Extreme ultraviolet frequency combs for precision
measurement and strong-field physics
byCraig Benko
BS, Santa Clara University, 2010

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
2016This thesis entitled:
Extreme ultraviolet frequency combs for precision measurement and strong-field physics
written by Craig Benko
has been approved for the Department of Physics

Prof. Jun Ye

Prof. John L. HallDate

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.

iii
Benko, Craig (PhD, Physics)
Extreme ultraviolet frequency combs for precision measurement and strong-field physics
Thesis directed by Prof. Jun Ye

The extreme ultraviolet (XUV) is a relatively unexplored spectral region for high-resolution
laser spectroscopy. Many atomic and molecular systems of fundamental interest lie in wait ofinvestigation, but the lack of highly coherent sources has forgone the ability to experiment. The XUV
frequency comb offers exciting new frontiers for fundamental physics and measurement science byenabling direct and highly coherent laser access to the XUV. Prior to 2012, our group demonstrated
the best levels of phase coherence in the XUV at the 10 MHz level and the most powerful XUV
light source originating from high-order harmonic generation with powers of 220 µW/harmonic.
The work in this thesis improves upon both of these metrics demonstrating coherence at the 62.5mHz level (eight orders of magnitude improvement) and power levels approaching 1 mW/harmonic(five times improvement). Our work shows that it is possible to produce XUV light with coherence
properties that rival that of visible light using the high-order harmonic generation process.
Leveraging XUV frequency comb technology, we also extend the work to probe strong field
physics in atomic and molecular systems. We use the phase stable light produced during high-orderharmonic generation to probe attosecond phenomena in atoms manifested in the intensity dependent
dipole phase. We also study strong-field light-matter interactions in molecular systems. Using
our femtosecond enhancement cavities, we perform field-free molecular alignment at unprecedented
repetition rates. This allows for a sensitive study of the strong-field interaction and allows thehigh-order harmonic generation process to be performed in an aligned molecular target.
As XUV frequency comb technology continues to mature, further gains in power levels are
anticipated. Additional applications in high-resolution spectroscopy, strong-field physics, solid-statephysics, and laser science will come to fruition.

Dedication

Briefly, you can only find truth with logic if you have already found truth without it.
–G. K. Chesterton
To my parents, for never ceasing to love and support me.
To my wife, for helping me keep my head on straight and always reminding me of what is truly
most important.
To Gram, for always being there for me (even when we were miles apart). I hoped dearly you would
see me finish, but I rest assured that you do indeed know and are at peace.

v

Acknowledgements

First and foremost, I need to thank my parents, Randy and Kay Lynn, for their continual
sacrifice on behalf of myself and my education. I am also very thankful when they did not panic
when I decided to switch from engineering to physics and for trusting me that I was making theright decision. I also have to thank them for not being too suspicious for passing up graduate schoolat a fancy east coast school to attend the University of Colorado. But, I think they would agree that
I made the right choice and I would do it again in less than a femtosecond (if you could possibly
make a decision that fast).
Santa Clara University made every effort to ensure my future success as a physicist. Whilethe entire physics department deserves my thanks, two in particular stand out. A special thanks to
Betty Young for giving me my first job in a lab. But possibly more important, she selflessly teamed
me up with, then a brand new, professor Chris Weber. I cannot thank Betty enough for setting
me up to work with Chris as it would mark the point at which I became addicted to working withlasers. And to Chris, thanks for showing me how to build an experiment and be a physicist. I still
do not completely understand how effortless it seemed for him to let me just start getting my hands
dirty building the lab’s first Ti:Sapphire and constructing the rest of the apparatus. I still try torecall how he guided me along when I find myself in similar situations. His patience and instruction
are nearly unrivaled. I am ever grateful for him showing me how cool lasers are, for stretching thelab budget to take me to a Gordon conference, for taking me up to see a beamtime at the ALS,
for all the help in applying for graduate school, and for being a great friend along the way. I also
have to thank my first lab mate, Dustin Ngo. It always seemed like he was a few steps ahead of

