High-Order Harmonic Generation Driven by Mid-Infrared
Laser Lightby
Benjamin R. Galloway
B.S., Colorado School of Mines, 2011
M.S., Colorado School of Mines, 2012

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
High-Order Harmonic Generation Driven by Mid-Infrared Laser Light
written by Benjamin R. Galloway
has been approved for the Department of Physics

Prof. Margaret Murnane

Prof. Henry Kapteyn

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

Galloway, Benjamin R. (Ph.D., Physics)
High-Order Harmonic Generation Driven by Mid-Infrared Laser Light
Thesis directed by Prof. Margaret Murnane

Light is a powerful tool for making observations of the physical world. In particular, light in
the extreme ultraviolet (EUV) and X-ray regimes enable unique and higher resolution measurements
than is possible using longer wavelengths. A relatively new technique called high-order harmonic
generation (HHG) provides a route for scientists to produce light in these useful spectral ranges,
starting with lasers operating at more accessible wavelengths. HHG has been successfully applied
to a number of applications including high resolution microscopy, spectroscopy, and measurementsof magnetism, thermal transport, and molecular structure.
This dissertation covers several illuminating studies of HHG in the temporal and spectral
domains when the process is driven by long wavelength, mid-infrared light. Interestingly, thecharacteristics of the harmonic emission are highly dependent on the driving laser parametersand geometries. As the driving laser wavelength is increased, the harmonic cutoff and bandwidth
naturally broaden, while the emitted pulse train reduces in length until a single isolated burst ofphase-matched harmonics with sub-femtosecond duration is achieved. This trend is experimentally
verified by performing an electric field autocorrelation of the harmonic emission. The resulting HHG
supercontinuum has particular utility in X-ray absorption fine structure spectroscopies, where thenanoscale lattice structure can be probed. These spectroscopies have been performed on polymer,
scandium, and iron samples using the broadest HHG bandwidths achieved to date, extending up
to 1.6 keV. Pushing this harmonic cutoff further would conventionally require the use of longer
wavelength drivers approaching the far-infrared regime. However, long driving wavelengths canalso result in relativistic effects, resulting in longitudinal Lorentz drifts that could cause the HHG
process to be inhibited. A theoretical accounting of all of the forces involved does not indicate
HHG would be shut off entirely, however, and it is possible for HHG to occur even with driving

iv
wavelengths beyond 10 µm and harmonic cutoffs in the hard X-ray regime. The use of cylindrical
vector beams or multi-beam geometries can also be used to compensate for relativistic effects, as wellas to create new phase-matching conditions for sum and difference frequency processes. Throughhigh-order difference frequency generation in a two-beam noncollinear geometry, it is predictedthat the conventional phase-matching limitations could be significantly exceeded, opening up the
possibility to use visible drivers to reach the soft X-ray regime or further. Pushing the limits of the
HHG spectral characteristics would inevitably enable new levels of capability for its applications.
The investigations presented here will follow a progression from shorter to longer wavelengths as drivers for the HHG process, starting with experiments using the most commonly used
Ti:sapphire wavelength of 800 nm, moving to 1.3 µm and 2.0 µm, then up to 3.9 µm, and ultimately arriving at theory for far-infrared drivers up to 20 µm. Furthermore, the conventional
single-beam driving configuration will be primarily investigated, but new capabilities are predicted
for multi-beam and multi-color geometries, which will be discussed.

Dedication

For Jenny - because if I didn’t have this, I’d still have you.

vi

Acknowledgements

First, I would like to thank Prof. Margaret Murnane and Prof. Henry Kapteyn for the
excellent guidance and constant support. The opportunities they opened up for me have taught
me so much and have made my Ph.D. journey enjoyable.
The entire KM group also deserves acknowledgment for all of the aid over the years. In
particular, a big thanks to Tenio Popmintchev, Dimitar Popmintchev, Ming-Chang Chen, Dan
Adams, Michael Gerrity, Seth Cousin, Amitava Adak, Chris Mancuso, Christian Gentry, Kevin
Dorney, Jennifer Ellis, Liz Shanblatt, Giulia Mancini, Chrissy Porter, and Michael Tanksalvala.
The collaboration with KM Labs has made my research more feasible, so I would like to thankSterling Backus, Xiaoshi Zhang, Daisy Raymondson, Jon Garlick, and Stephen Uhlhorn for theirlaser and production expertise. I appreciate Prof. Chip Durfee for sending me along to the KM
group and for teaching me the fundamentals of optics.
I would like to express my deep appreciation for the outstanding support from the Departmentof Energy, National Nuclear Security Administration, and Krell Institute through the Stewardship
Science Graduate Fellowship program - there are some very good people that manage the fellowship
to whom I owe a considerable amount. (This work was supported by the Department of Energy National Nuclear Security Administration Stewardship Science Graduate Fellowship program, which
is provided under grant number DENA002135.)
Finally, I would like to thank the support of my family and friends who kept me sane during
the many instances of laser trouble. I couldn’t have made it without the love and laughter. Many
thanks!

Contents

Chapter
1 Introduction to High-Order Harmonic Generation1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Microscopic Picture of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

Step 1: Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Step 2: Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3

Step 3: Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.4

Characteristics of Microscopic HHG Emission . . . . . . . . . . . . . . . . . . 19

1.3

1.4

Quantum Mechanical Picture of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.1

The Quantum Three Step Model . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.3.2

Quantum Diffusion of the Electron Wavepacket . . . . . . . . . . . . . . . . . 35

1.3.3

Quantum Mechanical Characteristics of HHG . . . . . . . . . . . . . . . . . . 39

Macroscopic Picture of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.4.1

Phase-Matching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.4.2

Phase-Matching in Fractionally Ionized Gases . . . . . . . . . . . . . . . . . . 42

1.4.3

Phase-Matching Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.4

Quasi-Phase-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1.4.5

Reabsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.4.6

Characteristics of Macroscopic HHG Emission. . . . . . . . . . . . . . . . . 60

viii
2 Phase-Matching Gating during Mid-IR Driven HHG

63

2.1

Grating Spectrometer Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2

Electric Field Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3

HHG Pulse Train and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4

Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.5

Field Autocorrelation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.6

Phase-Matching Gating Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.7

Phase-Matching Gating Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3 Water Window and Soft X-ray HHG Spectroscopies

80

3.1

XAFS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.2

NEXAFS in Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3

XAFS in Scandium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4

EXAFS in Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.5

XAFS Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 The Lorentz Force during HHG

92

4.1

The Full Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2

Field Components for Finite Laser Modes . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3

Recombination Probability with Lorentz Drift . . . . . . . . . . . . . . . . . . . . . . 97

4.4

RPR Scaling with Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5

Lorentz Drift Spatial Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6

Lorentz Force Compensation through Noncollinear Geometries . . . . . . . . . . . . 107

4.7

Experimental Signatures of the Lorentz Drift . . . . . . . . . . . . . . . . . . . . . . 113

4.8

Lorentz Drift Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Pushing HHG to Higher Photon Energies
5.1

HHG using Cylindrical Vector Beams

116. . . . . . . . . . . . . . . . . . . . . . . . . . 117

ix
5.2

HHG in Bragg Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3

Noncollinear High-Order Difference Frequency Generation . . . . . . . . . . . . . . . 125

5.4

Future Work in Mid-IR Driven HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Concluding Remarks

135

Bibliography

137

Appendix
A Hollow Waveguide Modes

148

x

TablesTable
1.1

Coherent X-ray light source capability comparison . . . . . . . . . . . . . . . . . . .

3

1.2

αBS values for atomic and ionic species . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3

Ionization potentials for atomic and ionic species . . . . . . . . . . . . . . . . . . . . 20

1.4

Absorption lengths for various photon energies in gases at 1 atm pressure . . . . . . 58

4.1

Lorentz displacement results for selected laser Wavelengths . . . . . . . . . . . . . . 99

5.1

Noncollinear HDFG phase-matching calculations . . . . . . . . . . . . . . . . . . . . 130

Figures

Figure1.1

Regimes of the electromagnetic spectrum and currently available ultrafast laser technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Coherent X-ray light source brightness comparison . . . . . . . . . . . . . . . . . . .

4

1.3

The photon picture of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

The first measurements of the HHG spectrum . . . . . . . . . . . . . . . . . . . . . .

7

1.5

The three step model of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.6

Ionization possibilities by intense laser light . . . . . . . . . . . . . . . . . . . . . . .

9

1.7Time-dependent ionization populations of Ar for low and high intensity cases . . . . 14

1.8

Electron trajectories for a variety of ionization phases φ0 . . . . . . . . . . . . . . . . 15

1.9

Duration τf of returning electron trajectories as a function of ionization phase φ0 . . 17

1.10 Kinetic energy of the recolliding electron as a function of ionization phase φ0 . . . . 17
1.11 Possible consequences of electron-ion recollisions . . . . . . . . . . . . . . . . . . . . 20
1.12 Qualitative characteristics of HHG in the time and frequency domains . . . . . . . . 23
1.13 Measured characteristics of HHG in the time and frequency domains . . . . . . . . . 24
1.14 Plots of various orders of dispersion resulting from the HHG three step model . . . . 25
1.15 Larmor radiation pattern for an electron recombining from the ŷ direction . . . . . . 27
1.16 A plot of the limiting cases (Ip >> Up and Up >> Ip ) of the quasi-classical action
S(p̄s , t, t0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.17 Total electron wavepacket calculated through numerical quantum simulation . . . . . 36

xii
1.18 A diagram of the phase-matching concept during HHG . . . . . . . . . . . . . . . . . 43
1.19 A qualitative illustration of the frequency-dependent refractive index . . . . . . . . . 44
1.20 Optimal driving laser intensities for efficient HHG in noble gases . . . . . . . . . . . 47
1.21 An illustration of a Gaussian free-focus and its characteristic parameters . . . . . . . 50
1.22 Geometric phase mismatch ∆kGouy + ∆k∇IL for a Gaussian free-focus with typical
HHG conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.23 Illustration of the hollow waveguide geometry . . . . . . . . . . . . . . . . . . . . . . 53
1.24 Various schemes to achieve quasi-phase-matching . . . . . . . . . . . . . . . . . . . . 57
1.25 Buildup of HHG flux as a function of longitudinal position . . . . . . . . . . . . . . . 59
1.26 Macroscopic scaling of the HHG cutoff photon energy with wavelength . . . . . . . . 61
1.27 Pressure and harmonic cutoff energy scalings with driving laser wavelength . . . . . 62
2.1

Experimental apparatus to measure the HHG field autocorrelation . . . . . . . . . . 69

2.2

Fourier transform method to extract field autocorrelation from crossed beam spatial
modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.3

HHG electric field autocorrelation results for various driving laser wavelengths andintensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.4

Dynamic HHG phase mismatch and coherence length for various driving laser conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.5

Advanced numerical simulations of dynamic phase-matching during HHG . . . . . . 79

3.1

Absorption spectra for various elements in the water window and soft X-ray regimes

3.2

Diagrams of electron interferences from solid lattices and molecules, resulting in

81

EXAFS and NEXAFS effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3

Soft X-ray HHG spectra produced using OPA and OPCPA light sources for XAFS
measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4

NEXAFS spectroscopy near the carbon K-edge for mylar and parylene . . . . . . . . 87

3.5

XAFS spectroscopy near the scandium L-edge . . . . . . . . . . . . . . . . . . . . . . 88

xiii
3.6

EXAFS spectroscopy near the iron L-edge . . . . . . . . . . . . . . . . . . . . . . . . 90

3.7

X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1

Transverse and longitudinal field components for Gaussian and EH11 modes . . . . . 96

4.2

Field components and longitudinal drift components in a Gaussian focus geometry . 100

4.3

Total Lorentz drift magnitude and diffused electron wavepacket width as a functionof driving laser wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4

Spatially-resolved Lorentz displacements and RP R for a Gaussian focus . . . . . . . 104

4.5

Spatially-resolved Lorentz displacements and RP R for a waveguide geometry . . . . 105

4.6

HHG photon energies emitted where Lorentz drift cancellation occurs . . . . . . . . 106

4.7

Spatially-resolved, phase-integrated RP R for Gaussian and waveguide geometries . . 108

4.8

Noncollinear bi-circular HHG geometry to cancel magnetic Lorentz drift effects . . . 109

4.9

Comparison of even and odd harmonics under the influence of Lorentz drift effects
in a noncollinear bi-circular geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.10 Recovery of high flux harmonic emission through a noncollinear bicircular geometrywhen Lorentz drift nondipole effects are present . . . . . . . . . . . . . . . . . . . . . 112
4.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1

Spatial modes for radially- and azimuthally-polarized cylindrical vector beams . . . . 118

5.2

Radial and longitudinal field strengths of a radially-polarized LG01 mode throughout
a tight focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3

Wavefront for a l = 2 OAM beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.4

Scanning electron microscope image of a Bragg fiber . . . . . . . . . . . . . . . . . . 124

5.5

Geometry for a two-beam noncollinear frequency upconversion process (HHG or
HDFG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.6

Experimental and theoretical HHG phase-matching cutoff energies including contributions from ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xiv
5.7

Velocity map imaging concept to measure the longitudinal Lorentz force duringstrong field ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.1 Illustration of the hollow waveguide geometry . . . . . . . . . . . . . . . . . . . . . . 148
A.2 Power coupled from linearly-polarized Gaussian modes into EH1q waveguide modes . 152
A.3 Plot of the LG01 mode intensity as a function of radius . . . . . . . . . . . . . . . . 153
A.4 Power coupled from azimuthally- or radially-polarized LG01 modes into TE0q (orTM0q ) waveguide modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Chapter 1

Introduction to High-Order Harmonic Generation

1.1

Background
Light is an incredibly powerful tool for measurement and excitation. From the casual use

of light by the human eye, to the implementation of X-rays to create shock waves in matter that
can initiate a fusion reaction - light truly is versatile. The various properties of photons - thefundamental particles that make up light - allow them to be utilized so flexibly. Photons carryenergy and momentum, and they can be superimposed to create electromagnetic fields with arbitrary spatial and temporal structure. The polarization and coherence properties of light give it
additional complexity that can be harnessed for a plethora of applications. In particular, laser lightmakes use of the coherent nature of photons to create a highly directional beam of light. Throughthe process of stimulated emission, lasers take single photons and coherently multiply them manyorders of magnitude, reaching electromagnetic intensities in the laboratory unavailable through anyother means.
Since the first demonstration of the laser by Theodore Maiman in 1960 [77], lasers have foundapplications in imaging [156, 153, 126, 150, 84, 127, 131], remote sensing [108, 82], spectroscopy [66,6, 35, 135], micro-machining [83, 31, 20, 168], nano-lithography [148, 54, 41], particle acceleration[119, 162, 151, 158], quantum mechanics [5, 80, 113, 114, 86], magnetics [157, 138, 159], fusion[88, 152], ultra-cold matter [67, 78], thermal transport [91, 133, 52], gravitational research [99, 125],and more. Each of these applications requires particular laser properties in terms of wavelength(color), bandwidth (pulse duration), peak and average powers, polarization, and coherence. In2particular, light in one region of the electromagnetic spectrum (Fig. 1.1) may be suitable fora particular application, but inappropriate for another. Alternatively, an application might beable to use of a wide range colors, but the sensitivity or fidelity of the result could be dependent
on the particular wavelength(s) used. For applications that require the coherence of laser light,
the most easily accessible colors are in the infrared and visible regions. The reason behind thislimitation lies in the atomic transition energies of the available materials that can be used in lasers
and amplifiers. Additionally, as the laser wavelength is decreased (toward the ultraviolet regime),
the upper state lifetime of the gain medium reduces. This makes it more difficult to maintainthe population inversion that is required for the laser to operate. As a result, conventional laser
oscillators and amplifiers in the extreme ultraviolet (EUV) and X-ray regions of the spectrum would
require a prohibitively large amount of (fast) supply power. However, there exist several alternative
approaches to generate laser-like beams of EUV and X-ray light: synchrotrons, free-electron lasers(FELs), and high-order harmonic generation (HHG).
Hair

Red Cell

10 μm
Sub-psLasers
Class

CO2 *

1 μm
Palitra*OPA

Ti:sapph

1 eV

Molecules

10 nm

SHG / THG

VisibleNear-IR0.1 eV

100 nm

1 nm

Atoms

0.1 nm = 1 Å

HHG

Yb/Nd
OP(CP)A
SHG / THG
doped
Tm/Ho/Cr SHG / THG
OPCPA
doped

Mid-IR

Far-IR

DNA

Virus

XFEL
Synchrotron

Soft X-ray

VUV
UV

Hard X-ray

EUV

10 eV

100 eV

1 keV

10 keV

Figure 1.1: Regimes of the electromagnetic spectrum and regions where ultrafast laser technology
is currently available [56, 171, 97, 65, 47, 139, 111]. ∗ The PalitraTM optical parametric amplifier
specifies a broad tunability range for ultrafast operation, but the conversion efficiency for highenergy models is less than 1% for idler wavelengths greater than 2.6 µm [118]. A CO2 amplifier canbe used to boost the idler at 10.6 µm, but at the cost of bandwidth and achievable pulse duration[104, 101].

3
Table 1.1: A comparison of the accessible range of photon energies Eph , pulse duration dt, coherence times τcoh , and degree of transverse spatial coherence rcoh /w for HHG, X-ray FEL, andsynchrotron light sources. The values listed are approximate records to date. ∗ Achieving ultrashortpulse durations or high levels of coherence with facility scale sources usually requires pulse-slicing,
bandpass filtering, or spatial filtering, each of which reduce the source brightness. For HHG, tuning
the driving laser wavelength can also be associated with trade-offs.

Eph range
dtmin
τcoh
rcoh /w

HHG
0.3 eV - 1.6 keV [161]∼300 as [18]
> 1 s [12]> 0.75 [155]

X-ray FEL
0.26 - 25 keV [34]100 as [61]1.5 fs [174]0.45 [174]Synchrotron
0.1 eV - > 300 keV [62]100 fs∗ [13]∼as∗ [62]0.008 - 0.38 [117]Synchrotrons and FELs are facility scale sources that are capable of producing very brightbeams of EUV/X-ray light with wavelength tunability and progressively improving levels of coherence and temporal duration (see Table 1.1 and Fig. 1.2). Despite the many advantages ofthese large-scale sources, their cost and accessibility can be prohibitive in cases. For example,
proposed experiments that have a high level of scientific risk might be denied time on the lightsource. If an experiment is approved, the allotted time could be insufficient to perform a thoroughstudy. Fortunately, HHG is a complementary technology that can provide more opportunities forexperimentation, but at the cost of source brightness and spectral ranges available to date.
High-order harmonic generation describes a frequency conversion technique in which an intense driving laser is focused in a medium, and as a result of the laser-matter interaction, harmonicfrequencies of the driving laser are emitted. In most cases, the HHG medium is a noble gas, butmolecules and solid targets have also been used [48, 116, 173]. At the core of the HHG process
is the coherent nature of the driving laser and the electrons in the target medium, which work
together to produce a highly coherent beam of harmonics. Figure 1.3 depicts HHG from a photon
point of view. A number of effects influence the characteristics of the harmonic output, both on
the microscopic and macroscopic scales. The remainder of Chapter 1 will elaborate on these fundamental details behind the HHG process, while the following chapters provide experimental andtheoretical approaches that have been and can be performed to provide insight into the limits of

PEAK BRIGHTNESSA[photons/s/mm 2 /mrad 2 1% BW]

4

1035

XFELs
HHG
Driving Laser

1030

Compressed HHG

Synchrotrons

1025

Coherent

1020
10

Generated Chirped
HHG

-1

10

0

B

1

Incoherent
2

10
10
PHOTON ENERGY [eV]

10

3

10

4

XFELs
PEAK POWER [W]

1010

105

HHG
Driving Laser

Compressed HHG
Generated Chirped
HHG

Dye
Synchrotrons

100 -1
10

100

101

102

PHOTON ENERGY [eV]

103

104

Figure 1.2: Plots of the (A) peak brightness and (B) peak power of coherent HHG X-ray sources(solid lines), compared to X-ray FELs and synchrotron sources (dashed lines). Driving lasers usedfor HHG shown in open circles - 3.9 µm, 2.1 µm, 1.5 µm, 0.8 µm, 0.4 µm, 0.27 µm). Data compiled
from [34, 27, 64].

5
HHG technology when driven by mid-infrared laser light.

hνL + hνL + hνL + hνL + hνL + ...
= hνqFigure 1.3: The photon picture of HHG. The driving laser with fundamental photon energy hνLcontributes q photons to generate a harmonic photon with energy hνq .

Nomenclature
In this dissertation, SI units are used unless otherwise noted. The following physical constants
will be used:• 0 is the permittivity of free space (8.854 × 10−12 F/m).
• µ0 is the permeability of free space (1.256 × 10−6 N/A2 ).
• e is the natural Euler number (2.718).
• c is the speed of light in free space (2.998 × 108 m/s).
h• ~ = 2π
is the reduced Planck constant (1.054 × 10−34 J · s/rad).

• qe is the charge of the electron (−1.602 × 10−19 C).
• me is the mass of the electron (9.109 × 10−31 kg, 511 keV ).
• rB is the Bohr radius (5.292 × 10−11 m)
2

qe
1−15 m)• re = 4π
2 is the classical electron radius (2.818 × 10
0 me c6• Na is the number density of atoms in a gas at standard temperature and pressure (2.687 ×1025 m−3 ).
Additionally, the following formulae will be implied:
• n=

q

µ
0 µ0 is the index of refraction of a medium.

• IL = c20 n |EL0 |2 is the intensity of a laser field with electric field strength EL0 .
• k = 2πnλ is the wavenumber of an electromagnetic wave with wavelength λ.
• ω = 2πc
λ is the angular frequency of an electromagnetic wave with wavelength λ.
• T = 2π/ω = λ/c is the period of an electromagnetic wave with wavelength λ.
• dt is the duration of a laser pulse, defined as the full width at half maximum (FWHM) of
the intensity profile.
• vφ = ωk is the phase velocity of an electromagnetic wave through a medium.
• vg = ∂ω∂k is the group velocity of an electromagnetic wave through a medium.

1.2

Microscopic Picture of HHG
The mechanism that facilitates HHG fundamentally resides at the atomic scale. In HHG,

driving radiation illuminates the atoms in the target medium, and energy is re-radiated by theelectrons bound to the atoms. According to Coulomb’s law, charged particles experience a force in
response to an electric field. Electrons, being 1,837 times less massive than protons, are accelerated
more readily by the oscillating electric field of the illuminating electromagnetic wave. The subsequent motion of the electron, an accelerating charged particle, gives rise to re-radiation. However,for frequency conversion to occur, the system must be nonlinear - that is, the electron must not
simply oscillate at the same frequency as the incident laser field. The Coulomb potential from theion allows the electron to take on nonlinear motion, thus facilitating frequency conversion.

7
The field of nonlinear optics is rich in processes where spectral/temporal and angular/spatial
characteristics undergo changes, where much of the observed phenomena can be explained using
perturbation theory. However, the spectral trends experimentally observed in HHG do not agreewith perturbative predictions. In the first report on HHG in 1987 [81], the observed harmonic flux
did not simply scale exponentially with harmonic order; there was a plateau at high orders (Fig.
1.4). This indicated a non-perturbative regime of frequency conversion that had not been seen inlower intensity experiments. A theoretical framework other than perturbation theory was needed
to explain these findings.
In 1993, a semi-classical three step model was proposed to describe the HHG process [26, 60],
and the predictions of that model [68, 70], including the non-perturbative characteristics in the
spectral domain, have held true experimentally. In the three step model, the electron starts boundto its atom, confined by the atomic Coulomb potential. Then, the electron is subjected to theelectric field of the laser, thus distorting the total potential that it experiences. If the laser field isintense enough, the electron can tunnel ionize through the distorted Coulomb barrier. Once free ofthe short-range potential of the ion, the electron propagates only under the influence of the laser
field. Initially, the electron accelerates away from the ion, but since the laser field is oscillatory,
the electron may reverse direction and return close to the ion. If the electron does re-encounterthe ion, there is a probability for the electron to recombine with it, returning to its ground stateand giving up any excess energy that it might have gained while under the influence of the laser
field. That energy is released in the form of a photon. Thus, the three step model consists of (1)ionization, (2) propagation, and (3) recombination. Figure 1.5 depicts this process, and the details
behind each step are conveyed in the following sections.

1.2.1

Step 1: Ionization

In the first step of HHG, the electron is liberated from its parent ion by the incident laser
field. This can occur if the electric field strength of the laser approaches the Coulomb field strengthof the ion. In the simplest case, the atomic Coulomb potential experienced by the electron is given

8

100

Relative Energy Conversion Ef7iciency

A

Ne
Perturbative
regime10-1

10-2

Non-perturbative
regime10-3(2s2p63p)

10-4

5
7
(25) (35)1517(75) (85)

1
10-1

Ar

10-2
10-3
10-4
10-5
10-6
10-7
3 7 11 15 19 23 27 31 35

Harmonic Order

1

Kr

10-1
10-2
10-3
10-4
10-5
10-6
3

7 11 15 19 23 27 31 35

Harmonic Order

Relative Intensity of Harmonics

Harmonic Order N (eV)Relative Intensity of Harmonics

Relative Intensity of Harmonics

B

9
11
13
(45) (55) (65)

Xe

1
10-1
10-2

Plateau

10-3
10-4

Cutoff10-5
3

7

11 15 19 23

27 31 35

Harmonic Order

Figure 1.4: The first reports on HHG, demonstrating perturbative behavior at low harmonicorders and non-perturbative behavior at high orders. (A) A plot of the relative harmonic conversion
efficiency in Ne, demonstrating the shift from perturbative behavior to non-perturbative behavior
at the 11th harmonic. Adapted from [81]. (B) Similar plots showing the ”plateau” and ”cutoff”
for harmonics in Ar, Kr, and Xe. Adapted from [36]. Typical error bars are shown for a single
harmonic.

9

Figure 1.5: Diagram of the three step model describing the HHG process. Adapted from [106].

10by:Vatomic (r) = −qe2
4π0 r(1.1)

The electric field of the laser introduces an additional potential felt by the electron:
Z r
Vlaser (r, t) = −qe EL (r 0 , t) · dr 0

(1.2)

0

where EL (r, t) is the electric field of the laser, which points in a direction ψ̂ corresponding to the
local polarization direction of the laser field 1 . In this section, consider a plane-wave electric fieldof the form EL (r, t) = EL0 cos(ωL t)ψ̂. The total potential experienced by the electron is then thesum of Eq. (1.1) and Eq. (1.2), which may result in one of several ionization possibilities depending on the strength of EL0 : multiphoton ionization, tunneling ionization, and barrier-suppressed
ionization. Each of these are depicted in Fig. 1.6.

Ip
Total Coulomb Potential

Multiphoton Ionization

Tunneling Ionization

Barrier-Suppressed
Ionization

Figure 1.6: Diagrams indicating the possible mechanisms for ionization under the influence ofintense laser light. The electric field strength of the laser increases from left to right. It is notedthat single-photon ionization is a possibility if the laser photon energy ~ωL is larger than theionization potential Ip of the atom, which is not considered here.

The parameters that determine the particular ionization regime are inherent to the laser
characteristics and the atomic ionization potential, as described by Keldysh [55]. In particular,
Keldysh introduced a single parameter that relates the ionization potential to the laser pondero1

The local polarization direction may vary as a function of time, as in the case of elliptically polarized light.
Furthermore, the polarization of the laser mode can vary spatially (e.g. cylindrical vector beams). The local fieldpolarization at the location of the atom is of primary importance for the microscopic picture of HHG.

11
motive potential, dubbed the Keldysh adiabatic parameter γKeldysh :s
IpγKeldysh =
2Up

(1.3)

where Up is the ponderomotive potential of the laser field:
Up =2
qe2 EL0
4me ωL2(1.4)

The three regimes of ionization can be determined through γKeldysh according to the following:• γKeldysh >> 1, Multiphoton Ionization: The laser ponderomotive potential Up is muchsmaller than the ionization potential Ip of the atom, thus distorting the total Coulomb
potential to a small degree. Only the perturbative mechanism of multiphoton ionization
can liberate the electron from its potential well, in which N photons from the driving
field impart the sum of their energies to the electron, liberating it from the potential well
(N · ~ωL ≥ Ip ). The ionization rate in this regime scales perturbatively as ILN [172].
• γKeldysh < 1, Tunneling Ionization: Up approaches Ip , significantly distorting the Coulomb
well and creating a finite barrier through which a portion of the electron wavefunction cantunnel. Only a portion of the electron wavefunction escapes the ion’s Coulomb potential,
while the remainder stays in a bound state. The ionization rate in this regime is related
to the proportion of the wavefunction that tunnels and can be modeled using AmmosovDelone-Krainov (ADK) theory [4], discussed later in this section.
• γKeldysh << 1, Barrier-Suppressed Ionization: Up greatly exceeds Ip , completely suppressing the Coulomb barrier that originally confined the electron. The electron leaks awayfrom the ion directly into continuum states, rapidly accelerating due to the strong laser
field. The ionization rate in this regime can be large enough to completely populate theion species within a few cycles of the driving laser.
Tunneling ionization is the relevant regime for the purposes of HHG, as will be evident by the
discussion in Sections 1.3 and 1.4.

