model

calculations of wavefunction overlaps for Ne, Ar, and Kr between ground states Pnl r with

21 This deﬁnition is accurate for photon energies less than 10 keV [120][133].

101 (a) (b)

Figure 41: Model calculations of atomic CMs. In (a), the ground state wavefunctions for Ne, Ar, and Kr are depicted along with their corresponding 0 d-waves. The ground state of Ne does not contain a radial node, whereas the ground states of Ar and Kr do. Thus Ar and Kr exhibit CMs. In (b), Rl1 are shown for all three atoms. The sign changes occur at ¤ 2 Ry in Ar and ¤ 2.5 Ry in Kr. Adapted from [114].

p-parity and outgoing zero-energy 0 d-waves P,l1r are shown in Figure 41(a). As is increased, the d-wave will “move in,” like an accordion, and at a particular energy there

will be a change in the sign of Rl1 . In Figure 41(b), the matrix elements Rl1 for

all three atoms are shown. Clearly, Rl1 does not change sign in the case of Ne, but it does change sign for Ar and Kr at photoelectron energies of ¤ 2 Ry ( photon energy ¤ 43

eV ) and ¤ 2.5 Ry ( photon energy ¤ 48 eV ), respectively.

From the above discussion, it can be deduced that for particular quantum numbers n

and l, the minimum position is determined. This implies that any molecule that retains

3p and 4p character will reﬂect the CMs of Ar and Kr. In reality, the situation is not

so simple in molecules. As an example, l is no longer a good quantum number, and the

continuum is often expressed as an inﬁnite sum of partial waves [135]. It is reasonable,

102 though, to expect molecules with lone-pair orbitals to largely retain atomic character, because it can be associated with a particular atom in a molecule. This is indeed the case in the linear combination of atomic orbitals (LCAO) method to calculate molecular orbitals.

For example, in the LCAO model, the lone-pair orbitals for S and Cl are 3p, analogous to Ar [135]. Additionally, features of molecular CMs were measured by Carlson et al. in

1981 [136], rigorously conﬁrmed in the following year [137][135], and then further explored by subsequent studies [122][123]. One of many conclusions of these papers is that Cl- and

S-containing molecules exhibit features of Cooper minima that are similar to those of Ar, with modiﬁcations depending on the exact molecular electronic structure.

As mentioned above, the cross section goes through a minimum due to a sign change in Rl1, which corresponds to a π spectral phase jump in Rl1 . That is to say,

there is a phase jump as a function of continuum photoelectron energy in Rl1 . As 2 one can see from Equation 105, though, the cross-section is a function of SRl1 S and

not Rl1 ; therefore, photoionization experiments are not able to directly measure the phase associated with this interference. Through HHS, however, we are able to access the

phase of the RDME, which is related to the photoionization dipole matrix element through

Equation 41 in Chapter 2. The next subsection will describe how a CM manifests in HHG

measurements and brieﬂy detail the history of HHS molecular CM measurements.

6.3.2 Cooper Minima in a Coherent HHS Measurement

As described in Chapter 2, HHS probes the RDME through measurements on the emit-

ted XUV. In the case of a p-parity ground state like Ar, both the s and d channels contribute

to the RDME, and so the XUV spectral intensity IXUV is a coherent sum of the electric

ﬁelds generated through the s and d channels, Es and Ed, respectively, or

2 IXUV ω SEs EdS . (107)

This means that HHS measures the coherent sum of the two channels, making separation

of the two channels in an experiment eﬀectively impossible. This coherent feature of HHG

103 bears consequences on HHS measurements of Cooper minima. In the above section, it was explained that the d channel for Ar goes through a minimum at ¤ 43 eV, but for most HHS measurements, the minimum is closer to 50 eV [138][10]. This is because interference with the s channel causes the minimum to appear at higher energies in HHS experiments, even though the minimum in the d channel is closer to 43 eV. This also means HHS does not measure the phase of the d channel, but rather measures the phase diﬀerence between the s and d channels. It was shown theoretically and experimentally by Schoun et al. [10] that in the case of Ar, the measured phase jump, and therefore the phase diﬀerence accumulated between s and d channels, is 2.6 rad.

6.4 HHS of a Cooper Minimum in CH3Cl : The Experiment

We will now detail our experimental results from HHS of the Cooper minimum in CH3Cl.

6.4.1 Experimental Details

All of the measurements detailed in this section were taken using the 1 kHz mid-infrared output of the OPA. The pulses used were centered at a central wavelength of 1650 nm with a pulse duration of ¤ 65 fs. Each pulse had an available energy of ¤1 mJ. For intensity scans, a waveplate-polarizer pair was inserted into the beam for controlling the pulse power, which was measured with a pyroelectric power meter. The spectral intensity and spectral

GD measurements were taken using the “attobeamline” apparatus, detailed in Section 5.4 of Chapter 5. All GD measurements were acquired using the Reconstruction of Attosec- ond Beating by Interference of Two-photon Transitions (RABBITT) method, discussed in

Chapters 3 and 5. The HHG-driving infrared beam was focused into the HHG gas source

(Even-Lavie d = 200 um) using an f = 400 mm Ag focusing mirror, and the emitted XUV was ﬁltered from the IR using a 200 nm Al ﬁlter. The interaction densities for all of the

HHG targets were estimated to be ¤ 1018 cm3. Lastly, Ne was chosen as the detection gas for all measurements.

104 6.4.2 High-Harmonic Spectral Intensity and GD Analysis

Similar to HHS measurements in atomic targets, the target-speciﬁc features of a molecular target are superimposed on the more general background HHG features. This is consistent with Quantitative Rescattering (QRS) theory [139], which showed that the complex HHG spectrum can be decomposed into the product of a normalized HHG spectrum,

HHGref , from a reference (atomic) system, an ionization rate Γ, for both the target and reference, and the target’s photoelectron scattering cross-section, σtarget, directly in the spectral domain:

Γtarget HHGtargetqω0 HHGref qω0 σtargetqω0. (108) Γref

Here, q is the harmonic order and qω0 is the XUV frequency. By normalizing HHGtarget

to an HHGref and accounting for the Γs, we are able to isolate σtarget, which carries the

information about the CM. In our case, we will use the spectrum of CH4 as our reference

for CH3Cl. Additionally, we will individually normalize each spectrum to unity in order to remove the eﬀects of the ionization rates Γ.

A similar referencing process can be done in the GD analysis in order to isolate the delay

associated with the target alone, τtarget. The GD of the HHG spectrum can be represented as

τHHG τref τtarget. (109)

In our experiment, we approximate τref ¤ τSFA (described in detail in Chapter 2). τSFA

depends on the target IP, the laser frequency ω0, and the laser intensity I0. While the ﬁrst two parameters are known to high precision, the intensity in the HHG focus is not. When

I0 is estimated by standard techniques, the resultant τSFA is too imprecise to be useful in

trying to isolate τtarget. Instead we developed a high precision algorithm to determine I0

and then proceeded with the removal of τSFA from τHHG in order to isolate τtarget.

In our experiment at 1650 nm, τSFA spans several femtoseconds. When τSFA dominates

105 over τtarget, τSFA can be treated as the sole contributor, and can be used to ﬁt to the precise value of the laser intensity. We perform this ﬁt using a least squares minimization procedure

given by τ qω τ qω τ Q HHG 0 SFA 0 offset I0 argmin 2 , (110) I0,τoffset q EB qω0

where EB qω0 is the measurement error bar for harmonic q, and is used to weight the degree of conﬁdence at each data point. The error bars in this experiment are constructed from

multiple ﬁts taken across the full-width at half max (FWHM) of each sideband, weighted

by their respective spectral intensities. τoffset accounts for the fact that GD retrieved with

the RABBITT method is measured up to an unknown constant. I0 is the only physically meaningful parameter in the ﬁt.

In practice, the ﬁtting algorithm in Equation 110 was implemented through the development of an in-house graphical user interface. This algorithm was largely written by Dr.

François Mauger, our collaborator at Louisiana State University, and was carried to com- pletion by me. The algorithm takes the following inputs: the generating laser wavelength, pulled directly from input data; the generation gas IP; the ﬁtting range; the number of

fitting steps; the filter type and thickness; the detection gas type; and the GD calculation method (FFT, Integrated FFT, Fit, or Integrated Fit). An image of the GUI and example plots from the fitting process are shown in Figure 42. At the end of the algorithm, a final intensity is exported along with the corresponding τSFA.

It is clear from Equation 110, that for smaller τtarget, the algorithm is able to more accurately extract I0. For some atoms and molecules, τtarget is on the order of τSFA, rendering the intensity extraction unusable. This problem can of course be circumvented by ﬁrst calibrating the intensity on a relatively featureless reference gas: a role ﬁlled by

CH4 in this experiment.

6.4.3 Results

Before applying our SFA attochirp removal procedure to the CM of CH3Cl, we ﬁrst conﬁrmed the method’s validity in the case that it was initially meant to describe: the

106 (a)

2 2 2 Minimum Intensity =5.571429e+13

1.5 1.5 1.5

1 1 1 Group delay (fs) Group delay (fs) Group delay (fs) 0.5 0.5 0.5

0 0 0 20 30 40 50 60 70 20 30 40 50 60 70 20 30 40 50 60 70 Photon energy (eV) Photon energy (eV) Photon energy (eV) (b)

Figure 42: SFA fitting program used to calibrate I0 and the corresponding τSFA. In (a), the in-house GUI is shown, and in (b) various frames are shown from the fitting process. In (b), the data are shown by the black curves. The τSFA calculations are shown as dashed lines from the lowest-intensity τSFA (blue) to the highest-intensity τSFA (red). The algorithm calculates the lowest-intensity τSFA first (left), then iteratively increases the intensity and recalculates τSFA (middle) until the preset limit is reached (right). In this case, the data 13 2 is from CH4 and the final fitted intensity is ¤ 5.57 10 W ~cm . The vertical dashed lines indicate the fitting region selected in this example.

107 ionization of an atomic target. Xenon was chosen due to its low IP compared to other noble gases like Ar and Ne, allowing us to achieve a more direct comparison to CH4 and

CH3Cl. We did this by measuring the GD in Xe at multiple laser intensities, and comparing the measured GD to the SFA GD. In Figure 43(a), we can see the τHHG results (circles) in Xe from a laser intensity scan at ﬁve intensities in the range 6.37 1013 W ~cm2 to

10.50 1013 W ~cm2. Each data set is separated by 0.2 fs for visual clarity. For all scans, the GDs from the Al ﬁlter and Ne detector were removed prior to the ﬁtting procedure.

Because the focal geometry and laser wavelength were unchanged between scans, the fitted laser intensities were expected to scale linearly with the pulse power. This was tested by only fitting the SFA intensity to the curve with the lowest input power, and, for the rest of the data sets, scaling the calculated intensity by the measured input power. The results of this fitting procedure are shown by the solid lines.

The calculated delay τSFA qualitatively agrees for all intensities. The best agreement is at the lowest energies, where the HHG signal is large and the error bars are small. At the highest energies, however, the HHG signal strength is weak so that the error bars are large and the agreement with the SFA calculations is worse. The agreement of the data with the scaling of the SFA can be quantized by calculating the ﬁt residual for each laser intensity, which was weighted by the experimental error bars. The residuals are plotted in

Figure 43(b), where the error bars represent a standard deviation of the error from the SFA

ﬁt. Here it is shown that the ﬁts are consistent with zero delay error for all measurements.

Though the error bars vary in size at the diﬀerent intensities, this plot shows that there are no systematic deviations from the SFA GD for the case of Xe.

Now that we have confirmed the validity of our I0 fitting method for the case of Xe, we can extend this method to the molecular case of CH4. We still expect this method to be valid, because there are no known interferences in the HHG spectra of CH4 in this spectral region [140]. Also, the spatial extent of the molecule is small relative to the size of the excursion amplitude (effective diameter of methane is 4 Å [141] and the electron excursion amplitude is on the order of 10 nm), further justifying the use of the SFA. The results of

108 0.15 5.0 Xe Fit Residuals 13 2 6.37x10 W/cm 0.1 7.38x1013 W/cm2 4.5 8.39x1013 W/cm2 13 2 9.40x10 W/cm 0.05 10.50x1013 W/cm2 4 0 3.5 -0.05

Group Delay (fs) 3 -0.1 Group Delay Mean Error (fs) 2.5 Xe -0.15 30 40 50 60 70 6 7 8 9 10 11 13 Photon Energy (eV) Intensity (W/cm2) x10 (a) (b)

Figure 43: Laser intensity scaling of the HHG group delay Xe. In (a), The measured group delays (open circles) were fitted using using the intensities indicated by the legend. The lowest-intensity curve was fitted for intensity using Equation 110, and the intensity was scaled to the measured laser input power for each additional scan (solid lines). Each data set and fit combination was separated by 0.2 fs for visual clarity. In (b), the SFA fit errors at each intensity are shown. Quantitative agreement, within the experimental error, is found at all intensities.

an intensity scan in CH4 are shown in Figure 44(a). The laser intensity was scanned from 4.9 1013 W ~cm2 to 8.51 1013 W ~cm2, and the same fitting criteria were used here as in the case of Xe. Again the plots are separated by 0.2 fs for visual clarity. It is clear from the plots that the fitted τSFA values qualitatively match the data at all intensities. Figure 44(b) confirms this, showing that the fit is consistent with nearly zero delay error across all intensity values.

Since we have established our SFA-based ﬁtting method, we can turn to CH3Cl. For this molecule, we expect the GD contribution of the CM to be large enough to potentially eﬀect

our ﬁt for τSFA; therefore, our ﬁtting method should not be directly applied to CH3Cl. We

have veriﬁed, though, that CH4 can be treated within the SFA, so we will use CH4 as a

reference for CH3Cl. This allows us isolate the CM contribution to the HHG spectrum, as we will show below.

We begin by analyzing the spectral intensities of the two molecules. In Figure 45(a),

109 4.98x1013 W/cm2 6.34x1013 W/cm2 5.43x1013 W/cm2 6.79x1013 W/cm2 13 2 13 2 5.89x10 W/cm 7.51x10 W/cm 0.15 8.51x1013 W/cm2 CH Fit Residuals 5.5 4 0.1 5 0.05 4.5

4 0

3.5 -0.05 Group Delay (fs) 3 -0.1 CH Group Delay Mean Error (fs) 2.5 4 -0.15 30 40 50 60 70 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 13 Photon Energy (eV) Intensity (W/cm2) x10 (a) (b)

Figure 44: Laser intensity scaling of the HHG group delay CH4. In (a), The measured group delays (open circles) were fit using using the intensities indicated by the legend. As with the Xe case, the lowest-intensity curve was fitted for intensity using Equation 110, and the intensity was scaled to the measured laser input power for each additional scan (solid lines). Each data set and fit combination was separated by 0.2 fs for visual clarity. In (b), the SFA fit errors at each intensity are shown. Within the experimental error, 6 of the 7 experimental intensities showed no deviation from the SFA calculations. The only outlier at 8.51 1013 W ~cm2 sits 1.25 standard deviations from zero.

110 the spectral intensities for CH4 and CH3Cl are shown. In order to directly compare the two molecules, the data shown here were taken under identical conditions, one after an-

other. Due to the discrepancy in IPs, the yields were not expected to be the same; to keep

conditions identical, HHG was optimized for CH3Cl to avoid saturating ionization in either sample. Each spectrum has been normalized to unity in order to remove the eﬀects of

Γtarget (CH3Cl) and Γref (CH4). The spectral envelopes in Figure 45(a) were obtained by ﬁnding harmonic peaks and smoothing the result with a second-order Savitzky–Golay ﬁlter.

For energies less than approximately 30 eV, the HHG signal from CH3Cl is very similar in

strength to that of CH4. Beyond 30 eV, however, the signal from CH3Cl sharply decreases

compared to CH4. The signal stays much smaller than that of CH4 until ¤ 55 eV, where the signals are again similar. The cutoﬀ positions are similar between the two molecules,

because the IPs are only separated by ¤ 1.5 eV.

Evident in Figure 45(a), the CH4 HHG spectrum is eﬀectively featureless, only resembling the basic plateau-like structure expected from the SFA model of HHG. This conﬁrms

its use as HHGref . So that we may isolate the CM features contained in σtarget, we plot

the ratio of the CH3Cl spectrum to the CH4 spectrum in Figure 45(b). Now the eﬀect of

the CM on the CH3Cl cross-section is obvious. The CM causes a broad minimum that is ¤ 20 eV wide at the full-width at half minimum, centered around ¤ 43eV. The depth, width,

and position of the CH3Cl CM are very similar to that of the previously-measured Ar CM spectral features [10]: a minimum that drops to ¤ 20 40% of unity, centered around 50 eV

with a width of ¤ 20eV. This is already strong evidence for the fact that we have isolated

the eﬀect of the CM in CH3Cl, but as we discussed before, we can go a step further and extract the GD of the CM as well.

Shown in Figure 46(a) are the GD measurements taken for CH3Cl and CH4 using the

same conditions as those for the spectral intensity plots. τSFA, shown by the solid red lines,

was ﬁt to the GD measured in CH4 (blue circles). While ﬁxing the intensity to the CH4

GD fit result, the CH3Cl GD was fit for only τoffset, defined in Equation 110. The CH3Cl data set and fit result are shifted by -0.2 fs for visual clarity. In the 30-50 eV region, the

111 CH 1.2 1 4 CH Cl/CH CH Cl 3 4 3 1 0.8 0.8 0.6

0.4 0.6

0.2 0.4

0 0.2 Normalized Counts (arb. units) -0.2 Ratio of Normalized Counts 0 20 30 40 50 60 70 20 30 40 50 60 70 Energy (eV) Energy (eV) (a) (b)

Figure 45: Inﬂuence of the CM on the spectral intensity of CH3Cl relative to CH4. In (a), the normalized spectra of CH4 (blue) and CH3Cl (green) were taken back-to-back in identical conditions. The envelopes of both spectra were retrieved through a second order Savitzky-Golay smoothing. Panel (b) shows the ratio of the CH3Cl envelope to the CH4 envelope. The minimum at ¤ 43eV is attributed to the presence of the CH3Cl CM.

GD of CH3Cl dips below τSFA, whereas the CH4 GD is a near exact ﬁt along τSFA into

the cutoﬀ. The behavior for CH3Cl is the same as that observed for the CM in Ar [10].

In Figure 46(b), τtarget for CH4 and CH3Cl are plotted, which are the same data in Figure

46(a) minus τSFA. Here the minimum in τtarget for CH3Cl is quite obvious. At its deepest, it is ¤ 120 as, and similar to the spectral intensity results, the width is approximately ¤ 20

eV wide. τtarget for CH4, on the other hand, is very ﬂat throughout the whole region, until

the cutoff where it deviates from τSFA. Deviations in the cutoff GD are not unexpected in these molecular HHG measurements. First, the harmonics in the cutoff region are created in

the highest intensity part of the IR driving ﬁeld. Because the intensity is higher here, there

is an increased likelihood to generate harmonics from lower lying orbitals [142][143][131].