vi
my understanding of things. I owe a lot of my success to his help. I will never forget some of ouradventures outside of lab, especially sitting on the top of Half Dome. It is also hard to believe
our first experiment worked as well as it did, considering a $50 chunk of copper almost broke thebudget.
It is hard to imagine where I would be if Jun Ye did not give me the opportunity to join his
group back in 2010. He graciously offered me a position on the strontium experiments even though
I might have had my heart set on the XUV frequency comb experiment. Walking in to Jun’s lab
for the first time was certainly intimidating, but Jun was very welcoming and enthusiastic. It wasvery easy to feel right at home amongst what might look like total chaos to the casual observer.
However, as I would quickly learn, it was far from chaos but a living and breathing lab battling at
the forefront of what is experimentally possible. The lab he has nurtured is unrivaled in breadth
and ability, a strong testament to his ability as a leader and a scientist.
I think one of the most important and beneficial decisions Jun has made on my behalf waswhen upon my arrival he partnered me with a senior graduate student Michael Martin. My earlydays in the lab were spent under Mike’s guidance. He was wonderfully patient and a great experimentalist. Hopefully some of it rubbed off, but, as Mike would probably agree, maybe not his lab
notebook practices. My first couple years were spent on the not-so-glamorous aspects of trying tobuild the most stable laser in the world, but it was certainly a good way to build up experimentalist
chops. And, as it would turn out, that laser ended up being pretty good with all due credit going toMike. I think my time working with Mike ultimately gave me the confidence to ask Jun if I could
take over the XUV frequency comb experiment, my first interest in Jun’s lab. Jun graciously agreed
and I would depart from my short tenure on the strontium experiment to venture off into the XUV.
I would be remiss if I did not acknowledge the fellow strontium-ites I worked with during mytime on the experiment. To Michael Bishof, for keeping the work lighthearted without ever losing
the ability to be productive. And for welcoming me into the Knights (you still owe me a sword
fight). To Matthew Swallows, for answering all my atomic physics questions almost effortlessly andwithout getting too grumpy. And to Yige Lin, for being one of the nicest people to work along side.

vii
I am only sorry I did not get to overlap with him more.
Making the jump from the strontium experiments to the XUV frequency comb could havebeen a disaster if it were not for Thomas Allison, Arman Cingöz, and Dylan Yost to welcome
me. Unfortunately, my time with Dylan was brief, but it was nevertheless productive. He wasable to explain things with great clarity and his ability approach an experimental problem andimmediately see the solution was uncanny. Even in his absence, I regularly benefited from his timeon the experiment. Arman was also a great labmate and probably the only one I have had thatthoroughly documents his electronics. He was always willing to patiently help me. I still do notthink I can wipe the cavity’s . . . well you know . . . as well as he could. I overlapped with Tom thelongest. I will not forget how we fought to his very last day in lab to see a heterodyne beat deep inthe XUV and sure enough we saw it on his last day. It certainly was not for lack of trying or lackof singing “Beat It” 1 to coax it out of the machine. Even after leaving JILA, Tom was always oneof my most useful resources and he was always available to help. For that, I am very grateful.
The second and third wave of post docs have also been a great help. François Labaye is
owed a great deal of thanks for battling with the fiber broadening and nonlinear pulse compression.
Linqiang Hua was always willing to work hard and learn the experiment. He certainly made datataking possible and he was the first one brave enough to start putting molecules in the cavity
for HHG. Nadine Dörre was a visitor for only a brief amount of time, but she was instrumental
in improving the XUV comb flux by nearly 5×. I think some of the tricks we learned will evenbecome commonplace in future XUV combs. The newest arrivals are Gil Porat, Stephen Schoun,and Christoph Heyl. The experiment will be in good hands as they push to turn the XUV comb
into a soft x-ray comb (and possibly come up with a more clever name).
A special thanks is due to our collaborators at IMRA. Ingmar Hartl, Axel Ruehl, and Martin
Fermann delivered an excellent laser that I had the pleasure of using extensively. Ingmar and Axel
continue to be a great resource and I am indebted to their hospitality during a visit to their new
home at DESY in Hamburg, Germany.
1