12
Keldysh calculated the ionization rate ωKeldysh for the hydrogen atom in the quasi-static
limit [55], which was later extended to more complex atoms and electronic states by Ammosov,
Delone and Krainov. The resulting ADK ionization rate ωADK of an atom or ion can be calculatedusing the following equations in atomic units (adapted from [4]):
Z̄qe2
n =
~

βADK =(1.5)

2(2E0 )3/2
3

(1.6)

E0 (2l + 1)(l + |m|)!
αADK =
2πn∗ 2|m| |m|!(l − |m|)!

ωADK (t) = αADK E(t)

r

me
2E0

∗



1+|m|−2n∗

2e
n∗

2n∗

∗

(3βADK )2n −|m−1|

(1.7)




βADK
exp −
∗ FBS (t)
E(t)

(1.8)

where in au, ~ = 1, me = 1, and qe = 1 (not to be confused with e, the natural Euler number). Z̄
is the charge state of the resulting ion, E0 is the ionization potential of the atom/ion (in hartree), l
and m are the orbital quantum number and its projection for the electron being ionized, E(t) is theelectric field strength of the laser (in au), and FBS is a modification factor that must be included
when the barrier-suppression regime is approached (γKeldysh << 1) [163]:


Z̄ 2 E(t)FBS (t) = exp −αBSE0 (2E0 )3/2

(1.9)

where αBS is a fitting parameter used by Tong and Lin in [163] to match calculated ionization rates
in the barrier-suppressed regime using the single-active electron approximation (see Table 1.2 forvalues of αBS ). Without FBS , the ADK ionization formula would overestimate the ionization ratein the barrier-suppressed regime and would therefore only be valid in the tunneling regime. The
addition of FBS maintains the simplicity of the ADK formulation while extending the applicablerange of the model.
Equation (1.8) represents the instantaneous ionization rate of the atomic or ionic mediumby the laser. For the case of neutral atoms being singly-ionized to the first ionic state, the time-

13
Table 1.2: Some values of the fitting parameter αBS for various atomic and ionic species, including
critical field strengths Ecr and ionization potentials E0 (in au). Adapted from Table 2 in [163].

αBS
Ecr
E0

Rb
6.0
0.006
0.154

H
6.0
0.063
0.500

He
6.0
0.204
0.904

Ar
9.0
0.084
0.579

Ar+
8.0
0.129
1.016

Ne
9.0
0.157
0.793

Ne+
8.0
0.285
1.506

dependent ionization population fraction ηADK (t) can be simply calculated through integration:

 Z tωADK (τ )dτηADK (t) = 1 − exp −(1.10)−∞For media that become multiply-ionized by a single laser pulse, one must use a more complex
calculation that accounts for the simultaneous population and depopulation of each ionic species.
To approach the multiple-ionization case, we define η0 (t) as the time-dependent population ofneutral atoms, while ηi (t) represents the population of ion species i. The rate at which electrons
are ionized from species i − 1 into species i is given by ωi (t) (each of which will depend on the
ith ionization potential Ei , the particular electron’s quantum numbers l and m, and the resulting
charge state Z̄). The neutral atom population can be calculated simply from the rate at which thefirst ionization species becomes populated: Z tη0 (t) = exp −ω1 (τ )dτ

(1.11)−∞

Thus, the neutral population starts at unity and decreases as electrons are removed to populate thesingly-ionized state η1 (t). However, the population and depopulation of the first ionization species
is more complicated since it depends not only on transitions from the neutral species, but also ontransitions into the second ionization species. For the first ionization species, the population is
given by an adjusted form of Eq.(1.10): Z t
η1 (t) = [1 − η0 (t)] exp −ω2 (τ )dτ−∞(1.12)

14
For an arbitrary species i, this equation can be generalized:
"
#

 Z t
i
X
ηi (t) = 1 −ωi+1 (τ )dτηi−k (t) exp −(1.13)−∞

k=1

This formula is self-referencing, since the definition needs to be re-implemented to solve for eachηi−k (t) until the neutral state (i = k) is reached. In other words, to solve for η5 (t), it is first necessary
P atomto solve for η4 (t), and so on. The total ionization population fraction ηtot (t) = Z
i=1 i ∗ ηi (t) can
also exceed 100%, indicating more free electrons than ions in the resulting plasma.

1.2.2

Step 2: Propagation

Once the electron is ionized, it quickly escapes the short-range potential of the ion (severalrB ) whereupon the laser potential dominates. The motion of the electron following ionization can
be classically modeled. Using a z-propagating plane-wave for the laser field with a polarization
direction along the y-axis, the electric field is:
EL (r, t) = EL0 cos(ωL t − kz + φ0 )ŷ(1.14)

where φ0 is the phase of the field at the time of ionization. The magnetic component of the laser
field takes on a similar form:
BL (r, t) = −EL0cos(ωL t − kz + φ0 )x̂c(1.15)As a result, the Lorentz force governing the motion of the ionized electron is:


dp(t)
v(t)
F (t) =
= qe [EL (t) + v(t) × BL (t)] = qe EL0 ŷ −× x̂ cos(ωL t − kz + φ0 )dt
c
The non-relativistic (|v(t)| << c) equations of motion at z = 0 are then:


qe
v(t)
a(t) =
EL0 ŷ −× x̂ cos(ωL t + φ0 )
me
c
qe
∼
EL0 cos(ωL t + φ0 )ŷ=mev(t) ∼=qe
EL0 [sin(ωL t + φ0 ) − sin(φ0 )]ŷ + v0
me ωL

(1.16)(1.17)(1.18)

15

Tunnel ionization of Ar

A

100%

10%

Ar

Ar+

97.5%
ηAr

7.5%

95%

5%

92.5%

2.5%
ηtot = 7.63%

90%
-20

-10

0

10

20

ηAr+

0%

Time (fs)B

Multiple-ionization of Ar
100%

Ar

100%

Ar+

2+

Ar

75%
η

50%

75%
50%

Ar3+

25%

ηtot = 222%

0%
-20

-10

0

10

20

25%
0%

Time (fs)Figure 1.7: Ionization populations for different ion species of Ar plotted as a function of time. The
driving laser pulse (shown in gray) has a wavelength of 1 µm and an 8-cycle pulse duration. (A)The low intensity case (IL = 1.8 × 1014 W/cm2 , γKeldysh = 0.69, E/Ecr = 0.85) where Ar becomeslightly tunnel-ionized; relevant for HHG from atoms. (B) The high intensity case (IL = 3.0 × 1015
W/cm2 , γKeldysh = 0.17, E/Ecr = 3.5) where Ar becomes multiply-ionized in the barrier-suppressed
regime; relevant for HHG from ion species [134].

16

x(t) ∼=−qe
EL0 [cos(ωL t + φ0 ) + ωL sin(φ0 )t − cos(φ0 )]ŷ + v0 t + x0
me ωL2(1.19)

Typically, the initial displacement of the electron x0 is assumed to be small (i.e. zero), but theinitial velocity v0 can be nonzero depending on the initial momentum state of the electron before
ionization. Note that in the above derivation, the magnetic component of the Lorentz force (which
would tend to accelerate the electron in the +z direction) was assumed to be negligible due to the
non-relativistic assumption. In cases where the electron velocity becomes comparable to the speedof light, this component cannot be ignored, as will be discussed in Chapter 4. However, for the
purposes of recollision processes like HHG, the non-relativistic assumption is semi-classically valid
for laser intensities < 1015 W/cm2 and wavelengths < 5 µm.
For the following analyses, assume the electron starts at rest at the origin. Its trajectory canbe plotted for a variety of ionization phases φ0 , as shown in Fig. 1.8. Only ionization phases within[0, π/2) give trajectories that return to the origin, leading to recollisions.
qe(× m ω E )
e

10

2
L

L0

ϕ0-dependent electron trajectories
-π/4
-π/10

5

ϕ0 = 0

y

0
π/10

-5
π/4
π/2

-10
0

0.5

1

1.5
t/T

2

2.5

3

Figure 1.8: Electron trajectories for a variety of ionization phases. The electron returns to its
starting position only for φ0 ∈ [0, π/2).

For returning trajectories, the amount of time the electron spends away from the ion (τf )17cannot be exactly solved for all values of φ0 , since setting Eq. (1.19) to zero forms a transcendental
equation with only the trivial solution t = 0. Of course, numerical methods may be used. However,for the special case of φ0 = 0, the exact solution is easily calculated to be τf = λ/c = T , the laser
period. Additionally, for φ0 = π/2, the solution is τf = 0. Approximate solutions can be found forionization phases close to 0 and π/2 (i.e. φ0 = ε and φ0 = π/2 − ε, where ε is small). In theseregimes, the sinusoidal terms may be Taylor expanded. Keeping only the first two terms in the
expansions for sine and cosine gives an equation with orders up to ε3 , which yields the following
approximate forms for τf (φ0 ):312π − φ0 −
τf (φ0 <π/30) ∼=p
φ60 − 24πφ0 (φ20 − 6)T
12π

(1.20)
3(π/2 − φ0 )∼
τf (φ0 >π/3) =T
2πThese approximations are valid with less than 1% error for the phase ranges listed, as evidenced
by their plots compared to the numerical solution for τf (φ0 ) shown in Fig. 1.9. Note that Taylor
expanding out to the first three terms of the sinusoidal functions (orders up to ε5 ) yields a much
more complicated solution that partially extends the phase ranges where the approximations arevalid, but does not impart any additional scientific insight.
With τf solved for, the electron’s velocity at the time of recollision can be calculated byinserting the values of φ0 and τf (φ0 ) into Eq. (1.18). The kinetic energy of the electron atrecollision is then:Ekinetic (τf ) =me |v(τf )|2= 2Up [sin(ωL τf (φ0 ) + φ0 ) − sin(φ0 )]22(1.21)

where the ponderomotive potential Up from Eq. (1.4) was substituted. Equation (1.21) is plottedin Fig. 1.10, showing a maximum value 2 of 3.17Up for an electron ionized around φ0 ≈ π/10. This
phase marks a division between electron trajectories labeled ”long” (0 < φ0 < π/10) and ”short”(π/10 < φ0 < π/2), simply due to the relative duration of their excursions with respect to thepeak energy scenario. Long and short trajectories do result in different quantum and macroscopic
physics, as will be discussed in Sections 1.3 and 1.4.
2

The electron can achieve a kinetic energy > 3.17Up during its trajectory (up to 8Up if φ0 ≈ ±π/2), but theelectron is far from its parent ion when it achieves these higher energies.

18
Electron return time

1
0.8

τf
T

0.6
0.4
0.2
π/30

0
0

π/3

π/16

3π/16

π/8

π/4
ϕ0

5π/16

3π/8

7π/16

π/2

Figure 1.9: The amount of time τf an electron spends away from the ion before returning to
its starting position, plotted as a function of ionization phase φ0 . Only phases within [0, π/2) areconsidered since all other phases result in non-returning trajectories. The solid black line is thenumerical solution, while the solid red and blue lines represent the approximate solutions from Eq.
(1.20). The shaded regions indicate the phases where the approximate solutions have less than 1%
error.

Returning electron kinetic energy
3.17
3
Long

Short

2.5
2
E kinetic
Up
1.5
1
π/10

0.5
0
0

π/16

π/8

3π/16

π/4
ϕ0

5π/16

3π/8

7π/16

π/2

Figure 1.10:
The kinetic energy Ekinetic of the recolliding electron, plotted as a function ofionization phase φ0 . A maximum value of 3.17Up occurs for φ0 ≈ π/10. Electrons ionized before
this phase (long trajectories) and those ionized after this phase (short trajectories) have reduced
energy upon recollision.

19
At the time of recollision, the electron can recombine with the ion and enter the groundstate, causing it to release its kinetic energy combined with the ionization potential Ip of the atom,
leading to the HHG single-atom cutoff energy rule:
ESAC = Ip + 3.17Up(1.22)

This key equation results from a purely classical analysis of the electron’s motion following ionization, and it represents the highest photon energy that can be emitted via the HHG process from asingle atom under monochromatic, linearly-polarized illumination.
Note that the above calculations assumed the electron started at rest. Had the electron been
ionized with residual momentum from its bound state, its motion would be identical except with alinear drift in time. This drift points in the direction of v0 , which can result in electrons returning
to the origin for ionization phases outside φ0 ∈ [0, π/2) if v0 k ŷ, the polarization direction.
Furthermore, there exist alternative driving field configurations that allow the electron to takereturning trajectories, such as bichromatic, bicircular driving fields used to produce circularlypolarized harmonics [154, 130, 129, 128].

1.2.3

Step 3: Recombination

As described in the previous section, the recombination step provides the mechanism for theelectron to release its kinetic energy through photon emission. Whether or not a photon is released
depends on a number of factors. The following must occur:(1) Ionization of the electron must occur.
(2) The electron must return to the vicinity of its parent ion.
(3) Upon recollision with the ion, the electron must recombine with the ion, re-entering the
ground state.
The first two items have been discussed in this semi-classical analysis, but the recombination
pathway is only one of several options that the electron can take. In general, a recollision can also

20
result in:• Above-threshold ionization: The electron absorbs more photons than necessary to become ionized, leaving it in a continuum state with non-zero kinetic energy [3, 9, 32].
• Elastic backscattering: The ionized electron recollides with the ion, scattering elastically
backward and leaving the system as an energetic free electron [95].
• Inelastic excitation: The ionized electron recollides with the ion and imparts some of its
kinetic energy to a remaining bound electron by scattering inelastically, thus exciting thebound electron to an excited state [29].
• Non-sequential double ionization: The ionized electron recollides with the ion at highenergy, imparting > Ip of its energy to a bound electron such that the second electron also
becomes ionized [75, 39].
Figure 1.11 depicts each of these processes in addition to the HHG pathway. The cross sections
for each of these processes vary; for HHG, a quantum mechanical treatment is required and will bepresented in Section 1.3. Should the electron take the HHG pathway and recombine with ion, its
total energy changes:
0Etot,0 = Ekinetic,0 + V0−→Etot,f = −Ip(1.23)

which leads to the emission of a photon with energy Eph = −(Etot,f − Etot,0 ), limited by the single
atom cutoff energy, presented in Eq. (1.22). Some ionization potentials are listed in au as E0 in
Table 1.2, and in electron-volts as Ip in Table 1.3.

1.2.4

Characteristics of Microscopic HHG Emission

The harmonic emission that results from the HHG three step process has properties andlimitations that are inherent to the microscopic picture. Already, a spectral limitation for HHG
has been presented through the derivation of the single-atom cutoff energy described in Eq. (1.22).

21

Possible consequencesof recollision- Energetic electron emission
by elastic backscattering ofelectron-- Energetic photon emission-

by electron recombining
with its parent ion (HHG)- Detachment of another electron:

τ0

non-sequential double ionization
Excitation of bound electrons
by inelastic collision

-

-

τf

+

x-ray

-

Free electron with non-zerokinetic energy: Above-thresholdionization-

Figure 1.11: Several processes observed in electron-ion recollisions following ionization. Only HHG
results in photon emission, while all other processes result in one or more free electrons and an ion
species. In the figure, τ0 and τf refer to the times when the electron is born and when it returns
to the ion, respectively. Adapted from [19].

Table 1.3: Ionization potentials Ip (in eV) for various gases and their ionic species. Data obtained
from [71]. Ionization potentials beyond first ionization can be important for HHG from ions, as
demonstrated in [134].

Ip (eV)
H
He
Ne
Ar
Kr
Xe
N

X → X+

X +→ X 2+

X 2+→ X 3+

X 3+→ X 4+

X 4+→ X 5+

X 5+→ X 6+

X 6+→ X 7+

13.6
24.59
21.56
15.76
14.00
12.13
14.53

54.42
40.96
27.63
24.36
21.21
29.60

63.45
40.74
36.95
32.1
47.45

97.11
59.81
52.5
46
77.47

126.21
75.04
64.7
57
97.89

157.93
91.01
78.5
82
552.06207.26
124.4
111
100
667.03

22
In essence, the maximum harmonic energy achievable from HHG is determined by the ionization
potential of the medium as well as the driving laser’s intensity and wavelength. An additional
property in the spectral domain is the scaling of harmonic flux with harmonic order; at low orders,the exponential scaling typical of perturbative effects is observed, but at large orders, there is nonperturbative behavior indicated by the plateau of harmonics extending to the cutoff energy (Fig.
1.4). Only the three step model of HHG explains these qualities.
With regards to the time domain, only the electron motion has been discussed. However,
the electron is at the core of the HHG process, and the harmonic output is therefore dependent
on its behavior. In particular, harmonic emission occurs at the time the electron recombines with
its parent ion - some time after the electron is first ionized. If the electron is ionized at a phaseφ0 ∈ [0, π/2), then it will recombine and emit a photon at a later phase φf ∈ (π/2, 2π]. Take notethat the laser field amplitude is periodic, such that the non-relativistic 3 physics that takes place
within a given half-cycle should be repeated for the following half-cycle, except with a change ofsign. This necessarily results in the possibility of harmonic emission occurring every half-cycle ofthe driving laser field. For an ensemble of emitters, the averaged microscopic result would be aλpulse train of harmonics with temporal spacing 2c. This pulse train would last for the duration

dt of the driving laser, while the intensity is large enough to tunnel ionize (γKeldysh < 1) and theionization level remains unsaturated (ηADK < 1). Each individual harmonic pulse would have a
λduration less than a half-cycle of the driving field (dtHHG < 2c).
λThe 2cperiodicity of the harmonic pulses in the time domain generates interferences inthe spectral domain. Without the interferences (i.e. considering the emission from only a single
half-cycle) the HHG spectrum would be a broad supercontinuum extending from the fundamental
frequency ωL to the single-atom cutoff energy (many harmonic orders higher). With the interference
from the neighboring pulses, however, the supercontinuum becomes modulated with peaks occurring
at every odd harmonic of the fundamental. The separation of 2ωL between each peak is a direct3

Note that for relativistic cases ignored in Section 1.2, the half-cyclic symmetry can be broken. These cases will
be investigated in Chapter 4

23
λresult of the 2cperiodicity in the time domain. The peaks occur at every odd harmonic (as opposedto every even harmonic) due to the relative phases of each pulse in the harmonic pulse train. The
emission from electrons ionized during an up half-cycle - φ0 ∈ [0, π/2) - is followed by emission
from electrons ionized during a down half-cycle - φ0 ∈ [π, 3π/2). The opposing signs of the fields
and trajectories result in a π phase shift between neighboring harmonic pulses, which in turnresults in the odd harmonics that are observed experimentally. Figure 1.12 provides a qualitative
depiction of these phenomena, while Fig. 1.13 contains measured data of HHG in the time andfrequency domains. The spectral and temporal characteristics of HHG can be modified either
through breaking of the half-cycle symmetries, or through adjusting the driving laser waveform in
the time domain. For instance, emission of an isolated harmonic pulse is possible by driving HHG
with a single-cycle laser pulse (dt ≈ λ/c) [22, 140, 137].
Another important property of the harmonic beam in the time domain is the chirp rate ofeach pulse. As a result of the three step model, electrons ionized at different times within thepulse will recombine at different times and emit harmonics of different energies. From the analyses
performed in Section 1.2.2, the spectral phase (and thus the various orders of chirp/dispersion)
can be calculated. Dispersive orders tend to increase the duration of a pulse, thus imposing alimitation on the pulse duration achievable via HHG without using compression techniques (which
can be difficult and inefficient in the EUV and soft X-ray regimes). Figure 1.14 contains plots of
2

3

d φ
d φ
the group delay dispersion (GDD = dω2 ) and third-order dispersion (T OD = dω 3 ). The GDD has

the largest effect on the pulse duration, while higher orders can be largely neglected.
Finally, the spatial and angular characteristics of the harmonic emission can also be deduced
from the electron motion. In this microscopic picture, the size of the photon source would obviously
be no larger than the size of the atom. As for the direction of photon emission, the electron’s
acceleration vector gives a clue. Accelerating charged particles radiate light, according to theLarmor formulation:dP (t)
q 2 sin2 (θ)a2 (t)
= e
dΩ
4πc
c2

(1.24)

24

Qualitative HHG pulse train

t

λL

FT

FT-1
Qualitative HHG spectrum

100

10-1

2ωL

Signal -2
10(arb.)

Plateau
10-3

Cutoff
10-4

1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
ω / ωL

Figure 1.12: Plots of the qualitative characteristics of HHG in the time and frequency domains.
The spacing of the pulses in time result in a 2ωL spacing of peaks in the spectrum. The absolute
positions of the spectral peaks is governed by the relative phases of neighboring pulses in the time
domain. Odd harmonics are naturally generated for non-relativistic HHG using monochromatic
drivers.

25

HHG autocorrelation driven by 2 μm

1.0
0.5
Signal(arb.)

0.0
-0.5
-1.0

-10

0
Time (fs)

-5

5

10

40

50

HHG spectrum driven by 2 μm

1.00.8
Signal 0.6(arb.) 0.4

200 nm Al �iltertransmission

0.2
0.0
0

10

20

30
Photon energy (eV)Figure 1.13: Plots of the measured signal in the time and frequency domains for HHG driven by
λ = 2 µm at low intensity and transmitted through an Al filter. An electric field autocorrelationof the HHG pulse train is presented for the time domain, whose Fourier transform directly gives
the spectrum (Chapter 2, [18]). Attosecond streaking methods are otherwise required to indirectly
measure the HHG pulse train, as opposed to its autocorrelation [144]. Note that macroscopic effects(Section 1.4) apply here.

26

A

Emitted photon energy versus time
3.17
2.5

E ph - Ip
Up

2

Long
Trajectories

Short Trajectories

1.5
1
0.5
0
0.25
(π/2)

0.5
(π)

0.75
(3π/2)

1(2π)

tf / T(ϕf )B

Natural GDD from the HHG process

0.4
0.2

GDD
ℏ T / Up

Short Trajectories

0.0Long Trajectories

-0.2

-0.4
0.0

C
TOD

Up 2

1.0

1.5
E ph - Ip
Up

2.0

2.5

3.0

Natural TOD from the HHG process

0.4
0.2

ℏ2 T

0.5

Long Trajectories

0.0

-0.2
Short Trajectories

-0.4
0.0

0.5

1.0

1.5
E ph - Ip
Up

2.0

2.5

3.0

Figure 1.14: Plots showing the dispersion of natural HHG emission. (A) A plot of the energy of theemitted photon for an electron returning to the ion at a final time tf , corresponding to a final phase
φf . (B) and (C) Plots of the normalized group delay dispersion (GDD) and third-order dispersion(TOD) that result from the three step model of HHG. These are used in Taylor expansions for the
spectral phase, and tend to increase the durations of the harmonic pulses unless compensated forthrough compression techniques. Note that the atom’s ionization potential has no effect on the
microscopic dispersion, but would only shift the curves uniformly along the energy axis.

27

P (t) =

2 qe2 a2 (t)3 c3(1.25)

Here, θ is the angle between the acceleration and observation vectors. Equation (1.17) gives theacceleration of the electron, but under the influence of the driving laser field only, which would
result in the electron radiating at the same fundamental frequency. It is the recollision with theion that causes the electron to radiate at higher frequencies, so it is the impulsive acceleration
upon recombination that must be considered. In terms of quantitave power radiated, the classicalacceleration upon recollision is difficult to calculate, since the rate at which the electron stops torecombine is not readily deduced. Scattering theory can be used, but gives a result that is highly
dependent on the impact parameter. The quantum mechanical treatment in Section 1.3 provides
a more intuitive and straightforward calculation for the quantitative power radiated. Fortunately,
the qualitative direction of the acceleration upon recollision is obvious, and therefore the emission
pattern would follow the sin2 (θ) scaling in Eq. (1.24), with θ being the angle measured from theŷ-direction. For a single atom, the harmonic emission is brightest in the x-z plane, while there iszero emission along the ŷ direction. Figure 1.15 depicts this microscopic radiation pattern. Despite
the broad angular distribution for single atom radiation, macroscopic effects reduce the total beam
divergence (see Section 1.4).

1.3

Quantum Mechanical Picture of HHG
The semi-classical analysis of microscopic HHG presented in Section 1.2 is a valid approachto derive the primary characteristics of the harmonic emission. A quantum approach should yield
the same physical conclusions, but can give deeper insight into the process from the viewpoint of
the electron wavefunction. Ensemble averages, which are relevant to realistic laboratory conditions
with many atoms, can be derived directly from the quantum perspective.

28

-

y

z

x

Figure 1.15: Depiction of the Larmor radiation pattern following the sin2 (θ) scaling from Eq.
(1.24). The power is primarily radiated along the x-z plane, with zero emission directly alongŷ. Microscopically, HHG would be emitted with this broad angular distribution, but macroscopic
effects enhance the emission along the laser propagation direction k k ẑ.

29
1.3.1

The Quantum Three Step Model

The quantum derivation for the three step model is found in [68] and is reproduced here. The
general approach is to solve the time-dependent Schrödinger equation while treating the electron’s
wavefunction as a sum of the bound state and ionized states in the continuum. By solving for thecontinuum states, the dipole radiation at the time of recombination can be solved for.
The electron’s wavefunction obeys the time-dependent Schrödinger equation:
i~


 2
∂ |Ψ(r, t)i−~ 2∇ + V (r, t) |Ψ(r, t)i ,
=
∂t
2me

(1.26)

where |Ψ(r, t)i = e−iEt/~ |ψ(r)i is the electron’s wavefunction, and V (r, t) is the total potential
experienced by the electron. |ψ(r)i satisfies the time-independent Schrödinger equation:



−~2 2
∇ + V (r, t) |ψ(r)i = E |ψ(r)i
2me(1.27)

Prior to the laser field, the electron is in the atomic ground state |ψ(r)i = |0i with energy E0 = −Ip ,confined by only the atomic Coulomb potential Vatomic (r), which in the simplest case is given by
Eq. (1.1).
In the presence of the laser field, an additional potential arises. For a linearly-polarized laser
field aligned along the ŷ direction:
Vlaser (r, t) = −qe EL0 cos(ωL t) y

(1.28)The time-dependent Schrödinger equation then becomes:
 2
∂ |Ψ(r, t)i
−~ 2
i~
=
∇ + Vatomic (r, t) − qe EL0 cos(ωL t) y |Ψ(r, t)i∂t2me(1.29)At this point, the following assumptions are necessary to proceed:(1) Only the |0i bound state contributes to the the evolution of the atomic system, while allother states’ contributions are neglected.
(2) The depletion of the |0i state is ignored, indicating ionization fractions η << 1.

30(3) Once the electron ionizes to a continuum state |pi, the short-range atomic potential Vatomic (r)no longer acts on it (strong-field approximation).
The validity of the assumptions are generally good for Keldysh parameters γKeldysh < 1, but for
intensities less than the saturation intensity. Note that the continuum states satisfy Eq. (1.26)
with the form:|Ψp i = e−iEp t/~ |pi = Aei(k·r−ωp t) = Aei(p·r−EP t)/~ ,(1.30)

2p
is its energy, k = p~ = λ2πD v̂ is its
where p = me v is the momentum of the electron, Ep = 2m
e
2

E

~k
wavevector, λD is the DeBroglie wavelength, and ωp = ~p = 2m
= λπvD .
eWith the assumptions, the total wavefunction of the electron can be decomposed into boundand continuum components as follows:|Ψ(r, t)i = e

iIp t/~

Z
3a(t) |0i + d p b(p, t) |pi(1.31)

where a(t) represents the time-dependent proportion of the wavefunction that stays in the groundstate, and b(p, t) represents the proportion that ionizes and enters the continuum state with momentum p. 4 Note that the phase eiIp t/~ is factored, shifting the energy of the ground state |0i to
2pzero and the energy of the continuum states up to 2m+ Ip . Ignoring the ground state depletionea(t) ∼= 1, the left-hand side of the time-dependent Schrödinger equation becomes:∂i~ |Ψ(r, t)i = −Ip |Ψ(r, t)i + i~eiIp t/~∂t

Z

d3 p ḃ(p, t) |pi(1.32)

With the substitution of Eq. (1.31) and using the fact that the atomic potential Vatomic (r) acting
on the phase-shifted continuum states results in −Ip |pi, the right-hand side of the time-dependent
Schrödinger equation is:
H |Ψ(r, t)i = − Ip |Ψ(r, t)i − eiIp t/~ qe EL0 cos(ωL t) y a(t) |0i

 2
Z
p
iIp t/~3+ed p+ Ip b(p, t) |pi
2meZiIp t/~−eqe EL0 cos(ωL t) y d3 p b(p, t) |pi
4
Here, a(t) is unitless, while b(p, t) has units of p−3 .