Additionally, because the signal is dropping rapidly above 60 eV, we expect the quality of

our ﬁts to decrease as well. This is not always reﬂected in the error bars, which, strictly

speaking, represent the spread in ﬁt phases across the width of the sideband rather than a

typical statistical error bar.

Lastly, in Figure 46(c), τtarget for each molecule has been integrated in order to retrieve

the phase associated with their respective RDMEs. CH4 has a slight negative slope, but

112 (CH ) 4

(a) (b)

(c)

Figure 46: Comparison of the GD and phase features between CH4 (blue circles) and CH3Cl (green circles). In (a) τSFA (solid red curve) is shown along with the high-harmonic delays (τHHG) from the experiment. The laser intensity was ﬁt using the CH4 GD, resulting in an 13 2 intensity of 5.5610 W ~cm , and then, this same intensity was used to calculate τSFA for the CH3Cl GD. The CH3Cl data is shifted by -0.2 fs for visual clarity. (b) Removal of τSFA isolates τtarget. By doing so, the CM in CH3Cl is revealed as a ¤ 120 as “dip” in the GD. (c) Group delays were then integrated to show a phase shift of ¤ 2.6 rad across the CM. Additionally shown are the results for Ar at 1.3 µm and 2.0 µm from Schoun et al. [10].

113 accrues very little phase across the harmonic spectrum. On the other hand, CH3Cl goes through a phase jump of ¤2.6 rad. This value is very similar to that of the Ar phase jumps extracted from the paper by Schoun et al. [10], which are shown in Figure 46(c) as well.

Our agreement with the previous Ar results supports our method’s ability to extract target- speciﬁc spectral intensity and phase information from HHS measurements in molecules.

6.5 Conclusions

In this chapter, we have rigorously demonstrated the ability of HHS to accurately extract target-specific information from molecular HHG spectra in intensity and phase, particularly the characteristics of the CH3Cl Cooper minimum. Using a novel fitting algorithm, we first tested the applicability of HHS GD techniques by verifying the SFA-like intensity-scaling of the GD for Xe and CH4. We then compared the spectral intensities of our CH4 reference

to that of the target CH3Cl, revealing a large minimum in the cross-section of CH3Cl.

Next, we used our ﬁtting algorithm and the CH4 GD to calculate a τSFA that could be

subtracted from the CH3Cl GD, revealing a target-speciﬁc ¤ 120 as dip in the GD. Finally,

we integrated our GD results to retrieve a ¤ 2.6 rad phase jump in CH3Cl, which is nearly

identical to the results of atomic Ar: thus fully characterizing the 3p character of the CH3Cl HOMO.

114 Chapter 7: Attosecond Spectroscopy of Autoionizing

Resonances in Ar and He

“In a very dark chamber at a round hole about one third part of an inch broad made in the shut of a window I placed a glass prism, whereby the beam of the sun’s light which came in at that hole might be refracted upwards toward the opposite wall of the chamber, and there form a coloured image of the Sun.” —Isaac Newton, Opticks Book I, Part I, Prop. II, Theor. II, Exper. 3

7.1 Introduction to Autoionizing Resonances

One of the most important tools to study physical phenomena is the scattering of light by matter. This interaction was most notably studied in 1905, when Albert Einstein used the photoelectric measurements of Max Planck to deduce the particle nature of light. Since that work, light-matter interactions have been used to build quantum mechanical models of atoms, molecules, and solid-state materials. Studies of light-matter interactions have become so widespread and well-understood that they are now even being used for fundamental tests of extensions to the standard model of physics [144].

A particularly ubiquitous light-matter interaction of importance is the autoionizing resonance, which has been observed in atoms, molecules, and solids [145]. An autoionizing resonance is the result of the interference between two quantum paths: direct ionization and excitation of a discrete state coupled to the continuum. A cartoon example of such a resonance in Ar is shown in Figure 47. Autoionizing resonances were ﬁrst discovered by

Beutler in 1935 [146], who noted their sharp, asymmetric absorption proﬁles. Their origin was later described by Fano in 1935 [147] and 1961 [148], which is why the resonances are sometimes referred to as “Fano” resonances. Fano was the ﬁrst to derive the scattering cross section σF of an autoionizing resonance, which is given by

q2 σ . (111) F 2 1

115 Figure 47: XUV photoionization of an Ar atom in the presence of an autoionizing resonance. In this case, there are two pathways to the continuum: a direct transition to the continuum and a transition to a bound state which decays into the continuum. Because there are two pathways to the same ﬁnal state, an interference in the photoionization spectrum occurs.

Here, q is referred to as the asymmetry parameter and 2E EF ~Γ is the reduced energy, where EF is the resonant energy and Γ is the width of the autoionizing state. q, which will be derived in the following section, is a ratio of the transition amplitude to the mixed bound-continuum state over the transition amplitude to just the continuum state.

Essentially, q tells us how strongly the ground state couples to the discrete bound state: smaller SqS implies weaker coupling to the bound state. Figure 48(a) shows how σF varies with q. The minimum and maximum values of σF can be derived from Equation 111 and

2 can be inferred from the plots: σF,min 0 at q and σF,max 1q at 1~q. Clearly, for diﬀerent values of q, the resonance can dramatically change the shape of σF from asymmetric to symmetric. It is also known that the corresponding dipole matrix element goes through a π phase jump across the resonance’s energetic region, which can be seen for the same values of q in Figure 48(b). Note that the center of the phase jump does not need to be centered on 0.

Autoionization is a particularly interesting phenomenon to study because it is entirely the result of electron-electron correlations. In fact, it is forbidden in the non-interacting- particle approximation [149]. This is why autoionizing resonances have been the targets of attosecond spectroscopy (AS) experiments22 hoping to resolve electron-correlation driven attosecond dynamics in the time domain [150][151][152][29][153][154]. In a typical AS ex-

22For a more in-depth discussion of the principles of AS, please refer to Chapter 1.

116 (a) (b)

Figure 48: Example calculations of autoionizing resonance parameters for a few values of the asymmetry parameter q. Panel (a) shows that the cross section can change from asymmetric to symmetric depending on q. Panel (b) shows that the position of the phase jump associated with the resonance depends on q and does not necessarily need to occur at 0. periment, extreme ultraviolet light (XUV) from high-harmonic generation (HHG) combined with an infrared ﬁeld are used to probe the time-dependent properties of autoionization.

In order to accurately reconstruct the time-domain properties of an autoionizing resonance through an AS experiment, it is important to not only characterize the spectral intensity of the resonant features, but also the phase. This is analogous to the characterization of femtosecond laser pulses, where a spectrum without the phase does not necessarily reﬂect an accurate pulse duration.

To date, three experiments have measured autoionization spectral phases. One measurement performed by Kotur et al. [29] tuned the HHG infrared (IR) laser central wavelength so that a single harmonic scanned across the 3s3p64p (q 0.389, Γ 87.4 meV [155])

resonance of Ar, ﬁnely sampling the phase in the neighboring sidebands. Another measure-

ment by Gruson et al. [153], using a single HHG-driving wavelength, was able to resolve

the He 2s2p (q 2.77, Γ 37 meV [156]) resonance in intensity and phase within a single

sideband. They took their measurement a step further than the measurement by Kotur and

used their results to reconstruct the evolution of the resonance in the time domain. Doing

so revealed that the ionized electronic wave packet begins symmetric and must build up

to its asymmetric line shape. Lastly, there was a recent study by Cirelli et al. [152] that

117 characterized the spectral phase of the Ar 3s3p64p resonance as a function of angle relative to the XUV polarization.

All of these pioneering phase measurements were done using the near-infrared (NIR) output of Ti:sapphire-based laser systems. Using this output, though convenient, limits the experiment because only resonances that fall very close to the energy of 800 nm harmonics can be probed. Even in the measurement by Kotur, the central frequency of the laser could only be tuned 50 nm. Thus, greater tunability of the HHG driving wavelength will open the way for experiments probing other resonances.

In this chapter, we will also probe the spectral intensity and phase of autoionizing resonances, but instead of using the ﬁxed output of a Ti:sapphire system, we will employ an optical parametric ampliﬁer (OPA). An OPA allows the central wavelength to be easily tuned over hundreds of nanometers in the mid-infrared (MIR).23 Using MIR wavelengths

has the additional advantage of an extended cutoﬀ and a more densely sampled harmonic

comb compared to NIR wavelengths. We will explore the Ar 3s3p64p resonance and the

tentatively-labeled 3s23p44s4p resonance of Ar, and we will also explore the 2s2p resonance

of He. We will carry out these studies in two ways. First, we will use odd-harmonic

XUV combs and the Reconstruction of Attosecond Beating by Interference of Two-Photon

Transitions (RABBITT) technique to study two-omega oscillations in the vicinity of these

resonances. This method will give us direct access to the two-photon autoionizing phase

structures. Next, we will use harmonic combs comprised of even and odd harmonics to study

similar omega oscillations.24 These additional measurements will act as an extra test of

theoretical predictions, and comparing the results of the two-omega and omega oscillations

to theory will allow us to thoroughly explore the behavior of continuum-continuum pathways

in the presence of a Coulomb ﬁeld.

This chapter is structured as follows. Section 7.2 will describe the theory of autoion-

ization in the presence of XUV-IR two-color ﬁelds. Section 7.3.1 details the theoretical

23For more on the operation and parameters of our OPA, please refer to Chapter 5. 24For a description of the origin of these two types of oscillations, please refer to the detailed derivations in Chapter 3.

118 methods developed by our collaborators in the Luca Argenti Group at the University of

Central Florida and the Joachim Burgdörfer Group at the Vienna University of Technology.

The remainder of Section 7.3 will discuss our experimental results and compare previous experiments and theory, and Section 7.4 contains the conclusions.

7.1.1 Statement of Contributions

Before describing the theory of autoionizing resonances, I would like to state my direct contributions to the experiments described in this chapter. The spectral intensity measurements of Ar and He were taken by Dr. Timothy Scarborough and me, and the data sets were analyzed by me. All of the Ar RABBITT measurements were performed by me with assistance from Dr. Scarborough. The He RABBITT data were collected by Dr. Scarbor- ough and Daniel Tuthill with assistance from me, and all of the analysis of that data was performed by me. The omega-oscillation experiments were conceived by me and carried out by Dr. Scarborough, Daniel Tuthill, and me with assistance from Dr. Guillaume Laurent and John Vaughan. Lastly, the wavelength scanning technique applied to the group delay measurements was created and developed by Dr. Scarborough and me.

7.2 Theoretical Description of Autoionizing Resonances in Two-Photon Ionization

Though it would be ideal to only measure the single-photon phase associated with the resonance, a secondary ﬁeld, typically IR, is necessary to extract the phase. In RABBITT measurements for example, an odd harmonic is incident on the resonance so that the resonant phase is imprinted onto the neighboring sidebands, which is depicted in Figure 49(a).

Unfortunately, this two-photon process comes with additional phase contributions dubbed

“continuum-continuum” (CC) phase, as discussed in Chapters 3 and 5. In order to understand how the addition of the IR ﬁeld aﬀects the phase retrieval, we will follow the work of

Jiménez-Gálán et al.[157] and Kotur et al. [29] to establish a theoretical description of autoionizing resonances in two-photon ionization, speciﬁcally in the context of the RABBITT

119 (a) (b)

Figure 49: Depictions of the RABBITT interaction in the presence of an autoionizing resonance for the case of the Ar 3s3p64p resonance. In (a), the phase of the electron ionized by the (q+1)th harmonic is imprinted onto the neighboring qth and (q+2)th sidebands. In (b), the speciﬁc notation used in the derivation is shown. Sge symbolizes the ground state, Ω is the XUV frequency, Sφe represents the resonant bound state with energy Eφ, SψαEe at T f energy E with angular momentum α and ψγEf at energy Ef with angular momentum γ represent the intermediate and ﬁnal states, respectively, connected by an infrared photon at frequency ω. interaction. We will build the theory around an example autoionizing state, the Ar 3s3p64p,

1 6 which appears in a series of P0 resonances leading to the 3s3p ionization threshold [157]. This framework is easily generalizable to the other resonances explored in this chapter.

The ionization dipole matrix element for two-photon ionization of an XUV photon with frequency Ω and the absorption or emission of an infrared photon with frequency ω from S e T f the ground state g (energy Eg) to a ﬁnal continuum state ψγEf with angular momentum

γ and energy Ef is given by

ˆ ˆ bψγ U d SψαEe `ψαES d Sge Ef MγE g lim Q o dE , (112) f ξ 0 α Eg Ω E iξ where SψαEe are the non-resonant intermediate states (not including the resonant bound state Sφe) in the discrete or continuum spectrum with angular momentum α and energy E, and ξ is an inﬁnitely small quantity. The integral sum is over all discrete and continuous intermediate states SψαEe, which are eigenfunctions of the unperturbed Hamiltonian, HˆX. Figure 49(b) depicts the nomenclature used in this derivation. As described in Chapter 3,

120 we can assume that the XUV is absorbed first to an intermediate state SψαEe, followed by T f the absorption or emission of an IR photon to the final state ψγEf . At first glance, Equation 112 seems daunting because of the infinite sum of intermediate states, but we can reduce the problem to a few pathways by considering which pathways the resonant bound state Sφe will decay through. We know that Sφe 3s3p64p has total

1 electronic angular momentum L 1, because it is part of the P0 series. By angular momentum conservation, the intermediate states must also have total electronic angular momentum L 1. In that case, Sφe can decay through the ¡3p1εs¦ and ¡3p1εd¦ L 1 L 1 channels, where -1 indicates the ionized orbital, α s or d, and ε indicates a continuum energy level. Then after decaying to the continuum, an IR photon can transfer the electron to three possible final states with L 0 or L=2 : ¡3p1εp¦ , ¡3p1εp¦ , or ¡3p1εf¦ . L 0 L 2 L 2 The specific final state depends on the intermediate state’s α value. These final state contributions to the photoionization spectrum are summed incoherently to calculate the total sideband intensity, because they lead to the same final energy but not the same final states [158][76].

Up to this point, we have not directly incorporated the Coulomb interaction (Vˆ ) between the bound 3s13p64p state and the continuum. We will do so in the following way. First, let the bound state be represented as Sφe with an energy Eφ. Next, to simplify the interaction, we will recast the s intermediate state SψαEe and the d intermediate state SψαEe into an interacting state SψI,Ee and non-interacting state SψNI,Ee, which are given by

VαE VαE SψI,Ee SψαEe SψαEe and (113) VE VE

VαE VαE SψNI,Ee SψαEe SψαEe , (114) VE VE

2 2 2 where SVES SVαES SVαES and Vα,αE `ψα,αES Vˆ Sφe. Then by construction `ψNI,ES Vˆ Sφe

0 and `ψI,ES Vˆ Sφe VE, which can easily be checked with Equations 113 and 114. We now

25 diagonalize the full Hamiltonian Hˆ Hˆ0 Vˆ in the new basis {Sφe, SψI,Ee, SψNI,Ee},

25For a similar, very detailed derivation of this diagonalization, please refer to the original paper by Fano [148].

121 producing the eigenstates {SΦe, SΨI e, SΨNI e}, which are given by

V Sψ e SΦe Sφe P o dE E I,E , (115) E E

sin ∆ E i∆E SΨI,Ee SΦe cos ∆E SψI,Ee e , and (116) πVE

SΨNI,Ee SψNI,Ee . (117)

In Equation 115, “P ” indicates the Cauchy principle value. ∆E is the phase shift of

ΨI,E relative to ψI,E and is given by

2 π SVES ∆E arctan , (118) E EΦ

where EΦ is the energy of the state SΦe and is given by

SV S E E P o dE E . (119) Φ φ E E

We can now formally deﬁne the asymmetry q and detuning , which were introduced in

Section 7.1: `ΦS dˆSge q ¡ and (120) ` S ˆS e πVE ψI,E d g

¡ E EΦ 2 , (121) π SVES

2 where EF from Section 7.1 is now EΦ and Γ ¡ 2π SVES . With these, the one-photon ionization dipole matrix elements can be written as

q `Ψ S dˆSge `ψ S dˆSge and (122) I,E I,E i

ˆ ˆ `ΨNI,ES d Sge `ψNI,ES d Sge . (123)

122 In this basis set, it is now clear that the matrix element with the interacting state is highly sensitive to q and . Then to calculate the one-photon cross section of the resonance, one must incoherently sum the two matrix elements. The resonance is contained in the interacting channel, whereas the non-interacting channel simply contributes a smooth background across the resonant feature. Finally, the two-photon matrix element to the ﬁnal state can be written as (after some algebra)

I q NI MγE ,g M M , (124) f γEf ,g i γEf ,g

where M I and M NI are the two-photon matrix elements with interacting and non- γEf ,g γEf ,g

interacting intermediate states, respectively. q and are calculated at the energy Eg Ω. It should be noted that by writing Equation 124 we have neglected IR transitions from

the resonant state to the continuum (dashed red arrow in Figure 49b). It is clear that the

ﬁrst term on the right-hand side of Equation 124 contains the resonant features and the

second term is a smooth non-interacting background. So through a two-photon process, we

not only measure the resonant-phase structure but also the phases from the non-resonant

terms.

Now that we have established a two-photon framework in which to understand autoion-

izing resonances, we can further motivate our two probing methods: using odd-harmonic

combs (Figure 50(a)) and even-and-odd harmonic combs (Figure 50(b)). We have so far

described how to probe autoionizing resonances with odd-harmonic combs of light, and we

mentioned that the IR probing ﬁeld adds extra CC phases to the measurement. Typically,

we calculate and remove these phases, as described in Chapter 5, but in the presence of the

resonance’s strong Coulomb interaction we can no longer treat the CC phase as separate

from the single-photon Wigner phase. By additionally using the even-and-odd harmonic

comb, we introduce separate pathways to the same ﬁnal state, i.e. through a direct XUV

photon transition. So in principle, by comparing the results of the two measurements to

theory, we can probe the phases accrued from diﬀerent pathways to the continuum in the

presence of an autoionizing resonance.

123 RABBITT + Fano Omega Oscillation + Fano q+3 q+2 q+2 q+1 q+1 q q

q-1

Energy Energy

(a) (b)

Figure 50: Illustrations of the two-photon interactions used to probe autoionizing resonances in this chapter. (a) and (b) show the “two-omega oscillation” RABBITT interaction and the “omega-oscillation” interaction, respectively. The phase in the RABBITT interaction is probed through the sidebands neighboring the resonant harmonic, and the phase in the “omega-oscillation” experiment is probed through the resonant harmonic and its two neighboring harmonics.

Lastly, and on a more fundamental level, spectral phase measurement methods for continuous XUV spectra or combs of even and odd harmonics are still developing, particularly for the latter. Measuring the phases of even-and-odd harmonic spectra is important to understanding the XUV spectral properties for two-color high harmonic generation [159][160][161] and for high-harmonic spectroscopy of oriented samples [37][34][162].

To date, only one paper by Laurent et al. [74] has attempted to reconstruct the XUV spectral phase of even-and-odd harmonic combs. Our measurements with even-and-odd- harmonic combs will serve as benchmarks for future phase measurements using nontrivial high-harmonic generation schemes.