While it was an earnest attempt to sing the popular song, the rendition certainly qualifies as a “Tom original”.

viii
Of course, I would like to thank the rest of the Ye lab team, especially the Mid-IR guys who
were the XUV team’s companions in the weekly group meeting. A special thanks to Jan Hall who
is probably the lab’s greatest treasure. He has helped the XUV experiment and myself in more
ways than one. He always brought a smile to lab (along with the Peanuts lunch box) and he wasalways willing to share his knowledge of all things lasers, RAM, and what often seemed like pure
wizardry. Attempting to mention everyone else that I have worked with will certainly end in failure.
Nevertheless, you know who you are and for all the help you have given me along the way, I amvery thankful.
A final acknowledgment is to my wonderful wife Monica. I owe a great deal of my sanity
to her constant love and support. It is funny how life happens, but I never anticipated moving toColorado for graduate school only to find my best friend out in Wyoming. I cannot wait for whatcomes next.

Contents

Chapter
1 Extreme ultraviolet frequency combs

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Optical frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.1

Carrier-envelope offset frequency detection and measurement . . . . . . . . .

10

1.2.2

Frequency comb stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2.3

Frequency comb performance characterization . . . . . . . . . . . . . . . . . .

18

1.3

Extreme ultraviolet frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.4

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.4.1

Spectroscopic applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.4.2

High-flux extreme ultraviolet source applications . . . . . . . . . . . . . . . .

25

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

1.5

2 Experimental apparatus

27

2.1

Amplifying ultrafast laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.2

The femtosecond enhancement cavity . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2.1

Cavity finesse, buildup, and contrast . . . . . . . . . . . . . . . . . . . . . . .

30

2.2.2

Gaussian beams and gaussian beam manipulation . . . . . . . . . . . . . . . .

35

2.2.3

Cavity geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2.4

Cavity mirror dispersion and the femtosecond enhancement cavity . . . . . .

43

x
2.2.5

Laser-cavity stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.2.6

Output coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

The laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.3.1

Nonlinear pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

2.4

Typical laser-cavity operating parameters . . . . . . . . . . . . . . . . . . . . . . . .

60

2.5

The chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.5.1

62

2.3

Ozone and plasma cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 High-order harmonic generation
3.1

3.2

3.3

69

Single atom response to a strong laser field . . . . . . . . . . . . . . . . . . . . . . . .

69

3.1.1

Ionization by an intense laser pulse . . . . . . . . . . . . . . . . . . . . . . . .

69

3.1.2

Simple model of high-order harmonic generation . . . . . . . . . . . . . . . .

72

3.1.3

The quantum description of high-order harmonic generation . . . . . . . . . .

74

3.1.4

Beyond the strong-field approximation . . . . . . . . . . . . . . . . . . . . . .

76

Macroscopic response and phasematching . . . . . . . . . . . . . . . . . . . . . . . .

79

3.2.1

Single-pass case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3.2.2

Enhancement cavity case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Power scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.3.1

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4 Interferometry in the extreme ultraviolet

96

4.1

Motivation and brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.2

The extreme ultraviolet interferometer . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.2.1

The acousto-optic modulator . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

4.2.2

Construction of the beam combiner . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.3

Beam combiner theory and scaling . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2.4

Extreme ultraviolet detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

xi

4.3

4.2.5

Extreme ultraviolet optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.6

The Seya-Namioka monochromator . . . . . . . . . . . . . . . . . . . . . . . . 111

Putting it all together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Phase noise and phase sensitive detection in the extreme ultraviolet

119

5.1

Optical phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2

Noise scaling of high-order harmonic generation . . . . . . . . . . . . . . . . . . . . . 124