(1.33)

31
Putting together the two sides of the time-dependent Schrödinger equation results in the cancellation of the terms −Ip |Ψ(r, t)i and eiIp t/~ . Additionally, the y operator can be expressed in
momentum space as i~ ∂p∂ y . The result is:Z
i~ d3 p ḃ(p, t) |pi = − qe EL0 cos(ωL t) y a(t) |0i

 2
Z
p
3
+ d p
+ Ip b(p, t) |pi
2meZ∂− i~qe EL0 cos(ωL t)d3 pb(p, t) |pi∂py
R 3 0 0Multiplying both sides of the equation by 5
d p hp |: 2
p+ Ip b(p, t)i~ḃ(p, t) = − hp| qe EL0 cos(ωL t) y |0i +2meZ∂− i~qe EL0 cos(ωL t)
d3 p0 hp0 |b(p, t) |pi∂py
The product rule is needed for the last term, resulting in:ZZ∂b(p, t)3 00 ∂b(p, t) |pi =d3 p0 hp0 |pi
d p hp |
∂py
∂py
Z
∂
+ b(p, t) d3 p0 hp0 ||pi∂py(1.34)(1.35)(1.36)

From Eq. (1.30), the derivative of |pi with respect to py causes the second term’s integrand to beodd, resulting in zero for the integral. Finally, the time-dependent Schrödinger equation for b(p, t)
takes the form of the Volkov equation:
 2

i
p
i
∂b(p, t)ḃ(p, t) = −+ Ip b(p, t) − qe EL0 cos(ωL t)+ qe EL0 cos(ωL t) hp| y |0i
~ 2me
∂py
~

(1.37)

Note that the term hp| y |0i = dy (p) is the atomic dipole matrix element for the bound-free transition parallel to the polarization axis ŷ. The Volkov equation can be solved exactly using theelectromagnetic vector potential, which takes the following form for this electric field in the radia:tion gauge E(t) = − dA(t)
dt


0




 EL0

A(t) = −

sin(ω
t)
L
ω


L


0

(1.38)


Here, we implicitly assume there is an additional factor with amplitude 1 containing units of p−3 . Thisimplicit factor will disappear when integrals over d3 p collapse, but would remain for the first term in Eq. (1.35) andits eventual conversion to dy (p), which persists through the calculation.
5

32
With the substitution for the canonical momentum (p̄ = p + qe A(t) = me v + qe A(t)), the solution
for b(p̄, t) is then:b(p̄, t) =

i
~

Z t

0

dt0 qe EL0 cos(ωL t0 ) dy (p̄ − qe A(t0 )) e−iS(p̄,t,t )(1.39)

0where the integral begins at time t = 0 (corresponding to the peak of the driving electric field),and S(p̄, t, t0 ) is the semi-classical action:S(p̄, t, t0 ) =

1
~

Z t
t0

dt00



(p̄ − qe A(t00 ))2
+ Ip
2me


(1.40)At this point in the derivation, we have arrived at a closed form solution for the continuum
state amplitudes as a function of time. Recall that the goal of the derivation is to arrive at an
equation for the harmonic dipole emission upon recombination. The harmonic electric field will be
proportional to the second time-derivative of the dipole:ZhΨ(r, t)| y |Ψ(r, t)i = h0| y |0i + d3 p d3 p0 b∗ (p0 , t) hp0 | yb(p, t) |piZ+ d3 p h0| yb(p, t) |piZ+ d3 p b∗ (p, t) hp| y |0i(1.41)

The first term contains an odd integrand and is therefore zero. The second term concerns transitions
between continuum states, which do not contribute to the harmonic emission. The remaining terms
are complex conjugates of each other. The component of the dipole that concerns harmonic emission
can therefore be written as:Zχ(t) =

3∗d p b (p, t) hp| y |0i + c.c. =Z

d3 p b(p, t)dy (p) + c.c.

(1.42)

Substituting b(p̄, t) from Eq. (1.39) 6 :i
χ(t) =
~

Z t
dt

0

Z

0

d3 p̄ qe EL0 cos(ωL t0 ) dy (p̄ − qe A(t0 )) e−iS(p̄,t,t ) d∗y (p̄ − qe A(t)) + c.c.

(1.43)

0Equation (1.43) can be physically interpreted in terms of probability amplitudes corresponding toeach component of the three-step model. First, the electron is ionized at time t0 from a bound6


Recall b(p̄, t) contains an implicit factor with units p−3 , which carries into χ(t).

33state into a continuum state with a probability amplitude: EL0 cos(ωL t)dy (p̄ − qe A(t0 )). Following
ionization, the free electron propagates in the laser field with a constant canonical momentum p̄,0acquiring a phase equal to the quasi-classical action: e−iS(p̄,t,t ) . Finally, the electron recombineswith the ion at time t, transitioning from a continuum state to the bound state with a probability
amplitude: d∗y (p̄ − qe A(t)). Note that Eq. (1.43) can be extended to arbitrary polarization andtemporal shape. For an electric field of the form EL (t), the dipole moment along direction n̂ is:i
χ(t) =
~

Z t
dt

0

Z

0

d3 p̄ qe EL (t0 ) · d(p̄ − qe A(t0 )) n̂ · d∗ (p̄ − qe A(t)) e−iS(p̄,t,t ) + c.c.

(1.44)

0

The dipole moment contains an integral over the canonical momentum where the terms in0

the integrand oscillate rapidly for many values of p̄. Because the term e−iS(p̄,t,t ) oscillates most
rapidly, the integral averages to zero except in regions where ∇p̄ S(p̄, t, t0 ) ∼= 0 (which can also be
stated as x(t) − x(t0 ) ∼= 0; i.e. only electrons with trajectories that return to the ion contributeto HHG). Therefore, the result of the integration can be approximated by finding the stationary
phases p̄s of S(p̄) and Taylor expanding about them:1
S(p̄) ∼= S(p̄s ) + ∇2p̄ S|p̄s (p̄ − p̄s )2 ,2where p̄s satisfies ∇p̄ S(p̄s , t, t0 ) = 0 →

Rt

00
00
t0 dt (p̄s − qe A(t )) = 0.

(1.45)
Lumping together the otherterms in the integrand as F (p̄), and recognizing that F (p̄) varies slowly compared to S(p̄), theintegral over the momentum becomes:−iS(p̄s )F (p̄s )eZid p̄ exp − ∇2p̄ S|p̄s (p̄ − p̄s )22
3

(1.46)

This integral is separable into integrals over the three Cartesian directions, each of which can besolved using the same substitution πξ 2 /2 = ∇2p̄ S|p̄s (p̄x − p̄s,x )2 . Equation (1.46) then takes thefollowing form:−iS(p̄s )F (p̄s )eπ2∇p̄ S|p̄s

3/2 Z ∞ 3dξ exp −iπξ /22(1.47)−∞

which contains a Fresnel-type integral with solution

R∞

√

2
2
−∞ dx (cos(πx /2) − i sin(πx /2)) = i

i.

34
0

τ

)
f
7 :
Also noting that ∇2p̄ S|p̄s = (t−tme ~ = me ~ , the dipole moment finally becomes1
χ(t) =
~

Z t
dt

0



0

iπme ~
τf3/2

qe EL (t0 ) · d(p̄s − qe A(t0 ))(1.48)× n̂ · d∗ (p̄s − qe A(t)) e−iS(p̄s ,t,t0 )+ c.c.

Calculating qe χ̈(t) will give the harmonic electric field in the time domain, which when Fouriertransformed will result in the HHG spectrum. The emission amplitude into the N th harmonic withR∞
frequency N ωL can be calculated by performing the integral −∞ dt qe χ̈(t)e−iN ωL t .
Every half-cycle of the driving laser contains two stationary phases p̄s corresponding to thelong and short electron trajectories, both of which should be considered for the total harmonic
dipole emission from a single atom. 8

For an electron ionized at time t0 and returning at timet = t0 + τf , the stationary value of the canonical momentum along the polarization direction is:p̄s (t, τf ) =qe EL[cos(ωL t) − cos(ωL (t − τf ))]ωL2 τf

The quasi-classical action can then be evaluated at the stationary points:


Z
00 2
1 t
0
00 (p̄s − qe A(t ))S(p̄s , t, t ) =dt+ Ip~ t−τf2me(Ip + Up )τf
Up G(τf )2Up−[1 − cos(ωL τf )] −cos (ωL (2t − τf )) ,
=
2
~
~ωL
~ωL τf(1.49)(1.50)

whereG(τf ) = sin(ωL τf ) −4sin(ωL τf /2)2 .
ωL τf(1.51)

Equation (1.50) is plotted in Fig. 1.16 for the limiting cases when Ip >> Up (γKeldysh >> 1)
and when Up >> Ip (γKeldysh << 1). For HHG, the tunneling regime is relevant (γKeldysh < 1,Up ∼ Ip ), so the curve for the quasi-classical action during HHG would lie between the two limitingcases. It is noted that the curve for the case Up >> Ip exactly matches the fully classical approach
Rt
where an integral is performed over the electron’s kinetic energy: SUp >>Ip (t, t0 ) = t0 dt00 me v(t00 )2 /~,with v(t) given by Eq. (1.18).

Noting the presence of the implicit factor containing units p−3 .
However, it is noted that macroscopic phase-matching effects can preferentially select either the long or short
trajectory to be generated brightly.
7

8

35

Ip + Up(× ℏ ω )2π

Electron quantum phaseL

3π/2
S(p s , t, τf )Ip ≫ Up

π

π/2
0
0

Up ≫ Ip

π/8

π/4
ϕ0

3π/8

π/2

Figure 1.16: S(p̄s , t, τf ) plotted as a function of ionization phase φ0 for the limiting cases Ip >> Up(red) and Up >> Ip (blue). For HHG in the tunneling regime, Up will be comparable to Ip , so theresulting curve will lie between the two curves shown.

36
The quasi-classical action depends linearly on the duration of the trajectory τf , causing thelong trajectories to develop significantly more phase compared to the short ones. S(p̄s , t, t0 ) also
depends linearly on the ponderomotive potential Up , resulting in a linear dependence with respect
to IL and ultimately a cubic dependence with respect to λL . The linear dependence of S(p̄s , t, t0 )on IL can result in a spatially-dependent harmonic phase for driving fields with nonzero ∇IL .
Additionally, the long trajectories have a stronger dependence on IL than the short trajectories.
dS(p̄s ,t,t0 )For the trajectories that emit photons with energy 2Up (φ0 ∼is= 3π/16),= π/32 and φ0 ∼dIL

5.3 times larger for the long trajectory than the short. The importance of this trajectory-dependent
phase will be discussed in Section 1.4.3.
Another characteristic to note about the oscillating dipole qe χ̈(t) is that it contains terms
related to the interaction between the bound state |0i and the continuum states |pi, specifically, thedipole matrix elements dy (p) combined with the relative phase difference S(p, t, t0 ). The physicalinterpretation is that the time-varying interference resulting from the overlap of these states gives
rise to the oscillating dipole, and thus, harmonic emission. Figure 1.17 shows a quantum simulationof the electron’s wavepacket, indicating the interference between the ground and continuum states.
Finally, the integral in Eq. (1.48) is usually limited to values of t ≤ T so that only electrons
ionized during the same cycle of the harmonic emission are considered. It is possible for electrons
ionized during previous cycles to recollide with the ion during the current cycle, but these freeelectrons’ wavepackets have spread out and reduced in amplitude as a result of quantum diffusion(discussed in Section 1.3.2). Therefore, it is usually valid to neglect the contributions from previous
cycles’ ionized electrons.

1.3.2

Quantum Diffusion of the Electron Wavepacket

Quantum diffusion of the free electron wavepacket ultimately reduces the harmonic dipole
strength. The longer the electron spends in the continuum, the more dispersed its wavepacket
becomes, and the smaller amplitude it will have in the region that overlaps with the bound state
wavefunction. This results in reduced harmonic emission for large values of τf , as can be seen in37

λL= 800 nm

λL= 1.3 µm

λL= 2.0 µm

Figure 1.17: Total electron wavepacket (2D projection, normalized to 1) calculated at the peak of
a three cycle laser pulse for three different wavelengths. The laser polarization is along the longer
dimension of the plots. Adapted from [106].

38
−3/2

the τfdependence of Eq. (1.48). In terms of the harmonic intensity or flux, the scaling fromquantum diffusion is τf−3 . These dependencies on τf are a direct result of quantum diffusion andcan be derived using a Gaussian approximation for the bound state |0i:
−3/2
|0i ∼
exp
= π −3/4 σ0



−r2
2σ02


,

(1.52)

where σ0 specifies the spatial width of the bound wavefunction, which should be on the order ofthe Bohr radius rB .
Upon ionization in the strong field approximation, some fraction of the bound wavefunction
enters the continuum states |pi, where the relative amplitudes of the individual unbound states
C(p) can be calculated through a simple decomposition of |0i into |pi:Z
|0i = d3 p C(p) |pi
ZC(p) = d3 r hp|0i(1.53)


Here, the amplitudes C(p) must have units of p−3 . From Eq. (1.30), it is clear that the amplitudes
C(p) are calculated through a simple Fourier transform of the bound state, resulting in a Gaussianin p-space:Z
C(p) =−3/2
d r π −3/4 σ0
exp
3

√
3/2
= 2 2π 3/4 σ0 exp



−r2
2σ02

2





× e−ip·r/~−p4~2 /σ02(1.54)

After the ionization step, the freed electron is subjected to the time-dependent oscillating
laser potential. The expectation value of the ionized electron’s position
Z
hri =

3

d r

Z

d3 p C ∗ (p) hp| r C(p) |pi(1.55)

should follow the motion described during the propagation step in Section 1.2.2. The electron’s
wavepacket will diffuse in addition to this motion, and it is assumed that the diffusion transverseto the polarization direction can be calculated separately from the average motion. This is allowed
since the Schrödinger equation is separable in the non-relativistic regime, giving rise to a freeparticle Hamiltonian for the transverse directions. Conveniently, each |pi has a propagator for

39
2

p
time development in the form of the phase e−iEp t/~ , where Ep = 2m
. Therefore, to propagate theewavefunction forward in time, a quadratic phase must be added:Z|Ψ(r, t)i =d3 p C(p) |pi exp

−ip2 t2me ~
(1.56)

Using Eq. (1.30) and Eq. (1.54), |Ψ(r, t)i takes the form of an inverse Fourier transform of a
Gaussian multiplied by a quadratic phase. The quadratic phase in momentum space is responsible
for broadening the wavepacket in position space, and the integral can be handled by treating thetotal integrand as a single Gaussian with a complex width in momentum space. Performing the
integration along one of the transverse dimensions yields the wavefunction along that dimension attime t after ionization:|Ψx (t)i = π −1/4 σ(t)−1/2 exp

−x2
2σ(t)2





me x2
exp i
exp(iΓ(t)),
2~R(t)(1.57)

where:
tD = σ02 me /~s 2t
σ(t) = σ0 1 +tD" 2 #tD
R(t) = t 1 +
t
 
1
t
Γ(t) = − tan−12
tD(1.58)

This form 9 derived from the free particle Schrödinger equation is valid for the two dimensions transverse to the polarization axis in the non-relativistic regime, but the presence of the time-dependent
laser potential prevents the polarization axis to be treated as a free particle. Additionally, theground state before ionization would not be expected to have the same width along the polarization axis as the transverse width σ0 . Indeed, for the transverse directions, σ0 is typically larger than
1/2
 √
3 2~
γ
was
rB due to the shifting of the electron in Vatomic (r) by the laser field. σ0 ∼= 8m
Keldyshe ωLpresented in [28], while σ0 ∼= 4 rB has been shown to agree with numerical studies [169, 170, 92].
9Note the similarity to Gaussian beam propagation (Section 1.4.3), which uses the same wave-decomposition ofa Gaussian electric field amplitude. For instance, the characteristic diffusion time tD is analogous to the Rayleighrange zR . The parabolic and Gouy phase terms are present as well.

40
Using this value, the characteristic diffusion time tD = 388 as, which for reference is comparable
to the return time τf of an electron ionized at φ0 = π/10 for a driving laser with wavelength 175
nm.
Despite the fact that the polarization direction cannot be treated equally, diffusion shouldstill occur in that direction, even if not exactly as σ(t). From Eq. (1.57) and the form for σ(t), it
can be assumed that the total probability density hΨ(t)|Ψ(t)i scales roughly as t−3 in amplitude,
for t > several tD . Therefore, the free electron’s overlap with the bound state will correspondingly
be reduced by this factor. This is in agreement with the statement at the beginning of this section:
the harmonic flux scaling from quantum diffusion is τf−3 , or λ−3 .
1.3.3

Quantum Mechanical Characteristics of HHG

The quantum mechanical analysis preformed in this section has revealed new physics beyond
the simple semi-classical approach from Section 1.2. This approach gives a more straightforward
and intuitive method for calculating the harmonic emission as opposed to classical scattering theory.
Of primary importance is the treatment of the ionized electron as a spreading wavepacket, which
accounts for the electron’s many possible initial momentum states and ultimately results in ananalytical scaling of τf−3 for the harmonic flux. The single-atom HHG flux scaling thus receives a−2
λ−3
L contribution from quantum diffusion, and a λL scaling from increasing harmonic bandwidth(Eq. (1.22)) for a fixed energy interval. Macroscopic effects can further limit this scaling, resulting−9
in a final HHG flux scaling ranging from λ−5
L to λL [161, 120, 42, 98].

Finally, the quantum mechanical explanation for the harmonic emission lies in the oscillating
dipole qe χ̈(t), where the harmonic frequency dependence arises from the phase S(p̄, t, t0 ). In short,the semi-classical action is responsible for the frequency conversion, while also describing the trajectory the electron takes. Thus, any perturbation of the electron’s trajectory can give rise to aphase shift in the harmonic dipole. Absolute phase shifts are of little importance in terms of theireffect on the harmonic spectrum except when the phase shifts are asymmetric across half-cycles ofthe laser. This phase asymmetry can emulate a linear phase in the time domain, mapping to a fre-

41
quency shift in the spectral domain. Therefore, it is possible to observe harmonics that are shiftedfrom their typical odd positions, provided there is a mechanism to break the half-cycle symmetry of
the electron trajectory. Relativistic effects (Chapter 4), quickly varying pulse envelopes (dt ≈ T ),and multicolor drivers are possible symmetry-breaking mechanisms.

1.4

Macroscopic Picture of HHG
To this point, only the single-atom picture of HHG has been discussed in detail. The physics

that governs the microscopic picture sets limits on the properties of HHG that are possible, andmacroscopic effects tend to further constrain those characteristics. Some of the additional constraints can be considered detrimental for certain applications. As will be shown, macroscopic
effects tend to reduce the spectral cutoff energy, which can limit spectroscopy experiments [135].
On the other hand, the presence of many atoms contributing to the final output leads to brightemission that is highly directional (as opposed to the Larmor radiation pattern of Fig. 1.15), allowing the use of smaller optics to steer the full harmonic beam. The harmonic pulse train in the time
domain can also be restricted to one or a few half-cycles, which can be beneficial for time-resolved
applications. Here, the details of phase-matching dynamics in the time-domain are left for further
discussion in Chapter 2.
The macroscopic treatment of HHG will assume a gaseous medium of length Lmed and density
ρ(z), where ρ(z) may vary as a function of location z along the medium. For simplicity, ρ(z) will
be equal to zero for z < 0 and for z > Lmed , and the final harmonic output will be defined at thelocation z = Lmed .

1.4.1

Phase-MatchingFirst, consider the 1D case where the gas medium has a constant density ρ(z) = ρ0 , and thedriving field is a monochromatic plane-wave with frequency ωL and wavenumber kL = n(ωL ) ωL /c.
The emission from an atom at location z will result in the following field strength at the medium

42
exit:
0
EHHG (Lmed ) = EHHG
eiΦ(z) eiγ(ωq )(Lmed −z) ,(1.59)

0where EHHG
eiΦ(z) is the microscopic amplitude and phase of the harmonic field at the time andlocation of emission, and γ(ωq ) = k(ωq ) + iκ(ωq ) is the complex wavenumber at the harmonicfrequency ωq = qωL . The harmonic emission phase Φ(z) is related to the phase of the fundamental
field at the time and location of emission. Considering the emission from two atoms at locations
z1 and z2 , the combined output will be given by:
2
X



0
EHHG (Lmed ) = EHHG
eiΦ(z1 ) eiγ(ωq )(Lmed −z1 ) 1 + ei(z2 −z1 )(q∗n(ωL )ωL /c−γ(ωq ))(1.60)zi

To achieve the maximum output amplitude for arbitrary z1 and z2 , the following must be true:k(ωq ) = q ∗ n(ωL )ωL /c. Since k(ωq ) is defined as n(ωq )ωq /c, the condition for maximum harmonic
field strength at the medium exit simplifies to an equivalence of refractive indices for the fundamental and harmonic frequencies: n(ωL ) = n(ωq ). Often, this is described as a matching of thephase velocities for the two frequencies vφ (ωL ) = vφ (ωq ), or as a wavevector mismatch equation:∆k ≡ k(q ωL ) − q k(ωL ) = 0(1.61)

Equation (1.61) represents the 1D phase-matching condition for the simple monochromatic, collinear
scenario. For an arbitrary number of driving fields with frequencies ωj and 3D wavevectors kj , thegeneral phase-matching equation is:∆k ≡ kQ (ωQ ) −

X
j

ωQ =

X

mj kj (ωj ) = 0(1.62)mj ωjjHere, mj is an integer describing the number of photons from driving field j that contribute to thefinal output field with frequency ωQ propagating along the direction of kQ (ωQ ). Sum frequency
generation (SFG) corresponds to values of mj that are all positive, whereas difference frequency
generation (DFG) can occur when one or more of the mj values are negative. More discussion on
high-order DFG is presented in Section 5.3.

43
Once again, Eq. (1.62) is the condition for the brightest harmonic field strength at the exit of
the HHG medium. Its physical interpretation is the conservation of momentum and energy for thefrequency conversion process. For |∆k| 6= 0, a coherence length Lcoh can be defined that specifies
the medium length over which the harmonic emission adds constructively:
Lcoh ≡π|∆k|(1.63)The effect of the coherence length Lcoh on the final harmonic output will be discussed further in
Section 1.4.5. A diagram of the phase-matching concept is shown in Fig. 1.18.

1.4.2

Phase-Matching in Fractionally Ionized Gases

Calculating ∆k requires knowledge of the medium and its effect on k(ωL ) and k(ωq ). Furthermore, the HHG geometry can contribute to the total phase mismatch. The wavevector can be
written as a sum of each contribution:k(ω) = kvacuum (ω) + katoms (ω) + kions (ω) + kplasma (ω) + kgeom (ω) + k∇IL (ω)(1.64)

The first four terms are simple to calculate for any experimental geometry. The vacuum
wavevector has magnitude kvacuum (ω) = 2πω/c. For collinear HHG where all driving beams areparallel to the output beam ( kj (ωj ) k k(ωQ ) ∀ j ), the vacuum contribution to ∆k vanishes.
However, this is not true for noncollinear geometries.
The neutral atom term can be determined from the dispersion of the neutral portion of themedium. The wavevector magnitude katoms (ω) is related to the refractive index:
katoms (ω) = [n(ω) − 1]ωc(1.65)

If the linear index of refraction of a neutral gaseous medium at 1 atm is n01 (ω), then the neutral
contribution to the linear index of refraction of a gas at arbitrary pressure (P = P̄ × 1 atm) andneutral population fraction η0 would be n1 (P, η, ω) = 1 + P̄ η0 (n01 (ω) − 1). 10 Furthermore, there is10

For clarity, η0 = 1 represents a fully neutral gas
P without ionization; ion species’ population fractions are
represented as ηi . The total ionization fraction ηtot = Z
i=1 i ∗ ηi

44A

Δk = 0Lcoh = ∞B

Δk ≠ 0

Lcoh

Figure 1.18: Phase-matching during HHG. (A) The wavevector mismatch ∆k = 0, corresponding
to an infinite coherence length Lcoh . All atoms emit harmonics in sync with each other, leading toa bright output. (B) The wavevector mismatch ∆k 6= 0, corresponding to a finite coherence length
Lcoh . The atoms shown span one coherence length, resulting in weaker (but nonzero) emission.
Note that Lcoh ∼ λL is an extreme case of phase mismatch.

45
a nonlinear index of refraction n02 from neutral atoms that can contribute to the total index if thelaser intensity is high. In HHG, only the driving fields have sufficient intensity for the nonlinear
refractive index to be significant. Ultimately, the equation for ∆katoms in the monochromatic,collinear case becomes:∆katoms = katoms (qωL ) − qkatoms (ωL ) = −q P̄ η0

2π(∆δ 0 + n02 IL ),λL(1.66)

where ∆δ 0 = n01 (ωL ) − n01 (qωL ) is the difference in the linear refractive index at the fundamental
and harmonic frequencies, and n02 IL is the nonlinear contribution to the refractive index from thedriving laser at λL and 1 atm. For fundamental and harmonic frequencies far from resonances, ∆δ 0
is positive, as can be deduced from Fig. 1.19. Thus, ∆katoms < 0.

n(ω)

1
0

Infrared

VisibleωIR

Ultraviolet

ωUV

X-ray

ωK ,L ,M

Frequency
Figure 1.19: A qualitative illustration of the real part of the refractive index across the electromagnetic spectrum. Resonances occur at atomic and molecular absorption edges, resulting in variationsin n(ω) via the Kramers-Kronig relations. For X-ray frequencies above all atomic resonances, thereis asymptotic tendancy toward n(ω) = 1. Figure adapted from [7].

The contribution to k(ω) from ion species can be approached the same way as the atom term.
Unsurprisingly, the form for ∆k is nearly identical:∆kions = −

Z
X
i=1

q P̄ ηi

2π(∆δ i + ni2 IL ),λL(1.67)

where the index i refers to quantities that pertain to ion species of charge state i. The parameters
ni2 and ∆δ i can be difficult to determine precisely since experimental measurements would need tooccur on ultrafast time scales and would be convoluted by the presence of plasma and multiple ion

46
species. Ab initio calculations can be performed to estimate the order of magnitude of the linear
and nonlinear refractive indices, as demonstrated in [134] for Ar species at 270 nm. Similarly toneutral atoms, ∆kions < 0.
The plasma term of Eq. (1.64) can also be determined through the refractive index. The
plasma resonance frequency ωp is given by:sωp =qe2 ne,0 me(1.68)

where ne = ηtot Na P̄ is the free electron number density, and Na is the number density of atoms
in a gas at standard pressure and temperature. The plasma resonance leads to a plasma refractive
index given by:rnplasma (ω) =1− ω 2
p

ω

s
=



1−


ne
,
nc (ω)(1.69)

where nc (ω) = 0 me ω 2 /qe2 is the critical plasma density above which light at frequency ω will bereflected by the plasma. The typical pressures and ionization levels used for HHG are such thatthe plasma density ne is much smaller than the critical density nc . 11 As such, a Taylor expansion
can be performed to approximate the plasma refractive index:
1  ωp 2
nplasma (ω) ∼=1−
2 ω(1.70)The plasma term of Eq. (1.64) has a magnitude:
kplasma (ω) = (nplasma (ω) − 1)

ωp2
ω
=−
c
2cω

(1.71)For the monochromatic, collinear HHG case, the plasma term for the phase mismatch is then:∆kplasma =ωp2 (q 2 − 1)
(q 2 − 1)(q 2 − 1)= ne reλL = ηtot Na re P̄
λL
2cωL
q
q
q

(1.72)

Here, the classical electron radius re was substituted. For large harmonic orders q & 10, the term(q 2 −1) ∼= q, allowing ∆kplasma to have the same q P̄ dependence as ∆katoms and ∆kions . For nonzeroqionization levels, ∆kplasma > 0.
11

However, high pressures can be necessary for phase-matching HHG when using longer wavelength drivers.
Therefore, plasma mirroring effects could introduce a limitation on HHG driven by wavelengths λL > 10 µm.