7.3 Results

7.3.1 Ab Initio Calculations of Autoionizing Resonance Properties

In order to better understand our autoionization results, we initiated a multi-university collaboration with two theoretical groups: the Argenti Group at the University of Cen- tral Florida, who developed a theoretical framework to study Ar, and the Burgdörfer group at the Vienna University of Technology, who developed a theoretical framework to

124 study He. Their approaches to calculating delay-dependent two-photon XUV-IR ionization of atoms in the presence of autoionizing resonances are based on solving the time- dependent Schrödinger equation (TDSE) [163] using the time-dependent close-coupling

(TDCC) scheme [164][165][166][167], in which the wavefunction is expanded on coupled spherical harmonics. Both models have some of the same key ingredients. First, the wavefunction of the complete atom (all N electrons), must be numerically represented from its initial bound state, through the intermediate stages promoted by the radiation where all the excited components of the atom interact, and finally to the asymptotic region in which the ionized electron and parent ion are well-separated. Secondly, the time-evolution of the system under the external field must be accurately determined. This usually requires that the time-dependent Schrödinger equation is numerically integrated using parallelized algorithms. Lastly, the wave packet that is generated at a finite time must be analyzed to determine its asymptotic form at large times. Typically, this is done by projecting the

N-electron wave packet at the end of the simulation onto multichannel scattering states of the atom, which must be separately calculated by solving the scattering problem for the correlated system.

For the Ar calculation, all of the above is implemented using the newstock suites of code, which are based on an extension of the multi-electron close-coupling Stockholm code

[155]. The current version of the code assumes the electrostatic and dipole approximations and is nonrelativistic. It should be noted that neglecting relativistic terms is not insigniﬁ- cant, seeing as the Ar+ parent ion exhibits a 0.18 eV spin-orbit splitting that is comparable to the Auger width of many relevant resonances in the photoionization spectrum of Ar

[168]. The multi-electron wave function is constructed from antisymmetric products of one- electron atomic orbitals. The reduced radial component of each one-electron atomic orbital is expanded in a B-spline basis, which is a ﬂexible set of compact-support functions that have proven ideally suited to represent bound, Rydberg, and continuum orbitals in atomic and molecular physics [169]. The splines have a nodal spacing of 0.4 a.u. and extend out to 500 a.u. A complex absorbing potential with a quadratic radial dependence is placed at

125 Rabs 400 a.u. in order to reduce reﬂections at the boundary. As for the time propagation, the TDSE is solved with time steps of 0.8 as.

For the He calculation, the electric fields are treated in the dipole approximation. The temporal propagation is handled by a short iterative Lanczos procedure with adaptive time- step control [170][171], and the spatial grid is discretized using a finite element discrete variable representation [172][173][174]. They employ an L-shaped box where the extension along the longer edge spans 3410 a.u. and the smaller edge spans 20 a.u. To avoid reflec- tions at the boundary, they use absorbing boundaries starting at 3325 a.u. and 15 a.u., respectively. Propagation effects are not considered since the probability for reabsorption of photons is negligible in the investigated energy region. Convergence of their calculations is tested by comparing velocity and length gauge results. With exact wavefunctions, the two gauges should produce the same result, but with approximate wavefunctions, the results may vary. Therefore, comparing the two methods acts as a convergence test of the calculations.

For the He calculations, IR full-width at half max (FWHM) pulse durations and XUV pulse trains of ¤ 20 fs were used. The corresponding IR and XUV pulse intensities were

2 109 W ~cm2 and 1 1010 1 1011 W ~cm2, respectively. For the Ar calculations, IR

FWHM pulse durations and XUV pulse trains of ¤ 50 fs were used. The corresponding IR and XUV pulse intensities were 1109 W ~cm2 and 11010 11011 W ~cm2, respectively. In the calculations modeling the RABBITT interaction, the theoretical photoelectron results are angle-integrated, just as in the experiment.26 When modeling two-photon interactions with the even-and-odd harmonic combs, they calculate the time-dependent asymmetry of ionization along the XUV/IR polarization, i.e. “up minus down over the sum” or “down minus up over the sum.” From this they extract the phase properties of the electron spectra.

For all calculations, the XUV and infrared intensities are well within the perturbative regime, which suppresses unwanted higher-order IR transitions that can complicate the interpretation of the phase results.

26Refer to Chapter 5 for a detailed description of the detection apparatus.

126 7.3.2 Experimental Details

For all of the measurements described in this chapter, we used the tunable mid-infrared output of the optical parametric ampliﬁer and the “attobeamline” apparatus described in detail in Chapter 5. The high harmonics were generated in either Ar, CH4, CO, or N2O with focal lengths of f=500 mm, f=400 mm, and f=300 mm, depending on the generation gas. Then the harmonics were refocused into the magnetic bottle electron spectrometer to photoionize the target gas, Ar or He. All measurements used an Al foil to ﬁlter the mid-infrared HHG driver. The generation gases were switched for two reasons: (1) to test whether or not important spectral features originated in the generation or target gases and

(2) to tailor the harmonic spectra to be suﬃciently intense in the spectral range of interest.

In order to probe the autoionizing resonances in intensity and phase, we have chosen to use our wavelength tuning method described in Chapter 5. Brieﬂy, in this method we tune the wavelength of the HHG mid-infrared driver so that a single harmonic scans across the resonance. For the case of odd harmonics only (RABBITT, two-omega oscillation), this imprints the autoionizing phase onto the neighboring sidebands. For the case of even and odd harmonics (omega oscillation), the autoionizing phase should be present in the resonant harmonic and the two neighboring harmonics.

In order to generate XUV spectra comprised of even and odd harmonics, we used the setup shown in Figure 51. In this setup, the HHG driving beam is doubled (Type I) using a beta barium borate (BBO) crystal, so that the 2nd harmonic has approximately 1% of the power of the fundamental and exits the BBO with perpendicular polarization relative to the fundamental. The two-color beam is then propagated through ¤ 0.2 mm of rotatable calcite in order to pre-compensate for the accumulated group delay between the two colors as they propagate through other transmissive optics. Finally, a half waveplate at the fundamental wavelength is used to rotate the fundamental 90 degrees so that the polarizations of the two colors are parallel. This leaves the two colors with horizontal polarization (along the time-of-flight axis), which is necessary based on the discussion in Section 5.6. For these measurements, a filter was inserted into the attobeamline “dressing” arm in order to filter

127 Figure 51: Diagram of the collinear frequency-doubling design used to generate high- harmonic spectra comprised of even and odd harmonics. The fundamental frequency ω is ﬁrst doubled using Type I second harmonic generation in a Beta Barium Borate crystal. The two colors are then copropagated through a rotatable calcite crystal for tunable delay pre-compensation between the two colors. Lastly, the polarization of the fundamental beam is rotated 90X to match that of the second harmonic polarization. out the second harmonic. This guaranteed that only a one-color ﬁeld interacted with the

XUV in the target ionization process.

We took great care to ensure that the two-color generating conditions were as similar as possible between scans. To achieve this, we focused on two controls. First, we tuned the calcite angle so that the subcycle delay between the fundamental and second harmonic was the same for all scans. This is done by observing the well-known oscillations between even and odd harmonics that depend on the relative delay between the two driving colors [161].

Secondly, we tuned the BBO to ﬁx the even harmonic intensity to ¤ 25% that of the neighboring odd harmonics in the spectral regions surrounding the autoionizing resonances. The intensity of the even harmonics are kept weaker than the odds because the fringe contrast of the even-harmonic omega oscillation increases with increasing amplitude disparity between the neighboring harmonics [74]. Lastly, it should be noted that the omega-oscillation measurements were largely recorded in collaboration with Professor Guillaume Laurent and his graduate student John Vaughan of Auburn University.

7.3.3 Ar and He Photoionization Spectra

We begin by exploring the resonant spectral intensity features of Ar and He. In Figure

52, we can see various results from previous experimental studies of Ar and He. In Ar, we are particularly focusing on the lowest-lying resonances of two separate series: the

3s3p64p resonance, which is in a series leading to the 3s3p6 threshold, and the 3s23p44s4p

128 resonance, which is in a series leading to the 3s33p4 threshold. Figure 52(a) shows the

Ar 3s3p64p photoionization cross section extracted from synchrotron-based photoelectron measurements by Berrah et al. [175]. The resonance is located at ¤ 26.6 eV with Γ ¤80 meV

and q ¤ 0.3.

An XUV absorption measurement of the higher-lying 3s23p44s4p resonance from a dif-

ferent synchrotron-based experiment is shown in Figure 52(b) (indicated by the red dashed

line) [176]. The resonance is centered around 33.8 eV, and the q-value has been estimated

to be positive, but otherwise q and Γ are not well-known [177] [176]. It was reported by

Madden et al. [177] that the cross section of this resonance is many times smaller than that

of the 3s3p64p, and that the 3s23p44s4p resonance overlaps with many other, sharper reso-

nances [176] [177]. Given these complications, it is diﬃcult to calculate exactly or to extract

speciﬁc parameters of this resonance, so the assignment of 3s23p44s4p parameters should

be taken as tenuous at best. The 3s23p44s4p resonance is, however, the widest and largest

in its respective energetic region, making it the most likely contributor to any structure in

the surrounding energetic region in our photoionization experiments.

Figure 52(c) shows the cross section of the He 2s2p resonance, which was calculated by

using resonance parameters retrieved from ﬁtting He transmission data from a synchrotron

source [178]. The resonance is centered at E 60.15 eV with q 2.6 and Γ 38 meV.

When comparing the He 2s2p cross-section to that of the Ar 3s3p64p, the eﬀect of the q

parameter is obvious. Because q is close to zero for Ar, the resonance is mostly symmetric

and generally suppresses photoionization in the energetic region. In contrast, the increased

magnitude of q for the 2s2p resonance leads to a far more asymmetric proﬁle, which also

indicates a stronger coupling of the ground state to the resonant state.

For comparison, in Figures 53(a) and 53(b), we can see our photoelectron spectra for Ar

and He, respectively, taken at diﬀerent HHG-driving wavelengths. The Ar photoelectron

spectra were taken with harmonics generated in Ar (the HHG source) with wavelengths

between of 1400 nm (purple) and 1520 nm (yellow) in 2 nm steps. The He photoelectron

spectra were taken with harmonics generated in CO with wavelengths between 1460 nm

129 (a) (b)

(c)

Figure 52: Spectral intensity structures of the autoionizing resonances in Ar and He. Panel (a) shows the photoionization cross section of the Ar 3s3p64p resonance measured by Berrah et al. [175]. In panel (b) is plotted an Ar absorption measurement recorded on a photographic plate measured by Baig et al. [176]. Decreased plate density implies increased (decreased) absorption (transmission). Panel (c) shows the He 2s2p cross section calculated by Morgan et al. [178] using parameters extracted from a ﬁt of XUV transmission measurements.

130 (purple) to 1510 nm (yellow) in 2 nm steps. Vertical lines mark the positions of the resonances that we are exploring. For Ar, a minimum is clearly evident near the 3s3p64p resonance, whereas there is no obvious feature at the position of the 3s23p44s4p resonance.

For He, there is an increase in intensity just below the 2s2p resonance, followed by a slight decrease in intensity.

These scans are somewhat difficult to interpret, however, because the HHG yields are not the same across different HHG driving wavelengths. To correct for this, we normalize the scans in the following way. First, we define a reference spectral envelope Sref to be the spectral envelope of a spectrum that has no resonant harmonics,

Sref qω ¡ SNR qω , (125)

where SNR is the nonresonant spectral envelop, q is the harmonic order, and ω is the fundamental driving frequency. For Ar, this means that SNR is extracted from a spectrum

6 that has no harmonic resonant with the 3s3p 4p state, and for He this means that SNR is extracted from a spectrum that has no harmonic resonant with the 2s2p state. SNR is extracted by peak-ﬁnding across the nonresonant spectrum and interpolating those har-

th monic peaks for an approximate envelope. We then assume that the envelope Si for the i generating wavelength, deﬁned by the harmonic peaks, can be scaled by a factor Ai such that

Si qω AiSref qω , (126)

where Ai is estimated by the spectral intensity average in a region near the resonance. This assumption relies on the fact that Sref and Si have essentially the same spectral shape, which is generally reasonable away from the HHG cut oﬀ region. Finally, we can calculate a normalized spectrum Ni such that

Ni qω Si qω ~Sref qω Ai 1, (127)

where “1” is added because SNR~Sref 1. The above normalization should remove most of

131 (a) (b)

Figure 53: Photoelectron spectra of Ar and He using XUV generated in Ar and CO, respectively, with diﬀerent HHG wavelengths. In (a), Ar photoelectron spectra recorded with HHG driving wavelengths between 1400 nm (purple) and 1520 nm (yellow) in 2 nm steps are shown. A slight decrease in signal occurs in the region around the 3s3p64p resonance (green vertical line, E =26.6 eV). No clear feature is seen around the 3s23p44s4p resonance (blue vertical line, E = 33.8 eV). In (b), He photoelectron spectra recorded with driving wavelengths between 1460 nm (purple) and 1510 nm (yellow) are shown. The signal increases just below the 2s2p resonance and then decreases quickly at the resonant position. Normalized spectra of Ar and He are shown in panels (c) and (d), respectively. Refer to the text for a detailed explanation of how the signals were normalized. After normalization, the Ar 3s3p64p resonance and the He 2s2p resonant structures are clearer and qualitatively similar to the corresponding cross sections in Figure 52. No obvious features can be see in the vicinity of the 3s23p44s4p resonance.

132 the HHG envelope eﬀects, leaving the sharp resonant features of the respective photoionization targets.

Normalized spectra for Ar and He can be seen in Figures 53(c) and 53(d). First, the eﬀect of the Ar 3s3p64p resonance is now obvious because of a ¤ 400 meV-wide minimum in the spectral region surrounding the resonance, which is qualitatively very similar to the

3s3p64p cross section in Figure 52(a). This is much wider, however, than the ¤ 80 meV width extracted from the photoionization experiments of Berrah et al. This is most likely due to the measurement’s convolution with the harmonic widths, which are much larger than the width of the resonance, and the magnetic bottle resolution. As discussed before, we expect the cross section of the 3s23p44s4p resonance to be much smaller than the 3s3p64p, and indeed no obvious spectral feature is apparent near 33.8 eV in Ar. The ﬂuctuations in normalized amplitude are too large to draw a solid conclusion.

The He 2s2p structure comes through quite clearly in the normalized spectrum. There is a large spike in the photoelectron signal just below 60.15 eV, followed by a shallow minimum.

This shape is qualitatively the same as the He 2s2p photoionization cross section extracted from the measurements by Morgan et al. [178]. Again, due to a convolution with our magnetic bottle resolution and our harmonic widths, the size of the resonant feature is wider than the synchrotron photoionization measurements, where our resonant width is at least four times as wide as the cross-sectional width calculated by Morgan et al.

The above photoelectron spectra verify our ability to qualitatively reconstruct the spectral proﬁles of the Ar 3s3p64p and He 2s2p resonance, while being unable to reveal the Ar

3s23p44s4p resonance. To further probe the resonances, we will now move on to describing

the phase measurement results using odd-harmonic combs and the RABBITT method.

7.3.4 Ar and He RABBITT: Two-Omega Oscillation Results

In Figure 24 of Chapter 5, we used a cartoon example to show how our tunable mid-

infrared laser is able to scan a single harmonic across a sharp resonance and record the

resonant phase into the neighboring sidebands. In Figure 54, we demonstrate the same

133 technique using real RABBITT phase measurements. In this example, harmonics are generated in Ar and then used to also photoionize Ar. Here the data has already been corrected for filter delay and the attochirp from the HHG process. In the first panel in the top left, one RABBITT scan is shown, which only reflects the more slowly evolving GD envelope. In the second panel to the right, a few more scans at different wavelengths are added, revealing some large delay peaks and minima. Finally, in the last panel in the bottom middle, the RABBITT results taken at many driving wavelengths are shown. Now there are two very prominent features in the spectrum: one centered around the 3s3p64p resonance and another centered around the 3s23p44s4p resonance. First, on either side of the 3s3p64p resonance (green solid line) are two mirror structures that resemble the shape of the calculated autoionizing phase jumps shown in Figure 48(b). In the sideband just below the resonance, the jump is negative, and in the sideband just above the resonance, the jump is positive and slightly reduced in size (to be explained later in the section). The change in sign of the jump is due to the fact that the emission and absorption of IR photons to the lower sideband and upper sideband have opposite signs, respectively. Another way to say this is that for a resonant harmonic at a given wavelength, the lower sideband phase and upper sideband phase will move opposite each other: i.e. one up and one down. For example, at

1460 nm, the phase of the lower sideband jumps up while the higher sideband phase dips down. In the sidebands on either side of the 3s23p44s4p resonance (blue solid line), there are also two mirrored structures. Upon careful inspection, one can see that these features also demonstrate a sign ﬂip between the lower and upper sidebands, evidence of a resonant feature based on the argument made for the 3s3p64p resonance.

In order to be sure that the phase features were due to the photoionization of Ar, we decided to vary the HHG and target gases. Shown in Figure 55(a) are Ar photoionization results using CH4 and Ar high-harmonic generators, along with Ne photoionization results using an Ar generator. The results have been converted to sideband 2ω phase, instead of group delay, because the features should more closely resemble the autoionizing phase

134 Ar Resonances Ar Resonances 0.1 0.1 1400 nm 1400 nm 1410 nm 1410 nm 1420 nm 1420 nm 1425 nm 1425 nm 1430 nm 1430 nm 1435 nm 1435 nm 0 1438 nm 0 1438 nm 1440 nm 1440 nm 1442 nm 1442 nm 1445 nm 1445 nm 1448 nm 1448 nm 1450 nm 1450 nm

Group delay (fs) 1453 nm

Group delay (fs) 1453 nm -0.1 1455 nm -0.1 1455 nm 1457 nm 1457 nm 1460 nm 1460 nm 1465 nm 1465 nm 1470 nm 1470 nm 1480 nm 1480 nm 1490 nm 1490 nm -0.2 1500 nm -0.2 1500 nm 24 26 28 30 32 34 36 38 40 24 26 28 30 32 34 36 38 40 Photon Energy (eV) Photon Energy (eV)

Ar Resonances 0.1 1400 nm 1410 nm 1420 nm 1425 nm 1430 nm 1435 nm 0 1438 nm 1440 nm 1442 nm 1445 nm 1448 nm Ar 3s23p 6 3s23p44s4p 1450 nm

Group delay (fs) 1453 nm -0.1 1455 nm 1457 nm 1460 nm 1465 nm 1470 nm 1480 nm Ar 3s23p6 3s13p64p 1490 nm -0.2 1500 nm 24 26 28 30 32 34 36 38 40 Photon Energy (eV)

Figure 54: An example of how resonances appear in group delay through our HHG wavelength-scanning technique. In the top-left panel, no obvious resonant phase features can be discerned with only one driving wavelength. As more phase measurements are recorded at diﬀerent driving wavelengths, mirrored resonant phase structures are revealed around the Ar 3s3p64p and 3s23p44s4p resonances. The mirrored structure is due to the fact that the resonant phase is imprinted into the two neighboring sidebands, but with opposite phases due to the diﬀerence in IR absorption and emission phases.