5.3

Testing the limits of coherence in the extreme ultraviolet . . . . . . . . . . . . . . . . 126

5.4

Measurement of intensity dependent dipole phase . . . . . . . . . . . . . . . . . . . . 130
5.4.1

Theory of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4.2

Measurement systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.4.3

Results and data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6 Strong-field physics with molecules

142

6.1

Impulsive stimulated Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2

Field-free molecular alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3

Field-free molecular alignment experiments . . . . . . . . . . . . . . . . . . . . . . . 148
6.3.1

Single-pulse case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3.2

Multi-pulse case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4

High-order harmonic generation with molecules. . . . . . . . . . . . . . . . . . . . . . 155

6.5

Future potential of cavity-based field-free molecular alignment . . . . . . . . . . . . . 155

7 Conclusion and Future Prospects
7.1

160

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

xii
Bibliography

162

Appendix
A Numerical Code Repository

173

B Publication list

175

Tables

Table
1.1

Ionization potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1

ABCD matrix table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

6.1

Molecular parameters for field-free molecular alignment . . . . . . . . . . . . . . . . . 148

Figures

Figure1.1

NIST frequency chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

High-order harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

Optical frequency comb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.4

Super continuum generation and f0 detection . . . . . . . . . . . . . . . . . . . . . .

13

1.5

Yb:fiber frequency comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.6

Frequency comb characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.7

HHG and frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.1

Fabry-Perót ring cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.2

Finesse measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.3

Cavity geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.4

Impact of mirror dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.5

Pound-Drever-Hall laser stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.6

Intensity servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.7

Output couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

2.8

80 W Yb:fiber frequency comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.9

Nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.10 Nonlinear pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

2.11 Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

xv
2.12 Vacuum chamber vibration isolation . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.13 Gas dump design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

2.14 in-situ plasma cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.1

Ionization probability by an intense laser pulse . . . . . . . . . . . . . . . . . . . . .

71

3.2

Simple man model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.3

Strong–field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.4

Beyond the strong-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.5

Absorption limited phase matching . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.6

Phase matching in a fsEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.7

HHG detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

3.8

Super sonic jet and cavity mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

3.9

Effects of steady state plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.10 Phase matching in a fsEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

3.11 Extreme ultraviolet power comparison . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.1

Mach-Zehnder interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.2

XUV Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3

AOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4

XUV beam combiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5

XUV beam combiner schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.6

XUV reflectivity for boron carbide and gold . . . . . . . . . . . . . . . . . . . . . . . 110

4.7

Harmonic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.8

XUV reflectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.9

Relative harmonic powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.10 High harmonic heterodyne beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1

Phase noise scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xvi
5.2

Phase modulated XUV beatnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3

Test of XUV comb coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4

Intensity dependent phase time domain

5.5

Measurement schematic of intensity dependent phase . . . . . . . . . . . . . . . . . . 133

5.6

Intensity dependent dipole phase summary . . . . . . . . . . . . . . . . . . . . . . . . 137

5.7

Data set from an intensity dependent dipole phase measurement

5.8

Lock-in amplifier data for dipole phase measurment . . . . . . . . . . . . . . . . . . . 140

6.1

Field-free molecular alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2

Field-free molecular alignment apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3

Impulsive stimulated Raman scattering single pulse . . . . . . . . . . . . . . . . . . . 150

6.4

Impulsive stimulated Raman scattering multiple pulses . . . . . . . . . . . . . . . . . 152

6.5

Impulsive stimulated Raman scattering spectral shifts . . . . . . . . . . . . . . . . . 154

6.6

HHG with aligned molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.7

High-order revivals, coherence decay, and spectral modulator . . . . . . . . . . . . . 159. . . . . . . . . . . . . . . . . . . . . . . . . 131. . . . . . . . . . . 139