47Before considering the two remaining terms relevant to phase-matching, some conclusions
can be drawn concerning the contribution from the fractionally-ionized medium. Under the mostcommon HHG conditions where a low level of tunnel ionization occurs (η1 << 1, η0 = 1 − η1 ),∆kions can be neglected. The medium’s contribution to the total phase-mismatch at high harmonicorder is then:


2π
0
0
∼
∼
∆kmedium = ∆katoms + ∆kplasma = q P̄ −(1 − η1 ) (∆δ + n2 IL ) + η1 Na re λL
λL(1.73)

Satisfying the phase-matching condition ∆kmedium = 0 for arbitrary driving wavelength and harmonic order gives a solution for the critical ionization level:λ2L Na reηc =+12π(∆δ 0 + n02 IL )−1(1.74)

Often, the nonlinear term n02 IL is small compared to ∆δ 0 , but ηc still depends on λL (and somewhat
on q through ∆δ 0 ). The critical ionization level ηc represents a zero crossing of ∆kmedium (η1 ). For
η1 < ηc −→ ∆kmedium < 0, and for η1 > ηc −→ ∆kmedium > 0. In the next section, the geometricalcontributions to ∆k will be shown to be positive. As a result, phase-matching is only possible forη1 < ηc (in the monochromatic, collinear HHG scenario).
Just considering the medium contribution to phase-matching, some conclusions can be drawn
regarding the driving laser intensities that are appropriate for phase-matched HHG. Assuming an
8-cycle driving laser pulse, ADK ionization levels can be calculated so that the critical ionizationlevel ηc is achieved at the peak of the pulse. This sets an optimal peak intensity level that will
maximize the efficiency of the HHG process, since exceeding this peak intensity level will cause
phase-matching to occur on the leading edge of the pulse where there are lower intensities, whilesmaller peak intensities will result in phase-matching on the trailing edge of the pulse (or not atall). Performing the ADK calculation results in Fig. 1.20, which shows the optimal intensities
to drive HHG in noble gases for driving wavelengths up to 20 µm. Also plotted are the optimal
intensities for a mode-averaged HHG signal.

48

Figure 1.20: Predicted laser intensity required to reach the critical ionization level at the peak ofan 8-cycle laser pulse on axis (solid line), and when modal averaging is taken into account (dashedline). Adapted from [106].

49
1.4.3

Phase-Matching Geometries

In this section, the geometrical contribution to ∆k will be discussed. For monochromatic,collinear HHG, there are two primary geometries: the Gaussian free-focus geometry and the waveguide geometry. Certainly, there are other geometries possible for multi-color, noncollinear schemes,
in which the full vector form of ∆k must be considered. The two-beam, noncollinear, bichromatic scenario will be discussed in Section 5.3. Here, only the Gaussian free-focus and waveguide
geometries will be considered.
Gaussian Free-Focus Geometry
Realistic laser spatial modes can be flat-topped, Bessel- or sinc-like as a result of amplifier
gain profiles, propagation in fiber waveguides, and clipping on apertures. However, these canbe approximated as Gaussian for simplicity, since it is straightforward to arrive at a theoreticaldescription of how Gaussian modes propagate. Gaussian modes are convenient to model because
they maintain their Gaussian shape throughout propagation. The derivation for a Gaussian mode
amplitude and phase in the paraxial approximation involves solving the source-free scalar wave
equation, and is analogous to the derivation of the free electron wavepacket |Ψ(t)i presented in
Section 1.3.2. A Gaussian mode takes the following scalar form in the transverse direction (r), asa function of propagation distance z from a focal point:




w0
−r2
ikr2
E(r, z) = E0
exp
exp
exp(iΓ(z)),
w(z)
w(z)2
2R(z)

(1.75)

where:
zR =

πw02
λL
s


z 2
w(z) = w0 1 +zR z 2 R
R(z) = z 1 +z z
Γ(z) = −tan−1zR(1.76)

Here, w(z) is the radius at which the intensity drops to 1/e2 of the central intensity (with value
w0 at the focal position), zR is called the Rayleigh range, R(z) represents the wavefront radius of

50
curvature, and Γ(z) is called the Gouy phase. Note that here, k = 2π
λ is the vacuum wavenumber.
Figure 1.21 depicts the qualities of a Gaussian free-focus.
To evaluate the wavevector kGaussian of the mode, the gradient of its phase is taken:
kGaussian = ∇Φ(r)




kr2
∂
ω  w02 + r2 → kGaussian =+ Γ(z) = −∂z 2R(z)c 2z 2 1 + z 2
R(1.77)

2zR

The wavefront curvature and Gouy phase ultimately reduce the Gaussian mode wavenumber compared to the plane-wave value in vacuum. Looking on-axis (r = 0):
kGaussian (r = 0) → kGouy =

−1


2

zR 1 + zz2

(1.78)

R

−1
∼
=
zR

(for z << zR )If HHG was possible far from the focus (z >> zR ), the mode wavenumber would become kGouy ∼= 0.
For the purposes of standard HHG in a free-focus, only regions near the focus where there is highintensity will be relevant. With the assumption that the harmonics are generated with a similar
confocal parameter, the geometric contribution to ∆k from the Gouy phase is:∆kGouy =(q − 1)zR(1.79)

Clearly, ∆kGouy > 0.
Off-axis, the effect of the radius of curvature is nonzero and varies quadratically with r:∆kRoC =

(q − 1) r2
zR w02(1.80)

Its contribution can result in a radially-dependent phase-mismatch where certain radii can beroughly phase-matched for a length ∼ zR while other radii contain a large amount of phasemismatch. Again, the wavevector mismatch is postive: ∆kRoC > 0. The total geometric wavevector
mismatch for a Gaussian mode at arbitrary radius is the sum ∆kGouy + ∆kRoC :(q − 1)∆kGaussian =
zR



r2
1+ 2
w0


(1.81)

51

zR
w(z)

Γ(z)

w0

R(z)

Intensity

Intensity

2

1
0.8

1

0.6
y/w 0

0.4
-1

-2
-2

0.2

-1

0
x/w

1

2

-3

-2

-1

0
z / zR1

2

3

0

Figure 1.21: An illustration of a Gaussian free-focus and its characteristic parameters.

52
Recall that there is one final term that contributes to the wavevector: k∇IL (ω). This termis related to the linear dependence of the intensity on the electron’s quantum phase, as discussed
in Section 1.3.1. Thus, any driving field that has a nonzero intensity gradient along the z-axis will
result in an additional phase-mismatch. The phase only applies to the harmonic wavevector, sothat the form for the wavevector mismatch is:∆k∇IL =

qe2 τf∇|E(r, z)|2
4~me ωL2(1.82)The gradient (along z) of |E(r, z)|2 from Eq. (1.75) is:∂(w(z)2 − 2r2 )λL|E(r, z)|2 = −2|E(r, z)|2∂zπw(z)4zzR(1.83)

It is evident that ∆k∇IL for a Gaussian free-focus varies with respect to z and can have a negative
magnitude. This fact can be exploited when also considering the geometric contribution ∆kGaussian .
Looking on-axis, it is possible for the sum ∆kGouy +∆k∇IL = 0 at certain locations along the z-axis.
Additionally, ∆k∇IL depends on τf , suggesting that different trajectories can be phase-matched at
different z-positions. Figure 1.22 illustrates this trajectory-dependent, z-dependent phase-matching
concept. The importance of this finding is that for a Gaussian beam focused through a short medium(Lmed < zR , i.e. a gas jet or gas cell), the trajectories that will be phase-matched will depend on thez-location of the gas jet with respect to the focal position. Long trajectories can be phase-matchedwhen the gas jet is close to the focus, whereas short trajectories would be phase-matched for a gas
jet placed just after the focus. This analysis assumes the phase-matching occurs along the axis,
but phase-matching can occur off-axis. L’Huillier and Balcou et al. [69, 8] discovered that theharmonic efficiency and beam divergence can be controlled by adjusting the focal spot with respect
to the gas jet. Long trajectories tend to phase-match in an off-axis ring, whereas short trajectories
phase-match on-axis.
For a Gaussian free-focus, the total phase mismatch equation on-axis is the sum of all the

53

10 000Trajectory-dependent phase mismatch
3.17U p8000

Driving laser parameters
IL = 4×1014 W/cm2
λL = 800 nm

Δ

ΔkGouy + Δk IL

6000
4000
2000

w0 = 50 μm
2U p (short)

0
-2000
-4000
-2

2U p (long)

-1

0
z/zR

1

2

Figure 1.22: A plot of the on-axis geometrical phase-mismatch ∆kGouy + ∆k∇IL for a Gaussian
free-focus as a function of axial position. Typical HHG driving conditions are used: IL = 4 × 1014
W/cm2 , λL = 800 nm, and w0 = 50 µm. Under these conditions, zR = 9.8 mm, and 2Up corresponds
to q ∼= 31 while 3.17Up corresponds to q ∼= 49. Ionization phases for the 2Up cases are φ0 = π/32
(long) and φ0 = 3π/16 (short).

54
contributions discussed here and in Section 1.4.2:∆k = ∆katoms + ∆kions + ∆kplasma + ∆kGouy + ∆k∇IL| {z } | {z } | {z } | {z } | {z }
<0

<0

>0

>0

+/−

(1.84)#
Z X
qe2 τf(q − 1)2π= q P̄ −ηi (∆δ i + ni2 IL ) + ηtot Na re λL ++∇|E(r, z)|2λL
zR
4~me ωL2"

i=0

The off-axis equation requires the full vector form of ∆k, which includes a nonzero contribution
from ∆kvacuum and ∆kRoC .
Waveguide Geometry
HHG in a Gaussian free-focus is limited to a region where the driving laser maintains highintensity. Additionally, the z-dependence of ∆k∇IL imposes a restriction on the length of medium
where phase-matching is possible. These limitations ultimately reduce the harmonic flux possible
in a free-focus geometry. Thus, an alternative geometry that can extend the interaction length for
efficient harmonic generation is desirable. The goal would employ radial confinement of the laser
energy over long distances, which is possible with waveguiding techniques. Plasma waveguides havebeen used [85, 16, 132], as well as gas-filled hollow capillaries [110, 106]. Here, the latter will be
explored in detail. Figure 1.23 depicts the hollow waveguide geometry under consideration.
yLε , µ0θrx2a

ε0 , µ0z

Figure 1.23: Illustrations of the hollow waveguide geometry.

Marcatili solved for the full electromagnetic field components for the hollow waveguide modesin [79]. Appendix A contains detailed information on the waveguide modes and a comparison toGaussian beams. Here, the mode that most closely resembles a linearly-polarized Gaussian mode(the EH11 mode) will be investigated (though the TE and TM ”doughnut” modes can be used to55
drive HHG, as will be discussed in Section 5.1). For the purposes of phase-matching, the complex
wavenumber γpq is of primary importance:1  upq 2
i (n2 + 1)√
γpq = k 1 −1−
2 ka
ka n2 − 1(1.85)

Here, Jp is a pth order Bessel function of the first kind, upq is the q th root of the equation Jp−1 (upq ) =p0, n = /0 is the complex refractive index of the waveguide material, k = 2π/λ, and a is theinside radius of the hollow waveguide.
For the EHpq mode, kwaveguide (ω) = Re[γpq (ω)]. Neglecting the small real contribution
 
resulting from Im n2 , the wavevector mismatch from the waveguide mode can be easily obtained:∆kwaveguide = kwaveguide (qωL ) − qkwaveguide (ωL ) ∼=qu2pq λL∼=
4πa2u2pq λL (q 2 − 1)4πqa2

(1.86)(for q >> 1)

Because the waveguide confines the light, each mode will propagate independently and without any change in shape or phase (barring any nonlinear effects that may exist during strong fieldprocesses like HHG). In the linear regime, there are no Gouy phase or radius of curvature effects.
For low loss modes (Im[γpq ] ∗ Lwaveguide << 1) and low levels of ionization, there would be a negligible longitudinal gradient in the intensity, so ∆k∇IL ∼= 0 on-axis. ∆kwaveguide would be the only
geometrical component. Generally, however, the imaginary component of γpq leads to a nonzero
gradient of |Epq (r, z)|2 along the propagation direction:∂|Epq (r, z)|2 = −2Im[γpq ] |Epq (r, z)|2∂z

(1.87)

Since |Epq (r, z)|2 decays exponentially as a function of z, so does ∆k∇IL for this geometry. Ultimately, the total wavevector mismatch on-axis for the hollow waveguide geometry is:∆k = ∆katoms + ∆kions + ∆kplasma + ∆kwaveguide + ∆k∇IL| {z } | {z } | {z } |{z} | {z }<0"= q P̄ −

Z 
X
i=0

<0

>0

#

>0

∼
=0

qu2pq λLqe2 τf2π
ηi (∆δ i + ni2 IL ) + ηtot Na re λL ++∇|Epq (r, z)|2λL
4πa2
4~me ωL2


(1.88)

56
Phase-matching in the waveguide geometry is possible by balancing the phase-mismatch
contribution from the medium with that from the waveguide: ∆kmedium + ∆kwaveguide = 0. Recall
that the medium contribution is either positive or negative depending on the ionization level.
Only for η1 < ηc can ∆kmedium = ∆katoms + ∆kions + ∆kplasma < 0. Furthermore, the medium
contribution depends linearly on the pressure. Thus, as long as the ionization level is below thecritical level, there is a route to achieving phase-matching by adjusting P̄ .
Perfect phase-matching (∆k = 0) during HHG creates restrictions on the driving conditionsof the process. Because the ionization level has an upper bound ηc , the driving laser cannot produce
an excess level of ionization within the medium due to its high intensity. For driving pulses withlong durations, the ionization fraction builds to high levels before the peak of the pulse arrives. In
this case, the peak intensity must be reduced such that phase-matching may occur at the pulse’s
peak (thus limiting the single atom cutoff that scales as IL λ2L ). Alternatively, the pulse duration
may be reduced to limit the accumulation of plasma before the center of the pulse, thus allowing
higher peak intensities to be achieved at the phase-matched ionization level. The pulse duration
can only be reduced so far (a single cycle), imposing a maximum intensity achievable for perfectphase-matched conditions. Single-cycle drivers can also have group velocity walkoff issues that
limit the medium length over which HHG is generated efficiently. Ultimately, the phase-matching
constraint effectively reduces the scaling law for the harmonic cutoff energy to the phase-matching
cutoff [160]:
EP M C ∝ λ(1.6−1.7)

1.4.4(1.89)

Quasi-Phase-Matching

Even if the driving laser pulse has enough intensity to result in very high single atom cutoff
energies, perfectly phase-matched emission would be impossible at any energies that exceed EP M C(using a standard single-beam geometry). It is possible to bypass the phase-matching cutoff using
a technique called quasi-phase-matching (QPM). Through QPM, the ionization limitation ηc canbe exceeded to result in a nonzero phase-mismatch (∆k 6= 0), resulting in a finite coherence length

57
Lcoh . Typically, this scenario would result in the coherent buildup of harmonic light during thefirst portion of the medium (up to Lcoh ), but in the next length of medium (Lcoh < z < 2Lcoh ) any
additional generated harmonic light would deconstructively interfere with the existing harmonic
emission. To reduce or avoid the cancellation effect in this second region of the medium, the phasemismatch ∆k can be altered or the harmonic emission can be shut off in this latter region. By
repeating this process in a periodic way, the total harmonic emission at photon energies that areconventionally unachievable can grow to moderately bright levels.
A simple model of the generation of harmonic light can be modeled as follows:
Z LmedEq ∝Aq e−i∆kz dz,(1.90)

0where Aq represents the transverse spatially- and time-integrated dipole emission strength per unit
length of the HHG process of order q. In QPM, an additional factor is included in the integral:d(z) = def f∞X

Gm eiKm z ,(1.91)m=−∞where Km = 2πm/Λ is the effective wavevector of the QPM scheme, m is the order of the QPM
process, and def f and Gm are amplitudes that allow d(z) to be a general periodic function. DefiningΛ = 2Lcoh and adding d(z) to Eq. (1.90) reveals that the brightest harmonic emission occurs for
K1 = ∆k and Gm = δm,1 . However, enhanced emission can occur for higher order QPM (oddvalues of m). Ultimately, the phase-mismatch equation is modified in QPM:
∆kQP M =π= ∆kmedium + ∆kgeometry − Km
Lcoh

(1.92)

Larger values of Km (smaller periodicity Λ) can compensate for a larger conventional ∆k, andtherefore higher levels of ionization within the medium.
There are various approaches to achieve QPM. Periodic modulation of the medium can beachieved through waveguides with longitudinally-varying radii and through pressure modulations
created by multiple gas sources (successive gas jets or alternating gas supply/pump holes in a
waveguide). Additional light waves can also be implemented to achieve QPM [24, 176]. These
schemes are shown in Fig. 1.24.

58

B

A

C

P1

P1

P2

D

P1

P2

P2

intense driving fs laser
weak counterpropagating CW laser

+

-

+

-

+

-

+

-

-

+

Emitted phase(only driving laser)

+

-

+

-

+

-

+

z
Emitted phase(driver + counterprop.)

Phase shift induced bycounterpropagating light+=z+ - + - + - + -

-

L coh

coherent zones

HHG signal+

HHG
signal+ - + - + - +Figure 1.24: QPM schemes for HHG. (A) Waveguides with modulated inner diameters [93, 136].
(B) Pressure modulation in waveguides. (C) Pressure modulation through successive gas jets [147].
(D) Grating-assisted phase-matching through counterpropagating CW laser light (adapted from[24]).

59
Table 1.4: A table of absorption lengths for light with various photon energies propagating througha number of gases relevant to HHG. The gas pressure is 1 atm in all cases.

E
30 eV
100 eV
300 eV
500 eV
1 keV

1.4.5

He
7.2 µm93 µm
2.3 mm
11 mm
1.0 m

Ne
4.5 µm
9.8 µm108 µm408 µm156 µmLabs,0 (E)Ar
1.5 µm
30 µm
13 µm
34 µm
181 µmKr
2.4 µm
17 µm
10 µm
21 µm
99 µm

Xe
5.2 µm
1.5 µm
24 µm
39 µm
19 µm

Air
9.3 µm201 µm37 µm
55 µm
231 µmReabsorption

In the macroscopic picture of HHG, the generation and coherent superposition of harmoniclight is only part of the full story. Harmonic light is also reabsorbed by the same medium thatgenerates it. To accurately predict the final harmonic flux from a medium, one must consider
generation, phase-matching, and reabsorbtion simultaneously. Constant et al. provide a simple
1-D approach to model the harmonic flux as a function of longitudinal position [25]. Their analysis
assumes a constant phase-mismatch, but can be extended to longitudinally-varying mismatch (i.e.
pressure profile) as is investigated here.
The transmission of harmonic light through an absorbing medium can be determined using
data from the Center for X-ray Optics (CXRO) database [63]. For a gas at a given pressure P
and light at a particular photon energy E, one can define an absorption length Labs (E, P ) for the
medium through which the transmission is 1/e. Table 1.4 lists absorption lengths for several gases
and photon energies at a pressure of 1 atm, which we define as Labs,0 (E) = Labs (E, 1atm). To
calculate the absorption length of a medium at a different pressure: Labs (E, P ) = 1 atm
Labs,0 (E).
P
To calculate the transmission of a gaseous medium with length Lmed at pressure P :T (E, P, Lmed ) = e−Lmed /Labs (E,P )(1.93)The simplest 1-D approach to calculate the harmonic flux level at the exit of a uniform gaseous
medium is straightforward; the electric field output is simply the integrated sum of the emission for
all points along the medium, where each differential position has a local emission amplitude ρAq

60
and propagation factor that simultaneously accounts for phase-mismatch and reabsorption:
 
exp −



1
+ i∆k (z − Labs )2Labs (E, P )(1.94)

Ultimately, the harmonic flux is:ωq4c0 ~Z Lmed 
ρAq exp −

0



2
1
+ i∆k (z − Labs ) dz
2Labs (E, P )(1.95)For a medium with constant pressure and driving laser, the integral evaluates to:
2

ρ A2q








4L2abs
Lmed
πLmed
Lmed
1 + exp −− 2 cos
exp −1 + 4π 2 (Labs /Lcoh )2
Labs
Lcoh
2Labs

(1.96)

Harmonic buildup
1.

L coh >> L abs

No absorption
0.8

0.8

L coh = 10 L abs

HHG
0.6
Flux
(a.u.)
0.4

L coh = 5 L abs

0.2
0.
0

1.

0.6
0.4
0.2

L coh = L abs
5

10

0.
15

z / L abs
Figure 1.25: Plots of the accumulation of HHG flux as a function longitudinal position within the
HHG medium. The ratio between the absorption length Labs and coherence length Lcoh are varied.
For perfect phase-matching, the HHG flux saturates after ≈ 10Labs .
Equation (1.96) is useful in understanding the general behavior of the harmonic buildup fora variety relative values for Labs , Lcoh , and Lmed . Figure 1.25 shows various regimes of harmonic
generation along the longitudinal direction. However insightful, Eq. (1.96) is valid for constantmedium and driving conditions. In reality, the medium will have a varying pressure profile, and61
the driving laser can change in intensity. Maintaining the 1-D dimensionality of the problem but
adding the longitudinal variations, Eq. (1.95) can be modified:ωq
4c0 ~Z Lmed0 2z − Lmed
ρ(z)Aq (z) exp −+ iφq (z)dz
2Labs (E, z)(1.97)

Here, Labs (E, z) represents the average absorption length for the remainder of the medium from z
to Lmed , and φq (z) is the exit phase of the harmonic emission from position z. Numerical evaluationof Eq. (1.97) is straightforward to perform in a step-wise manner. The details of the calculation
involve keeping track of the total harmonic flux in each differential section dz, adding to it theemission from that section, developing its phase according to the local index, and reducing its
magnitude as a result of reabsorption from that section.

1.4.6

Characteristics of Macroscopic HHG Emission

The importance of phase-matching during HHG cannot be underestimated. Applicationsof the HHG light source typically require a high level of photon flux for sufficient signal-to-noise
ratios in the experiment. Phase-matching must be satisfied, so the limitations associated with thephase-matching constraint are unavoidable in standard, single-beam HHG geometries. Equation(1.89) describes one macroscopic limitation. The cutoff photon energy cannot scale as favorably asin the single-atom picture ∼ IL λ2L . Instead, the phase-matching constraint reduces the scaling to∼ λ(1.6−1.7) [160]. This scaling has been experimentally verified, as demonstrated in Fig. 1.26. It
is possible to overcome the macroscopic cutoff energy limitations through multi-beam geometries,
which will be further investigated in Section 5.3.
Another macroscopic effect is the ability to preferentially select particular trajectories in a
Gaussian free-focus geometry. Depending on the location of the gas jet with respect to the focus,
long or short trajectories can be preferentially phase-matched, as described in Section 1.4.3.
The effect of reabsorption within the HHG medium limits the harmonic flux that can beachieved. Even in perfect phase-matching conditions, the harmonic signal saturates after a distanceof ∼ 10Labs . Furthermore, realistic experimental conditions will have pressure profiles that vary

62

VUV-EUV-Soft & Hard X-rays

50
10
5
1
0.5
0.1
0.05

■■
0.01
0.05

He
Ne
Ar
Kr
Xe

EPMC ~ λL (1.6 - 1.7)
Ti:Sapp

Hard X-rays
Soft X-rays
EUV
VUV

Upconverted Photon Energy (keV)

Phase Matching Cutoffs

○
○

EPMC
Full
Phase Matching

IP

■■■

0.1

Region0.5
15Laser Wavelength λL (μm)

1050Figure 1.26: Experimentally-verified scaling of the macroscopic HHG cutoff photon energy EP M C
as a function of driving laser wavelength. Open circles represent the predicted harmonic cutoff
without reabsorption limitations that are present during experiment. Adapted from [160].

63
longitudinally. Even if a constant pressure region can be created for perfect phase-matching, there
will be a subsequent transition region before vacuum is reached. The transition region will havereabsorption without phase-matched generation, again limiting the practical flux achievable.
Finally, the phase-matching constraint has two main contributions that must cancel: ∆kgeometryand ∆kmedium . We have shown that the medium contribution can be positive or negative dependingon the ionization content. As the driving wavelength is increased, the index of refraction n(λL )for the driving field generally approaches unity (far from resonances). This results in a decreased
value of ∆δ 0 . Since the atom contribution is proportional to ∆δ 0 as well as λ−1
L , while the plasma
contribution is proportional to λL , the critical ionization level decreases for increasing λL . Anotherconsequence is that for a given geometry and ionization level η < ηc , there is a route to achievephase-matching through the gas pressure P̄ . However, the geometric term tends to increase inmagnitude with λL , which ultimately results in the phase-matching pressure P̄P M increasing with
λL . Figure 1.27 indicates this pressure dependence on wavelength.

Figure 1.27: Pressure and harmonic cutoff energy scalings with driving laser wavelength. The
predicted optimal pressure shows quadratic growth with λL . Adapted from [106].

Chapter 2

Phase-Matching Gating during Mid-IR Driven HHG

1Once the conceptual and mathematical framework of the microscopic and macroscopic

physics of HHG had been developed, the scaling laws revealed a clear route to pushing the technology to higher photon energies. More dramatic frequency conversion could occur according thescaling of the cutoff harmonic photon energy with driving laser wavelength: EP M C ∝ λ(1.6−1.7) .
Experiments using long wavelength drivers have been designed in order to investigate the spectral
capabilities and limitations of the HHG process.
Due to the need for ultrafast pulses for phase-matched HHG, and due to the lack of conventional laser oscillators that are ultrafast and operate beyond the near-infrared region, one of themost practical approaches to achieve appropriate mid-infrared drivers is through optical parametricamplification (OPA). In OPA, an ultrafast pulse with near infrared or visible wavelength is used topump a second-order nonlinear frequency conversion process. The energy of a single pump photon
gets split between two photons with tunable wavelengths. Thus, an input laser pulse with shortwavelength can be converted into two output laser pulses with longer wavelengths (signal and idler
beams). In this chapter, an experiment will be discussed in which two OPA output wavelengths
were utilized. Ultrafast laser light with wavelengths between 1.2 µm and 2.4 µm were achieved
through the use of an 800 nm-driven OPA (Light Conversion HE:TOPAS). Signal light at 1.3 µm
and idler light at 2.1 µm were chosen for this study.
1

The experiment and results described in this chapter are published in [18].

65

2.1

Grating Spectrometer Limitations
As described in Section 1.2.4, HHG emission in the temporal domain has a number of char-acteristics. First, the half-cycle symmetry of the HHG process results in bursts of harmonic pulses
every half-cycle of the driving laser. Thus, a pulse-train of HHG is emitted in the microscopic
picture. Consequently, the spectral domain contains signatures of the interferences between theindividual pulses in the time domain, indicated by the odd harmonic spectral peaks. In many
low-order spectroscopy experiments, observation of these odd harmonics is common. However,the spectrometer in the experiment must have sufficient spectral resolution to identify individual harmonics, as opposed to measuring a broad continuum. For visible and near-infrared driven
HHG, this resolution requirement is not an issue; the moderate-energy driving photons cause theseparation between odd harmonics to be relatively large, while the phase-matching cutoff energy
makes the total HHG spectral range to be relatively small. Thus, CCDs with 1024 pixels along
one dimension can easily span the spectral range while resolving individual orders. Assuming anearly-linear dispersion of the spectral orders, the requirement for a CCD to adequately resolve
individual harmonics while spanning the entire HHG bandwidth is given by:
EP M C
Ns
≤
,
~ωL
2

(2.1)

where Ns is the number of sample points in the spectral domain (the number of transverse pixels
on a CCD).2 Additionally, a grating spectrometer’s ultimate resolving power depends strongly on
the number of illuminated grating grooves, the apertures, and any aberrations added to the beamby optical elements.
Moving to longer wavelength drivers in the mid-IR causes the HHG spectral range to increase
while the harmonic orders become less separated. To then satisfy Eq. (2.1), the number of spectral
sampling points Ns must increase. Using the scaling law described in Eq. (1.89), Ns scales as(2.6−2.7)

λL

2. In practice, satisfying this requirement is difficult. As demonstrated in [161] where 3.9

It is possible to alleviate this constraint by performing a panoramic spectrum measurement, where the spectral
area of interest is scanned. This measurement, however, may contain complications if there is a need to use multiple
gratings with varying spectral efficiencies in different spectral ranges.