135 instead of the autoionizing group delay.27 Clearly, in both of the cases where Ar is photoionized, the features centered around the two resonances are present. There is a slight diﬀerence, however, in the 3s13p64p resonance structures between the two generation gases, indicating phase structure from one of the HHG sources. We know from Chapter 6 that the

2ω phase from generating in CH4 is featureless (i.e., essentially only attochrip) and so is the

Ne atomic ionization 2ω phase, which indicates that the phase discrepancies between CH4 and Ar generation are likely due to the Ar generator. Indeed, in the Ne results, we see that

group delay around 26.6 eV has a similar overall structure to that of the Ar photoionization

results with an Ar generator, sans the resonant features. The Ne results have a spike in

the lower sideband, and a slope across the higher sideband, which support the fact that

Ar generation imprints additional phase structure onto the harmonics. This explains the

discrepancy in phase excursions between the lower and upper sidebands when generating

in Ar and measuring photoelectrons from Ar, mentioned earlier in the section. Turning to

the higher-lying 3s23p44s4p resonance, Ne does not exhibit the same phase structures that

are present in the two Ar photoionization measurements. This indicates that the resonant

phase features around 26.6 eV and 33.8 eV must be the result of photoionizing Ar.

Now that we have conﬁrmed that the resonant phase structures are a result of Ar

photoionization, we can compare our experimental results to theory. Shown in Figure

55(b) is a comparison of the Ar 3s3p64p experimental results and the ab initio theory described in the Section 7.3.1. Good agreement is found between the theory and experiment, indicating that indeed these phase structures can be interpreted as belonging to the Ar

3s3p64p resonance. The size of the phase jump is less than π, which is expected and

was previously explained [29] as the result of measuring both the interacting and non-

interacting channels in the measurement. Furthermore, these results are in near-quantitative

agreement with the size of the phase jump measured by Kotur et al. [29]. Unfortunately,

the theoretical model was unable to reproduce the 3s23p44s4p phase results, due to the

many other overlapping resonant pathways, but because the 3s23p44s4p resonance is the

27See Chapter 5 Section 5.2.2 for a discussion on this point.

136 CH4-Ar Ar-Ar Ar-Ne

(a) (b)

Figure 55: (a) Ar experimental RABBITT results with diﬀerent HHG generation gases, CH4 (blue) and Ar (purple). Also shown are Ne photoionization results with an Ar HHG generator (green). The resonant features survive changes in generation gases, but are no longer present with a Ne target; therefore, the resonant features belong to Ar. (b) Compar- ison of the Ar photoionization phase results to the ab initio theoretical calculations. The experiment has good agreement between the resonant shapes predicted by theory. deepest and broadest resonant feature in the energetic region, we can tentatively assign the resonant phase features around 33.8 eV to it. More calculations or supplemental work are required to draw any further conclusions.

Now we focus our attention on the RABBITT results for He, shown in Figure 56(a).

These results were taken with an N2O generator, rather than the CO generator used to get the amplitudes because we found that N2O provided a considerably higher photon yield than most generators in the 60 eV region. Additionally, when using a Ne target gas with a harmonic cutoﬀ signiﬁcantly beyond 60 eV, N2O was found to be featureless in phase around 60 eV.

For He, a similar resonant behavior to that of Ar is observed: if the phase in the lower sideband is “up,” then the phase of the higher sideband is “down.” Figure 56(b) shows the He results converted to sideband phase against the theoretical calculations. Besides a slight energetic shift between the experiment and theory, there is excellent agreement. This shift could be due to blue-shifting of the high-harmonic spectrum, which would lead us to mistakenly assign harmonics to lower energies in our energetic calibration. Also, according to discussions with Professor Argenti, the theory is prone to calibration errors along the

137 (a) (b)

Figure 56: (a) He 2s2p experimental RABBITT results. The sideband just below the resonant energy sees an increase in phase where as the sideband just above see a decrease. This is qualitatively similar to the results recorded using the so-called “Rainbow RAB- BIT”, which spectrally resolves the resonance within one sideband [153]. (b) Comparison of the He experimental results to the theoretical calculations. Here there is excellent agreement between experiment and theory, verifying the model’s ability to reproduce two-photon ionization phases of RABBITT. energy axis, which could also be a source of the energetic mismatch between theory and experiment.

With the above results, we have conﬁrmed our wavelength scanning technique as a valid method for probing the phase of autoionizing resonances, and we veriﬁed that the results can be reproduced by cutting-edge ab initio theoretical calculations, with the exception of the Ar 3s23p44s4p resonance. In the following section, we will extend the experiment and theory to the case of attosecond pulse trains comprised of even and odd harmonics.

As discussed in Section 7.2, using even and odd harmonics with an infrared probing ﬁeld allows for diﬀerent pathways to the continuum, giving rise to an omega oscillation in the delay-dependent two-photon ionization spectra. Comparing the theory and experiment in this case serves as another test of the ab initio calculation.

7.3.5 Ar and He with Even and Odd Harmonics: Omega Oscillation Results

Figure 57 depicts an Ar photoelectron spectrum created with harmonics generated in

Ar using a 1515 nm and 757.5 nm, two-color HHG beam. In the region surrounding the

138 Ar-Ar Even and Odd Harmonics 102 1515 nm

100

10-2

Photoelectron Signal (arb. units) 10-4 20 30 40 50 Photon Energy (eV)

Figure 57: A photoelectron spectrum resulting from photoionization with an XUV comb comprised of even and odd harmonics. The even harmonics in the region around the 3s3p64p resonance are kept at 25% of the intensity of the neighboring odd harmonics. The XUV light was generated by a 1515 nm driver and its spatiotemporally-overlapped second harmonic. The second harmonic power is ¤ 1% that of the fundamental.

3s13p64p resonance, the even harmonics are at ¤ 25% of the intensity of the neighboring odd harmonics. This ratio was ﬁxed for all omega scans. Figure 58(a) shows the omega phase results for many diﬀerent fundamental wavelengths using harmonic combs generated in Ar with the properties just described. For these results, a single odd harmonic is resonant with the 3s13p64p resonance. No resonant features were observed in the region around the 3s23p44s4p resonance, most likely due to the considerably more complex properties of the omega phase. Figure 58(a) panel (1) shows both the even and the odd omega phases.

Because we measure the omega phase up to an overall constant, there is an unknown phase shift between different omega scans. In order to account for this, the even-harmonic phase results for each wavelength have been shifted to a nonresonant even-harmonic reference, and the same was separately done for the odd harmonics. The shifting is done by simply using a nonlinear least squares minimization to fit for the error-bar-weighted offset between each data set and an arbitrarily-chosen reference data set in an energetic region near the resonance. Separately shifting the evens and odds is a small change, slightly modifying a ¤ π phase shift that was always observed between consecutive even and odd harmonics. Because we are not interested in the phase structures across multiple harmonics, this should not affect

139 our interpretation, and shifting the data as such allows us to better resolve the autoionizing structures within each even and odd harmonic. As an aside, this π phase shift has been reported before [74][179] and has been used to calculate that the XUV light has ¤ π~2 phase jumps between consecutive harmonics. As discussed in Chapter 3, the omega phase is a complicated mixture of the XUV properties and the atomic ionization characteristics, so this does not strictly guarantee that the actual XUV spectral phases between odd and even harmonics are π~2 out of phase for our measurements. Additionally, due to systematic errors caused by the transmission function of our magnetic bottle electron spectrometer, we were unable to reliably reconstruct the XUV phases from the omega oscillation (discussed in

Chapter 3). Due to these complications, the XUV phase is unknown in these measurements.

Continuing with our analysis of the omega phases, we turn to panels (2) and (3) of Figure

58(a). In (2), we see that the phase of the resonant odd harmonic has some structure to it, but this structure is not much larger than the error bars of the individual data points. On the other hand, the neighboring even harmonics have pronounced phase structures that are on the order of 0.5 rad. These features have been reproduced using diﬀerent combinations of the following parameters: a 50% even-to-odd harmonic intensity ratio, a resonant even harmonic instead of a resonant odd harmonic, and using diﬀerent generating gases including

CH4. Additionally, these features were not observed when a Ne target gas was photoionized instead of Ar.

When we turn to the theoretical calculations, however, we see drastically diﬀerent results from the experiment. First, a large phase jump is observed in the resonant harmonic oscillation, where the experiment does not see this. Secondly, the phase jumps in the neighboring harmonics are 3-4 times larger than the phase jumps recorded in the experiment and are generally diﬀerent in shape. Lastly, the theory does not observe π jumps between the omega oscillations of neighboring harmonics because the XUV pulse train was chosen to be Fourier limited in the calculation.

From an experimental point of view, Ar is a simple gas target to study, due to its large photoionization cross section compared to other noble gases. When it comes to the

140 (a)

(b) (c)

Figure 58: Experimental and theoretical results for the omega-oscillation phases around the Ar 3s3p64p resonance, where an odd harmonic is resonant. Panel (a) shows that the even and odd harmonic phases are roughly separated by π. In the resonant odd harmonic (green dashed box), there is not much structure. In contrast, the neighboring even harmonics (blue dashed box) exhibit pronounced structures much larger than the measurement error bars. In the lower even harmonic, the phase increases, then goes through a sharp decrease. In the upper even harmonic, the phase decreases sharply and then recovers within a few hundred meV. The theoretical results shown in (b) (odd) and (c) (even) are dramatically diﬀerent from the experimentally retrieved phases. At the time this dissertation was written, these diﬀerences were not yet resolved.

141 ab initio TDSE calculations though, Ar is considerably more diﬃcult to model than a simpler atom like He. This is because Ar has many more electrons, which drastically increases the complexity of the electron-electron correlations involved in photoionization.

So in order to shed some light on the discrepancies between the Ar experiment and theory, we also investigated two-photon omega oscillations in He, the results of which can be seen in Figures 59(a) and 59(b). For He, just like in Ar, and odd harmonic is resonant and the even-to-odd ratio is kept at ¤ 25%. The resonant harmonic does not exhibit any clear phase structure, whereas the neighboring even harmonics do. Interestingly, the shapes of the even-harmonic phase structures look much like the RABBITT two-omega phase structures.

Unfortunately, the theoretical model does not resemble the experimental results. Again, just like Ar, the model predicts phase jumps that are much larger than the ones observed in experiment, particularly in the resonant odd harmonic. The shapes of the even-harmonic phase structures are similar between experiment and theory, though. Where the lower- energy even harmonic increases in phase as a function of energy, so does the theory, and vice versa for the higher-energy even harmonic. One possible reason for the discrepancies between theory and experiment is that we are not able to characterize the spectral phases of pulse trains comprised of even and odd harmonics. If we were able to properly characterize these phases, then we would be able to better understand their impact on the autoionizing phase measurements.

142 (a) (b)

Figure 59: Experimental and theoretical results for the omega oscillation phases around the He 2s2p resonance. An odd harmonic is resonant in this case. In panel (a) the odd harmonic phases do not exhibit any signiﬁcant structures larger than the measurement error. In panel (b), however, the neighboring even harmonics exhibit structure: an increase in phase for the lower even harmonic and a decrease in phase for the upper even harmonic. Panels (c) and (d) show the theoretical results for He, which again do not agree with our experimental results. The origin of this discrepancy was not understood at the time this dissertation was written.

143 7.4 Conclusions

In this chapter, we used attosecond spectroscopy to study autoionizing resonances in

Ar and He. First we explored the spectral intensities of the resonances by scanning the mid-infrared HHG driver so that it tuned a single harmonic across the resonance. With this method, the retrieved spectral intensity structures of the Ar 3s13p64p and He 2s2p resonances qualitatively agreed with previous synchrotron measurements of the same features.

We were unable to observe, however, the intensity modulation due to a suspected 3s23p64s4p resonance in Ar. We then discussed two-omega phase measurements in both atoms, which revealed phase structures in the energetic regions of all three resonances, including the

3s23p64s4p resonance. Ab initio TDSE calculations were able to reproduce the two-omega phase structures for the Ar 3s3p64p and He 2s2p resonances, but were not able to reproduce the 3s23p64s4p features. Lastly, we explored a more complicated omega-oscillation that again revealed phase structures due to the Ar 3s13p64p and He 2s2p resonances. Inter- estingly though, these features are drastically diﬀerent in size from the predictions made by the ab initio models. These results will pave the way for extending attosecond spectroscopy to more complex electron correlation studies, and our results using combs of even and odd harmonics will particularly serve as benchmarks for future measurements of XUV spectra from oriented molecules and two-color HHG.

144 Chapter 8: High-Harmonic Spectroscopy of Two-Center

Interference in CO2,N2O, and OCS

“The eﬀect of substituting oxygen for one of the sulfurs has been devastating.”

—T. A. Carlson [123]

8.1 Two-Center Interference Introduction

For nearly a decade and half, molecular high-harmonic spectroscopy (HHS) has been used towards the study of electron dynamics in molecules. Much of the work began with the study by Itatani et al. [180], in which it was claimed that the HOMO of N2 could be reconstructed from molecular-frame high-harmonic spectra. The result from Itatani relied on the assumption that the returning electron wavepacket could be described in the plane wave approximation. Other studies found, though, that this assumption was invalid due to excluded Coulomb eﬀects [181] and multi-electron eﬀects [131][142] in high-harmonic generation (HHG). It was concluded that proper tomographic reconstruction required a measurement of not only the spectral intensity, but also of the phase.

Other studies soon followed to extend the tomographic technique to molecules like

CO2 (carbon dioxide), while making measurements on the phases of harmonic spectra [182][183][129]. A dominant feature of these papers was the presence of a pronounced interference that was attributed to two-center interference, wherein the de Broglie wavelength of the recolliding electron is resonant with the separation of the two oxygen-centered

HOMO orbital lobes, causing constructive or destructive interference in the HHG spectra.

In the CO2 case, the interference is destructive, so a minimum occurs along with a phase jump in the HHG spectrum. Using the Reconstruction of Attosecond Beating by Interfer- ence of Two-photon Transitions (RABBITT) method, Boutu et al. [183] directly measured a positive phase jump in the spectral phase, φE, of high-harmonics generated with an 800 X nm driver in CO2 aligned at 0 relative to the laser polarization, where E is the harmonic

145 energy.28 Additionally, Zhou et al. [182] and Vozzi et al. [129], made measurements of

φ θ, the angle-dependent phase at a given harmonic, at 800 nm and MIR wavelengths, respectively. Both groups reported an angle-dependent phase jump in their measurements, and Vozzi further veriﬁed that the measured interference position was intensity-independent.

Additionally, Vozzi et al. used calculations that either assumed or demonstrated that the spectral phase jump φE was positive. It is not clear from the paper which is true.

The interpretations of the 800 nm results were later brought into question, though, with measurements by Worner et al. [184] in CO2 and Rupenyan et al. [143] in N2O. These studies found that similar interferences at near-infrared (NIR) wavelengths were intensity-

dependent, which they attributed to multi-orbital eﬀects between HOMOs and lower-lying

orbitals. Clearly then, the interferences measured with an 800 nm driver by Boutu et al.

and Zhou et al., and the interference measured with mid-infrared (MIR) wavelengths by

Vozzi et al. cannot both be attributed to the same two-center interference. Thus, there is

no experimental consensus on the spectral properties of two-center interference in CO2. In this chapter, we will use HHS to study the intensity and phase features of two-center

interference in CO2 (Ionization Potential (IP)=13.77 eV) with NIR and MIR driving wavelengths, and we will compare to theoretical calculations from time-dependent density func-

tional theory (TDDFT). Our motivation for this study is twofold. First, with comparisons

to theory and measurements at NIR and MIR wavelengths, we will accurately characterize

two-center interference, which we unequivocally separate from the eﬀects of multi-electron

interferences. Secondly, we extend these new results to characterize and compare two-center

interferences using MIR generating wavelengths in a linear π-series: CO2, OCS (carbonyl

sulﬁde, IP=11.18 eV), and N2O (nitrous oxide, IP=12.89 eV). As discussed and shown in Chapter 6, Cl- and S-containing molecules exhibit molecular Cooper minima, so long as

they retain the atomic 3p character of Cl and S. It is known that OCS retains some 3p

character in its HOMO [185][136][137], but interference from the O 2p in the photoioniza-

tion spectrum dramatically alters the shape and appearance of the S Cooper minimum.

28The positive phase jump was strictly measured against a krypton reference generation gas.

146 Figure 60: A cartoon depiction of two-center interference. Two center interference occurs when an electron with de Broglie wavelength λde Broglie is incident on a molecule with lobe separation R and, for the case of destructive interference, satisﬁes the criterion of Equation 132. Isosurfaces of the HOMO orbitals for each molecule are represented with red and blue, which indicate opposite phases. The interference can be aﬀected by the recombination dipole phase of the respective centers, for example if one center is of 2p character and the other is of 3p.

We will use this attribute of OCS to compare to two-center interferences in the largely

2p-based HOMOs of CO2 and N2O [186]. Our results demonstrate that HHS of two-center interference is a sensitive tool to study electron ionization dynamics.

8.1.1 Statement of Contributions

Before describing the theory of two-center interference, I would like to state my direct contributions to the experiments described in this chapter. The CO2 NIR measurements were taken by Dr. Timothy Scarborough and me, and the analysis of those results was

done by me. The MIR measurements of CO2,N2O, and OCS were performed by me with assistance from Dr. Scarborough. The analysis of those results was performed by me with

assistance from Dr. Scarborough. Lastly, the toy model used to verify our interpretations of

two-center interference in CO2 and OCS was originally conceived by Dr. François Mauger with assistance from me, and the model was advanced and implemented by Dr. François

Mauger.

147 8.2 Theoretical Description of Two-Center Interference

Before discussing the theoretical and experimental results, we begin with a quantum- mechanical description of two-center interference in the context of HHG from aligned molecules, following Etches et al. [82]. The HOMO of an oriented molecule, Sψ, θe, may be expanded as the linear combination of atomic orbitals (LCAO), Sψn, θe, by summing over atomic centers in the following way:

Sψ, θe Q cn Sψn, θe , (128) n

where ψ represents the wavefunction, θ represents the orientation of that wave function

relative to the HHG laser polarization, and cn is the amplitude associated with the nth center. Recall from Chapter 2 that in the case of HHG, the electron is tunnel-ionized and

returned along the laser polarization. This means that θ also represents the angle of the

recolliding electron relative to the nuclear axis. The recombination dipole velocity matrix

29 element between Sψ, θe and a recombining plane wave Skee is given by

ike Rnφnk,θ `keS vˆdip Sψ, θe Q e Tcn `keS vˆdip Sψn, θeT , (129) n

where ke 2π~λdB is the wavevector of the recolliding electron with de Broglie wavelength

λdB, Rn is the position of the nth center, and φn k, θ may be interpreted as the intrinsic

recombination phase associated with the atom located at the nth center. By inspection

of Equation 129, we can see that two types of phases contribute to the dipole: ke Rn, which we will refer to as the “geometric phase,” and φn k, θ, which we will refer to as the “recombination dipole” (RD) phase.