Chapter 1

Extreme ultraviolet frequency combs

1.1

Introduction
Optical frequency combs are a remarkable laser technology technology. They seamlessly inte-

grate the disciplines of ultrafast lasers with precision continuous wave (CW) laser stabilization andfrequency metrology. The technology is so powerful, it was recognized by a Nobel Prize awarded toJohn L. Hall and Theodor Hänsch in 2005 [1, 2]. That same year, the extreme ultraviolet (XUV)
frequency comb, an extension of the optical frequency comb to wavelengths below 100 nm, wasdemonstrated by the Ye group and the Hänsch group virtually concurrently [3, 4]. This marked thefirst step to performing laser spectroscopy at wavelengths previously inaccessible to traditional laser
technology. With a few exceptions [5], high-resolution laser spectroscopy below 100 nm is a vastly
uncharted territory. To a large degree, the motivation behind the XUV frequency comb is enablinghigh-resolution spectroscopic exploration in this part of the electromagnetic spectrum. Nevertheless, the XUV comb bridges the worlds of strong-field physics with that of precision measurementin much the same way as the original optical frequency comb bridged ultrafast science and precision
metrology.
To put the XUV frequency comb into the broader context of laser technology and the work inthis thesis, it is useful to start at the laser’s origin, the maser. Gordon and Townes were the first to
produce the ammonia maser [6, 7]. To characterize the maser, two versions were built to compare
using heterodyne interferometry, a trick we will also use extensively in this thesis. By comparison of
two such masers, it was immediately apparent that the spectral purity of the microwave signal and2the stability were quite remarkable. Schalow and Townes followed to show how this demonstration
could be feasibly extended to the infrared and optical domain [8]. The promise of being able toperform high-resolution spectroscopy at optical frequencies was certainly tantalizing. In 1960, thefirst laser was created by Maiman [9]. It was pulsed source that used ruby as a gain medium. Thiswas followed in 1961 with the demonstration of the first CW laser using a helium-neon mixture
as the gain medium [10]. These two historical breakthroughs in technology essentially lead to,among many other applications, the birth of nonlinear optics and laser spectroscopy, two seemingly
unrelated fields of research that will coalesce to give optical and XUV frequency comb technology.
As laser technology rapidly advanced, it became apparent that there was a need to be able
to measure an optical frequency in an absolute sense. The current definition of the SI unit thesecond, and hence frequency, is based on a hyperfine transition in Cesium-133 at 9,192,631,770 Hz
exactly. Optical frequencies are typically at ∼ 400 THz, which is much too high for direct electronic
measurement. Before the late 1990’s, the only way to bridge this frequency gap was a large,
magnificently complex system of multiple lasers and microwave sources spanning the electromagnetic
spectrum compared at radio frequencies. An example of these heroic efforts is shown in Fig. 1.1.
The device that rendered such frequency chains virtually extinct was the femtosecond opticalfrequency comb1 . Ultrashort laser pulses were routinely produced starting in the 1980’s. However,it was not until the first demonstration of passive, Kerr-lens modelocking that lasers with femtosecond pulse durations became robust [12]. It became apparent that the train of femtosecond pulses
produced by a modelocked laser creates a comb structure in the frequency domain provided thatthe repetition rate of the pulses is stable and provided the phase relationship between the successive
pulses’ electric field is stable or slipping regularly from pulse to pulse. Under these conditions, thefrequency νn of each tooth of the comb can be identified by a simple equation νn = n × fr + f0 ,where n is an integer, fr is the repetition rate of the pulses, and f0 is the carrier-envelope offset
frequency which is defined as the rate the electric field slips in phase from pulse to pulse. The
1

For this thesis, we will use “femtosecond optical frequency comb”, “optical frequency comb”, and “frequency comb”
virtually interchangeably to refer to frequency combs generated by modelocked, femtosecond pulse duration lasers.
There are various other frequency comb technologies and if we are referring to them, it will be clear from context.