66µm drivers generated soft X-ray light up to 1.6 keV, the very broad measured spectrum could
not resolve individual harmonic orders. Therefore, it is unclear whether the harmonics in thatexperiment truly possessed a broadband super-continuum spectrum (and thus a single, isolated
burst of HHG), or whether individual harmonics were indeed present (indicating a pulse-train ofharmonics in the time domain). In response to this uncertainty, a new experiment was designedthat would investigate the temporal and spectral domains simultaneously. This would be possible
through measurement of the electric field autocorrelation of the HHG signal for a variety of driving
laser conditions.
One powerful technique for characterizing the ultrafast temporal structure of a laser pulse isthrough Frequency-Resolved Optical Gating (FROG). FROG describes an intensity autocorrelation
measurement that relies on the nonlinear response of an optical medium to measure both theamplitude and phase of the laser pulse, both of which are required to obtain a full characterizationof the temporal structure of the pulse [164]. This technique requires a nonlinear medium with hightransmission at the laser wavelength. Unfortunately, such media do not exist for EUV and X-ray
light, making standard FROG measurements for HHG impractical. Alternative approaches exist
that can indirectly measure the amplitude and phase of HHG light, such as attosecond streaking[144]. However, for the purposes of determining whether a train of HHG pulses exists (and acorresponding spectral supercontinuum) the full amplitude and phase are not absolutely necessary.
Instead, a high-resolution measurement of only the spectral amplitude can be adequate when pairedwith knowledge of the coherent nature of the HHG driver and its emission. The first componentof this formula can be obtained through an electric field autocorrelation measurement of the HHG
emission.

2.2

Electric Field Autocorrelation
In an electric field autocorrelation, a laser pulse is interfered with a delayable duplicate

of itself. When the two pulses are not overlapped in time, the incoherent sum of their energies
is measured on the detector. When overlapped, the interference between the electric fields will

67
increase or decrease the measured signal. Mathematically, the measured intensity is:
Z tI /2
IM (τ ) =|E(t) + E(t − τ )|2 dt(2.2)−tI /2

Here, tI is the integration time of the detector, and τ is the temporal delay between the two laser
pulses (tI >> τ ). Expanding Eq. (2.2) yields:
Z tI /2
IM (τ ) =|E(t)|2 + |E(t − τ )|2 dt−tI /2

Z tI /2+

E(t)E ∗ (t − τ )dt +−tI /2

Z tI /2

E ∗ (t)E(t − τ )dt(2.3)−tI /2= 2I0 + A(τ ) + A∗ (τ )Thus, the measured signal on the detector is the sum of the pulses’ average intensities (I0 ) plus thefield autocorrelation function A(τ ) and its complex conjugate. According to the Wiener-Khinchintheorem, the Fourier transform of A(τ ) is the power spectrum of E(t) (though the spectral phasecannot be extracted through A(τ )). Measurement of A(τ ) is the basis for Fourier transform spectroscopy.
The electric field autocorrelation of the HHG signal provides a solution to the challenge of aspectral measurement with simultaneously high resolution and range. These characteristics of themeasurement do not rely on a physical number of CCD pixels, but instead rely on the number andspacing of delays τ . For a measurement with Nτ delay positions, each spaced δτ apart (giving atemporal range ∆τ = Nτ δτ ), the spectral range ∆E and spacing δE that can be achieved are:
2π~
δτ
2π~
2π~
δE =
=
∆τ
Nτ δτ

∆E =

2.3

(2.4)

HHG Pulse Train and Coherence
For a particular half-cycle of the driving laser pulse, electrons will tunnel ionize, propagate,

and recombine during the following half-cycle. In the quantum picture of HHG, emission resultsfrom two parts of the electron wavefunction interfering with themselves. Because both componentsof the electron wavefunction stem from the same initial state, there is a strong coherence between

68
the two portions and the quantum phase upon recombination should develop in a consistent way.
Aside from possible scattering from adjacent atoms, there is no reason to believe decoherence
should occur during the three step model. Additionally, the electron dynamics during the threestep model are inherently coupled to the driving laser dynamics, so it follows that the HHG emission
will be coupled to and coherent with the driving laser. Finally, for a monochromatic driver, themicroscopic picture of HHG should be consistent between neighboring half-cycles (aside from asign-flip). As such, there is no reason to believe that a burst of harmonics originating from a given
half-cycle should have a random phase offset compared to a neighboring harmonic burst. Therecan be phase shifts associated with a temporally-varying driving envelope or with chirped driving
pulses, but these phase shifts would not be random. Therefore, there would be some amount ofcoherence between the different pulses within the train of HHG emission. In fact, the characteristic
HHG spectrum that contains harmonics separated by 2ωL demonstrates that coherence must be
maintained across different pulses in the HHG train (since random phase variations would wash
out the modulations observed in the spectral domain).
This conclusion pertains to the measurement of the electric field autocorrelation since delaying
the HHG pulse train with respect to itself will inevitably result in pulses overlapping and interferingwith different pulses within the train. Consistent interference signatures would not be measurable if
there were a mechanism for random phase decoherence between neighboring pulses. As an example,
consider a train of N + 1 pulses:

Etrain (t) =

N/2
X

 

En exp −t − n λ2cL

2 

dt2

n=−N/2 i(ωq t+φn )
e

(2.5)

Here, each pulse has its own field amplitude En and absolute phase φn . Inserting this into Eq.
(2.3):
IM (τ ) = N/2X

n=−N/2

I0,n + An (τ ) + A∗n (τ ) + 

X∗
Xnm (τ ) + Xnm
(τ )

(2.6)

n6=m

where An (τ ) is the individual field autocorrelation function for pulse n (neglecting all other pulses),

69
and:
Z tI /2
Xnm (τ ) =−tI /2 
En Em exp −t − n λ2cL

2


2 
+ t − τ − m λ2cL i(ω τ +φn −φm )
dt
e q
dt2

(2.7)

The interaction between different pulses within the train (inter-pulse cross-correlations) is contained
within Xnm , which will result in a nonzero contribution for delays near τ = l λ2cL , where l is an integer
spanning from −N to +N , excluding 0. Considering the contributions from An (τ ) and Xnm (τ ),PM (τ ) will result in a total of 2N + 1 interference signatures (for N + 1 pulses in the original train).
From a temporal perspective, the HHG field autocorrelation can provide a direct measurementPof the number of pulses within the HHG pulse train. By removing the DC contribution ( n P0,n )and performing a Fourier transform, the HHG power spectrum is acquired. Thus, measuring theelectric field autocorrelation is a powerful approach to probe temporal and spectral aspects of the
HHG source with high resolution in both domains, despite the lack of temporal phase information.
Although the field autocorrelation can measure spectral bandwidths (and thus transform-limited
pulse durations), it cannot be used to measure the actual HHG pulse duration, which may bebroadened due to chirp.

2.4

Experimental Apparatus
To drive the experiment, a 1-kHz Ti:sapphire laser is used to supply near-infrared 0.8 µm light.

It is also used to pump a three-stage optical parametric amplifier (Light Conversion HE:TOPAS) to
generate 1.3 and 2.0 µm light. All three wavelengths are then used to drive HHG in a 2-mm-long,
Ar-filled gas cell in a Gaussian focus geometry (Fig. 2.1). Second-harmonic FROG was used tomeasure the pulse durations for all three driving wavelengths, and these were adjusted to ensure
consistent ∼ 10 cycle pulses (24 fs at 0.8 µm - 9.5 cycles, 35 fs at 1.3 µm - 8 cycles, and 90 fs
at 2.0 µm - 13.5 cycles). The resulting harmonic cutoff energies were used to estimate the laser
peak intensities in the focus, using the single-atom cutoff scaling from Eq. (1.22). Optimal phasematching was obtained via longitudinal translation of the focusing lens, which adjusts the focal
position with respect to the medium (Section 1.4.3).

70

Figure 2.1: Experimental apparatus used to measure the HHG field autocorrelation. Phasematched harmonics are generated in an Ar-filled cell, transmit through thin metal filters (eliminating the residual fundamental beam), refocused by a Kirkpatrick-Baez mirror pair, spatially split
and temporally delayed by a beam separator, and finally measured on an X-ray CCD. Adaptedfrom [18].

71
To perform the electric field autocorrelation, the HHG output was sent through a glancing
incidence interferometer consisting of refocusing optics (a Kirkpatrick-Baez cylindrical mirror pair)
and an in-line partially-split-mirror spatial beam separator. The separator is able to delay one
half of the beam with respect to the other half with a temporal resolution of ∼ 1.5 as (Fig. 2.1).
After some propagation, the two halves of the HHG mode overlap and interfere at an X-ray CCD(Andor), where the field autocorrelation of the HHG pulse is acquired. Spatial fringes arise fromthe crossing of the two replicas of the HHG beam at the CCD. Because the beam is focusing as it
reflects from the beam separator, each half of the mode will have opposing transverse wavevectors~k1,x = −~k2,x . At the CCD, the measured intensity is:
ICCD (x, y) = E1 (x, y)ei(k1,z z+k1,x x−ωq t) + E2 (x, y)ei(k2,z z+k2,x x−ωq (t−τ ))

2

(2.8)
2

2

= |E1 (x, y)| + |E2 (x, y)|+ E1 (x, y)E2∗ (x, y)ei(2k1,x x−ωq τ ) + c.c.

The third term and its complex conjugate result in a spatial modulation factor cos(2k1,x x − ωq τ ).
As τ is varied, the absolute phase within the cosine will change, shifting the observed fringes along
the x-direction.
To extract the autocorrelation, the amplitude at a single pixel on the CCD could be monitored
as a function of delay, yielding a signal of the form described in Eq. (2.6). However, the selection ofa single pixel is fairly arbitrary and does not make use of the information gathered by other pixels.
An effective alternative would use all of the modulation information, which can be obtained throughthe use of a spatial Fourier transform of the CCD image. The result will be an image in the spatial
frequency domain, where there will be a large DC peak corresponding to F{|E1 (x, y)|2 +|E2 (x, y)|2 },as well as two side peaks corresponding to F{cos(2k1,x x − ωq τ )}. By extracting the amplitude
and phase of one of these side peaks as a function of τ , the field autocorrelation function A(τ )is immediately obtained. Figure 2.2 shows an example of this procedure using real data. This
technique can improve the fidelity of the autocorrelation because the side peaks represent the
integrated modulation signal across the entirety of the overlap area of the crossed beams.

72

Spatial frequency domainqxF{|E1|2 + |E2|2}

F

F{
E

1 *

E

2 e

-i(

2k

x–

ω

τ)

}

Measured on CCD

x–
kx
(i 2

τ) }
ωq

* e

E
F{

E2

1

Figure 2.2: Spatial interference measured on a CCD from an HHG beam crossed with a replica ofitself for a single delay τ . Performing a Fourier transform yields three peaks: one corresponding to
the DC signal and two from the modulation signal. A single side peak contains information about
the field autocorrelation function A(τ ).

73

2.5

Field Autocorrelation Results
Figure 2.3 shows the measured HHG field autocorrelations and the resulting spectra (via

Fourier transforms) using low and high laser intensity conditions for each of the three driving laser
wavelengths mentioned above. The HHG cutoff energy and flux from Ar was optimized throughthe driving intensity, which identifies the high-intensity case that is plotted. Reducing the intensity
to shift away from optimal HHG conditions then gives the results for the low-intensity case. For
a 0.8 µm driver, the low and high intensities are 1.5 × 1014 and 2.6 × 1014 W/cm2 , which resultin 15 and 9 X-ray bursts observed in the autocorrelation, respectively. For a 1.3 µm driver, thelow and high intensities are 1.3 × 1014 and 2.1 × 1014 W/cm2 , which result in a faster reduction
in measured X-ray bursts: 9 to 4. Finally, for a 2 µm driver, we measure a single isolated burstof harmonics for the high intensity case of 1.6 × 1014 W/cm2 . Furthermore, the single burst case(high intensity) corresponds to higher HHG flux than the HHG pulse train case (low intensity).
The HHG spectrum from the 2 µm, high-intensity driver spans from 90 to 180 eV with a FWHM
bandwidtih of ∼ 60 eV. This spectrum has a transform-limited pulse duration of ∼ 35 as, verified by
the measured electric field autocorrelation that has a 70 as FWHM (as expected for Gaussian-like
pulses). As discussed through Fig. 1.14, the HHG emission is expected to be emitted with linear
chirp, giving a predicted pulse duration of ∼ 300-as (discussed below). To our knowledge, ourtemporal/spectral measurement of the 2 µm, high-intensity driven HHG has, to date, the highest
photon energy and broadest bandwidth of any measured isolated attosecond pulse [112]. Following
suggestions by previous experiments [123, 143], we also have concluded that the HHG spectrum
and number of pulses are not dependent on the carrier-envelope phase (CEP) of the driving laser,
since the CEP of the 2 µm laser was varied in our experiment with no observed change.
We assert that as long as the harmonic emission maintains pulse-to-pulse coherence (asdescribed in Section 2.3 and being consistent with previous studies of HHG [94, 90]), then thefield autocorrelation measurements presented here are sufficient to support the wavelength andintensity scalings of a phase-matching gating effect in the temporal domain, as detailed in the next

74

Figure 2.3: Comparison of the measured HHG field autocorrelation data from Ar, driven by ∼ 10
cycle laser pulses at wavelengths of (A) 0.8 µm, (B) 1.3 µm, and (C) 2 µm under high and low laser
intensity conditions (red and blue lines) described in the text. (Left) Field autocorrelation of the
HHG field and enlarged view near time 0 with the coherence time of the central pulse envelope.
The temporal phase-matching window is highlighted in yellow. Note that the bandwidth-limited
pulse duration is half of this coherence time. (Right) HHG spectra obtained from the FFT ofthe field autocorrelation traces (filled-area plots), showing excellent agreement with the spectra
obtained using a Hettrick grating spectrometer (black dotted lines). The low-intensity spectra areenhanced to see their harmonic structure. The predicted phase-matching cutoffs are indicated bygreen dashed lines. Adapted from [18].

75
section. This assertion, reliant on the temporal coherence of the harmonic emission, is further
supported by the high level of spatial coherence measured across the entire beam of phase-matched
HHG driven by mid-IR lasers [149, 161, 155]. Additionally, simulations have been performed torepresent the unphysical case of drastically different chirps between neighboring HHG bursts. Evenin these simulations, the electric field autocorrelation would contain indications of multiple pulses
interfering at the detector.
Presented as an alternative to attosecond streaking, the field autocorrelation gives a direct
measurement of the number of pulses in the HHG train (i.e. a total of 2N + 1 fringes will bemeasured if there are N + 1 bursts in the pulse train). This can be useful in providing fast feedbackin the experiment when optimizing the HHG source. Observing a supercontinuum spectrum froma grating spectrometer could be considered for fast feedback, but the spectrometer’s resolution
would need to be able to sufficiently resolve individual harmonics. Due to the limitations of grating
spectrometers explained in Section 2.1, the field autocorrelation could be more effective, especiallyfor harmonics driven by long wavelength, low individual photon energy lasers. Through the fast
Fourier transform (FFT), the spectral resolving power is governed by Heisenberg’s uncertainty relation: δE = 2π~/∆τ , where δE is the FFT spectral resolution and ∆τ represents the maximum
temporal delay range. Our spatial beam separator is capable of a delay range of ∆τ ∼ 300 fs,
corresponding to an impressive spectral resolution of δE ∼ 0.01 eV (2.0 µm driven odd harmonics
would be separated by 0.62 eV). This level of spectral resolution is necessary to unambiguously
determine whether the HHG spectrum contains individual harmonics or is supercontinuous, particularly since mid-IR driven HHG can contain orders up to 5,000 [161]. Finally, this technique
directly probes the temporal domain, giving insight into the dynamic phase-matching conditionsof the HHG process and its trends with driving intensity and wavelength.

2.6

Phase-Matching Gating Theory
The decreasing trend in number of pulses within the HHG pulse train as the driving wave-

length and intensity are increased can be explained through a dynamic picture of phase-matching.

76
Phase-matching is supported only during a narrow temporal window within the laser pulse. On theleading edge of the laser pulse, the laser phase velocity vφ < c (speed of light) due to the dominanceof neutral atoms. In contrast, on the trailing edge, the laser phase velocity vφ > c due to ionizationof the gas beyond the critical ionization level, which terminates phase matching [160]. In the caseof mid-IR driving lasers, three factors lead to a much shorter phase-matching window comparedwith 0.8 µm lasers: the harmonic order is higher, the period of the driving wave is longer, andthe phase-matching pressure and ionization density are higher. Each of these factors contributes
to a larger phase shift within each half-cycle and, therefore, to a shorter phase-matching window(Fig. 2.4). In contrast, for laser intensities below the critical ionization limit, optimal pressure
and ionization are low, the harmonic order is low, and phase matching extends over many laser
cycles. The practical consequence of this physical scaling is that for any driving laser wavelength,
the temporal phase-matching window can be narrowed by increasing the driving laser intensity andgas pressure. However, for longer driving laser wavelengths, the temporal narrowing effect is muchstronger, making isolated attosecond pulse generation more robust and natural.
The phase-matching theory detailed in Section 1.4.1 supports these findings when calculated
dynamically. The terms of the phase-mismatch equation that vary with respect to time are thosethat have dependence on the ionization level. Thus, ∆kmedium is the only dynamic component toconsider, while the geometric terms remain constant. Taking a temporal derivative of Equation(1.73) yields:∂∆k(t)
2π 0
dη1 (t)≈ q P̄
∆δ + Na re λL ·∂t
λLdtdη1 (t)∝ q P̄ λL
dt

(2.9)

The rate at which ∆k(t) sweeps past zero (compared to the period of the driver) provides anindication for the number of half-cycles that can be phase-matched. Note the neutral gas dispersion
contribution in Eq. (2.9) is negligible compared with the free-electron dispersion contribution,
especially for long-wavelength driving lasers. At low laser intensities, the harmonic order q anddη1 (t)/dt are small, which suppresses phase-matching gating, in agreement with the data of Fig. 2.3

77(low intensity autocorrelation). Near optimal phase-matching, the pressure P̄P M , and the central
harmonic order qP M scale by the laser wavelength as P̄P M ∝ λ2L and qP M ∝ λ2.7
L , as have been
observed experimentally [160, 149, 161]. The scaling of qP M arises from two contributions: a factorof 1/2·λ1.7
L because the central energy of the attosecond burst is half the phase-matching cutoff [160],
and an additional factor of λL from the fundamental laser photon energy. Thus, under optimalphase-matching conditions, Eq. (2.9) can be approximated by ∂∆k(t)/∂t ∝ λ5.7
L · dη1 (t)/dt, whichscales strongly with the wavelength of driving laser. The combined effects of higher pressures,
higher harmonic orders, and stronger free-electron dispersion for long-wavelength driving lasers
creates a large phase-mismatch jump between adjacent half-cycles of the driving laser. Note thatthe larger separation between half-cycles for long wavelength driving lasers further isolates theattosecond bursts. Consequently, a longer wavelength driving laser can much more easily induce
strong phase-matching temporal gating. Figure 2.4 depicts the time-gated phase-matching effectusing the conditions of the experiment.

Figure 2.4: Calculated phase mismatch ∆k (blue line), Lcoh (red line and yellow highlight), andinstantaneous HHG cutoff photon energy (green) for HHG in Ar driven by 10-cycle laser pulses:(A) 0.8 µm, 2.42 × 1014 W/cm2 ; (B) 1.3 µm, 1.87 × 1014 W/cm2 ; and (C) 2.0 µm, 1.5 × 1014 W/cm2 .
The temporal window during which phase matching occurs is highlighted in yellow. Adapted from[18].

In addition to the analytic approach to understand the phase-matching gating effect, anadvanced 3D numerical calculation was performed to simulate the microscopic generation of HHG
and its macroscopic propagation to the detector. The gas medium is discretized into individual

78
radiators, whose emission is individually propagated to the detector:
|rd − rj |qj sˆdEj (rd , t) = 2sˆd × aj t −c |rd − rj |c(2.10)

where qj is the charge of an electron at position j, sˆd is the unitary vector pointing from thatelectron to the detector, and rd and rj are the position vectors of the detector and of the elementary radiator, respectively. Each radiation source has a dipole acceleration aj that is calculatedusing SFA+, an advanced strong-field approximation (SFA) method [98]. aj gives rise to coherent
radiation that propagates at a phase velocity c to the far-field detector, where the coherent sum of
all of the emitters’ contributions is calculated. Macroscopic effects are incorporated into the calculation through the fundamental field, which exhibits time-dependent phase- and group-velocities.
Dispersion from plasma and neutral atoms are considered, and Beer’s law accounts for the reabsorption of the harmonics by the gas in the 2 mm argon gas cell assuming uniform density, for avariety of pressures. The 2 µm fundamental field was modeled using an 8-cycle FWHM Gaussianenvelope in the time domain (53 fs). A peak intensity of 1.2 × 1014 W/cm2 was chosen to satisfy
phase-matching (∆k = 0) at the center of the pulse.
In Fig. 2.5, the HHG emission for single-cycle and multicycle drivers (1.5-cycle and 8-cycle
pulses, respectively) are contrasted for low, moderate, and high pressures. Group velocity walk-off
effects are evident in the single-cycle results, whereas strong phase-matching gating is observed
for the multicycle case at high pressure. Figure 2.5 A-F show the fundamental laser field at theentrance and exit of the gas medium. For higher pressures and multicycle drivers, the neutral atom
contribution is large for the front of the pulse, creating a phase shift at early times. The timevarying plasma content in the middle and end of the pulse creates a chirp for later times. These
effects confine the temporal window over which phase-matching occurs, down to a suboptical cycle
duration (300 as) near the center of the pulse. Figure 2.5 H reveals that an isolated attosecond
HHG pulse is emitted with linear chirp (group delay dispersion ∼ 0.005 fs2 ). For high pressures and1.5-cycle drivers, the group velocity walk-off causes the fundamental laser field envelope to shift
away from the initial phase-matching window. Through propagation, the effective phase-matching

79
window is nonexistent, thus preventing bright HHG for the single-cycle case.
Group velocity walk-off does occur for longer duration drivers (Fig. 2.5 F), but the envelope
varies slowly enough in intensity that the walk-off does not significantly reduce the phase-matchingconditions at the center of the pulse. In both the low pressure (5 Torr, Fig. 2.5 G) and highpressure (600 Torr, Fig. 2.5 H) cases, the positive slope of the emission structure in chronocyclic
space indicates the short electron trajectories are phase-matched (recall Fig. 1.14). Despite thechirp that stretches the HHG pulse to ∼ 300 as, the phase-matched HHG bandwidth obtained from
Fig. 2.5 H (∼ 60 eV FWHM) corresponds to a transform-limited pulse duration of ∼ 35 as. These
spectral and temporal characteristics agree with the empirical measurements to a high degree.

2.7

Phase-Matching Gating Summary
In summary, we present an alternative method to measure the temporal and spectral charac-

teristics of HHG emission through the use of an electric field autocorrelator. To our knowledge, wehave observed the first isolated attosecond pulses measured in the soft X-ray region, with sufficient
temporal resolution (1.5 as) to resolve individual harmonic bursts (and harmonic orders). Throughphase-matching gating, we demonstrate the benefit of mid-IR lasers to naturally and robustly produce bright isolated attosecond pulses up to 180 eV. This also represents, to our knowledge, thefirst experimental verification of the time-dependent balance of plasma and neutral atom dispersions through a direct measurement of the HHG emission in the time domain. We present a newunderstanding of the wavelength, pressure, and intensity scalings of the phase-matching dynamics,
which surprisingly indicate that efficient generation of bright isolated bursts of harmonics in the
EUV necessitate multicycle driving pulses to avoid group velocity walk-off effects. Additionally,
CEP-stabilization and few-cycle laser technologies are not necessary to produce stable, isolated
attosecond pulses. It is predicted that this approach can be extended to generate single digit
attosecond and even zeptosecond pulses in the keV region [105, 161]. Our results provide further insight into the attosecond physics of high intensity laser-matter interactions and frequency
conversion driven by mid-IR laser light.

80

Figure 2.5: SFA+ calculations of microscopic and macroscopic HHG emission, demonstratingphase-matching gating and group velocity walk-off effects for 1.5-cycle and 8-cycle drivers at low(5 Torr), moderate (100 Torr), and high (600 Torr) pressures. For a 2 mm argon gas cell, entrance(red) and exit (blue) laser fields are plotted in A-F. Phase-matching intensities of 1.3 × 1014 W/cm2
and 1.2 × 1014 W/cm2 are used for the single- and multicycle fields, respectively. The driving lasermode was assumed to be a Bessel profile of radius 60 µm. The phase-matching gating window ishighlighted in yellow. The HHG emission in chronocyclic space is plotted for the 8-cycle driverat (G) 5 Torr and (H) 600 Torr. Isolation of a 300 as chirped pulse occurs for the high pressure
scenario (H). Phase-matching over an extended distance is not possible for a 1.5-cycle driver due
to group velocity walk-off effects. Adapted from [18].

Chapter 3

Water Window and Soft X-ray HHG Spectroscopies

1 Aside from the complex temporal aspects of mid-IR driven HHG, the spectral characteristics

have been shown to follow the macroscopic cutoff scaling law from Eq. (1.89): EP M C ∝ λ(1.6−1.7) .
Mid-IR drivers are especially suitable for generating broad supercontinua with utility in spectroscopy applications. In this chapter, we demonstrate the use of table-top HHG sources for the
spectroscopic interrogation of thin samples with absorption edges in the water window (defined by
the K-edges of C and O: 284–541 eV) and soft X-ray regimes (> 1 keV) (Fig. 3.1). The 1.2 - 2.4µm output from the HE:TOPAS described in Chapter 2 is used, in addition to 3.9 µm idler lightfrom a 1.064 µm pumped, 1.460 µm seeded OPCPA [139, 161]. Aside from HHG spectroscopies
being useful in determining the elemental composition of unknown samples, the nanoscale atomic
structure of pure samples can be revealed through a measurement called X-ray Absorption Fine
Structure (XAFS). Three experimental XAFS measurements using the two aforementioned mid-IR
HHG drivers will be detailed here.

3.1

XAFS Theory
Atomic absorption spectra contain sharp features corresponding to energetic transitions.

From a single-atom perspective, electron transitions govern the spectral locations of the observed
absorption edges. Photon absorption can result in electron transitions between bound orbitals aswell as photoionization to continuum electron states. In molecules, elemental absorption edges and
1

The experiment and results described in this chapter have been submitted for publication [135].

82

Absorptance from 50 nm thick foils
1.0C

Absorptance

0.8

N

Al

O

Water Window

Ti

Soft X-rays

Zr

0.6

Cr
Fe

0.4

Sc

0.2
0.0

C
100

200

1000

2000

Photon Energy (eV)Figure 3.1: Absorption spectra for various elements in the water window and soft X-ray regimes.
A material thickness of 50 nm is used for all curves. Data retrieved from the CXRO database [63].

83
the surrounding spectral structure can be modified due to the presence of neighboring atoms andthe chemical bonds that are formed. Several molecular factors can influence the near-edge absorption structure (NEXAFS), such as oxidation and magnetic states, or localized charge to specificatoms in the molecule or lattice. Spectroscopic measurements near the absorption edge can giveinformation about the electronic band structures, as well as orbital and spin ordering phenomena.
The molecular orientation with respect to the incident light wave’s polarization can also influence
the NEXAFS measurement.
In ordered solids, atoms form a crystallographic lattice with consistent inter-atomic spacing.
Even with one type of element present, the single-atom absorption spectrum can be modified due tothe presence of neighboring atoms. This phenomenon can be understood in a three-step quantum
model, where an atom absorbs an incident X-ray, causing it to eject a core-shell photoelectron.
Scattering of the photoelectron from neighboring atoms leads to quantum interferences between
the outgoing and incoming electron waves (Fig. 3.2). The interferences produce modulations in
the X-ray absorption spectrum past the absorption edge, and can extend to significantly larger
photon energies. The information contained within the interference structure is not limited to only
single rescatterings, but can also be influenced mulitple scattering events. In the electron scattering
picture, an XAFS function χ(k) can be used to model the absorption probability of the incident
X-ray photon:
X Ni fi (k)µ − µ02 2
sin(2kRi + δi (k))e−2k σi −2Ri /λ(k)= χ(k) ∼
=
2
∆µ0
kRi

(3.1)

i

Here, the left hand side is measured experimentally; µ is the total absorption coefficient for thelattice or molecule, while µ0 corresponds to the absorption from an isolated atom. The right handside of Eq. (3.1) is the theoretical formula used to fit the data; different electron scattering paths
labeled by i generate different modulation frequencies in the XAFS absorption spectra. fi (k) isthe backscattering amplitude, Ni is the number of scattering atoms in a particular shell (Fig. 3.2),
δi (k) is the total phase shift from both absorption and rescattering, λi (k) is the electron mean
free path, Ri is the half path length between the absorbing atom and the i coordination shell of

84scattering atoms (for single scattering), σi is a disorder parameter, and k is the electron wave
hip−1
number: k Å
≈ 0.512 (E − Eedge )[eV ].
Due to the tight packing of atoms within the lattice, the interferences occur on very shorttime scales. This results in long periodicity modulations in the spectral domain, requiring a broad
spectral probe that extends well past the absorption edge to sufficiently resolve the modulations.
Thus, measuring the fine structure of solid samples with Å-resolution is labeled extended-edge
XAFS (or EXAFS). Figure 3.2 depicts the electron interferences and chemical environments that
result in EXAFS and NEXAFS effects.