For the case of an aligned molecular sample, the wavefunction is comprised of orienta-

29Here we really do mean “dipole velocity,” and not the “velocity gauge.” Harmonic spectra may be calculated using expectation values of the dipole, dipole velocity, and dipole acceleration in either the length or velocity gauges: a total of 6 pathways [187].

148 tions θ and θ π. The dipoles for the two orientations are related by

`keS vˆdip Sψ, θ πe `keS vˆdip Sψ, θe . (130)

It follows that the recombination dipole velocity matrix element for an aligned sample is

given by taking the real part of Equation 129:

` S S e ` S S e ke vˆdip ψ, θ aligned Re ke vˆdip ψ, θ (131) Q cos ke Rn φn k, θ Tcn `keS vˆdip Sψn, θeT . n

Thus, destructive interference in an aligned sample occurs for values of ke that minimize the sum in Equation 131. To a ﬁrst approximation, we can ignore the dipole amplitudes

Tcn `keS vˆdip Sψn, θeT. Under this approximation, the criteria for destructive interference with two centers is given by

SkeS R cos θ φ2 ke, θ φ1 ke, θ 2m 1π (132)

for any integer m, where R SR2 R1S. For molecules with π-symmetry HOMOs like CO2

(1πg), N2O(2π), and OCS (3π), one can expect that Sφ2 φ1S ¤ π. It should be noted that

although we have used the nuclear positions to deﬁne Rn, in reality R is the mean distance between the two orbital lobe centers. In Equation 132, we have a qualitative description of

two-center interference, which we will now analyze.

First, though we have derived Equation 132 for the explicit case of aligned molecules,

the same interference criterion can be derived from the oriented recombination dipole as

well [82][188]. This is important because impulsive molecular orientation typically produces

weakly oriented samples, and methods for high degrees of orientation are experimentally

challenging [34]. On the other hand, impulsive molecular alignment is easier to implement

compared to molecular orientation and can produce high degrees of alignment with a simpler

30 2 optical design. Secondly, because the electron’s return energy is proportional to SkeS and

30Refer to Chapter 4 for more on impulsive molecular alignment.

149 HHG photon energies are directly proportional to electron return energies, we expect the

2 minimum position in HHG spectra caused by SkeS R cos θ to be proportional to 1~ cos θ. Thirdly, the interference minimum is laser intensity- and wavelength- independent. This would mean that any movement in the interference minimum position as a function of laser intensity or wavelength would demonstrate the interference to be the result of a different mechanism. Lastly, Equation 132 says that two-center interference is affected by the RD phase difference between centers φ2 ke, θ φ1 ke, θ. Through this term, which we will refer to as the “electronic structure term”, we can compare the recombination dipole matrix elements (RDME) properties between OCS, CO2, and N2O, pictured in Figure 60. To date, no experiments have explored the impact of the electronic structure term φ2 ke, θ

φ1 ke, θ on two-center interference measurements. It was shown by Schoun et al. [98] and in Chapter 6 that measurements of the group delay (GD) of the RDME are able to

uncover complex electron dynamics in molecules. In Section 8.4.3, we will extend this idea

to measuring the interplay of electronic structure φ2 ke, θφ1 ke, θ with simple geometric

two-center interference SkeS R cos θ, particularly how the interference of the S 3p and the O 2p in OCS alter the measured two-center interference.

8.3 Time-dependent Density Functional Theory Calculations

In order to understand spectral features that one should expect from HHG in aligned

CO2, our collaborators in the Mette Gaarde and Ken Schafer Groups at Louisiana State University have developed a TDDFT framework against which to compare our measure-

ments. The framework was primarily constructed by their graduate student Paul Abanador

using the OCTOPUS open-source software [189]. In principle, one could describe such

multi-electron strong ﬁeld processes in HHG by solving the time-dependent Schrödinger

equation (TDSE) for a multi-electron wavefunction ψt. In practice, however, solving the

TDSE comes with a high computational cost that renders it impractical. By instead solv-

ing the more computationally-friendly Kohn-Sham (KS) equations in TDDFT [190], their

framework provides a practical approach for describing the dynamics in simple molecular

150 Short

Long

X X Figure 61: Gabor Analysis of aligned CO2 TDDFT calculations for (a) 15 and (b) 45 . The forward tilting spectra (dashed purple line shown in ﬁrst half-cycle), are the short trajectory responses in CO2. These are clearly more dominant than the backwards tilting (dashed white line shown in ﬁrst half-cycle) long trajectories. A minimum clearly appears around 45 eV at 15X systems with multiple active orbitals.

Briefly, the calculation details are as follows. They model the HHG emission from the interaction of a strong infrared field (IR) with a single-molecule system of CO2 by solving the time-dependent KS equations. The exchange-correlation potential is approximated using the local density approximation (LDA) [191], including the Perdew-Zunger formulation for the correlation functional [192], which is supplemented with the average density self- interaction correction (ADSIC) [193]. In the calculation, the molecular geometry of CO2 is fixed, with bond lengths RCO 1.16 Å. Furthermore, it is assumed that only the 16 outermost electrons participate in the dynamical response.

The time-dependent KS equations are initialized in the ground state and then propagated with the applied laser field. The laser field is a “flat-top” pulse with a two-cycle ramp up. All of the calculations are done on a Cartesian grid with an absorbing edge. The dimensions are SxS B 195 a.u. and SyS SzS B 30a.u., where the laser is polarized along the x-direction. This means that in their calculations, θ is referenced to the x-axis instead of the z-axis, which is used in standard polar coordinates. The spatial grid spacing is set to

0.4 a.u. and the propagation time step is set to 0.05 a.u. (1 a.u.=24 as ).

151 The harmonic spectral properties, intensity and phase, are extracted from the above calculations by performing a sub-cycle frequency analysis of the time-dependent electron density, also referred to as a Gabor transform analysis [194]. It is known, though, that on a single-molecule scale, multiple trajectories contribute to the sub-cycle HHG signal.

This is problematic, because experiments are able to preferentially select the short trajectory by way of macroscopic phase-matching eﬀects (see Chapter 2). Macroscopic TDDFT calculations would involve many hundreds of single-molecule calculations that are not computationally feasible. To circumvent this problem, a weak attosecond pulse train (APT) comprised of a few harmonics is added to seed ionization of short trajectories in the strong

IR ﬁeld. The APT energies are selected to generate photoelectron energies near the ionization threshold. With an appropriate choice of the IR-APT time delay, this scheme has been theoretically and experimentally demonstrated to generate short-trajectory-dominated

HHG spectra[195][196][197]. In this case, the delay is set to 0.17 laser cycles before the zero crossing of the IR ﬁeld.

Although the laser ﬁeld in the experiment had a central wavelength of 1300 nm and an intensity of ¤ 1 1014 W ~cm2, similar parameters in the calculation led to too much

IR ionization relative to the APT. With no expected change to the interference features of the HHG spectrum, the wavelength in the calculation was increased to 1500 nm and the intensity was dropped to 0.6 1014 W ~cm2. This allowed the calculation to reproduce a similar cutoﬀ, while reducing the overall background from direct IR ionization.

In Figure 61, we can see the results of a Gabor analysis for the CO2 harmonic signal parallel to the laser polarization with the molecule at 15 degrees and 45 degrees. First, we can see that at both angles, the harmonic spectrum is indeed dominated by the short trajectories: the “forward-tilting” portions of the spectra. We also see a pronounced minimum in the short trajectory contribution to the harmonic spectra around 45 eV. This minimum is attributed to the two-center interference in CO2 and is of course expected from the references discussed above.

Shown in Figure 62 are the angle-dependent harmonic spectra normalized to the angle-

152 80 0.8

70 0.6

0.4 60

0.2 50

0.0 40

Energy (eV) -0.2

30 -0.4

20 -0.6

10 -0.8 -60 -40 -20 0 20 40 60 Angle (deg)

Figure 62: Alignment-dependent harmonic spectral yields in CO2 at 1500 nm with an intensity of 0.61014 W ~cm2. The color bar indicates log scale. These spectra are extracted from the time signal during the ﬁrst half-cycle after the ﬁeld ramp up. Each angle is normalized to the angle-averaged signal. This is done for a more direct comparison to the experimental results. Two-center minimum positions are indicated by the white diamonds, and the dashed line indicates 1~ cos2 θ dependence to guide the eye. averaged signal. Clearly, the two-center minimum, indicated by white diamonds, occurs at

0X near ¤ 45 eV. The minimum moves to higher energies in a fashion very similar to the

IP α~ cos2 θ trend expected from Equation 132, where α is a constant of proportionality.

Upon closer inspection, however, we observe a slower angular evolution of the geometric two- center interference feature, and empirically ﬁnd a better match with IP α~ cos θ. This

deviation from the prediction of Equation 132 is consistent with previous ﬁndings that plane-

wave approximations of RDMEs often gives poor quantitative predictions [181][131][142].

The modiﬁed θ-dependence will be used in a toy model at the end of this chapter.

Let us now consider the spectral phase. Plotted in Figure 63 are the spectral intensity X and phase of CO2 at 5 relative to the laser polarization. The total intensity and phase are shown with a black dashed line and the x- and y-components are shown in red and blue,

respectively. A deep minimum appears in the total signal and in the x-component very near

45 eV. This minimum is accompanied by a negative phase jump, which corresponds to a dip

in the GD. Though the absolute value of the spectral phase is arbitrary, its derivative is not,

153 X Figure 63: Calculated HHG spectral intensity and phase for CO2 at θ 5 . In (a), the total spectral intensity (black dashed) and its x-component (red) and y-component (blue) are plotted. A deep minimum appears in the total spectral intensity and it’s x-component. In (b), the corresponding spectral phases are plotted. A negative phase jump occurs at the energy of the spectral minimum in both the total spectral intensity and in the x-component. The negative phase jump would correspond to a minimum in the GD. In order to help guide the eye, the green dashed line indicates the expected position of the minimum based on Equation 132 . meaning that the sign of the phase jump should be measurable in our HHS experiments. As mentioned above, though, the only other measurement of aligned CO2 at 800 nm indicated a positive phase jump in a similar energy range.

With the above theoretical predictions for the GD in CO2, we now have an additional criterion to separate two-center interference from other interferences. This will be explored in the following section.

8.4 Results

8.4.1 Experimental Details

For all of the measurements around 1300 nm, we used the output of the OPA described in Chapter 5, and a focal length of 400 mm. For the 785 nm CO2 measurements, a similar

154 apparatus to the HHG-alignment interferometer described in Chapter 5 was used with one main diﬀerence; instead of using the OPA depleted pump, we rerouted the pump before the

OPA for use in the experiment. The power and spot sizes of each beam are adjusted with variable apertures. The RABBITT measurements at 785 nm were taken with a 400 mm focal length. This focal length was later switched to 300 mm for the laser intensity-scaling measurements, but should have no bearing on the outcome of the measurements. Similar intensity-scan results were found at 785 nm using a 400 mm focal length by adjusting the laser intensity with a variable aperture.

The generation intensity of the HHG driver was estimated to be approximately 10.2

1014 W ~cm2 for all measurements, except in the case of the intensity scans. A polarizer-

waveplate pair was used to control the intensity of the HHG arm for these scans. As for

the kicking pulse, its intensity was always kept below that of the HHG intensity, in order to

reduce the overall ionization of the sample. Too much intensity was gauged by a sharp drop

in the HHG signal that occurs with ground state depletion. After seeing this drop in HHG

signal, the intensity of the kicking pulse was decreased back under this limit by adjusting a

variable aperture. For molecular-frame GD measurements, the HHG-alignment pulse delay

was ﬁxed at the half revival, and the alignment pulse was rotated by a zero order half

waveplate. For molecular-frame spectral intensity measurements, the data were recorded

from the delay-dependent molecular-frame spectra at a ﬁxed alignment pulse polarization,

and also by rotating the polarization of the alignment pulse at a ﬁxed HHG-alignment delay.

Due to the low vapor pressure of OCS, it was necessary to seed it at 10% in a He

buﬀer gas in order to prevent any eﬀects of clustering. Doing so required the use of a

200-µm-nozzle-diameter Even-Lavie pulsed valve. Backing pressures between 15 and 20 bar

were used with opening times between 23 and 30 µs, depending on day-to-day pulsed valve

operation, in order to achieve interaction gas densities required for proper phase-matching.

It was conﬁrmed that no signiﬁcant XUV light was generated from the He carrier gas, by

testing a neat He sample under the same generating conditions. CO2 and N2O, however, were able to be delivered neat and with a continuous 200-µm-nozzle-diameter gas nozzle,

155 using typical backing pressures of half a bar. To rule out any dependencies on the gas nozzle or gas pressure, CO2 and N2O were tested at high pressures with the pulsed valve, and N2O was further tested in at 10% in an He carrier gas to rule out clustering. No significant differences were found, so these results will not be shown here. For all GD measurements, the integrated fit method described in Chapter 5 was used. The error bars of each sideband measurement are given by the fit error. For each GD measurement, the

Al ﬁlter delay and Ne detection atomic delay are removed before ﬁtting for and removing the attochirp. For a more in-depth discussion of the removal of the attochirp, please refer to the method developed in Chapter 6.

8.4.2 CO2 Interferences at 785 nm

Before discussing our results of two-center interference around 1300 nm, it is important to compare experimental results at 785 nm to previous measurements [183] and to the predictions of the TDDFT results at MIR wavelengths. In Figure 64, the molecular-frame measurements of CO2 at 785 nm are plotted. Figure 64(a) shows the GD measurements X X X X for CO2 alignment angles of 0 , 45 , and 90 . Relative to the 90 measurement, both the 0X and 45X measurements exhibit a positive “bump” in GD. Most obviously at 0X, the GD

sharply spikes around ¤ 38 eV (24th harmonic). This positive increase in GD corresponds

to a positive phase jump, which is qualitatively similar to the previous measurements of

Boutu et al. When compared to the TDDFT calculations, this is opposite to the predicted

sign.

The HHG spectrogram, corrected for the Al ﬁlter transmission and the Ne detector

cross-section, is shown in Figure 64(b). Around 0X the harmonic intensity decreases at

lower energies and increases at higher energies, with an eﬀectively extended cutoﬀ. At

90X, the spectral intensity increases for all harmonics, but with a slightly reduced cutoﬀ

compared to 0X.

In Figure 64(c), we have normalized the same spectrogram to the quasi-unaligned spec-

156 0° 90°

(a) (b)

8 620 mW 560 mW 6 500 mW 440 mW 4 380 mW

Log of Enhancement 0

-2 20 25 30 35 40 45 50 Photon Energy (eV) (c) (d)

Figure 64: CO2 molecular-frame HHG GD and intensity, generated with a 785 nm driver. In panel (a), the angularly-resolved GD results are shown for 0X, 45X, and 90X. At 0X, the GD bumps up compared to 90X, opposite that predicted by the TDDFT calculations. In (b) the delay-dependent spectrogram (corrected for Al transmission and Ne cross-section) is shown, and the same data normalized to the quasi-unaligned region is shown in (c). Both plots exhibit a minimum at 0X(dashed green) that is gone by the time the molecule reaches 90X (dashed blue). Panel (d) shows the laser intensity-dependence of the 0X minimum. The interference minimum moves from high energies to low energies as the laser intensity is decreased.

157 trum that is far from a revival structure.31 Doing so focuses on the eﬀects of the interference, rather than the overall spectral proﬁle. The energy axis has been binned to harmonic order to remove artifacts from the normalization process occurring in the gaps between harmonic orders. Additionally, the delay dimension has been smoothed with a 3rd order Savitzky-

Golay filter to remove delay-dependent noise. In this figure, the signal enhancement or suppression due to the interference is clearer. Specifically, the minimum at 0X is more pronounced. There may be a ¤ cos2 θ dependence to the minimum position, but this data is far too pixelated to be sure, due the large energy separation between harmonic orders.

It was discussed before that if this enhancement structure was truly due to two-center interference as described by Equation 132, then the minimum position should be independent of the laser intensity. Plotted in Figure 64(d) are lineouts at 0X of the enhancement results from a laser intensity scan. It is obvious from this ﬁgure that the minimum position moves by as much as 5 eV in a smooth fashion towards lower energies as the laser intensity is decreased. This result is also seen in the unnormalized spectra and is not an artifact of the normalization process.

The above results for HHG at 785 nm in CO2 have shown that the interference measured by Boutu et al. and our group at NIR wavelengths has an opposite 0X phase jump compared to theory, and that the minimum position associated with this phase jump is intensity-dependent. Therefore, this interference cannot be due to two-center interference, as previously concluded. It is more than likely that these spectral features are due to the interference of HHG from the HOMO (IP=13.77 eV) and HOMO-2 (IP = 18.1 eV) of CO2, which was concluded in the previous experiments by Wörner [184] and Rupenyan [143]. For such dynamical interferences, the minimum position is expected to vary with laser intensity.

Further theory is needed to correctly identify this interference, but the added measurement of the GD should greatly assist in its characterization.

158 100 CO 2 N O 2 OCS

10-5 Signal (arb. units)

CO 2 0.2 N O 2 OCS 0.1

-0.1 Group Delay (fs) -0.2

-0.3 20 30 40 50 60 70 Photon Energy (eV)

Figure 65: Shown are HHS results of unaligned CO2,N2O, and OCS at. In (a) are the spectral intensities at 1300 nm (each normalized to unity) and in (b) are the GD wavelength scans centered around 1300 nm. The spectra and GDs have been corrected for the eﬀects of the Al ﬁlter and Ne detection gas. Additionally, the attochirp has been removed from the GD measurements. These results demonstrate the unique aspects of OCS, including a sharp drop in signal after ¤ 43 eV and a GD dip centered at the same position.

159 8.4.3 Two-Center Interference in CO2,N2O, and OCS in the MIR

We now consider our measurements at 1300 nm. As a reminder, the purpose of these measurements is twofold. First, we would like to determine the true sign of the phase jump in CO2 in a wavelength regime that Vozzi et al. [129] demonstrated an intensity- and wavelength-independent interference minimum. Secondly, we want to explore the eﬀect of electronic structure term φ2φ1 on the intensity and phase of two-center interference minima by comparing the Cooper-like 3p-2p interference of OCS to that of the 2p-2p interferences of CO2 and N2O, which do not contain any large phase features over the relevant region.

First let us discuss the results for unaligned samples of OCS, CO2, and N2O. In Figure 65(a), results of unaligned HHG spectra taken at 1300 nm are shown. These spectra are

taken without the alignment pulse. Each spectrum is normalized to unity and has been

corrected for the Al ﬁlter transmission and the Ne detector photoionization cross-section.

In the unaligned samples, two-center interferences are not expected to be visible, due to

angular averaging. Indeed, the spectra for CO2 and N2O are relatively featureless until the

cut oﬀ of N2O is reached and until the aluminum edge is reached for CO2. On the other hand, OCS begins to dramatically depart from the spectra of the other two around 30 eV

and descends rapidly until a “kink” at ¤43 eV, indicated by the vertical dashed line. After

this point, the spectrum exhibits a ﬂat, plateau-like structure until the cutoﬀ is reached.