3

Figure 1.1: This diagram is of the NIST frequency chain used to directly link an iodine transitionat ∼ 520 THz to the Caesium-133 standard at ∼ 9 GHz. Reproduced from [11].

4beauty of this equation is that fr and f0 are both radio frequencies and can be easily measured withconventional electronics. The optical frequency comb provides a radio frequency (RF) indexed ruler
for a large bandwidth of optical frequencies. With this laser, there is a way to directly relate an
optical frequency to an radio frequency, the essence of the NIST frequency chain show in Fig. 1.1.
Direct frequency comparison could now be accomplished with a single modelocked laser emitting
ultrashort laser pulses. By making a heterodyne beat between the laser under investigation and theoptical frequency comb (in a way analogous to Gordon and Townes original maser demonstration),
one can precisely know the frequency of a laser to very high degree and be able to directly compare
it to an absolute standard like the cesium atomic fountain clock operating in the microwave domain(or various other atomic clock technologies).
Spectroscopy in the XUV (10 nm – 100 nm) is a difficult business [13] and it would be great
if we could make it easier with the laser technology that has been developed in the visible and nearinfrared parts of the electromagnetic spectrum. However, building a laser in the XUV is no simple
task. The problem is essentially two-fold [14]. The power required to maintain a population inversion
and achieve lasing scales ∼ 1/λ4 . Since XUV wavelengths are typically an order of magnitude shorter
than their visible counterparts, ∼ 104 times more power is required for their operation. Secondly,
solid materials are typically not transparent below 100 nm, limiting gain media to gaseous samples.
There is also a very severe problem in that high reflectivity mirrors are extremely rare in the XUV.
Despite the apparent difficulty in producing laser light below 100 nm, there is still strong scientificinterest in overcoming this challenge. A quintessential example is atomic hydrogen spectroscoy [15].
Moving spectroscopic investigation beyond hydrogen to hydrogen-like systems, or even helium-likesystems, will offer new insights to bound-state QED because the corrections scale favorably with
the nuclear charge Z [16]. Using the Bohr model as a reference, the energy levels of a hydrogen-likeion areEn = Z 2

RE
,
n2

(1.1)

5
where RE is the Rydberg energy of 13.6 eV and n is the principle quantum number. Simply movingfrom hydrogen to singly-ionized helium (a system similar to atomic hydrogen), the n = 1 → n = 2transition is already at 40.8 eV, deep into the extreme ultraviolet and well out of the reach of
traditional laser technology. Ultimately we are seeking a source of light that has laser-like properties
but we are flexible as to how it is produced. Therefore, we can take advantage of nonlinear frequency
conversion techniques. Immediately following Maiman’s ruby laser demonstration [9], the field of
perturbative nonlinear optics was born with the production of optical harmonics [17]. The highpeak power of the pulsed ruby laser source was used to create a nonlinear polarization in a material
and thus produced harmonics of the driving laser frequency. We can understand such phenomena
by expressing the polarization induced by the laser field in a material or gas by a simple power
series expansion expressed as [18]P = χ(1) E + χ(2) E 2 + χ(3) E 3 + . . . ,(1.2)

where P is the polarization, χ(k) is the k th order susceptibility, and E is the electric field. The χ(1)term is describes ordinary, linear optics. The χ(2) and χ(3) susceptibilities give rise to now common
nonlinear phenomena such as second and third harmonic generation, four-wave mixing, and selfphase modulation. These nonlinearities are routinely exploited to produce optical pulses with nearly
single-cycle durations (∼ 1 − 5 fs) and down to wavelengths around 200 nm [19]. One might expectthat as the intensity of the laser is increased, higher and higher optical harmonics will be produced.
To a certain degree, this is true. However, our understanding based on the perturbative expansion of
Eq. 1.2 will quickly run into trouble once the electric fields of the laser pulse become comparable tothe fields that hold the atom together. If we ignore bound-free transitions (ionization) it is possible
to relate the k + 1 order susceptibility to the k order of an atomic system with a resonance at ω0
by using perturbation theory to obtain [19]χ(k+1) E k+1
µE
eEaB
≈
≈
~∆
~∆
χ(k) E k(1.3)