Figure 3.2: (Left) Illustration showing the scattering paths of photoelectrons from the first (path1) and second shell (path 2), as well as secondary rescattering (path 3). (Center) Photoelectron
scattering and interference of the electron-wave packet for a 2D planar lattice. (Right) XAFS
sensitivity to the polarization ~ and final ~λf orbital direction. Adpated from [135].

3.2

NEXAFS in Polymers
To measure the NEXAFS signatures at the carbon K-edge near 284.2 eV, water-window HHG

light is produced using an OPA pumped by a 19.5 mJ, 20 fs, 1 kHz repetition rate Ti:Sapphire laser.
Because of the high peak intensities achievable with these laser specifications, nonlinear effects in
air can prove detrimental to the experiment and OPA process. Thus, the 20 fs Ti:Sapphire pulses
are stretched to > 150 fs to reduce the peak intensity for beam paths before the OPA. The pulse isthen recompressed just before each stage of the OPA using chirped mirrors. This allows the beam
to maintain excellent mode quality and pulse shape for efficient generation of several-cycle mid-IR

85
light. The OPA output at the idler wavelength of 2.1 µm is set to an energy of 2.75 mJ and a
duration of ∼ 26 fs (four cycles). The signal at 1.5 µm wavelength is produced with an energy of
4.5 mJ and a duration of ∼ 24 fs (five cycles). These driving pulses are then coupled into hollow
dielectric waveguides (Appendix A) with inner diameters ranging from 150 and 400 µm and lengths
between 1 and 5 cm. ∼ 10 − 20 atm of helium gas is supplied to the waveguide for phase-matched
HHG at these mid-IR wavelengths. The signal driver results in a HHG supercontinuum spanningphoton energies up to 360 eV, while the idler produces a supercontinuum up to 550 eV. A photon
flux of > 109 photons/s/1% bandwidth was measured in the water window region: three orders ofmagnitude larger than previously demonstrated [149, 121].
Following the HHG medium, our experimental beamline consists of an X-ray spectrometer(Hettrick) with a resolution in the EUV and soft X-ray regions of λ/∆λ > 1000. The XAFS
samples and metal filters are placed between the spectrometer grating and the X-ray CCD camera(Andor). Metal filters are used to reject the fundamental driver from the HHG supercontinuum,
while also allowing calibration of the spectrometer using characteristic absorption edges of foil
filters. Differential measurements with and without the XAFS samples allow the determination of
the spectrally-resolved absorptivity with high accuracy. The data is then analyzed with Athenaand Artemis software [107].
The HHG spectrum resulting from the 1.5 µm driver is plotted in pink in Fig. 3.3. This
broadband source was used to illuminate two polymers, mylar and parylene, which contain similar
carbon structures but different neighboring elements. Figure 3.4 shows the experimental results,
demonstrating the ability of HHG spectroscopy to differentiate between chemical compounds using
NEXAFS, despite the degree of chemical similarity and the common probe of the carbon K-edge.
Our measurements have a spectral resolution 0.2 eV, which is sufficient to identify a 0.2 eV differencein the ∼ 285 eV absorption peak between the mylar and parylene samples. In both samples, thispeak near 285 eV corresponds to electron transitions from the 1s core shell of carbon to the molecularantibonding π ∗ orbital for unsaturated C=C bonds (sp2 hybridization). The energy shift between
the two samples is clear for this absorption peak. There is an additional peak near ∼ 289 eV86Figure 3.3: Broadband soft X-ray supercontinua generated using phase-matched HHG. Photon
energies up to 0.36 keV, 0.55 keV, and 1.6 keV are generated in He gas using mid-IR driving lasers
at 1.5 µm (OPA signal), 2.1 µm (OPA idler), and 3.9 µm (OPCPA idler), respectively. Adaptedfrom [135].

87
corresponding to electron transitions from two contributions: the 1s core shell to the antibondingσ ∗ molecular orbital, as well as the 1s core shell to the Rydberg-like 3p orbitals of the aromaticring (benzene) [76]. One consideration for this measurement is the relative angle between the
HHG polarization (~ε) and the molecular orientation (final orbital direction ~λf ), an angle ϑ~ε,~λ thatfintroduces an absorption probability factor of cos2 (ϑ~ε,~λ ). The polarization ~ε in our experimentflies within the plane of the samples.

3.3

XAFS in Scandium
Using the 2.1 µm OPA idler to drive HHG results in the purple HHG spectrum in Fig.

3.3, which covers a significant range of photon energies beyond the scandium L-edge. Figure 3.5
shows the experimentally extracted XAFS function at the L-absorption edge of a monoatomic
hexagonal-close-packed (hcp) Sc thin foil at the 398 eV L-absorption edge at room temperature,both in k-space and in real space. By taking a Fourier transform of the XAFS function in k-space(Fig. 3.5B), the interatomic spatial distributions can be obtained (Fig. 3.5C). Using the Athenaand Artemis software [107], the spatial distribution data was fit to Eq. (3.1) to yield first- andsecond-shell distances of 3.25 ± 0.05 Å(six-fold degenerate) and 3.30 ± 0.05 Å(six-fold degenerate),
respectively. Our spectral measurements agree very well with the CXRO database [63], and theextracted shell distances agree well with the expected interatomic lengths for a Sc hcp bulk lattice.
Thus, our thin film samples (∼ 150 nm) are confirmed to have lattice constants similar to bulk
values.

3.4

EXAFS in Iron
To investigate the extended-edge XAFS signal (EXAFS) of a material, significant source

spectrum is required beyond an absorption edge. Investigating a magnetically relevant material
such as iron would then require HHG signal well beyond its L-edges near 0.7 keV. To achieve such a
broadband spectrum, an OPCPA with an idler wavelength of 3.9 µm was used [139]. The OPCPA
operated at a repetition rate of 20 Hz with 80 fs pulse duration, ultimately producing an HHG

88Figure 3.4: Experimental NEXAFS spectroscopy near the carbon K-edge using a 1.5 µm-driven
HHG supercontinuum up to 360 eV. A) NEXAFS of C-containing polymers (mylar (violet) andparylene (pink)), at the carbon K-edge. Also plotted is the experimental absorption spectrum of acarbon thin film (black) for reference. B) Close-up view showing peaks due to transitions from 1s
to both π ∗ and σ ∗ orbitals. C) Molecular structure of mylar and parylene rendered for trimers (Catoms in brown, O in red, H in white). Adapted from [135].

89

Figure 3.5: Experimental XAFS spectroscopy near the Sc L-edge, using a 2 µm-driven HHG
supercontinuum up to 550 eV. (A) The normalized absorption coefficient of Sc near the L-absorption−1edge at 398 eV. (B) k 2 -scaled EXAFS function in k-space extending to > 5Å . (C) Experimental
R-space XAFS function (blue), not corrected for δ phase shifts, and fitted (red) to Eq. 3.1. The
real space XAFS function exhibits a broad peak. Inset shows a Sc hcp unit cell. Adapted from[135].

90
supercontinuum spanning the EUV and into the soft X-ray spectral region up to 1.6 keV (∼ 7.7 Å)with ∼ 2 × 106 photons/s/1% bandwidth at ∼ 1 keV (orange spectrum in Fig. 3.3).
Figure 3.6 shows the k-space and real space XAFS function obtained at the L-absorption edges
of a room-temperature monoatomic body-centered-cubic (bcc) Fe thin foil. The fitting procedureyields first- and second-shell distances of 2.48 ± 0.03 Å(eight-fold degenerate) and 2.87 ± 0.03 Å(sixfold degenerate), respectively. The latter distance is the lattice constant for a bulk bcc Fe unit cell.
Figure 3.7 shows X-ray diffraction measurements on the same sample, independently confirming
the bcc crystal structure with a lattice constant of 286.68 ± 0.03 pm.
It can be challenging to distinguish features from multiple rescattering paths, so we also
decompose the XAFS function terms using a continuous Cauchy wavelet transform (CCWT) (Figure
3.6A [89]). This 2D plot can help discriminate shell distances by providing information regarding
the k and R ranges that each nearest neighbor shell can contribute to (Figure 3.6C).

3.5

XAFS Summary
Through the use of state-of-the-art OPA and OPCPA laser technologies, broadband HHG

spectra were obtained with unprecedented photon energies and fluence. Utilizing these broad HHG
sources for XAFS spectroscopies yielded chemically- and structurally-sensitive data that agree withtheory and independent measurements. This work represents a new benchmark for broadband,static spectroscopic applications, and can easily be extended to dynamic spectroscopies through theinherently ultrafast nature of the HHG source (currently under investigation). Coherent, ultrafast
light sources with broader spectral bandwidths than achieved here are limited, and HHG beyond
the soft X-ray regime is yet to be observed. The theoretical limits of long wavelength-driven HHG
will be investigated in Chapter 4, revealing the feasibility of generating higher cutoff harmonics
that could be used for further spectroscopic applications.

91

Figure 3.6: Experimental EXAFS spectroscopy of a Fe thin film using a 3.9 µm driving laser(orange soft X-ray spectrum from Fig. 3.3). (A) CCWT analysis (modulus) near the Fe L-edge−1(0.7 keV); (B) k 2 -scaled EXAFS function in k-space extending to 12 Å . (C) Experimental realspace EXAFS function (blue), uncorrected for δ phase shifts, with the first peak fitted to thetheoretical Eq. (3.1) (red). Inset shows a Fe bcc unit cell. Adapted from [135].

92

Figure 3.7: X-ray diffraction data showing the first few diffraction peaks for the Fe thin film sample.
The data indicates a body-centered-cubic (bcc) structure with a lattice constant of 286.68 ± 0.03
pm. Adapted from [135].

Chapter 4

The Lorentz Force during HHG

1To date, HHG has been demonstrated with energies up to a record level of 1.6 keV using

3.9 µm drivers [161]. This achievement was possible according to the phase-matching cutoff scaling(1.6−1.7)

law: EP M C ∝ λL

. To extend the HHG photon energy further, increasing the driving laser

wavelength is an obvious route (although more complex geometries described in Chapter 5 and HHG
from ions using ultraviolet drivers [134] are alternative approaches). One consideration that might−(5−9)

limit HHG driven by long wavelengths is the unfavorable flux scaling of λL[161, 120, 42, 98].

Fortunately, many more emitters can be present in very high pressure gases that satisfy longwavelength phase-matching conditions, which can compensate for the low single-atom HHG yield.
However, acquiring a driving laser that has both ultrafast pulse duration (a requirement for phasematched HHG) and central wavelength beyond 4 µm proves to be a challenge. To our knowledge,
there are currently no commercially available broadband gain media that would enable ultrafast
oscillators and amplifiers at mid-infrared wavelengths beyond 4 µm. OPA and OPCPA technologies
are possible solutions using appropriate frequency conversion media in multi-stage configurations[111], or possibly a single stage pumped using a relatively new ultrafast laser technology operatingnear 2.5 µm, Cr:ZnS and Cr:ZnSe [166]. Spectral broadening through self-phase modulation using
10 µm CO2 lasers provides another possibility, but this approach can require further compression
or might result in a multi-pulse structure [104, 101, 100].
Despite the present difficulty in performing HHG experiments with suitable long wavelength1

The work presented in this section is published in [43] and has been submitted for publication [102].

94
drivers, a theoretical investigation of the fundamental physics can be performed. Section 1.2 covers
the zero-order physics of HHG, providing an explanation of the mechanism and predictions thatare consistent with observed phenomena to date. However, several assumptions have been madeduring the derivations that hold true only for short wavelengths, non-relativistic intensities, andplane-wave driving fields. Testing the validity of these assumptions for the case of long wavelength
drivers and realistic laser modes is an important step toward understanding the spectral limitationsof HHG technology. A primary effect that might limit HHG at long wavelengths is the trajectoryof the ionized electron and how it might be influenced by the full Lorentz force.

4.1

The Full Lorentz ForceAs previously described by Eq. (1.16), the full Lorentz force that the electron experiences

once ionized is:
F (t) =dp(t)= qe [EL (t) + v(t) × BL (t)]
dt

(4.1)

By neglecting the magnetic component of the Lorentz force (assuming non-relativistic motion), one
arrives at Eqs. (1.17–1.19). These equations of motion are true to zero-order, and can be used asa basis to which relativistic (and non-plane-wave) perturbations can be applied. The zero-orderequations of motion are reproduced here (assuming the ionized electron is initially stationary at
the origin):qe
EL0 cos(ωL t + φ0 )ŷ
me

(4.2)

qe
EL0 [sin(ωL t + φ0 ) − sin(φ0 )]ŷ
me ωL

(4.3)

qe
EL0 [cos(ωL t + φ0 ) + ωL sin(φ0 )t − cos(φ0 )]ŷme ωL2(4.4)

a0 (t) =v0 (t) =x0 (t) = −95
The equations of motion can then be rewritten in terms of the zero-order solutions and first order
perturbations:a(t) = a0 (t) + a1 (t)qev0 (t)= a0 (t) +EL1 (t) − EL0 cos(ωL t + φ0 )
× x̂
me
c

(4.5)

Z t

a1 (t0 )dt0v(t) = v0 (t) + v1 (t) = v0 (t) +
0
Z t

2
q
E
qe
e
2
0
0
L0
EL1 (t )dt += v0 (t)+[sin(ωL t + φ0 ) − sin(φ0 )] ẑme 0
2me ωL2 c

(4.6)

Z t

x(t) = x0 (t) + x1 (t) = x0 (t) +v1 (t0 )dt00#!
"Zt Z t00
2
q
E
qe
e L0
EL1 (t0 )dt0 dt00 += x0 (t)+3 Υ(t, φ0 )ẑme 08m
e ωL c0(4.7)

where:
Υ(t, φ0 ) = 4ωL t + 4 sin(ωL t + 2φ0 ) − 2ωL t cos(2φ0 )−(4.8)− 4 sin(ωL t) − 3 sin(2φ0 ) − sin(2ωL t + 2φ0 )Here, we have included a perturbation to the electric field EL1 (t), which will be necessary when
considering non-plane-wave driving fields.
Equations (4.5–4.8) clearly contain two perturbing contributions. The first is from the firstorder electric field EL1 (t), and the second is from the magnetic Lorentz force arising from v0 (t) ×
BL (t). Already, the perturbative effect of the magnetic Lorentz force can be calculated to yield av × B drift at the time the electron would typically recollide with its parent ion, which we defineas:δv×B =

2
qe2 EL0Υ(τf , φ0 )ẑ
8m2e ωL3 c(4.9)

where the displacement is evaluated at the return time t = τf . From Eq. (4.9), it is clear that themagnetic Lorentz force causes the electron to drift along the z-direction, and its magnitude scalesas δv×B ∝ IL λ3L for a constant value of φ0 . For all returning trajectories φ0 ∈ [0, π/2), Υ(τf , φ0 ) is

96
positive, meaning the magnetic Lorentz drift is always in the positive z-direction. This is true forelectrons ionized during an up-cycle of the electric field or a down-cycle.
Previous work has considered the effect of the magnetic Lorentz force during strong field
ionization and recollision processes [72, 167, 49, 87, 57, 21], including a work by Walker et al.
in which recollision was suggested to be greatly reduced for driving lasers with λL ≥ 6 µm [92].
However, the laser intensities used in these calculations were much larger than are relevant for
HHG, and a plane-wave driving field was used (i.e. the field’s transverse extent is infinite andE1 (t) = 0). In the following sections, the full Lorentz drift during HHG-relevant conditions will beconsidered.

4.2

Field Components for Finite Laser Modes
By removing the assumption of a plane-wave driver, additional field components arise that

can influence the trajectory of the ionized electron. This is a consequence of satisfying Maxwell’sequations in free-space using finite transverse spatial modes. In particular, a longitudinal electricfield component necessarily arises in linearly-polarized beams:
Ez (t) =i ∂Eψ (t),k ∂ψ

(4.10)

where ψ represents the polarization direction (for all derivations here, ψ = y). This holds true
for the first-order field equations for waveguide modes expressed in Eq. (A.2) in Appendix A, andis also true for Gaussian modes, as described in [23]. For both a Gaussian mode and the EH11
waveguide mode, the longitudinal electric field Ez (t) qualitatively takes on a two-lobed amplitude
profile with opposing signs for each half, and has a π/2 phase shift with respect to Ey (t) and Bx (t).
Importantly, Ez (t) = 0 for all locations with y = 0, which includes the center of the mode. Figure
4.1 depicts the amplitude profiles for these geometries.
Equation (4.10) indicates that the temporal dependence of Ez (t) is the same as that of Ey (t),except for a π/2 phase shift. The amplitude of Ez (t) is proportional to EL0 λL /w for a Gaussianmode, or EL0 λL /a for the EH11 waveguide mode. Thus, it is straightforward to show that the97

EH11 Mode

Gaussian Mode
Ey , B x

Ey , B x

Ez

Ez

Figure 4.1: Transverse (top) and longitudinal (bottom) field components for Gaussian (left) andEH11 (right) modes. The axis ranges for the two geometries are chosen to give comparable mode
sizes. Scale bars are normalized.

98
perturbation to x(t) resulting from Ez (t) is:
Z t
qe
δEz =
me 0

Z t00

!
Ez (t0 )dt0

dt00

0

(4.11)−qe Ez (x, y)=[ωL t cos(φ0 ) − sin(ωL t + φ0 ) + sin(φ0 )]me ωL2where Ez (x, y) represents the spatially varying amplitude of Ez (t) and contains the EL0 λL /w
proportionality. Ultimately, the displacement from the Ez component of the driving field scales as
1/2

δEz ∝ IL λ3L w−1 .
For both longitudinal drifts, the scaling with τf , or more fundamentally φ0 , cannot be described in a simple form; however, the magnitudes of these drifts generally decrease as φ0 increases
from 0 to π/2. The total longitudinal drift resulting from the Lorentz force will be the sumδL = δv×B + δEz , and the interplay between the two drift components can result in large or small
total Lorentz drift depending on the relative signs of the two drifts, which varies spatially as discussed in Section 4.5.

4.3

Recombination Probability with Lorentz Drift
If the electron is treated classically, one would expect there to be greatly reduced probability

for the electron to recombine with its parent ion if there is a longitudinal drift larger than a Bohrradius or so. However, due to the quantum nature of the electron, the electron wavepacket diffuses
transversely as it travels in the continuum, as described in Section 1.3.2. Thus, for any finite
spatial overlap between the diffused wavepacket ψd and the wavefunction of a bound electron ψb ,the probability PR of recombination and emission of a photon is nonzero and modeled by:ZPR =d3 rψd∗ (r, t = τf )ψb (r)(4.12)
R

One may also describe the harmonic yield as originating from a time-varying dipole moment, arisingfrom the interference of ψd (r, t) with ψb (r, t). In either case, the photon yield will be proportional
to PR as well as the ionization rate at the time the electron was ionized ωADK (r0 , t = 0) [68, 72, 30].
When ψd (r, t) is evaluated for even the shortest wavelength considered here (0.8 µm), it
becomes evident that the integral describing RP can be reduced. The diffused wavepacket ψd (r, τf )

99
returns with a width that is orders of magnitude larger than the width of ψb (r). For the purposesof integration, ψb (r) can be approximated as a delta function. Instead of applying the Lorentz
drift δL to ψd it is straightforward to apply it to ψb , representing it as a shifted delta function:ψb (r) = δ(r −δL ẑ). Considering only the wavefunction amplitude, the form of PR can be rewritten:
PR = |ψd (δL , τf )|. Note that the drift-induced quantum phase dynamics of the electron can shift
the absolute positions of the harmonic peaks in the spectral domain [102], but should not change
the overall single-atom spectral shape. Only amplitude effects will be considered here.
To determine how the Lorentz drift influences the high harmonic flux, a comparison must
be made between the case when the Lorentz drift is included, and the case when it is assumed
to be zero. The relevant comparison is the ratio of recombination probabilities for the two cases,
expressed as a recombination probability ratio (RP R) defined as:
RP R = PR (δL )/PR (0) =|ψd (δL , τf )||ψd (0, τf )|(4.13)The RPR is defined for a single electron ionized during a particular half-cycle of the driving laser,and has a maximum value of unity corresponding to zero displacement. In the following section, wequantify how this recombination probability ratio scales with wavelength, indicating a drift-induced
reduction in HHG fluence as the wavelength is increased.

4.4

RPR Scaling with Wavelength
In order to study Lorentz displacement effects under conditions applicable to HHG, the inten-

sity of the driving laser must be chosen to achieve bright, phase-matched HHG flux. Constraining
the intensity to the wavelength also simplifies the investigation by reducing the parameter space.
We assume the target gas is helium and the peak intensity of the driver is optimized for maximal,
mode-averaged, phase-matched HHG flux driven by an 8-cycle pulse, as depicted in Figure 1.20.
With this assumption, and using reasonable values for the laser mode size (w ∼ 50 µm), it becomes
evident that the Lorentz displacement produces a non-negligible effect for wavelengths larger than
∼ 6 µm. Table 4.1 summarizes the relevant wavelength-dependent parameters and the resulting

100
Table 4.1: Lorentz displacement results for selected laser wavelengths. The driving laser mode isassumed to be 100 µm in diameter, and the ionization phase φ0 = 18◦ . Maximum values within
the spatial mode are presented for |δv×B | and |δEz |, whereas |δL |symm represents the total Lorentz
displacement at the spatial location along the polarization axis where |δv×B | = |δEz |. The reported
RP Rsymm values are for the same location.
λL (µm)
0.8
3.0
6.0
8.0
10.0
15.0
20.0

IL (×1014 W/cm2 )
7.58
4.20
3.26
2.97
2.77
2.45
2.25

|δv×B | (Å)
0.372
10.9
67.7
146
266
794
1,731

|δEz | (Å)
0.293
11.5
81.1
183
346
1,100
2,507

|δL |symm (Å)
0.455
15.0
103
223
406
1,213
2,648

σ(τf ) (Å)
19.1
71.1
142
189
237
355
474

RP Rsymm
0.99970.97790.76710.50000.22980.002910−7displacements and RPR values. The net Lorentz drift magnitude |δL | varies across the transverse
spatial dimensions, but a symmetric net drift |δL |symm can be defined at the location where δv×B
is the same magnitude as δEz . This symmetric net drift is listed in Table 4.1, and provides a metricto estimate when the field-driven electrons will be displaced far enough to miss the ion during one
half-cycle, while trajectories during the next half-cycle (when δv×B and δEz have opposite sign)
directly recollide with the ion. It is important to note that δEz has a magnitude ≥ δv×B whenusing the harmonic cutoff ionization phase φ0 = 18◦ . Even when considering the Lorentz force inultra-strong field experiments, the driving laser typically must be tightly focused to achieve relativistic intensities (IL > 1018 W/cm2 ), in which case the contribution from Ez is large and cannot
be ignored. Figure 4.2 serves as a visual aid for the field and drift components in a Gaussian focus
geometry.
Although we have previously derived a λ3L scaling for each of the longitudinal drift compo−0.4
nents, the optimal intensity for HHG flux scales close to λL. Thus, the final scaling of the total

Lorentz drift will be close to λ2.6
L . Figure 4.3 shows the magnitude of the net Lorentz drift |δL |symm
plotted as a function of wavelength, where its measured slope of 2.62 verifies the wavelength scaling.
The width of ψd upon return to the parent ion σ(τf ) scales linearly with wavelength, and is alsoplotted in Fig. 4.3. The inset of Fig. 4.3 shows the effect of |δL |symm on the RP R for adjacent

101

δL

Electron
Wavepacket

δEz
δv×B

Ion

δL

σ

δL
δv×B

δEz

Figure 4.2: Field components of a Gaussian free focus (in the waveguide case, the fields are
similar but the guided beam does not diverge). The Gaussian mode envelope is depicted in red,
the transverse fields Ey and Bx oscillate like the purple regions, and the longitudinal field Ez is
depicted by the green regions. The longitudinal displacements due to the v × B force (δv×B ),the longitudinal electric field (δEz ), and the total Lorentz force (δL ) are shown in two locationsin the focus, where the net displacement changes in magnitude. Top-right: Visualization of theelectron wavepacket drift. Quantum diffusion of the electron wavepacket enables recollision in spiteof significant Lorentz drift.

102
half-cycles. For one of the half-cycles δv×B cancels δEz , whereas they add in the next half-cycle.
Thus, the RP R alternates between unity and RP Rsymm .
1.62 faster than the diffused wavepacket
The Lorentz displacement |δL |symm increases at a rate λL

width. Accounting for the Gaussian form of ψd , the resulting scaling of RP Rsymm (and the total
HHG flux attenuation from the Lorentz drift) is given by:"   #" 
3.24 #
 1.62 !2
δL 2
δ
λ
[µm]
L
L
 ∼ exp −∼ exp −RP Rsymm ∼ exp −σ
σ
8.7(4.14)

The fitting parameter D was calculated to nicely fit the RP Rsymm trend using the data in Table 4.1.
Fitting to the mode- and φ0 - integrated RP Rsymm for harmonics > 2Up causes the D parameterto increase to ∼ 13.3, with some underestimation at longer wavelengths. The super-Gaussian formof Eq. (4.14) encapsulates the initially slow onset of the Lorentz displacement, followed by a steep
reduction of RP Rsymm for λL > 6 µm. Note that this flux scaling only accounts for the Lorentz−(5−6)

displacement and will act in addition to the λL

scaling from quantum diffusion. Additionally,

it is only true for half-cycles where the two drift components have the same sign. Interestingly,
the flux attenuation due to quantum diffusion is dominant even up to 20 µm drivers, indicatingthat the Lorentz drift is not the limiting factor with respect to high harmonic flux driven by longwavelengths.

4.5

Lorentz Drift Spatial Profiles
In addition to attenuating the total high harmonic flux, the Lorentz displacement also hasa spatial dependence that influences the HHG mode. Figure 4.4 shows the Lorentz displacements
and RP R profile using a Gaussian focus geometry for a single half-cycle of a 10 µm driving laser,
assuming the electron is ionized at the harmonic cutoff phase φ0 = 18◦ . The energy of an emitted
harmonic will be 3.17 times the local ponderomotive potential (Up,loc ), which depends on the local
intensity. This local energy can be rewritten in terms of the maximum harmonic energy 3.17Up ,occurring at the center of the mode where the intensity is the highest (IL ). In Fig. 4.4(d), rings areplotted where the local cutoff energies correspond to 1 and 2 times the maximum Up . Note that

103

Figure 4.3: Total Lorentz drift magnitude (red) and diffused electron wavepacket width (blue)
as a function of driving laser wavelength. The electron is assumed to be ionized at 18◦ , andthe laser intensities used are those that maximize phase-matched HHG flux in Helium. |δL |symm
is calculated at the spatial location along the polarization axis where |δv×B | = |δEz |. The lines
intersect at λL = 7.4 µm and 175 Å, after which the Lorentz drift exceeds the wavepacket width. The
inset shows the RP R for adjacent half-cycles where |δL |symm is calculated, indicating a harmonic
efficiency that is modulated in the time domain with λL -periodicity.