This feature has been found to be independent of laser intensity and extend as far as 60

eV for higher intensities at 1300 nm. Additionally, it was found to be independent of laser

wavelength using 1500 nm, 1700 nm, and 2000 nm drivers. Across the 40 - 50 eV region, the

OCS spectrum is approximately two orders of magnitude smaller than those of the other

two molecules. The results shown in Figure 65(a) are in qualitative agreement with the

results of photoionization experiments performed in all three molecules[137][185][198][199],

demonstrating our general sensitivity to the S-O interference in OCS that is not present in

CO2 or N2O. In contrast to those earlier photoionization experiments, we are able to additionally

31For more on revival structures, please refer to the chapter on impulsive molecular alignment, Chapter 4.

160 characterize the spectral GD. In Figure 65(b), the combined results of GD wavelength scans are shown for all three unaligned molecules. Scanning the driving wavelength and recording multiple GD measurements allows for a higher eﬀective sampling rate.32 Driving wavelengths in the range 1270 - 1330 nm with 10 nm steps were used. The GD results are only shown in the energy range of 28-60 eV because it was found that below ¤28 eV the

Ne detection gas atomic delay calculation introduced systematic artifacts. Above 60 eV we were not able to retrieve high-quality RABBITT measurements for all three molecules, so those results have been excluded as well. The OCS results demonstrate a large dip of

¤140 as in group delay around 43 eV, coincident with the spectral kink in the unaligned

OCS spectral intensity. Over the same spectral region, CO2 and N2O do not demonstrate a similar dip in group delay. Though there is a consistent decrease in GD for N2O near the cut oﬀ, this feature’s energetic position was found to scale with the laser intensity, indicating that it is not a structural feature that ﬁts within the scope of this chapter, so it will be ignored henceforth. The position and size of the OCS GD dip are very similar to that of our previously-discussed measurements of CH3Cl and the measurements of Schoun et al. [10], where minimum positions were found to be between 40 and 50 eV with GD dip

sizes between 100 and 200 as. This indicates the CM-character present in OCS. According

to Carlson et al. [137], however, the minimum in OCS cannot be thought of as a pure

Cooper minimum, but instead should be thought of as "Cooper-like minimum" because the

additional 2p contribution to the RDME. With these results we have ﬁrmly established our

ability to fully characterize the unaligned spectra of all three molecules.

To explore the θ-dependence of the interferences, let us ﬁrst turn to the spectral intensity

spectrograms (corrected for Al ﬁlter and Ne detector contributions) plotted against photon

energy and HHG-alignment delay for all three molecules. The plots are shown in Figure

66, panels (a), (c), and (e) for CO2,N2O, and OCS, respectively. Each spectrogram shows the results of a delay scan between the HHG and alignment pulses across the half revival.

Before directly analyzing the interference minima, let us discuss a few other obvious features

32For reference, this method was described in Chapters 5 and 7.

161 101 CO 2

100

10-1 quasi-unaligned 5 angle average 0° 22.5° Signal (arb. units) -2 10 45° 67.5° 90° 10-3 20 30 40 50 60 70 Photon Energy (eV) (a) (b)

102 N O 2

100

quasi-unaligned -2 10 5 angle average 0° 22.5° Signal (arb. units) 10-4 45° 67.5° 90°

10-6 20 30 40 50 60 70 Photon Energy (eV) (c) (d)

101 OCS quasi-unaligned 5 angle average 0 10 0° 22.5° 45° -1 10 67.5° 90°

10-2 Signal (arb. units) 10-3

10-4 20 30 40 50 60 70 Photon Energy (eV) (e) (f)

Figure 66: Molecular-frame spectral data for all three molecules at 1300 nm, extracted from the HHG-alignment delay-dependent and alignment polarization-dependent signals. All plots have been corrected for the Ne cross-section and the Al filter transmission, and the spectrograms (a, c, e) have been normalized to unity in order to compare the different spectral profiles on the same color scale. In panel (a) are the results for CO2 versus HHG-alignment delay and in (b) are shown the spectra obtained by rotating the alignment polarization at a fixed delay. The same is done for N2O in (c) and (d) and for OCS in (e) and (f).

162 and diﬀerences in the three spectrograms. First, the signal of OCS remains small at high energies for all molecular angles when compared to CO2 and N2O. Also, OCS does not exhibit an obvious extension of the cutoﬀ near 0X and 90X, which is otherwise obvious in

N2O. The same cutoﬀ extension occurs in CO2, but is obscured by the Aluminum L2,3 edge.

Shown in Figure 66 panels (b), (d), and (f) for CO2,N2O, and OCS, respectively, are similar curves acquired by rotating the alignment pulse polarization between 0X and 90X

in 5 steps, thereby rotating the orientation of the molecule relative to the HHG driver.

Additionally plotted is a lineout from the quasi-unaligned region far from the revival (when

both polarizations are parallel) and a curve representing the incoherent average of the

ﬁve selected angles. Both CO2 and N2O exhibit minima that move to higher energies with increasing angle. In OCS, however, it is not clear in these plots whether or not the minimum

moves to higher energies. Instead, the interference appears to be largely constrained to a

small region around 43 eV.

When comparing the ﬁve-angle incoherent average to the quasi-unaligned curve for CO2, the two curves are very similar to each other until energies above ¤ 45 eV, where the ﬁve-

angle average is larger than the unaligned curve. The same behavior is evident in N2O, but occurs at a higher energy, ¤ 60 eV. Again, the case is not the same in OCS, where instead

the two curves are very similar everywhere except around 43 eV. Because the angle-averaged

curve is created by an incoherent sum and the quasi-unaligned curve represents the coherent

sum, comparing the two curves hints at where interferences are present in the unaligned

spectrum: i.e. where the phase must be included to get the correct unaligned spectrum.

This will be revisited in a toy model at the end of this chapter.

Next, we reconsider the same spectrograms, but now normalized to the quasi-unaligned

regions. These plots are shown in Figure 67, panels (a) (CO2), (c) (N2O), and (e) (OCS). Given the very diﬀerent cross-section of OCS relative to the other two molecules, normalizing

to the quasi-unaligned spectral intensities allows for a more direct comparison of the θ-

dependent interference eﬀects in all of the molecules.

At 0X for all three molecules, there is a minimum at low energies between 40 and 55 eV,

163 (a) (b)

(e) (f)

Figure 67: Molecular-frame spectral intensity enhancements relative to unaligned samples for all three molecules at 1300 nm, extracted from the HHG-alignment delay-dependent and alignment polarization-dependent signals. In panel (a) are the enhancements for CO2 versus HHG-alignment delay and in (b) are shown the enhancements obtained by rotating the alignment polarization at a ﬁxed delay. The same data sets are shown for N2O in (c) and (d) and for OCS in (e) and (f). CO2 and N2O both demonstrate similar enhancement behaviors: deepest minimum at 0X with large enhancements at higher energies for large angles. For OCS, however, the deepest minimum is between 22.5X and 45X with positive enhancements localized to an energetic region around 43 eV.

164 corresponding to their respective destructive two-center interferences. The 0X interference position of CO2 is at ¤ 45 eV, which is in agreement with the TDDFT calculations. As the delay is increased, thereby increasing θ, the interferences for all three molecules move to higher energies, much like one would expect from the ﬁrst term on the left-hand side of

Equation 132.

The same angle-dependent spectral behavior was generally observed when the HHG- alignment delay was set to the “0X delay” and the alignment pulse polarization was rotated, shown in panels (b) (CO2), (d) (N2O), and (e) (OCS). In (b) and (d), some diﬀerences between CO2 and N2O are clearer than in the spectrograms, a particular diﬀerence being X X that the minimum does not move in CO2 between 0 and 25 , whereas it is clearly shifted X in N2O. Also, these curves show that the 0 minima for all of the molecules are consistent

with their center-to-center nuclear axis lengths: ROCS 2.71 Å, RCO2 2.32 Å, and

RN2O 2.31 Å.

Where CO2 and N2O share many qualities, OCS markedly deviates from the other two molecules in the behavior of its molecular-frame interference. In panel (f), the enhance-

ments in OCS relative to unaligned are largely constrained to the 30-50 eV region. The

most suppression (deepest minimum) is between 22.5X and 45X, whereas the maximum en- X hancement is at 90 . For comparison, both CO2 and N2O show large enhancements for nearly all angles when the energies are higher than the destructive interference minimum.

Additionally, they are most suppressed at 0X, in contrast to OCS.

It was shown before that the interference in CO2 at 785 nm was intensity-dependent, thereby disqualifying it as a structural two-center interference. It is also important to

answer this question at 1300 nm, where two-center minima are expected to be intensity-

independent [129]. Figure 68 shows the enhancements relative to unaligned for all three

molecules at diﬀerent laser intensities. These plots show that there is no obvious intensity-

dependence to the minima positions. At the lowest intensity for N2O, a slight shift is observed but this is more than likely due to the normalization procedure. The 0X spectrum

experiences a cutoﬀ extension relative to unaligned, which means that the cutoﬀ region at

165 (a) (b)

(c)

Figure 68: Enhancement curves at 0X for all three molecules as the laser intensity is varied. All of the intensity scans demonstrate the intensity-independence of the minima positions. This supports the interpretation that the minima are due to the electronic structure of the respective molecules.

166 0X does not have a high-statistics region to normalize against. So as the cutoﬀ approaches the interference, normalizing to the unaligned spectrum obscures the resonant feature. This problem is not seen in the other two molecules because their interferences are farther from the cutoﬀ. All of the above results for CO2 and N2O are in agreement with those of earlier studies [129][143]; therefore, it is reasonable to conclude that the measured angular

variation of the interferences in CO2 and N2O are primarily due to the SkeSR cosθ term of Equation 132. In contrast, the localized nature of the OCS interference enhancement

indicates a structural interference in OCS that has additional angular variation from the

phase diﬀerence φ2 ke, θφ1 ke, θ: not inconsistent with a Cooper-like minimum. Lastly, due to the lack of intensity dependence at 1300 nm, it is also reasonable to conclude that

only the HOMO is signiﬁcantly involved in the HHG process for all three molecules at this

wavelength.

To go a step further than previous studies, we now focus our attention on the angle-

dependent GD measurements shown in Figure 69, which were taken at a ﬁxed delay corre-

sponding to 0X, while rotating the polarization of the kick pulse. The attochirp for all three

molecules was ﬁt using the 90X GD measurement for each molecule, respectively. Shown in X panel (a) are the angle-dependent GDs for CO2. At 0 , the GD “dips” near the interference position, corresponding to a negative phase jump. This is in agreement with the TDDFT

calculations discussed above. These CO2 spectral GD results, coupled with the fact that

the CO2 interference minimum position is intensity-independent, allow us to unequivocally

identify this interference as the two-center interference in CO2, predicted in theory and only previously measured in amplitude. X Continuing with the CO2 GD measurements, at 22.5 the dip has not appreciably moved in position relative to 0X, but it is shallower. This is in agreement with the spectral minimum X for CO2 at 22.5 in Figure 67(b) and also not inconsistent with the minimum predictions from TDDFT. For angles at 45X and above, the GD dip is not present. The case is similar

for the N2O GDs shown in Figure 69(b), where again, the GD dips near the minimum associated with the destructive interference. It moves to higher energies at 22.5X and is out

167 (a) (b)

(c)

Figure 69: Molecular-frame GD measurements for (a) CO2, (b) N2O, and (c) OCS. For each measurement, the alignment pulse polarization was rotated and RABBITTs were then recorded. The attochirp, ﬁlter delay, and Ne atomic delay have been removed from all measurements. The most striking features are at 0X, where OCS bumps up in GD (a positive phase jump), and CO2 and N2O dip down (a negative phase jump).

168 2 CO 2 1.5 N O 2 OCS 1

0.5

0 Group Delay (fs)

-0.5

-1 35 40 45 50 55 60 Photon Energy (eV) Figure 70: 0X GD wavelength scans for all three molecules. The above results show that OCS goes through a smooth “bump up” in GD, where as the other two molecules “dip” down. The results for CO2 are in agreement with the TDDFT predictions. The OCS GD has been shifted by 300 as for visual clarity. of the measurable spectral range for angles of 45X and above.

Figure 69(c) shows the angle-resolved GD measurements of OCS. In stark contrast to the other two molecules, OCS has a “bump up” in GD for 0X and 22.5X, indicative of positive phase jumps. Then at 45X, the GD returns to a dip and then ﬂattens out for 67.5X and

90X. The localized-nature of the GD features in OCS is consistent with the character of the enhancements shown in Figure 67(f).

Because OCS deviates in such a drastic manner from the other two molecules at 0X, it

is important to perform a careful characterization of the 0X GD features with a wavelength scan, much like was done for the unaligned samples. Shown in Figure 70 are three wavelength scans from 1270-1300 nm in 10 nm steps at 0X for all three molecules. The results for OCS have been shifted by 300 as for clarity. With the combined results of the wavelength scans, it is easy to see that OCS has a smoothly-varying bump up in GD, whereas CO2 and N2O both dip downwards in GD.

169 8.5 Model Calculations

In the above 1300 nm data, we were able to qualitatively reproduce the CO2 two-center interference features predicted by TDDFT. The same theoretical framework could in principle be used to reproduce the results of OCS, but because it contains a Cooper-like minimum, it is unlikely that TDDFT can reproduce the features measured here. It is known that in order to theoretically reproduce Cooper minima, it is necessary to have the exact continuum state wavefunctions [138]. These are not available to TDDFT, and therefore TDDFT cannot be expected to reproduce the features of OCS. So to better understand our results at

1300 nm, we developed a toy model of two-center interference with our collaborators in the

Gaarde and Schafer Groups at Louisiana State University. The calculations were carried out by Dr. François Mauger. The goals of the model are to generally reproduce the main features in CO2, which TDDFT was able to predict, and also reproduce the main features in OCS.

The following is a description of the model. The building block of the model is a reference spectral amplitude Fref , upon which we impose structures from an angle-dependent two- center interference and an assumed angle-independent Cooper minimum in the case of OCS.

The reference spectrum is constructed by calculating the SFA HHG spectral amplitude, X FSFA, for both molecules, respectively, and then scaling FSFA to the respective 90 spectra from the experiment such that Fref is given by

ABν Fref ν e FSFA ν , (133) where A and B are the scaling factors, and ν is the XUV photon energy. The 90X spectrum was chosen because it should in principle be void of two-center interference. The laser parameters and gas IPs used in the SFA calculation are shown in Table 3. The two-center interference position νmin θ is chosen using the following relation:

ν 0 IP ν θ IP min . (134) min Scos θS

170 Target IP (eV) λ(nm) I(W ~cm2) 14 CO2 13.77 1300 1 10 OCS 11.18 1300 8.5 1013

Table 3: SFA parameters used in toy model.

This position is used for both the amplitude and GD/phase positions in the model. The two-center interference GD was modeled using a Gaussian function

2 ν νmin θ 2σ2 GD2C θ, ν αGDe GD , (135)

where αGD is actually imposed from knowledge of the experimental phase jump. The spectral amplitude is modeled with

2 ¿ ν νmin θ Á ÀÁImin 2σ2 F2C θ, ν 1 1 e I , (136) Iref where Imin~Iref is the relative intensity minimum. Putting it all together, the theoretical two-center interference formula is given by

ν F θ, ν Y ield θ F θ, ν exp i S GD θ, n dn F ν . (137) 2C 2C 2C ref

The factor “Y ield θ” is an angle-dependent yield deduced from the amplitudes of the harmonic spectra at energies below the two-center interferences. At these energies, it is a fair assumption that the angle-dependent amplitude changes are independent of the two- center interferences and instead are due to changes in other molecular-frame properties such as the ionization rate, etc. To approximate the Cooper minimum in OCS, we used the unaligned GD and spectral intensity results for OCS. Explicitly, terms similar to GD2C and

F2C were added to Equation 137 in order to account for the spectral features of the Cooper minimum. Those terms were assumed to be angle-independent, due to the fact that they appear in the unaligned measurement. Also, in order to match the results of OCS, it was

171 100

reference 10-2 Intensity (arb. u.) unaligned

20 30 40 50 60 70 Photon energy (eV) Figure 71: Comparison of the theoretical reference spectrum to the theoretical unaligned spectrum for the case of CO2. The unaligned spectrum was formed by coherently averaging angular spectra calculated with Equation 137. The unaligned spectrum exhibits a drop in signal for energies above the 0X two-center interference, due to the superposition of interferences from all angles. assumed that the OCS 0X two-center interference phase jump was positive. The remaining free parameters in the model are Imin, σGD, and σI . We start by comparing the reference spectrum to the theoretical angle-averaged result for CO2, dubbed “unaligned,” plotted in Figure 71. Here we can see that the unaligned spectrum dips below the reference right at the energy of the 0X two-center interference in

CO2. This is consistent with the experimental results for CO2 shown in Figure 66, where the quasi-unaligned signal dips below the 90X signal. Additionally, it conﬁrms our interpretation that the decrease in the unaligned sample relative to our incoherent angle-average (shown in Figure 66, panels (b) and (d)) is caused by the coherent superposition of the various two-center interferences.

In Figure 72, we can see the molecular-frame results from the model. For both molecules, we see fair qualitative agreement. In CO2, the intensities and GDs are well-reproduced except for the features at 22.5X, where the minimum and GD dip have moved in the model, but have not in the measurement. Good agreement is found between the model and the experimental results in OCS. In the spectral intensities, the minima appear to be largely constrained to a single energetic region. In the GDs, we have near quantitative agreement between this toy model and the OCS results. In the model and experiment, the GD goes from a bump to a dip between 22.5X and 45X. This is probably the point at which the two-

172 (a) (b)

Figure 72: Toy model comparison against results for (a) CO2 and (b) OCS. The dotted lines represent the experimental data and the solid lines represent the model calculations. Qualitative agreement is found for both molecules. The results indicate that the two-center interference of OCS must be positive, opposite that of the other two molecules.

173 center interference has moved to higher energies and is no longer in the energetic region, much like the way the minima behave in the other two molecules. Our results in OCS tell us that it is reasonable to conclude that the two-center phase jump in OCS is positive, opposite that of CO2 and N2O. Furthermore, the complex features in OCS are most likely due to the interplay of a positive phase jump from two-center interference and a negative phase jump from the Cooper-like minimum. These results emphasize the impact of electronic structure contained in RDMEs on two-center interference measurements.

8.6 Conclusions

In this chapter, we comprehensively explored minima appearing in the HHG spectra of CO2,N2O, and OCS. We attributed minima appearing in CO2 HHG spectra generated with NIR drivers to contributions from multiple orbitals in the HHG spectrum. Previously, the origin of this interference was not completely understood. Additionally, we conﬁrmed

TDDFT predictions of the features associated with two-center interference at MIR wavelengths in CO2, particularly that the minimum is associated with a negative spectral phase jump. We then compared the MIR CO2 results to those of N2O and OCS. N2O showed very similar behavior to that of CO2, whereas OCS demonstrated unique properties. We attributed these new features in OCS to its Cooper-like minimum, which is the result of the addition of the sulfur 3p orbital. These Cooper-like features were not observed in CO2

and N2O because their HOMOs are dominantly constructed with atomic orbitals of 2p character.

174 Chapter 9: Conclusions

“Nothing ends, Adrian. Nothing ever ends.”