104
the plots would be reflected about the x-axis for the next half-cycle, due to the sign flip of δEz but
not δv×B .
Figure 4.5 shows the same plots as Fig. 4.4, but for a waveguide geometry with an inside
diameter (I.D. = 2a) resulting in an equivalent mode size (w = 0.6435a, Fig. A.2). Note that the
separate components of the Lorentz drift δEz and δv×B add in one half of the mode, while they
cancel in the other half. While this is, from a flux perspective, detrimental for one half of the mode,the other half can benefit from the drift cancellation. If the flux for only the highest harmonics(> 2Up ) is to be optimized, the drift cancellation should occur as close to the center of the modeas possible. By definition, the Ez field is zero at the center, so its resulting drift cannot cancel thev × B drift, which is maximal at the center. To shift the location of drift cancellation closer to
1/2

the center, one can simply inspect the drift scaling laws (δv×B ∝ IL λ3L and δEz ∝ IL λ3L w−1 ) toconclude that the driving laser mode size should be reduced. By changing the relative magnitudeof δEz with respect to δv×B , one can control the spatial position of drift cancellation. In fact, byspecifying a driving wavelength and constraining the intensity to be optimal for HHG, the only freeparameter to control the RPR mode profile is the mode size.
In Fig. 4.6, the HHG photon energy at the location of drift cancellation is plotted as afunction of wavelength and mode size. Generally, the RP R approaches unity at a higher harmonicenergy in waveguides than in a Gaussian focus, for comparable mode sizes. For a 20 µm driverwith optimal intensity of 2.77 × 1014 W/cm2 in a 250 µm I.D. waveguide, the HHG energy that has
RP R = 1 is approximately 2Up which is ≈ 17 keV. This shows that even very high energy harmonics
in the hard X-ray region can be produced through HHG without the Lorentz drift shutting off therecollision process. Furthermore, Fig. 4.6 can be used as a guide to select the mode size necessary
for drift cancellation and efficient generation of a desired HHG photon energy.
Although the single-phase harmonic cutoff results in Figs. 4.4–4.6 are informative, the total
HHG mode emitted from a single half-cycle would result from electrons ionized at all phases within[0, π/2). Ionization phases closer to the peak of the field (φ0 ∼ 0) have the highest ionization
rate (ωADK ), but these electrons spend more time (τf ) free from the parent and have a lower

105

(a)

400

(b)

300

300

250

200
200

100
0

150

-100

100

-200
50

-300

0

-400

(c)

500

(d)

1.0
0.9

400

0.8

300

0.7

200
100
0
-100

0.6

2Up

0.5

1Up

0.4
0.3
0.2

-200

0.1

-300

0

Figure 4.4: Spatially-resolved Lorentz displacements and resulting recombination probability ratio(RP R) for a Gaussian focus with λL = 10 µm (IL = 2.77 × 1014 W/cm2 , w = 50 µm, φ0 = 18◦ ). (a)δEz , (b) δv×B , (c) δL , (d) RP R. The outer dashed circles represent the inside wall of a waveguidethat this Gaussian mode would couple to optimally. The middle and inner dashed circles in (d)represent the radii within which 1Up and 2Up HHG can be generated, corresponding to 2.5 keV
and 5.1 keV, respectively.

106

(a)

400

(b)

300

300

250

200
200

100
0

150

-100

100

-200
50

-300

0

-400

(c)

500

(d)

1.0

400

0.9

300

0.8
0.7

200
100
0
-100
-200

2Up
1Up

0.6
0.5
0.4
0.3
0.2

-300

0.1

-400

0

Figure 4.5: Spatially-resolved Lorentz displacements and resulting recombination probability ratio(RP R) for a EH11 waveguide mode with λL = 10 µm (IL = 2.77 × 1014 W/cm2 , I.D. = 2a = 150µm, φ0 = 18◦ ). (a) δEz , (b) δv×B , (c) δL , (d) RP R. The outer and inner dashed circles in (d)represent the radii within which 1Up and 2Up HHG can be generated, corresponding to 2.5 keV
and 5.1 keV, respectively.

107

2.5

10

5 keV

(x Up)
3.17
3.0

10 k
eV

keV

5k

1 keV

2.5

(b)
1 keV

(x Up)
3.17
3.0

192
eV

(a)

2.0
1.5

20 k

20 k

eV

eV

2.0
1.5

1.0

1.0

0.5

0.5

Figure 4.6: Energy of the harmonic emitted at the location of drift cancellation, corresponding
to RP R = 1. The color scale corresponds to energies in terms of the ponderomotive potential
Up ∝ IL λ2L , while the dashed lines indicate energy contours in keV. The cutoff ionization phase
φ0 = 18◦ is used. The Gaussian focus geometry is shown in (a), while the waveguide geometry is
shown in (b). The vertical axes of both plots give comparable mode sizes.

108
probability of recombining. For later phases, the low field strength results in reduced ionization
rates, and thus, reduced harmonic emission. Performing a weighted integral of the recombination
probability ratio over the entire phase range can provide an approximation of the total half-cycle
HHG mode. The appropriate weighting function f (φ0 ) to incorporate the phase-dependent effects
follows the harmonic yield scaling described in [53] Eqs. (5)-(7):f (φ0 ) = ωADK (φ0 ) × τf−3 (φ0 )/EL2 (φ0 ),(4.15)

Figure 4.7 shows the phase-integrated RP R for all harmonic energies in both the Gaussian focus(a) and waveguide (b) geometries. Figures 4.7(c) and 4.7(d) are identical to 4.7(a) and 4.7(b),but only HHG energies above 1Up contribute. Primarily due to differences in mode shape, there
is a larger area with nonzero RP R in Fig. 4.7(d) than in Fig. 4.7(c), indicating a mode-averaged
flux advantage for the waveguide case over the Gaussian focus case. It is important to note thatdifferent ionization phases will have different spatial profiles for the individual drifts, and thus,
the final RP R profile. As a result, different spectral regions will be emitted with different spatial
modes, giving rise to complicated spectral-spatial dynamics when propagation and phase-matching
are considered.

4.6

Lorentz Force Compensation through Noncollinear Geometries
The analysis in the previous sections relies on the interplay between the two components of

the Lorentz drift in finite laser modes, δEz and δv×B . However, the magnitude of δEz can only becomparable to δv×B when the mode size w approaches λL . If drift cancellation is desired in a large
mode scenario (w >> λL ), then it is impractical to rely on the inherent Ez field of the single-beammode. Additional fields would then be necessary.
Recent work in HHG has been performed using two-beam noncollinear geometries with linear
or bi-circular polarizations, which in itself has interesting phase-matching and selection rules [51,145, 96, 94, 40, 14, 130, 129]. Here we propose a similar method that can generate a forwards
field ellipticity that can act in the same direction as the magnetic Lorentz force, thereby enabling109

Hollow Waveguide

Gaussian Focus
(a)

1.0

(b)

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

2Up

0.5

1Up

0.4

(c)

0.6

2Up
1Up

0.5
0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

1.0

>1Up

(d)

1.0

>1Up
0.8

2Up

0.6

1Up

0.8

2Up
1Up

0.6

0.4

0.4

0.2

0.2

0

0

Figure 4.7: Spatially-resolved, phase-integrated recombination probability ratio (RP R) for all
harmonics vs plateau and cutoff harmonics. Plots (a) and (c) correspond to a Gaussian focus
geometry, while plots (b) and (d) correspond to a waveguide geometry. All harmonic energies
contribute in (a) and (b), while only energies above 1Up contribute in (c) and (d). For all cases,
the driving laser has wavelength λL = 10 µm, spot radius w = 50 µm (I.D. = 2a = 150 µm), andpeak intensity IL = 2.77 × 1014 W/cm2 .

110drift cancellation. The experimental geometry uses two noncollinear, counter-rotating, circularly
polarized beams of equal intensity and wavelength, as shown in Fig. 4.8(a). In a collinear geometry,adding counter-rotating circular polarizations with the same amplitude and frequency yields a linear
polarization, where the polarization angle depends on the relative phase between the two added
beams; in a noncollinear geometry, however, there is a projection of the fields along the forward
direction defined by the bisecting centerline between the two crossing beams (defined as the zdirection here). Certain locations within the overlapping region will result in the constructive
addition of these z-field projections, which when combined with the transverse field results in a
forwards ellipticity. In other words, the major axis of the elliptical polarization will lie within thex-y plane, while the minor axis is along the z-direction.

Figure 4.8: Two noncollinear, circularly polarized beams (a) produce a polarization gradient acrosstheir focus in the transverse direction (b), including points with forwards ellipticity. This makes
the usual linear trajectories (c) take paths that miss the ion (d), which when added to the effectof the magnetic Lorentz force (e) achieve one recollision per cycle (f), re-enabling the harmonic
emission. Adapted from [102].

Because the z-directed force from the forward ellipticity oscillates in sign, the electron will
experience an alternating forward and backward drift each half-cycle (Fig. 4.8(d)). Even in thenoncollinear beam case, the magnetic v ×B drift will be along the forward z–direction (Fig. 4.8(e)).
Again, the electron will experience additive and canceling drift components in neighboring halfcycles (Fig. 4.8(f)). As a result of the half-cycle symmetry breaking of the electron’s trajectories,

111
the resulting harmonic emission from neighboring half-cycles can be viewed as an unbalanced
interferometer, giving a high sensitivity measure of how the magnetic Lorentz force influences theionized electron during the HHG process.
A quantum mechanical analysis of the harmonic emission can be performed for the noncollinear, bi-circular geometry following similar steps as presented in Section 1.3.1. In this geometry, the laser vector potential is modified from the plane-wave form of Eq. (1.38), with the following
form (at z = 0 and small crossing angle): cos(kx sin(θ)) 

EL0 


A(r, t) ≈ −sin(kx sin(θ)) cos(ωL t)

ωL 


0

(4.16)

Although the small angle approximation results in near-zero amplitude for Az , larger crossing of
the two beams results in a periodic modulation along the z-axis. A maximal forwards-ellipticity isachieved when kx sin(θ) = π/2, in which case:




0




EL0 

A(r, t) = − cos(ωL t) 

ωL 


sin(θ) sin(ωL t)(4.17)

Because there is an experimental mechanism to adjust the forward ellipticity (crossing half-angleθ), it can be varied to control the amount of drift cancellation with δv×B as well as measureit. Because of the half-cycle asymmetry of the total drift, the returning electron’s wavefunction
overlap with the bound electron can be reduced for large drift (Eq. (4.12)), but small drifts can still
produce electron phase asymmetries, leading to the possibility to generate even harmonics. The
strength of the even harmonics should depend on the forward ellipticity, which varies spatially. As
a result, the far-field emission pattern of the even harmonics is predicted to be influenced by thisspatially-varying pattern. Recent experiments have used beam half-angles θ on the order of a few
degrees [130], resulting in ellipticities  = sin(θ) ≈ 9% (a sufficient amount to counteract significant
magnetic drifts).

112
With the vector potential from Eq. (4.16), the quantum action S(p, t, t0 ) can be calculated,leading to a nonzero harmonic dipole for even and odd harmonic frequencies. For a single atom
located in a maximal ellipticity region, the dipole emission strength was calculated and plottedfor various driving parameters in Figures 4.9 and 4.10. For IL = 1017 W/cm2 (nearly relativistic
intensity) 2 and a relatively short wavelength of 800 nm (Fig. 4.10), the magnetic Lorentz drift isshown to significantly reduce the harmonic emission from the ideal dipole scenario [21]. Partiallycompensating for the magnetic component by introducing a slight crossing (θ = 2◦ ), the harmonic
emission largely recovers and could be improved further.

Figure 4.9: Nondipole calculations of odd (black) and even (blue) harmonics produced in helium
for driving wavelengths 800 nm and 1.6 µm, intensities of 3.2 × 1014 and 1015 W/cm2 , and acrossing half-angle of θ = 4◦ . The arbitrary vertical scale is logarithmic, and z-polarized harmonics
are removed from the result. Even and odd harmonics are emitted with intensity ratios of ∼ 10−3
for 800 nm drivers and ∼ 10% for strong mid-IR fields at 1.6 µm and 1015 W/cm2 . Adapted from[102].

The importance of these findings is that the half-cycle symmetry breaking in the full quantum calculation predicts the presence of even harmonics at much lower intensities and shorter
wavelengths than the semi-classical, amplitude-only approach would suggest. Even harmonics resulting from nondipole effects should be detectable (within 2–3 orders of magnitude of the odd
harmonics) even for 800 nm drivers, and certainly for 1.6 µm drivers at 1015 W/cm2 . This new
result from the full quantum calculation indicates that the harmonic spectrum has a sensitive dependence on the quantum phase asymmetry of the returning electron. It is predicted that using
2

Note that this toy problem only considers microscopic emission, and macroscopic effects could prohibit phasematched HHG in such high intensity regimes, where the ionization levels would be large. Possible exceptions aredescribed in Section 5.3.

113

Figure 4.10: Calculations of the above-threshold harmonic emission from the Ne6+ ion for an 800
nm, IL = 1017 W/cm2 driver. The first pair of quantum orbits and the uniform approximation
were used for the calculation, and z-polarized harmonics were discarded. For zero crossing angle(linear polarization), nondipole effects (v × B) reduce the emission by two orders of magnitude
compared to the emission using the dipole-approximation. With θ = 2◦ , the resulting forward
ellipticity causes the harmonic emission to partially recover. Adapted from [102].

114
current OPA technology to achieve moderate mid-IR wavelengths, nondipole effects during noncollinear, bi–circular HHG should be measurable through the spectrum. Importantly, the reductionof total HHG flux as a result of Lorentz drift effects is not expected to be significant until driving
wavelengths ∼ 6 µm are approached (using optimal intensities) as discussed in Section 4.5.
In addition to being nonzero, the even harmonics can be easily distinguished from the typical
odd harmonics through their natural angular separation in the noncollinear geometry. As demonstrated in [130] for odd harmonics, momentum and energy conservation dictates that for an odd
photon to be emitted, an odd number of driving photons must be added together. Further, conservation of spin angular momentum for bi-circular drivers dictates that n photons are taken fromone beam and n + 1 taken from the other, giving the harmonic a net transverse momentum of±~kx = ±~k sin(θ) [103, 122]. For even harmonics, an even number of photons are added throughparametric conversion via the tensor operator r̂ ⊗ p̂ : ∇A. Ultimately, the even harmonic canabsorb -2, 0, or 2 units of ~kx transverse momentum, giving linear or circular polarizations. As aresult, the odd and even harmonics will be emitted at different angles and should be distinguishable
at a detector without the need for a grating spectrometer.

4.7

Experimental Signatures of the Lorentz Drift
There are a number of experimental indicators that can confirm the Lorentz drift effects

predicted in this chapter. One of the most important challenges when using mid-IR drivers is the
reduction in total HHG flux. It may be experimentally possible to distinguish the super-Gaussianscaling due to the Lorentz drift in Eq. (4.14) from the polynomial scaling due to quantum diffusion,
but would likely prove difficult. The quantum diffusion scaling is dominant and would obscure theLorentz drift contribution.
The Lorentz displacement is also expected to affect the HHG spatial mode. It has been
demostrated that as the laser wavelength is increased, the HHG emission from only one half-cycleof the driver is fully phase-matched, generating a single X-ray burst even for a multi-cycle driver(Chapter 2, [18]). Therefore, for sufficiently long driving wavelengths and ideal phase-matching115
conditions, an isolated HHG pulse should be produced with an asymmetric mode profile along thepolarization direction, as indicated by Fig. 4.7. Further, varying the intensity or carrier-envelopephase (CEP) of the driving pulse can cause the particular half-cycle that is phase-matched tochange, thus causing the asymmetric HHG mode to flip about the x-axis. Imaging this mode
onto an X-ray CCD camera and adjusting the intensity or CEP could reveal a signature of thespatially-dependent Lorentz drift. In contrast, for the lower photon energy range where multiple
half-cycles contribute to the HHG signal, a more symmetric mode is expected that could obscure
the Lorentz drift signature, but a full spatio-temporal attosecond pulse characterization techniquecould be employed to reveal the spatio-temporal asymmetry [58].
When multiple half-cycles of the driver contribute constructively to HHG, individual harmonics are spectrally resolved. Using single-color drivers, only odd order harmonics are observed dueto the emission of HHG bursts every half-cycle. This typical twice-per-cycle emission in the time
domain causes the energy spacing of harmonics in the spectral domain to be twice the fundamental
photon energy. Even harmonics are, therefore, not allowed. However, the Lorentz displacement
can break the half-cycle symmetry in some spatial regions and thus give rise to even harmonics.
Section 4.6 provides a method for observing even harmonics when using a noncollinear bicirculargeometry. For single beam drivers with finite spatial modes, even harmonics can also be observed.
Looking off-axis in Fig. 4.7, the recombination probability ratio is close to unity in the lower halfof the mode (i.e. δEz cancels δv×B ). In the next half-cycle, this spatial location will not experience
the same drift cancellation due to the sign flip of δEz but not δv×B . At this spatial position, there
will be bright emission, followed by weak emission during the next half-cycle, and so on. A portionof this signal will have the usual λL /2 periodicity, which produces odd harmonics. Additionally,
there will be a portion that has λL periodicity that will give rise to every harmonic, including theeven orders. By analyzing the asymmetry of Fig. 4.7, it is possible to predict the mode profile ofthe even and odd harmonics. By using an imaging spectrometer that can separate the individual
harmonics of a 10 µm driver near 1Up , one would expect to observe a signal similar to that shown in
Fig. 4.11. Here, the peak intensity of the even harmonics is close to 30% that of the odd harmonics,

116
but this ratio approaches unity as the driving wavelength and harmonic energy are increased (i.e.
20 µm and 2Up ). Again, note that this result includes only the modulations in amplitude, and that
half-cycle asymmetries in the electron’s quantum phase can shift the absolute spectral positionsof the harmonics [102]. Experimentally, resolving the individual harmonics with such long driving
wavelengths and large harmonic energies would prove to be difficult, but might be possible withthe use of a Fourier transform spectrometer, similar to the one used in Chapter 2, [18].
1½0Figure 4.11: Predicted HHG spectrum near 1Up using a driving laser with λL = 10 µm, w = 50
µm, and IL = 2.77 × 1014 W/cm2 . The moderate spatial and temporal asymmetry of the Lorentz
drift gives rise to even harmonics with a two-lobed structure along the polarization direction (y).

4.8Lorentz Drift Summary
The theory presented in this chapter has provided a deeper investigation of the full Lorentz

force and its effect during strong field ionization and recollision-based processes such as HHG.
From the analysis, it is clear that the Lorentz drift should not significantly affect the probability ofrecollision for driving laser wavelengths less than 6 µm and moderate intensities IL < 1015 W/cm2 .
For longer wavelengths, the magnetic Lorentz force can result in non-negligible electron drifts, but
this drift can be canceled by the influence of the longitudinal electric field Ez that arises in atight focus or in crossed beam geometries. The interplay between the two components produces aspatially-varying drift that is zero in some locations of the mode. Thus, the Lorentz drift should
not completely shut off the HHG process, even for hard X-ray HHG driven by 10 µm wavelengths
or longer.

Chapter 5

Pushing HHG to Higher Photon Energies

In Chapter 2, the temporal and spectral properties of HHG emission were investigated whenlong wavelength, mid-infrared drivers are used. To date, the hard X-ray regime has been inacces(1.6−1.7)

sible via HHG. Although there is an obvious route to this spectral region through the λL

macroscopic scaling law, ultrafast driving lasers with wavelength greater than 10 µm would benecessary. The unavailability of such high-power, ultrafast lasers has limited efforts in this regard.
It is possible to generate far-infrared light with ultrafast pulse durations using OPA or OPCPA
technologies. Additionally, self-phase-modulation of 10 µm CO2 laser light and pulse compression
could result in sufficient bandwidth to support several-cycle pulses at the necessary wavelengths.
Furthermore, the full Lorentz force and the associated longitudinal drifts are not expected to completely shut off the HHG process, as determined in Section 4.5. More detrimental to the process−(5−9)

would be the unfavorable flux scaling of λL[161, 120, 42, 98]. This scaling has been overcomein the past by taking advantage of the increased phase-matching pressure for long wavelengths,
which results in a larger emitter-density [160]. However, phase-matching pressures quickly reachsolid densities when such long wavelengths are considered, introducing further limitations such as
the solid medium’s damage threshold.
Achieving hard X-ray harmonics can be approached through alternative techniques that do(1.6−1.7)

not rely on increasing the driving laser wavelength. The cutoff photon energy scaling of λL

is a macroscopic scaling using a conventional single-beam geometry. The single-atom cutoff scales
as IL λ2L , which indicates that there exists an alternative route to increasing the harmonic cutoff:

118
through the peak intensity. Of course, the phase-matching requirement cannot simply be neglected,
but quasi-phase-matching (Section 1.4.4) is an option that can be considered. Additionally, multi(1.6−1.7)

beam geometries could bypass the typical λL

scaling. A noncollinear geometry that makes

use of high-order difference frequency generation (HDFG) is one possible approach that will be
discussed in Section 5.3. Alternative modes or polarization states could also alter the typical
phase-matching laws, and should be investigated for possible utility in hard X-ray HHG.

5.1

HHG using Cylindrical Vector Beams
Driving HHG with linear polarization has been a successful approach to frequency conversion.

The three step model of HHG (Section 1.2) makes use of the 1-D trajectory of the electron to ensure
a high probability of recollision and harmonic emission. Motion in non-polarization directions canbe detrimental to the HHG process, such as in the presence of significant magnetic Lorentz forces(Chapter 4) or driving fields with elliptical polarizations [154, 130, 129]. Bichromatic, bi-circulardrivers can result in high recollision probabilities, but each harmonic pulse in the HHG pulse train
has been shown to be locally linear, with an average HHG polarization ellipticity when measured
over the entire pulse train [128].
Aside from linear and circular, there are other polarization states that can be explored.
Cylindrical vector beams (CVBs) contain unique polarization states that can continuously varyfrom radial to azimuthal polarizations (Appendix A). Additionally, multiple modes of CVBs can be
combined to produce complex, spatially-varying polarization states. The microscopic and macroscopic high harmonic process can possibly be altered under these driving fields. Figure 5.1 shows
the spatial intensity distributions and polarization directions for the simplest radial and azimuthal
polarization modes (LG01 ):LG01 = √222
r e−r /w(z)2π w(z)(5.1)

For radially- and azimuthally-polarized CVBs with the ”doughnut” modes shown in Fig. 5.1,
the microscopic three step model should hold true. Although the polarization varies spatially, it119Figure 5.1: Intensity distributions and polarization directions for the radially- and azimuthallypolarized Laguerre-Gauss LG01 modes (Appendix A).

120
is locally linear on the length scale of the electron trajectories (on the order of nanometers fornear-infrared wavelengths). Thus, harmonic emission will also be locally linear. Overall, the HHG
mode will take on a similar shape and polarization as the driving field, though the nonlinear natureof the harmonic process will increase the amplitude contrast of the HHG mode.
CVBs, like Gaussian beams, exhibit interesting field properties when tightly focused. As
explained in Section 4.2, a longitudinal electric field component Ez arises when the mode size w
approaches the wavelength λ. In the linearly-polarized Gaussian case, the Ez component takes on a∂E

two-lobed spatial structure due to its proportionality with ∂ψψ , where ψ represents the polarization
direction. For the radially-polarized LG01 mode, this derivative is largest in the center of themode. Since the primary polarization component Er is zero in the center, Ez can be the strongest
component in a small region around the center of a tightly focused beam. Figure 5.2 depicts thefield strengths of the radial and longitudinal components throughout a tight focus.

Figure 5.2: Radial and longitudinal field strengths of a radially-polarized LG01 mode throughout
a tight focus. The longitudinal electric field Ez dominates over the radial component Er at thecenter of the mode in the focus.

A possible consequence of this strong longitudinal field is that Lorentz drift compensation

121
could be performed when driving with long wavelength CVBs. Another possibility is that harmonics could be generated from centrally-localized electrons that take on primarily longitudinal
trajectories. Ultimately, the single-atom harmonic emission will take on the Larmor radiation pattern from Fig. 1.15. Looking in the far field but close to the z-axis, this emission pattern takes ona similar ”doughnut” mode profile with radial polarization. Furthermore, macroscopic effects cancause the near-axis emission to be enhanced, provided that phase-matching is achieved. Otherwise,phase-matched HHG may be optimized at larger angles off-axis. Ultimately, radially-polarized
HHG could be possible by directly driving electrons along the radial directions in a loose focus, oralternatively by driving them in the z-direction in a tight focus.
Because there is roughly zero intensity at the center of a loosely focused CVB mode, it ispossible for there to be a phase singularity at that location. This is true for CVBs that containorbital angular momentum (OAM), which manifests as a continual phase shift along the azimuthal
direction: eilφ . That is, following a path along φ̂ from φ = 0 to φ = 2π, the phase of the driving beam
can shift a total of 2πl, with positive or negative integer l corresponding to the OAM topologicalcharge. Figure 5.3 illustrates the wavefront for such beams. 1

Locally, the field components areunchanged and develop in time according to the laser frequency. Thus, HHG is possible on the
microscopic scale. Macroscopically, the harmonic emission would inherit the spiraling phase of thedriver. However, due to the nonlinear nature of the process, the integer l would be multiplied by
the harmonic order q, resulting in a much more rapidly varying phase and much higher OAM forthe harmonic beam [50]. In [59] and [44], multiple OAM driving beams are combined to allow
tunability of the OAM contained within the HHG output.
In all of the HHG experiments performed to date using CVB drivers, focusing geometries ingas jets/cells were used. However, gas-filled hollow waveguides are valid candidates for CVB HHG
since there exist radially- and azimuthally-polarized guided modes: the T E0q and T M0q modes,
respectively (Appendix A). Future work could investigate HHG driven by these guided modes inhollow waveguides, which would result in an HHG flux benefit from the larger pressure-length1

Recently, fractional orbital angular momentum beams have been theoretically investigated [165].

122

Figure 5.3: Wavefront and intensity shading for a l = 2 OAM beam.

123
products achievable in a guided geometry. Finally, HHG driven by long wavelength CVBs could
show advantages in terms of cancellation of the magnetic Lorentz force by the strong Ez fieldcomponent that is present on-axis in tightly focused CVBs. Therefore, CVB-driven HHG could bea route to reaching the hard X-ray regime.

5.2

HHG in Bragg Waveguides
To date, the highest photon energy measured from an HHG source is in the soft X-ray regime

at 1.6 keV, resulting from the use of a long wavelength 3.9 µm driver [161]. The HHG geometry
used for this demonstration was a high pressure gas-filled waveguide with an inner diameter of 200µm and length of 5 cm. The waveguide was supplied with a gas pressure up to 80 atm. Due toharmonic reabsorption concerns (Section 1.4.5), vacuum must immediately follow the high pressuresection. Necessarily, there exists a large pressure differential over a relatively short distance after
the HHG medium. From basic principles in fluid dynamics, the P V throughput (Q) through a pipeof length L and radius R in the laminar regime is given by: 4d(P V )πR
P1 + P2
Qpipe =
= Cpipe (P2 − P1 ) =(P2 − P1 )
dt
8ηL
2

(5.2)

Here, η is the dynamic viscosity of the fluid. It is clear from Eq. (5.2) that the gas flow rate scales
strongly with the pressures P1 and P2 . If the lower pressure is at vacuum P1 ≈ 0, then the flowrate Q ∝ P22 . Thus, handling the high gas flows resulting from phase-matching pressures up to 80
atm requires a considerable effort in pumping speed and geometry.
Long wavelength HHG drivers that require higher pressures (Fig. 1.27) represent a significant
experimental challenge. The gas flow problem can be partially alleviated by using smaller radius
waveguides since the conductance scales as R4 . This solution has the additional advantage ofrequiring less pulse energy for the driving laser to reach equivalent peak intensities, since theintensity of the driver scales as R2 (assuming all else held constant). This is especially advantageous
in relaxing the front-end laser requirements for long wavelength drivers, since these sources generally
require complicated frequency conversion schemes. However, long wavelengths guided in small

124
diameter capillaries can suffer from significant propagation loss. As described in Appendix A, thecomplex propagation constant γpq has an imaginary component that depends on the ratio λ/a,and this imaginary component is responsible for propagation loss. Even in the case described in[161], 3.9 µm light guided through a 200 µm diameter waveguide for 5 cm will only transmit amaximum of 72% of the light coupled in. Decreasing the waveguide diameter further would resultin significantly more propagation loss that would influence the ability to ionize and phase-matchthe HHG process over centimeter length scales.
Eliminating the detrimental effects of high propagation loss is desirable for efficient, phasematched HHG driven by long wavelengths, while maintaining a manageable gas flow rate. One
approach to reduce the propagation loss is to use Bragg waveguides. These waveguides are similar tosimple hollow capillaries, except that the inside wall of the waveguide contains a multilayer coating(Fig. 5.4). This multilayer is effectively designed to provide high reflectance of the guided light,
similar to Bragg mirrors. Bragg mirrors utilize quarter-wave optical thicknesses for alternatingly
high and low index materials, resulting in constructive interference for the reflected beam. The layer
thicknesses can be designed to reflect a band of wavelengths, and they must be adjusted to accountfor the angle of incidence. For a guided EH11 mode, the angle of incidence is not immediately
apparent since the mode has a flat wavefront and propagates along the length of the waveguide.
However, a transverse wavenumber k⊥,σ can be determined by taking the standard deviation of
the mode’s spatial frequency content. The effective incidence angle can then be calculated: θσ =cos−1 (k⊥,σ /k). The transverse wavenumber k⊥,σ is inversely proportional to the waveguide radiusa, so the effective incidence angle θσ is less glancing as the ratio a/λ decreases. Bragg layers forthe inside wall of the waveguide can therefore be designed with θσ in mind.
Of course, this simplistic effective-angle approach is a first order approximation. A fullelectromagnetic solution for the mode and its propagation constant can be numerically calculated,as demonstrated by Badding et al. in [17] through the use of COMSOL Multiphysics® software.
In their work, alternating layers of silicon and silica are deposited through high-pressure chemical
vapor deposition (Fig. 5.4). Silicon and silica are ideal materials for the Bragg layers due to their

125

Hollow Core
SilicaSi
SilicaSi
SilicaSi
SilicaSi
SilicaSi

SilicaCladding

Figure 5.4: Scanning electron microscope image of a Bragg fiber. Alternating Si and Silica layers
are visible on the inside wall of the capillary.

126high index contrast and high damage thresholds. These Bragg waveguides are ideal for handlingthe high peak powers and intensities that are present during HHG conditions, and their ability toprovide a spectral band of high transmission (even for low values of a/λ) makes them useful for HHG
driven by mid-IR light in small diameter capillaries. Experiments are currently being performed togenerate and characterize mid-IR driven HHG in small diameter Bragg waveguides. Success in thiscurrent effort could provide a bright, stable light source in the water window regime (284–541 eV),lending itself to high resolution microscopies and spectroscopies of biologically-relevant samples.

5.3

Noncollinear High-Order Difference Frequency Generation
In Section 1.4.1, the equations governing phase-matching were presented. Before the standard

single-beam scenario was investigated in detail, the general vectorial phase-mismatch was written
in Eq. (1.62) as:∆k ≡ kQ (ωQ ) −X

mj kj (ωj ) = 0

j

ωQ =

X

(5.3)

mj ωj

j

Here, we see conditions that ensure momentum and energy conservation for the HHG process,regardless of how many beams are involved or what their frequencies and propagation vectors
are. However, the integers mj are not constrained to be positive, indicating that for multi-beamgeometries, a high-order difference frequency generation (HDFG) process is possible. In this section,
consider two plane-wave driving fields propagating in different directions (k̂1 and k̂2 ) in the x-z
plane, each with its own frequency (ω1 and ω2 ) as depicted in Fig. 5.5. Assume these fields areinteracting in a partially ionized gas medium. In this situation, Eq. (5.3) can be written as:∆k = k(ωQ ) − m1 k1 (ω1 ) − m2 k2 (ω2 )(5.4)
ωQ = m1 ω1 + m2 ω2
To satisfy the phase-matching condition in this vectorial situation, both the harmonic angle
θQ and magnitude kQ (ωQ ) ≡ |kQ (ωQ )| must match the angle and magnitude of the sum [m1 k1 (ω1 )+

127

k1

ω1

ωQ

Θ1/2

x

kQ

ΘQ
z

Θ1/2

ω2

k2

Figure 5.5:
Geometry for a two-beam noncollinear frequency upconversion process (HHG or
HDFG). Note that for HDFG (m1 > 0 and m2 < 0), the harmonic emission angle lies outside thesector spanned by the two driving vectors: θQ > θ1/2 .

128
m2 k2 (ω2 )]. These can be considered separately:θQ = tan−1
m1 k1 (ω1 ) − m2 k2 (ω2 )tan(θ1/2 )m1 k1 (ω1 ) + m2 k2 (ω2 )kQ (ωQ ) = m1 k1 (ω1 ) cos(θ1/2 − θQ ) + m2 k2 (ω2 ) cos(θ1/2 + θQ )

(5.5)

(5.6)

The magnitudes for each of the wavevectors can be calculated by summing the vacuum, atom, ion,
plasma, and geometrical contributions described in Section 1.4.1. Solving Eq. (5.5) and Eq. (5.6)
simultaneously can be performed numerically, allowing straightforward exploration of the large
parameter space of the problem {ω1 , ω2 , m1 , m2 , θ1/2 , P̄ , η}.
HHG in two-beam noncollinear geometries have been experimentally investigated for multiple colors [14] and even bi-circular polarization states [130]. Further, HDFG has been observed
experimentally [51, 145]. One benefit to HDFG is that phase-matching can occur at ionizationlevels that exceed the conventional critical level for single-beam HHG. This is possible by taking
advantage of the negative value for m2 and the large harmonic emission angle, both of which reduce kQ (ωQ ) to a greater degree than k1 (ω1 ) or k2 (ω2 ). This was predicted and experimentally
verified by L’Huillier et al. in [51], where a strong fundamental beam at 800 nm was crossedwith a weak (5%) second harmonic beam (400 nm) in an Argon gas cell. The crossing angle wassmall (θ1/2 ≈ 1 mrad), the pulse duration was ∼ 40 fs, and the peak intensity was approximately
2 × 1014 W/cm2 . They observed HDFG photon energies up to ∼ 40 eV, and a maximum of 2
second harmonic photons subtracted (m1 = 22–24, m2 = −2). Although HDFG was demonstrated
in this work, the maximum harmonic energy observed (∼ 40 eV) did not exceed the single-beamphase-matching cutoff (∼ 50 eV for 800 nm in Ar). This could be due to the low peak intensity
used, which would give a single-atom cutoff of ∼ 54 eV. It is possible the low crossing angle used
did not allow phase-matched HDFG up to the single-atom cutoff. However, the ionization level
under these phase-matched conditions was approximately 2–3 times the critical ionization level for
Ar (ηc ≈ 4.5%), indicating that HDFG can bypass the conventional phase-matching limitations for
single-beam, collinear geometries.

129
HDFG was again performed in [145], where slightly larger crossing angles θ1/2 ∼ 25 mradwere used. For the HDFG case presented, a strong 385 nm beam and weaker (10%) 790 nm beam
were focused into an Ar gas jet. The pulse duration was 45 fs, and the combined intensity was
3 × 1014 W/cm2 . They report a theoretical maximum HDFG order of 29 (for the 790 nm beam),
representing a photon energy of 43 eV, though the maximum measured energy for the strong 385
nm experimental case is not clear from the text. In any case, the conventional phase-matching
cutoff in Ar at 400 nm is ∼ 29 eV, or ∼ 50 eV for 790 nm. Despite phase-matching HDFG in thepresence of high ionization levels (4 × ηc ≈ 56%), the maximum photon energy observed did notexceed the single-beam phase-matching cutoff for one of the wavelengths used (790 nm: EP M C ∼ 50eV), when HDFG should allow phase-matching significantly beyond either of the traditional cutoffs.
It is surprising that even though higher ionization levels can be present during phase-matched
HDFG, no experiment has demonstrated photon energies that exceed the typical single-beam phasematching cutoff EP M C for the contributing wavelengths. This can most likely be attributed to thesmall crossing angles, low pulse energies, and choice of gas used in the two experiments describedabove. Increasing the crossing angle θ1/2 allows for more extreme ionization levels to be present
while phase-matching HDFG, which would allow higher peak intensities (and therefore higher
single-atom cutoffs) to be used. Due to the linear scaling of the single-atom cutoff with intensity,
much larger pulse energies might have been necessary to noticeably exceed the conventional phasematching cutoff. It is not specified in the past work whether significantly increased driving pulse
energies were attempted. It is possible that the choice of low-Ip Ar gas for the harmonic medium
would cause highly dynamic multiple-ionization that would limit the temporal window over which
phase-matched HDFG could occur.
To thoroughly investigate whether HDFG could significantly bypass the restrictive singlebeam phase-matching cutoff, a new experiment could be performed with the following conditions:(1) Large crossing angles θ1/2 that allow phase-matching of HDFG processes even at 100%
ionization levels.

130(2) Large enough peak intensities for a single-atom cutoff that is significantly greater than theconventional EP M C .
(3) A harmonic medium with high first- and very high second-ionization potentials (He) suchthat there is a significant portion of time where the ionization level is stabilized close to100% despite the very high peak intensities present.
The third point can be important for ensuring multiple half-cycles of the driving lasers contributeto the emission, which increases the emitted flux and can eliminate the need for carrier-envelopephase (CEP)-stable driving lasers. Electrons that contribute to the HDFG signal in this ∼ 100%
ionization case would be given from the He+ ion, resulting in doubly-ionized He while the electron
is in its propagation step (HHG from ions is considered in [134]).
As an example, numerical calculations were performed to determine the phase-matchingconditions when two crossed driving beams generate extreme ionization levels up to ∼ 100% in
He. It is assumed that there is a strong primary driver at 800 nm and a weak secondary beam
(to simulate the situation where m2 < 0 and |m1 | >> |m2 |, which results in θQ ≈ θ1/2 ). The
geometric phase-mismatch terms were ignored, which is reasonable for loose focusing geometries ina gas jet/cell. Table 5.1 shows the calculated phase-matching conditions and the resulting harmonic
outputs. Note that for the temporally stable case of ∼ 100% ionized He, there is a range of pulse
durations dt that can give the desired ionization level at the peak of the pulse. This is not true forthe cases where η < 100%, since the ionization level (and ∆k) changes rapidly past ∆k = 0.
The driving wavelengths used in Table 5.1 were chosen since ultrafast laser technology atthese wavelengths has reached impressively high peak power levels, enabling the most extreme
HDFG configurations listed. Long wavelength drivers in the mid-IR regime could be considered,provided that a stable, high power, ultrafast OPA or OPCPA is available [47, 139]. With such adriving source, the single-atom cutoff can increase dramatically, possibly bringing the hard X-ray
regime within reach through phase-matched, mid-IR driven, noncollinear HDFG.

131
Table 5.1: Calculations for several example driving conditions during noncollinear HDFG wherephase-matching is achieved. The gas medium is He. The left side of the table specifies the conditions
that satisfy phase-matching. The middle section gives the characteristics of the phase-matched
HDFG output. The right side states the minimum laser peak intensity to achieve the the HDFG
photon energy ESAC = EQ , and the corresponding pulse duration that results in ionization level ηat the center of the pulse.
λ1 (nm)
800
800
800
800

5.4

λ2 (nm)
800
800
400
400

m1
250
600
400
1000

m2

θ1/2

-1
-10
-5
-8

14◦
19◦
10◦
20◦

P̄
0.15
0.5
0.23
0.5

η

θQ

40%
100%
90%
100%

14.1◦
19.6◦
10.5◦
20.6◦

EQ (eV)IL (W/cm2 )

dt

386
915
605
1,526

1.9 × 1015

20
30–10020
20–40

4.7 × 1015
3.1 × 1015
7.9 × 1015

Future Work in Mid-IR Driven HHG
Continuing to push the HHG photon energy and flux to higher levels has a number of ben-efits for materials and light science applications. For instance, coherent diffractive imaging (CDI)
is a powerful form of optical microscopy. Using HHG or other X-ray light sources, CDI can obtain the complex transmission/reflection profile of samples at the nanometer scale. In CDI, theachievable spatial resolution achievable is limited by the experimental geometry (NA), the probe
wavelength (λ), and its spectral bandwidth (∆λ). To achieve better spatial resolution, a smaller
probe wavelength can be used. Extending the HHG probe energy from the EUV (13 nm) into thewater window (2.3–4.4 nm) would be a natural next step, particularly because light in the water
window region transmits highly through water, making it a strong candidate for performing CDI on
biological samples suspended in water/ice. Furthermore, there is a strong need for high resolution
lithography and microscopy for quality control in the semiconductor industry, which is continually
taking steps to reduce feature sizes.
One difficulty with using a standard HHG light source for water window CDI is that the conventional approach to reaching the water window requires mid-IR drivers, which results in naturally
broad spectral bandwidths as discussed in Chapter 2. With such a large value for ∆λ/λ, there will
be a detriment to the spatial resolution achievable through CDI. It is possible to narrow the spectral bandwidth through the use of narrow bandpass dielectric mirrors and an appropriate selection

132
of foil filters. An X-ray monochromator can also be designed using glancing incidence gratingsand mirrors. Unfortunately, these approaches reduce the HHG flux that would be incident on thesample, thus requiring long integration times and high levels of stability. This effect, combined−(5−9)

with the unfavorable HHG flux scaling law λL, makes mid-IR driven HHG CDI a challenge,

though not impossible. Efforts are currently being made to use the aforementioned methods toachieve new levels of spatial resolution through water window CDI. In particular, the use of long
Bragg waveguides designed for mid-IR wavelengths (Section 5.2) can allow a greater level of HHG
flux to build up due to the increased pressure-length product.
Furthermore, alternative approaches can be devised to avoid some of the limitations fromthe conventional HHG bandwidth and flux. For instance, the traditional HHG cutoff energy scalesas λ(1.6−1.7) according to the phase-matching constraints. This scaling has held true for infrared
wavelengths, but has been overcome using ultraviolet drivers to perform HHG in multiply-ionizedplasmas. In [134], HHG from ions was demonstrated, resulting in a previously unexpected harmonic
cutoff > 280 eV using 270 nm drivers (Fig. 5.6). Furthermore, the HHG flux for these high energy
regions were equal to or greater than the flux from longer wavelength drivers, which is predictablefrom the reduced amount of quantum diffusion. Additionally, the HHG emission was observedin discrete harmonics, as opposed to a supercontinuum, making this UV-driven light source astrong candidate to perform water window CDI. As discussed in Section 5.3, noncollinear highorder difference frequency generation could also be performed to produce high-flux, monochromatic
harmonics at photon energies beyond the typical phase-matching cutoff.
HHG at high photon energies, regardless of the driving configuration, is useful for a plethora
of applications, ranging from high resolution CDI microscopy to broadband XAFS spectroscopy.
The attosecond time-scales of the HHG pulses makes this light source unique and well-suited for
dynamic measurements, such as dynamic XAFS. By pumping a thin lattice with visible light, andsubsequently interrogating with a broadband HHG probe, the perturbation to the lattice from thepump can be determined. Acoustics, thermal transport, and the density of states for phonons[91, 133, 52] can all be probed on ultrafast time scales in this experiment. Efforts are currently

133

Figure 5.6: Bright HHG from atoms and ions for UV to mid-IR driving lasers. The experimental
data are plotted as circles. The solid lines plot the theoretical full phase-matching limits, including
only the index of neutral atoms. The dashed lines also include the refractive indices of ions, which
extend UV-driven HHG effective phase matching into the soft x-ray region. Adapted from [134].

134
being made toward this goal of dynamic XAFS probed by mid-IR driven HHG.
Finally, driving with much longer wavelengths beyond 10 µm can bring hard X-ray HHG
within reach, but the three step model will be influenced by relativistic Lorentz drift effects (Chapter 4) that should be noticeable in the HHG amplitude for driving wavelengths beyond 6 µm.
Experimental verification of the Lorentz drift can be obtained through the approaches detailed in
Section 4.7. Furthermore, it is possible to directly measure the Lorentz drift through measurementof the ionized electrons’ momenta. Through a technique called velocity map imaging (VMI), ionizedelectrons can be imaged in a transverse momentum space. A strong laser ionizes atoms in a gas
jet, after which the ionized electrons are accelerated by an external electrostatic potential. The
potential is designed to form a momentum-space image of the electrons at a microchannel plate
detector [33]. For short wavelength drivers, the atomic and molecular states can be probed without the influence of relativistic effects from the laser field. However, long wavelength drivers will
generate these effects that could then be observable in the VMI measurement [73]. The magnetic
component of the Lorentz force would tend to shift the electron momentum distribution along theforward z-direction of the laser. The Ez component would tend to broaden the distribution due toboth the forward and backward contributions. 2

However, if the gas jet size is small comparedto mode (and if the mode is sufficiently small for a significant Ez component) then the jet could
be translated across the mode, thus sampling different regions of the mode (Fig. 5.7). Thus, theEz component would vary as a function of the jet’s position, giving a direct measurement of theEz field profile along one dimension. This measurement could impart empirical verification of thetheory presented in Section 4.7.

2

Single-cycle drivers may be necessary to probe just the Lorentz force that exists during the three step model,
since the electrons that make it to the VMI detector will inevitably be subjected to the remainder of the field following
ionization. The influence of multiple cycles after ionization could distort the desired signal.

135

Ez+
v ×B
Ez-

Figure 5.7: Scanning VMI concept to measure the longitudinal Lorentz force during long wavelength strong field ionization in a spatially-resolved way. A small diameter gas jet can be translated
across the laser mode to sample the Ez and v × B contributions separately. A predicted signal at
the VMI is shown for three locations of the scan.

Chapter 6

Concluding Remarks

Light is, indeed, an incredibly powerful tool for making observations of the physical world.
In particular, light in the EUV and X-ray regimes enable unique and higher resolution measurements than is possible using longer wavelengths. High-order harmonic generation provides a route
to produce light in these useful spectral ranges starting with lasers operating at more accessible
wavelengths. Free-electron lasers and synchrotrons are complementary technologies that can exceed the capabilities of HHG in some aspects, but have certain drawbacks where HHG can fill thegaps. Investigating the physical mechanisms and limitations of the HHG source can reveal newroutes for improving the technology, thus advancing its ability to be applied in highly sensitive
experiments. Some examples of applications where HHG has shown unique utility are coherent
diffractive imaging, dynamic measurement (Chapter 2), and broadband spectroscopies (Chapter
3).
This dissertation has covered several illuminating studies of HHG in the temporal and spectral domains when the process is driven by long wavelength, mid-infrared light. Interestingly, thecharacteristics of the harmonic emission are highly dependent on the driving laser parameters andgeometries. As the driving laser wavelength is increased, the harmonic cutoff and bandwidth naturally broadens, while the emitted pulse train reduces in length until a single isolated burst ofphase-matched harmonics with sub-femtosecond duration is achieved. The resulting supercontinuum has particular utility in X-ray absorption fine structure spectroscopies, where the nanoscalelattice structure can be probed. These spectroscopies have been performed using the broadest HHG

137
bandwidth achieved to date, extending to 1.6 keV. Pushing this harmonic cutoff further would
conventionally require the use of longer wavelength drivers approaching the far-infrared regime.
Gaussian focus HHG geometries would typically be necessary for these long wavelengths, but theuse of novel Bragg waveguides could enable high flux, high cutoff HHG in waveguides. However,long driving wavelengths can also result in relativistic effects during the three step model of HHG,
resulting in longitudinal Lorentz drift effects that could cause the electron to miss its parent ion.
Accounting for all of the forces involved does not indicate HHG would be shut off entirely, however,and it is possible for HHG to occur even with driving wavelengths beyond 10 µm and harmonic
cutoffs in the hard X-ray regime. The use of cylindrical vector beams or multi-beam geometries canalso be used to compensate for relativistic effects, and also to create new phase-matching conditions
for sum and difference frequency processes. Through high-order difference frequency generation ina two-beam noncollinear geometry, it could be possible to significantly exceed the conventional
phase-matching cutoff, opening up the possibility to reach the soft X-ray regime or further whensimply using the intense, highly accessible, and highly HHG-efficient colors from Ti:sapphire lasers(800 nm). Pushing the limits of the HHG spectral characteristics would certainly enable new levelsof capability in CDI microscopies, X-ray spectroscopies, and more.
In the past decade, HHG technology and theory have progressed at tremendous rates, andthere are many more directions in which HHG can grow. The future looks promising for frequency
conversion, especially as driving laser technologies advance to higher intensities, shorter pulse durations, and longer wavelengths. Using next-generation driving lasers in novel geometries, withthe guidance of high-level computations, and through collaboration within the expanding HHG
science community, it is very likely that hard X-ray harmonics (among other new capabilities) will
be achieved within the near future.

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Appendix A

Hollow Waveguide ModesyLε , µ0θrx2a

ε0 , µ0z

Figure A.1: Illustrations of the hollow waveguide geometry.

Waveguides have a different set of boundary conditions than free space, resulting in a different
set of basis modes. In free-space, the basis modes are plane waves, which can be added together toconstruct any arbitrary field profile. For cylindrically-symmetric hollow capillaries, Bessel modes
form a basis. Marcatili solved for the full electromagnetic field components for the hollow waveguidemodes in [79] (neglecting powers of λ/a larger than 1). Three categories of modes are discussed:
circular/transverse electric (TE), circular/transverse magnetic (TM), and hybrid (EH). The TE andTM modes have azimuthal and radial polarizations, respectively. They also contain a ”doughnut”
intensity profile, with zero amplitude at the center of the mode. HHG can be possible using thesedriving modes, as discussed in Section 5.1. Here, the linearly-polarized modes will be investigatedin detail. The general EHpq mode has the following field components within the hollow region of
the waveguide:150•Internal field components for the EHpq mode in cylindrical coordinates:
iupq p 2
Epq,r = Jp−1 (ki r) +n − 1Jp (ki r) sin(pθ)ei(γpq z−ωt)2kr"#
iu2pq p 20
Epq,θ = Jp−1 (ki r) +n − 1Jp (ki r) cos(pθ)ei(γpq z−ωt)
2pka
upq
Epq,z = −i
Jp (ki r) sin(pθ)ei(γpq z−ωt)kar
0Hpq,r = −
Epq,θ
µ0
r
0
Epq,r
Hpq,θ =
µ0r
0Hpq,z = −Epq,z cot(pθ)
µ0

(A.1)

For a EHpq mode to be linearly-polarized along the y-direction, it is easy to show that p must be
equal to 1, resulting in the following field components:•

Internal field components for the y-polarized EH1q mode in Cartesian coordinates:
E1q,x = 0
 2
iu1q p 2
u1q x2 0y
E1q,y = J0 (ki r) +J1 (ki r) +J1 (ki r) ei(γ1q z−ωt)
n −1
2kr2
r
a
u1q  y 
E1q,z = −i
J1 (ki r)ei(γ1q z−ωt)
ka r
r
0
H1q,x = −
E1q,y
µ0

(A.2)

H1q,y = 0
r  
0 x
H1q,z = −
E1q,z
µ0 y
Here, B = µ0 H, Jp is a pth order Bessel function of the first kind, upq is the q th root of the equationpp
Jp−1 (upq ) = 0, n = /0 is the complex refractive index of the waveguide material, r = x2 + y 2 ,q
2 , and:k = 2π/λ, ki = k 2 − γpqi (n2 + 1)
1  upq 2√
γpq = k 1 −1−
2 ka
ka n2 − 1

(A.3)

These solutions are valid for λ << a. Under this condition, the z-component of the mode is small,as are the latter terms in the square brackets (terms containing higher order Bessel functions).

151

Furthermore, the boundary conditions at r = a allow the replacement: J0 (ki r) → J0 u1q ar . With
these assumptions, the linearly-polarized modes have the following electric field component along
the y-direction:r  i(γ1q z−ωt)E1q,y ∼e
= J0 u1qa(A.4)

The EH1q modes form a basis that can represent any linearly-polarized mode profile withcylindrical symmetry. Thus, one can consider an input laser mode incident on the entrance of ahollow waveguide. The amount of light coupled into each of the waveguide modes can be calculated
through mode decomposition, as investigated in [1]. Consider an input Gaussian mode with power
normalization:s
Einput (r) =
Z ∞

Z 2π
dθ
0

2
2
2
e−r /w(z) ŷ
πw(z)2(A.5)

dr r |Einput (r)|2 = 1

0

The normalized basis functions for the decomposition are:√2rĒ1q (r) =J0 u1qŷ
aJ1 (u1q )a
Z 2π Z adθdr r |E1q,y (r)|2 = 1
0

(A.6)

0

The input field can be represented as a sum of the basis functions:
Einput (r) =X

c1q Ē1q (r),

(A.7)

q

where
Z 2π

Z a

dr r Ē1q (r)∗ Einput (r)00Z 2π Z a2r  −r2 /w(z)2dθ
dr r J0 u1qe=√a
π aw(z)J1 (u1q ) 0
0

c1q =

dθ

(A.8)

The power coupled from the input Gaussian beam into each mode is simply P1q = |c1q |2 . Note that
this derivation assumes a flat wavefront for the input field (z = 0, w(z) = w0 ). If, for instance, there
is wavefront curvature (indicating a Gaussian mode that is focusing or defocusing at the waveguide
entrance), then there would be an additional phase term inside the integral of Eq. (A.8) taking the

152
form:

exp

ikr2
2R(z)





r2
= exp i2πN 2
a

(A.9)

Here, R(z) represents the wavefront radius of curvature, and N is the number of waves between
the center of the waveguide and the wall at r = a. Figure A.2 shows the power coupled into thefirst five EH1q modes for input Gaussian modes with varying radii and wavefront curvature.
Because the complex wavenumber γpq is dependent on the mode indices p and q, each mode
will propagate at a different phase velocity, and with a different propagation loss. Therefore, any
input field that couples into more than one waveguide mode will change its spatial profile during
propagation. Beating effects will result in modulation of the peak intensity along the longitudinal
direction. For long propagation distances, the modes with the lowest loss (those with the lowest
values of p and q) will dominate.
The analysis performed for the linearly-polarized Gaussian modes coupled to the hybrid
waveguide modes can be replicated in the case of radially- and azimuthally-polarized modes. In
this scenario, the relevant waveguide modes are the TE0q and TM0q modes from [79]. As mentioned
previously, these have ”doughnut” intensity profiles. Using the TE0q mode to represent this profile,the basis modes have the following normalized amplitudes:
TE0q = Ē0q (r) = √1r
J1 u2qaπ aJ0 (u2q )(A.10)

As for the free-space modes with radial/azimuthal polarizations, the solutions are Laguerre-Gauss
beams (or cylidrical vector beams) as described in [175]. The lowest order mode is the LG01 mode,
which has the following normalized amplitude:
LG01 = √222
r e−r /w(z)2π w(z)(A.11)

Note that the LG01 mode contains the same spot width parameter w(z) as for Gaussian modes, but
it does not correspond to the radius at which the intensity drops by 1/e2 . Instead, the maximum√
intensity of the ”doughnut” mode occurs at a radius r = w/ 2. The intensity at r = w is equal
to the maximum intensity multiplied by (2/e), where e = 2.718 is the natural Euler number (Fig.

153

A

P 11qqPower coupled from Gaussian mode into EH
EH11qq modes

1
0.8

EH11

0.6

EH12

0.4

EH13

w0/a = 0.6435

EH14

0.2
0
0

EH15
0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

w0/a

B

1
0.8

P 11qqPower coupled from (de)focusing Gaussian mode into EH
EH11qq modes
w/a = 0.6435

EH11
EH12

0.6

Focusing

0.4

Defocusing

EH13
EH14

0.2
0
-5

EH15
-4

-3

-2

-1

0

1

2

3

4

5

N (# of waves from center to wall)

Figure A.2: The power coupled from Gaussian modes into the first five EH1q waveguide modes.
(A) Coupling of Gaussian modes with flat wavefront and varying mode radii. The optimal couplingof the Gaussian mode into the EH11 mode occurs for w0 /a = 0.6435, with 98% coupling efficiency.
(B) Coupling of Gaussian modes with optimal radii but with varying wavefront curvature.

154A.3). Following the same modal decomposition, the power coupled from the LG01 mode into the
TE0q modes can be calculated, and these are plotted as a function of w0 /a in Fig. A.4.
LG01 mode

1.0

Intensity

0.8

I/Imax = 2/e

I/Imax = 2/e

0.6
0.4
0.2
0.0
-2

r/w = 1/ 2
= 0.707

-1

0

1

2

r/w

Figure A.3: Plot of the LG01 mode intensity as a function of radius. The beam spot widthparameter w does not carry the same meaning as it does for a Gaussian beam, despite it following
the same w(z) equation through propagation (Eq. (1.76)).

155

Power coupled from LG01 mode into TE
TE00qq modes

1

P00qq

0.8

TE01

0.6

TE02
TE03

0.4

0
0

TE04

w0/a = 0.5642

0.2

TE05
0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

w0/a

Figure A.4: The power coupled from azimuthally-polarized LG01 modes into the first five TE0q
waveguide modes. The plots are the same for radially-polarized LG01 modes coupling to the TM0q
modes. The optimal coupling of the LG01 mode into the TE01 mode occurs for w0 /a = 0.5642,
with 96.7% coupling efficiency. The wavefront is assumed to be flat for this calculation.