—Alan Moore, The Watchmen

The following summarizes the completed work presented in this dissertation and the research outlook moving forward.

9.1 Summary

Chapter 1 introduced the ﬁeld of attosecond science, particularly two methods for probing attosecond electron dynamics: high-harmonic spectroscopy (HHS) and attosecond spectroscopy (AS). Chapter 2 discussed the underlying theory of high-harmonic generation

(HHG), explaining the HHG process classically and quantum-mechanically. This chapter laid the theoretical ground work for the experiments described in Chapters 6 and 8. After that, Chapter 3 described two-photon ionization theory for weak infrared ﬁelds and XUV pulse trains. This theory is the basis of the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) method used in all of the experiments, and it is also the basis for the AS experiments described in Chapter 7. Chapter 4 described the theory of impulsive molecular alignment which was used extensively to study molecular-frame high-harmonic spectra in Chapter 8. Chapter 5 introduced the experimental apparatus, acquisition methods, and analysis methods. We then reached the experimental portion of the dissertation, beginning with Chapter 6, which described HHS of a methyl chloride Cooper minimum. This chapter also introduced our new method for ﬁtting and removing the attochirp from HHS group delay measurements. Chapter 7 detailed two-photon ionization studies of auto-ionizing resonances in Ar and He. Lastly, Chapter 8 explored the interplay of two-center interference and electronic structure in HHG.

175 9.2 Outlook

Recent years have seen an explosion of new attosecond techniques: high-harmonic spectroscopy of solids [200], attosecond transient-absorption spectroscopy of solids[201], and new variations of attosecond photoionization spectroscopy of atoms [30], molecules [202], and solids [203]. Additionally, high-harmonic generation has spread to other ﬁelds including

XUV diﬀraction spectroscopy [204][205], surface science [206], and stable XUV frequency combs [207]. As the ﬁeld continues to expand, the complexity of attosecond measurements will assuredly increase, which is where I envision the experiments detailed in this dissertation will be of great value. The experiments in this dissertation carefully characterized well-known features from photoionization experiments (Cooper minima, autoionizing resonances, and two-center interference), but also included attosecond techniques to probe these features in intensity and phase. Detailed knowledge of the simple atomic or molecular features may be used in the future to monitor attosecond electron dynamics. For example, consider the methyl chloride Cooper minimum (CM) studied in Chapter 6. In future experiments studying attosecond dynamics in methyl chloride, researchers may study time-dependent absorption or photoionization spectra. If there were electron dynamics in the valence shells of methyl chloride, then there would necessarily be modulations of the

CM, which could be monitored to study the dynamics. These same types of scenarios can be envisioned using the other features studied in Chapters 7 and 8. Still, more experiments are necessary to carefully understand how attosecond electron dynamics manifest in measurements, but hopefully this dissertation will serve as a foundation for future studies.

176 Bibliography

[1] Paul, P. M., Toma, E. S., Breger, P., Mullot, G., Auge, F., Balcou, P., Muller, H. G.,

& Agostini, P. Science 292, 1689–1692 (2001).

[2] Hentschel, M., Kienberger, R., Spielmann, C., Reider, G. A., Milosevic, N., Brabec,

T., Corkum, P., Heinzmann, U., Drescher, M., & Krausz, F. Nature 414, 509–513

(2001).

[3] Gaumnitz, T., Jain, A., Pertot, Y., Huppert, M., Jordan, I., Ardana-Lamas, F., &

Wörner, H. J. Opt. Express 25, 27506 (2017).

[4] Li, J., Ren, X., Yin, Y., Zhao, K., Chew, A., Cheng, Y., Cunningham, E., Wang, Y.,

Hu, S., Wu, Y., et al. Nat. Commun. 8, 186 (2017).

[5] Bucksbaum, P. H. Nature 421, 593-594 (2003).

[6] Vrakking, M. J. J. Phys. Chem. Chem. Phys. 16, 2775 (2014).

[7] Leone, S. R., McCurdy, C. W., Burgdörfer, J., Cederbaum, L. S., Chang, Z., Dudovich,

N., Feist, J., Greene, C. H., Ivanov, M., Kienberger, R., et al. Nat. Photon. 8, 162–166

(2014).

[8] Xie, X., Doblhoﬀ-Dier, K., Roither, S., Schöﬄer, M. S., Kartashov, D., Xu, H., Rathje,

T., Paulus, G. G., Baltuška, A., Gräfe, S., et al. Phys. Rev. Lett. 109, 243001 (2012).

[9] Zeidler, D., Staudte, A., Bardon, A. B., Villeneuve, D. M., Dörner, R., & Corkum,

P. B. Phys. Rev. Lett. 95, 203003 (2005).

[10] Schoun, S. B., Chirla, R., Wheeler, J., Roedig, C., Agostini, P., Dimauro, L. F.,

Schafer, K. J., & Gaarde, M. B. Phys. Rev. Lett. 112, 1–5 (2014).

[11] Haessler, S., Boutu, W., Stankiewicz, M., Frasinski, L. J., Weber, S., Caillat, J.,

Taïeb, R., Maquet, A., Breger, P., Monchicourt, P., et al. J. Phys. B 42, 134002

(2009).

177 [12] Agostini, P. & DiMauro, L. F. Rep. Prog. Phys. 67, 813–855 (2004).

[13] Krausz, F. & Ivanov, M. Rev. Mod. Phys. 81, 163–234 (2009).

[14] Calegari, F., Sansone, G., Stagira, S., Vozzi, C., & Nisoli, M. J. Phys. B 49, 062001

(2016).

[15] Pazourek, R., Nagele, S., & Burgdörfer, J. Rev. Mod. Phys. 87, 765–802 (2015).

[16] Gallmann, L., Cirelli, C., & Keller, U. Annu. Rev. Phys. Chem. 63, 447–469 (2012).

[17] Cederbaum, L. S. & Zobeley, J. Chem. Phys. Lett. 307, 205–210 (1999).

[18] Kuleﬀ, A. I., Lünnemann, S., & Cederbaum, L. S. Chem. Phys. 414, 100–105 (2013).

[19] Kuleﬀ, A. I. & Cederbaum, L. S. J. Phys. B 47, 124002 (2014).

[20] Falke, S. M., Rozzi, C. A., Brida, D., Maiuri, M., Amato, M., Sommer, E., De Sio,

A., Rubio, A., Cerullo, G., Molinari, E., et al. Science 344, 1001–1005 (2014).

[21] Hassan, M. T., Luu, T. T., Moulet, A., Raskazovskaya, O., Zhokhov, P., Garg, M.,

Karpowicz, N., Zheltikov, A. M., Pervak, V., Krausz, F., et al. Nature 530, 66–70

(2016).

[22] Cavalieri, A. L., Fritz, D. M., Lee, S. H., Bucksbaum, P. H., Reis, D. A., Rudati, J.,

Mills, D. M., Fuoss, P. H., Stephenson, G. B., Kao, C. C., et al. Phys. Rev. Lett. 94,

114801 (2005).

[23] Schoenlein, R. W. Science 287, 2237–2240 (2000).

[24] Inubushi, Y., Tono, K., Togashi, T., Sato, T., Hatsui, T., Kameshima, T., Togawa,

K., Hara, T., Tanaka, T., Tanaka, H., et al. Phys. Rev. Lett. 109, 144801 (2012).

[25] Mourou, G. A. & Tajima, T. Opt. Photonics News 22, 47 (2011).

[26] Nuhn, H.-D. "LCLS Parameters". (2002) https://www-ssrl.slac.stanford.edu/

lcls/lcls_parms.html (Accessed: 2018-05-08).

178 [27] Chang, Z., Corkum, P. B., & Leone, S. R. J. Opt. Soc. Am. B 33, 1081 (2016).

[28] Schoun, S. B. Attosecond High-Harmonic Spectroscopy of Atoms and Molecules Using

Mid-Infrared Sources. Ph.D. thesis, The Ohio State University (2015).

[29] Kotur, M., Guenot, D., Jimenez-Galan, A., Kroon, D., Larsen, E. W., Louisy, M.,

Bengtsson, S., Miranda, M., Mauritsson, J., Arnold, C. L., et al. Nat. Commun. 7,

10566 (2015).

[30] Cattaneo, L., Vos, J., Bello, R. Y., Palacios, A., Heuser, S., Pedrelli, L., Lucchini, M.,

Cirelli, C., Martín, F., & Keller, U. Nat. Phys. 14, 733–738 (2018).

[31] Isinger, M., Squibb, R. J., Busto, D., Zhong, S., Harth, A., Kroon, D., Nandi, S.,

Arnold, C. L., Miranda, M., Dahlström, J. M., et al. Science 358, 893–896 (2017).

[32] Le, A. T., Morishita, T., & Lin, C. D. Phys. Rev. A 78, 1–5 (2008).

[33] Wong, M. C. H., Le, A.-T., Alharbi, A. F., Boguslavskiy, A. E., Lucchese, R. R.,

Brichta, J.-P., Lin, C. D., & Bhardwaj, V. R. Phys. Rev. Lett. 110, 033006 (2013).

[34] Kraus, P. M., Baykusheva, D., & Worner, H. J. Phys. Rev. Lett. 113, 1–5 (2014).

[35] Manschwetus, B., Lin, N., Rothhardt, J., Guichard, R., Auguste, T., Camper, A.,

Breger, P., Caillat, J., Géléoc, M., Ruchon, T., et al. J. Phys. Chem. A 119, 6111–

6122 (2015).

[36] Langenhorst, F., Langenhorst, F., Petersen, F. L., Roddy, D. J., Pepin, R. O., Merrill,

R. B., Grieve, R. A. F., Roddy, D. J., Pepin, R. O., Merrill, R. B., et al. Recherche

229, 1681–1685 (2008).

[37] Kraus, P. M., Mignolet, B., Baykusheva, D., Rupenyan, A., Horný, L., Penka, E. F.,

Grassi, G., Tolstikhin, O. I., Schneider, J., Jensen, F., et al. Science 350, 790–795

(2015).

[38] Boyd, R. W. Nonlinear Optics 3rd ed. (Academic Press, San Diego, 2008).

179 [39] McPherson, A., Gibson, G., Jara, H., Johann, U., Luk, T. S., McIntyre, I. A., Boyer,

K., & Rhodes, C. K. J. Opt. Soc. Am. B 4, 595 (1987).

[40] Li, X. F., L’Huillier, A., Ferray, M., Lompré, L. A., & Mainfray, G. Phys. Rev. A 39,

5751–5761 (1989).

[41] Schafer, K. J., Yang, B., Dimauro, L. F., & Kulander, K. C. Phys. Rev. Lett. 70,

1599–1602 (1993).

[42] Corkum, P. B. Phys. Rev. Lett. 71, 1994–1997 (1993).

[43] Lewenstein, M., Balcou, P., Ivanov, M. Y., L’Huillier, A., & Corkum, P. B. Phys.

Rev. A 49, 2117–2132 (1994).

[44] Bellini, M., Lyngå, C., Tozzi, A., Gaarde, M. B., Hänsch, T. W., L’Huillier, A., &

Wahlström, C.-G. Phys. Rev. Lett. 81, 297–300 (1998).

[45] Chang, Z. Fundamentals of Attosecond Optics 1st ed. (Taylor & Francis Group, Boca

Raton, 2011) pp. 165–175.

[46] Becker, W., Grasbon, F., Kopold, R., Milošević, D., Paulus, G., & Walther, H. In

Advances in Atomic, Molecular and Optical Physics Vol. 48 (2002) pp. 35–98.

[47] Tate, J., Auguste, T., Muller, H. G., Salières, P., Agostini, P., & DiMauro, L. F. Phys.

Rev. Lett. 98, 013901 (2007).

[48] Dietrich, P., Burnett, N. H., Ivanov, M., & Corkum, P. B. Phys. Rev. A 50, R3585–

R3588 (1994).

[49] Doumy, G., Wheeler, J., Roedig, C., Chirla, R., Agostini, P., & Dimauro, L. F. Phys.

Rev. Lett. 102, 093002 (2009).

[50] Shiner, A. D., Trallero-Herrero, C., Kajumba, N., Bandulet, H.-C., Comtois, D.,

Légaré, F., Giguère, M., Kieﬀer, J.-C., Corkum, P. B., & Villeneuve, D. M. Phys.

Rev. Lett. 103, 073902 (2009).

180 [51] Eich, S., Stange, A., Carr, A., Urbancic, J., Popmintchev, T., Wiesenmayer, M.,

Jansen, K., Ruﬃng, A., Jakobs, S., Rohwer, T., et al. J. Elec. Spec. Relat. Phenom.

195, 231–236 (2014).

[52] Smirnova, O. & Ivanov, M. Multielectron High Harmonic Generation: Simple Man on

a Complex Plane. In Attosecond and XUV Physics (Wiley-Blackwell, 2014) Chap. 7,

pp. 201–256.

[53] Smirnova, O., Spanner, M., & Ivanov, M. J. Mod. Opt. 54, 1019–1038 (2007).

[54] Perelomov, A. M., Popov, V. S., & Terent’ev, M. V. J. Exp. Theor. Phys. 23, 924

(1966).

[55] Keldysh, L. V. J. Exp. Theor. Phys. 20, 1307–1314 (1965).

[56] Lai, Y. H., Xu, J., Szafruga, U. B., Talbert, B. K., Gong, X., Zhang, K., Fuest, H.,

Kling, M. F., Blaga, C. I., Agostini, P., et al. Phys. Rev. A 96, 063417 (2017).

[57] Ammosov, M. V., Delone, N. B., & Krainov, V. P. J. Exp. Theor. Phys. 64, 1191–1194

(1986).

[58] Delone, N. B. & Krainov, V. P. Phys.-Uspekhi 41, 469 (1998).

[59] Torlina, L. & Smirnova, O. Phys. Rev. A 86, 043408 (2012).

[60] Zimmermann, T., Mishra, S., Doran, B. R., Gordon, D. F., & Landsman, A. S. Phys.

Rev. Lett. 116, 233603 (2016).

[61] Starace, A. F. Atomic Photoionization. In Fundamental Processes in Energetic Atomic

Collisions edited by Lutz, H. O., Briggs, J. S., & Kleinpoppen, H. (Springer US, Bost,

MA, 1983) pp. 69–109.

[62] Le, A.-T., Della Picca, R., Fainstein, P. D., Telnov, D. a., Lein, M., & Lin, C. D. J.

Phys. B 41, 081002 (2008).

181 [63] Le, A.-T., Lucchese, R. R., Tonzani, S., Morishita, T., & Lin, C. D. Phys. Rev. A 80,

013401 (2009).

[64] Wigner, E. P. Phys. Rev. 98, 145–147 (1955).

[65] Varjú, K., Mairesse, Y., Carré, B., Gaarde, M. B., Johnsson, P., Kazamias, S., López-

Martens, R., Mauritsson, J., Schafer, K. J., Balcou, P., et al. J. Mod. Opt. 52, 379–394

(2005).

[66] Gaarde, M. B., Tate, J. L., & Schafer, K. J. J. Phys. B 41, 132001 (2008).

[67] Murakami, M., Mauritsson, J., & Gaarde, M. B. Phys. Rev. A 72, 023413 (2005).

[68] Mauritsson, J., Johnsson, P., López-Martens, R., Varjú, K., Kornelis, W., Biegert, J.,

Keller, U., Gaarde, M. B., Schafer, K. J., & L’Huillier, A. Phys. Rev. A 70, 021801

(2004).

[69] Chipperﬁeld, L. E., Robinson, J. S., Tisch, J. W. G., & Marangos, J. P. Phys. Rev.

Lett. 102, 2–5 (2009).

[70] Rae, S. C., Burnett, K., & Cooper, J. Phys. Rev. A 50, 3438–3446 (1994).

[71] Chen, M.-C., Arpin, P., Popmintchev, T., Gerrity, M., Zhang, B., Seaberg, M., Pop-

mintchev, D., Murnane, M. M., & Kapteyn, H. C. Phys. Rev. Lett. 105, 173901

(2010).

[72] Popmintchev, T., Chen, M.-C., Popmintchev, D., Arpin, P., Brown, S., Alisauskas,

S., Andriukaitis, G., Balciunas, T., Mucke, O. D., Pugzlys, A., et al. Science 336,

1287–1291 (2012).

[73] Ruchon, T., Hauri, C. P., Varjú, K., Mansten, E., Swoboda, M., López-Martens, R.,

& L’Huillier, A. New J. Phys. 10, 025027 (2008).

[74] Laurent, G., Cao, W., Ben-Itzhak, I., & Cocke, C. L. Opt. Express 21, 16914 (2013).

182 [75] Dahlström, J., Guénot, D., Klünder, K., Gisselbrecht, M., Mauritsson, J., L’Huillier,

A., Maquet, A., & Taïeb, R. Chem. Phys. 414, 53–64 (2013).

[76] Toma, E. S. & Muller, H. G. J. Phys. B 35, 306 (2002).

[77] Muller, H. G. Appl. Phys. B 74, s17–s21 (2002).

[78] Mairesse, Y. & Quéré, F. Phys. Rev. A 71, 011401 (2005).

[79] Chini, M., Gilbertson, S., Khan, S. D., & Chang, Z. Opt. Express 18, 13006 (2010).

[80] Vozzi, C., Calegari, F., Benedetti, E., Caumes, J. P., Sansone, G., Stagira, S., Nisoli,

M., Torres, R., Heesel, E., Kajumba, N., et al. Phys. Rev. Lett. 95, 1–4 (2005).

[81] De, S., Znakovskaya, I., Ray, D., Anis, F., Johnson, N. G., Bocharova, I. A., Ma-

grakvelidze, M., Esry, B. D., Cocke, C. L., Litvinyuk, I. V., et al. Phys. Rev. Lett.

103, 1–4 (2009).

[82] Etches, A., Gaarde, M. B., & Madsen, L. B. Phys. Rev. A 84, 1–7 (2011).

[83] Goban, A., Minemoto, S., & Sakai, H. Phys. Rev. Lett. 101, 013001 (2008).

[84] Chatterley, A. S., Karamatskos, E. T., Schouder, C., Christiansen, L., Jørgensen,

A. V., Mullins, T., Küpper, J., & Stapelfeldt, H. J. Chem. Phys. 148, 221105 (2018).

[85] Ortigoso, J., Rodriguez, M., Gupta, M., & Friedrich, B. J. Chem. Phys. 110, 3870–

3875 (1999).

[86] Hamilton, E., Seideman, T., Ejdrup, T., Poulsen, M. D., Bisgaard, C. Z., Viftrup,

S. S., & Stapelfeldt, H. Phys. Rev. A 72, 043402 (2005).

[87] Qin, M., Zhu, X., Li, Y., Zhang, Q., Lan, P., & Lu, P. Opt. Express 22, 6362 (2014).

[88] Herzberg, G. Electronic spectra and electronic structure of polyatomic molecules (Van

Nostrand, New York, 1966).

[89] Maroulis, G. & Menadakis, M. Chem. Phys. Lett. 494, 144–149 (2010).

183 [90] Hansen, J. L., Holmegaard, L., Nielsen, J. H., Stapelfeldt, H., Dimitrovski, D., &

Madsen, L. B. J. Phys. B 45, 015101 (2012).

[91] Kraus, P. M., Rupenyan, A., & Wörner, H. J. Phys. Rev. Lett. 109, 233903 (2012).

[92] Yun, S. J., Kim, C. M., Lee, J., & Nam, C. H. Phys. Rev. A 86, 051401 (2012).

[93] Rosenberg, D., Damari, R., Kallush, S., & Fleischer, S. J. Phys. Chem. Lett. 8, 5128–

5135 (2017).

[94] Torres, R., de Nalda, R., & Marangos, J. P. Phys. Rev. A 72, 023420 (2005).

[95] Haessler, S., Caillat, J., & Salières, P. J. Phys. B 44, 203001 (2011).

[96] Herring, G. C., Dyer, M. J., & Bischel, W. K. Opt. Lett. 11, 348 (1986).

[97] Cryan, J. P., Bucksbaum, P. H., & Coﬀee, R. N. Phys. Rev. A 80, 063412 (2009).

[98] Schoun, S. B., Camper, A., Salières, P., Lucchese, R. R., Agostini, P., & DiMauro,

L. F. Phys. Rev. Lett. 118, 1–6 (2017).

[99] Xu, N., Wu, C., Huang, J., Wu, Z., Liang, Q., Yang, H., & Gong, Q. Opt. Express

14, 4992–4997 (2006).

[100] Broege, D. W., Coﬀee, R. N., & Bucksbaum, P. H. Phys. Rev. A 78, 035401 (2008).

[101] Miller, D. R. In Atomic Molecular Beam Methods edited by Scoles, G., Bassi, D.,

Buck, U., & Laine, D. (Oxford University Press, New York, 1988) Chap. 2, pp. 14–53.

[102] Hagena, O. F. & Obert, W. J. Chem. Phys. 56, 1793–1802 (1972).

[103] Park, H., Wang, Z., Xiong, H., Schoun, S. B., Xu, J., Agostini, P., & DiMauro, L. F.

Phys. Rev. Lett. 113, 263401 (2014).

[104] Strickland, D. & Mourou, G. Opt. Commun. 55, 447–449 (1985).

[105] Brabec, T., Spielmann, C., Curley, P. F., & Krausz, F. Opt. Lett. 17, 1292 (1992).

184 [106] Chirla, R. C. Attosecond Pulse Generation and Characterization. Ph.D. thesis, The

Ohio State University (2011).

[107] Luria, K., Christen, W., & Even, U. J. Phys. Chem. A 115, 7362–7367 (2011).

[108] Balcou, P., Sali‘eres, P., L’Huillier, A., & Lewenstein, M. Phys. Rev. A 55, 3204–3210

(1997).

[109] Henke, B., Gullikson, E., & Davis, J. At. Data Nucl. Data Tables 54, 181–342 (1993).

[110] Rakić, A. D. Appl. Opt. 34, 4755 (1995).

[111] Roedig, C. Application of Strong Field Physics Techniques to X-Ray Free Electron

Laser Science. Ph.D. thesis, The Ohio State University (2012).

[112] Kruit, P. & Read, F. H. J. Phys. E 16, 313–324 (1983).

[113] Saha, S., Mandal, A., Jose, J., Varma, H. R., Deshmukh, P. C., Kheifets, A. S.,

Dolmatov, V. K., & Manson, S. T. Phys. Rev. A 90, 053406 (2014).

[114] Cooper, J. W. Phys. Rev. 128, 681–693 (1962).

[115] Mauritsson, J., Gaarde, M. B., & Schafer, K. J. Phys. Rev. A 72, 013401 (2005).

[116] Dahlström, J. M., L’Huillier, A., & Maquet, A. J. Phys. B 45, 183001 (2012).

[117] Powell, Forbes R., Vedder, Peter W., Lindblom, Joakim F., Powell, S. F. Opt. Eng.

29, 614 (1990).

[118] Marangos, J. P. J. Phys. B 49, 132001 (2016).

[119] Vampa, G., Hammond, T. J., Thiré, N., Schmidt, B. E., Légaré, F., McDonald, C. R.,

Brabec, T., & Corkum, P. B. Nature 522, 462–464 (2015).

[120] Manson, S. T. Phys. Rev. A 31, 3698–3703 (1985).

[121] Ditchburn, R. W., Tunstead, J., & Yates, J. G. Proc. Royal Soc. A 181, 386–399

(1943).

185 [122] Carlson, T. A., Fahlman, A., Krause, M. O., Whitley, T. A., & Grimm, F. A. J.

Chem. Phys. 81, 5389–5394 (1984).

[123] Carlson, T. A., Krause, M. O., Svensson, W. A., Gerard, P., Grimm, F. A., Whitley,

T. A., & Pullen, B. P. Z. Phys. D : At., mol. clusters 2, 309–318 (1986).

[124] Higuet, J., Ruf, H., Thiré, N., Cireasa, R., Constant, E., Cormier, E., Descamps, D.,

Mével, E., Petit, S., Pons, B., et al. Phys. Rev. A 83, 1–12 (2011).

[125] Shiner, A. D., Schmidt, B. E., Trallero-Herrero, C., Corkum, P. B., Kieﬀer, J.-C.,

Légaré, F., & Villeneuve, D. M. J. Phys. B 45, 074010 (2012).

[126] Strelkov, V. V., Khokhlova, M. A., Gonoskov, A. A., Gonoskov, I. A., & Ryabikin,

M. Y. Phys. Rev. A 86, 1–10 (2012).

[127] Wong, M. C. H., Brichta, J.-P., & Bhardwaj, V. R. Phys. Rev. A 81, 061402 (2010).

[128] Ren, X., Makhija, V., Le, A. T., Troß, J., Mondal, S., Jin, C., Kumarappan, V., &

Trallero-Herrero, C. Phys. Rev. A 88, 1–5 (2013).

[129] Vozzi, C., Negro, M., Calegari, F., Sansone, G., Nisoli, M., De Silvestri, S., & Stagira,

S. Nat. Phys. 7, 822–826 (2011).

[130] Diveki, Z., Camper, A., Haessler, S., Auguste, T., Ruchon, T., Carré, B., Salières, P.,

Guichard, R., Caillat, J., Maquet, A., et al. New J. Phys. 14, 023062 (2012).

[131] McFarland, B. K., Farrell, J. P., Bucksbaum, P. H., & Guhr, M. Science 322, 1232–

1235 (2008).

[132] Holland, D. M. P., Powis, I., Öhrwall, G., Karlsson, L., & von Niessen, W. Chem.

Phys. 326, 535–550 (2006).

[133] Fano, U. & Cooper, J. W. Rev. Mod. Phys. 40, 441–507 (1968).

[134] Fano, U. Phys. Rev. A 32, 617–618 (1985).

186 [135] Carlson, T. A., Krause, M. O., Grimm, F. A., Keller, P., & Taylor, J. W. J. Chem.

Phys. 77, 5340–5347 (1982).

[136] Carlson, T. A., Krause, M. O., Grimm, F. A., Allen, J. D., Mehaﬀy, D., Keller, P. R.,

& Taylor, J. W. J. Chem. Phys. 75, 3288–3292 (1981).

[137] Carlson, T. A., Krause, M. O., & Grimm, F. A. J. Chem. Phys. 77, 1701–1709 (1982).

[138] Wörner, H. J., Niikura, H., Bertrand, J. B., Corkum, P. B., & Villeneuve, D. M. Phys.

Rev. Lett. 102, 1–4 (2009).

[139] Le, A. T., Lucchese, R. R., & Lin, C. D. Phys. Rev. A 87, 013401 (2013).

[140] Rustgi, O. P. J. Opt. Soc. Am. 54, 464 (1964).

[141] Mao, Z. & Sinnott, S. B. J. Phys. Chem. B 105, 6916–6924 (2001).

[142] Smirnova, O., Mairesse, Y., Patchkovskii, S., Dudovich, N., Villeneuve, D., Corkum,

P., & Ivanov, M. Y. Nature 460, 972–977 (2009).

[143] Rupenyan, A., Kraus, P. M., Schneider, J., & Wörner, H. J. Phys. Rev. A 87, 1–5

(2013).

[144] Baron, J., Campbell, W. C., DeMille, D., Doyle, J. M., Gabrielse, G., Gurevich,

Y. V., Hess, P. W., Hutzler, N. R., Kirilov, E., Kozyryev, I., et al. Science 343,

269–272 (2014).

[145] Miroshnichenko, A. E., Flach, S., & Kivshar, Y. S. Rev. Mod. Phys. 82, 2257–2298

(2010).

[146] Beutler, H. Z. Phys. 93, 177–196 (1935).

[147] Fano, U. Il Nuovo Cimento (1924-1942) 12, 154–161 (1935).

[148] Fano, U. Phys. Rev. 124, 1866–1878 (1961).

187 [149] Connerade, J.-P. Highly Excited Atoms Cambridge Monographs on Atomic, Molecular

and Chemical Physics (Cambridge University Press, Cambridge, 1998).

[150] Doughty, B., Haber, L. H., Hackett, C., & Leone, S. R. J. Chem. Phys. 134, 094307

(2011).

[151] Wickenhauser, M., Burgdorfer, J., Krausz, F., & Drescher, M. Phys. Rev. Lett. 94,

1–4 (2005).

[152] Cirelli, C., Marante, C., Heuser, S., Petersson, C. L., Galán, Á. J., Argenti, L., Zhong,

S., Busto, D., Isinger, M., Nandi, S., et al. Nat. Commun. 9, 955 (2018).

[153] Gruson, V., Barreau, L., Jiménez-Galan, Á., Risoud, F., Caillat, J., Maquet, A.,

Carré, B., Lepetit, F., Hergott, J.-F., Ruchon, T., et al. Science 354, 734–738 (2016).

[154] Kaldun, A., Blättermann, A., Stooß, V., Donsa, S., Wei, H., Pazourek, R., Nagele,

S., Ott, C., Lin, C. D., Burgdörfer, J., et al. Science 354, 738–741 (2016).

[155] Carette, T., Dahlström, J. M., Argenti, L., & Lindroth, E. Phys. Rev. A 87, 1–13

(2013).

[156] Domke, M., Schulz, K., Remmers, G., Kaindl, G., & Wintgen, D. Phys. Rev. A 53,

1424–1438 (1996).

[157] Jiménez-Galán, Á., Argenti, L., & Martín, F. Phys. Rev. Lett. 113, 263001 (2014).

[158] Guénot, D., Kroon, D., Balogh, E., Larsen, E. W., Kotur, M., Miranda, M., Fordell,

T., Johnsson, P., Mauritsson, J., Gisselbrecht, M., et al. J. Phys. B 47, 245602 (2014).

[159] Kim, I. J., Kim, C. M., Kim, H. T., Lee, G. H., Lee, Y. S., Park, J. Y., Cho, D. J., &

Nam, C. H. Phys. Rev. Lett. 94, 2–5 (2005).

[160] Dudovich, N., Tate, J. L., Mairesse, Y., Villeneuve, D. M., Corkum, P. B., & Gaarde,

M. B. Phys. Rev. A 80, 011806 (2009).

188 [161] Dudovich, N., Smirnova, O., Levesque, J., Mairesse, Y., Ivanov, M. Y., Villeneuve,

D. M., & Corkum, P. B. Nat. Phys. 2, 781–786 (2006).

[162] Frumker, E., Hebeisen, C. T., Kajumba, N., Bertrand, J. B., Wörner, H. J., Spanner,

M., Villeneuve, D. M., Naumov, A., & Corkum, P. B. Phys. Rev. Lett. 109, 113901

(2012).

[163] Feist, J., Nagele, S., Pazourek, R., Persson, E., Schneider, B. I., Collins, L. A., &

Burgdörfer, J. Phys. Rev. A 77, 043420 (2008).

[164] Colgan, J. & Pindzola, M. S. Phys. Rev. Lett. 88, 173002 (2002).

[165] Laulan, S. & Bachau, H. Phys. Rev. A 68, 013409 (2003).

[166] Hu, S. X., Colgan, J., & Collins, L. A. J. Phys. B 38, L35–L45 (2005).

[167] Pindzola, M. S., Robicheaux, F., Loch, S. D., Berengut, J. C., Colgan, J., Foster, M.,

Griﬃn, D. C., Balance, C. P., Badnell, N. R., Witthoeft, M. C., et al. J. Phys. Conf.

Ser. 88, 012012 (2007).

[168] Becker, U., Wehlitz, R., Hemmers, O., Langer, B., & Menzel, A. Phys. Rev. Lett. 63,

1054–1057 (1989).

[169] Bachau, H., Cormier, E., Decleva, P., Hansen, J. E., & Martín, F. Rep. Prog. Phys.

64, 1815–1943 (2001).

[170] Park, T. J. & Light, J. C. J. Chem. Phys. 85, 5870–5876 (1986).

[171] Leforestier, C., Bisseling, R., Cerjan, C., Feit, M., Friesner, R., Guldberg, A., Ham-

merich, A., Jolicard, G., Karrlein, W., Meyer, H.-D., et al. J. Comput. Phys. 94,

59–80 (1991).

[172] Schneider, B. I. & Collins, L. A. J. Non. Cryst. Solids 351, 1551–1558 (2005).

[173] McCurdy, C. W., Horner, D. A., & Rescigno, T. N. Phys. Rev. A 63, 022711 (2001).

189 [174] Rescigno, T. N. & McCurdy, C. W. Phys. Rev. A 62, 032706 (2000).

[175] Berrah, N., Langer, B., Bozek, J., Gorczyca, T. W., Hemmers, O., Lindle, D. W., &

Toader, O. J. Phys. B 29, 5351–5365 (1996).

[176] Baig, M. A., Ahmad, S., Connerade, J., Dussa, W., & Hormes, J. Phys. Rev. A 45,

7963–7968 (1992).

[177] Madden, R. P., Ederer, D. L., & Codling, K. Phys. Rev. 177, 136–151 (1969).

[178] Morgan, H. D. & Ederer, D. L. Phys. Rev. A 29, 1901–1906 (1984).

[179] Laurent, G., Cao, W., Li, H., Wang, Z., Ben-Itzhak, I., & Cocke, C. L. Phys. Rev.

Lett. 109, 1–5 (2012).

[180] Itatani, J., Levesque, J., Zeidler, D., Niikura, H., Pépin, H., Kieﬀer, J. C., Corkum,

P. B., & Villeneuve, D. M. Nature 432, 867–871 (2004).

[181] Walters, Z. B., Tonzani, S., & Greene, C. H. J. Phys. Chem. A 112, 9439–9447 (2008).

[182] Zhou, X., Lock, R., Li, W., Wagner, N., Murnane, M. M., & Kapteyn, H. C. Phys.

Rev. Lett. 100, 073902 (2008).

[183] Boutu, W., Haessler, S., Merdji, H., Breger, P., Waters, G., Stankiewicz, M., Frasinski,

L. J., Taieb, R., Caillat, J., Maquet, A., et al. Nat. Phys. 4, 545–549 (2008).

[184] Wörner, H. J., Bertrand, J. B., Hockett, P., Corkum, P. B., & Villeneuve, D. M. Phys.

Rev. Lett. 104, 233904 (2010).

[185] White, M. G., Leung, K. T., & Brion, C. E. J. Elec. Spec. Relat. Phenom. 23, 127–145

(1981).

[186] Mulliken, R. S. J. Chem. Phys. 3, 720-739 (1935).

[187] Han, Y. C. & Madsen, L. B. Phys. Rev. A 81, 063430 (2010).

[188] Zhu, X., Zhang, Q., Hong, W., Lan, P., & Lu, P. Opt. Express 19, 436 (2011).

190 [189] Andrade, X., Strubbe, D., De Giovannini, U., Larsen, A. H., Oliveira, M. J. T.,

Alberdi-Rodriguez, J., Varas, A., Theophilou, I., Helbig, N., Verstraete, M. J., et al.

Phys. Chem. Chem. Phys. 17, 31371–31396 (2015).

[190] Runge, E. & Gross, E. K. U. Phys. Rev. Lett. 52, 997–1000 (1984).

[191] Kohn, W. & Sham, L. J. Phys. Rev. 140, A1133–A1138 (1965).

[192] Perdew, J. P. & Zunger, A. Phys. Rev. B 23, 5048–5079 (1981).

[193] Legrand, C., Suraud, E., & Reinhard, P.-G. J. Phys. B 35, 1115–1128 (2002).

[194] Debnath, L. & Shah, F. A. In Wavelet Transforms and Their Applications (Birkhäuser

Boston, Boston, MA, 2015) pp. 243–286.

[195] Schafer, K. J., Gaarde, M. B., Heinrich, A., Biegert, J., & Keller, U. Phys. Rev. Lett.

92, 023003 (2004).

[196] Gademann, G., Kelkensberg, F., Siu, W. K., Johnsson, P., Gaarde, M. B., Schafer,

K. J., & Vrakking, M. J. J. New J. Phys. 13, 033002 (2011).

[197] Azoury, D., Krüger, M., Orenstein, G., Larsson, H. R., Bauch, S., Bruner, B. D., &

Dudovich, N. Nat. Commun. 8, 1453 (2017).

[198] Brion, C. E. & Tan, K. H. Chem. Phys. 34, 141–151 (1978).

[199] Hitchcock, A., Brion, C., & van der Wiel, M. Chem. Phys. 45, 461–478 (1980).

[200] Ghimire, S., Ndabashimiye, G., DiChiara, A. D., Sistrunk, E., Stockman, M. I.,

Agostini, P., DiMauro, L. F., & Reis, D. A. J. Phys. B 47, 204030 (2014).

[201] Schultze, M., Bothschafter, E. M., Sommer, A., Holzner, S., Schweinberger, W., Fiess,

M., Hofstetter, M., Kienberger, R., Apalkov, V., Yakovlev, V. S., et al. Nature 493,

75–78 (2013).

[202] Vos, J., Cattaneo, L., Patchkovskii, S., Zimmermann, T., Cirelli, C., Lucchini, M.,

Kheifets, A., Landsman, A. S., & Keller, U. Science 360, 1326–1330 (2018).

191 [203] Locher, R., Castiglioni, L., Lucchini, M., Greif, M., Gallmann, L., Osterwalder, J.,

Hengsberger, M., & Keller, U. Optica 2, 405 (2015).

[204] Sandberg, R. L., Paul, A., Raymondson, D. A., Hädrich, S., Gaudiosi, D. M., Holt-

snider, J., Tobey, R. I., Cohen, O., Murnane, M. M., Kapteyn, H. C., et al. Phys.

Rev. Lett. 99, 098103 (2007).

[205] Kﬁr, O., Zayko, S., Nolte, C., Sivis, M., Möller, M., Hebler, B., Arekapudi, S. S.

P. K., Steil, D., Schäfer, S., Albrecht, M., et al. Sci. Adv. 3, (2017).

[206] Husek, J., Cirri, A., Biswas, S., Asthagiri, A., & Baker, L. R. J. Phys. Chem. C 122,

11300–11304 (2018).

[207] Benko, C., Allison, T. K., Cingöz, A., Hua, L., Labaye, F., Yost, D. C., & Ye, J. Nat.

Photon. 8, 530–536 (2014).

192