6where aB is the Bohr radius, the bound-bound transition detuning is ∆ = |ω − ω0 |, and µ is thebound-bound transition dipole. This is not an extremely rigorous equation, but it gives us a sense
for the orders of magnitude with which we are dealing. For a detuning of about 1 eV, the ratio
approaches unity when E is on the V/Å level. This corresponds to laser peak intensities on order1014 W cm−2 . These sorts of field strengths will cause the expansion in Eq. 1.2 to diverge as thelaser field is clearly no longer a perturbation. As it turns out, recent theoretical advances haveshown that this expansion starts to break down at much lower laser intensities [20, 21].
At these intensities, it is impossible to ignore bound-free transitions (ionization) making the
perturbative treatment even further unjustified. Intense electric fields lead to the ability for a boundelectron to quantum mechanically tunnel through the potential barrier and be ionized. There is also
the possibility of multiphoton absorption that will also lead to ionization. It is interesting to notethat the first strong-field laser-atom, multiphoton absorption experiment was performed at JILA
in 1965. An intense laser pulse incident was focused on a beam of I− and the subsequent ionizedelectrons were measured [22]. After ionization, the freed electron propagates in the electromagnetic
field. A useful metric for understanding the transition to the strong-field regime was introduced in1965 by Keldysh [23]. The Keldysh parameter is given by
s
γ=

Ip
,
2Up

(1.4)

where Ip is the ionization potential of the atom, see Table 1.1 for a list of common atomic targets
and the values for Ip . The Up is the pondermotive energy defined asUp =e2 E 2,4me ω 2

(1.5)

where e is the charge of the electron, me is the mass of the electron, and ω is the laser frequency.
The Keldysh parameter can be understood as a ratio between the laser frequency and the tunnelingrate. For γ >> 1, multiphoton absorption is dominant because the tunnel rate is slow compared to
the laser frequency. This is the case for laser pulses with short wavelengths and high intensities. By

7lowering the laser frequency, thereby increasing the laser wavelength, it is possible to make γ ≈ 1.
This is where the tunnel ionization rate and laser frequency are comparable, a situation found inmany strong-field experiments and where the process of high-order harmonic generation is typically
performed. For γ << 1, the electron can begin to topple over the potential barrier or tunnel ionize.
While this description has its usefulness, a more detailed review and criticism will be mentioned
later in Sect. 3.1.
Atom
He
Ne
Ar
Kr
Xe

Ip [eV]24.5874
21.5645
15.7596
13.999612.1298

Table 1.1: A table containing the ionization potentials of the noble gases used for HHG.

When an electron is tunnel ionized in an oscillatory field, to first order, the electron willpropagate in the electromagnetic field like a free, charged particle. At first, the electron will be
driven away from the ion core. Since the field is oscillatory, eventually the field will change sign andthe electron can return to the ion core it originally left. At this point, there is a small chance theelectron will recombine with the ion and emit a very high energy burst of light. This is essentiallythe three step semi-classical model of high-order harmonic generation (HHG) [24]. This process is
schematically shown in Fig 1.2. The three step model predicts that the maximum photon energy(the cutoff) that can be emitted is~ωmax = Ip + 3.17Up .

(1.6)

Since the HHG process is an ionization-related phenomena, it is best performed with pulse durations comparable to the ionization rates. The ionization rates can be calculated using the theoryof Amosov, Delone, and Krainov (ADK) [25]. For the noble gases assuming a near infrared laser,
it is easily possible to completely ionize a gas target in less that 100 fs if the peak intensity is

8Figure 1.2: Schematic of the three step model of high-order harmonic generation. Step 1 is ionization by an intense electric field. Step 2 is the acceleration of the free electron in the electromagnetic
field. The oscillatory nature of the electromagnetic field will cause the electron to return to theion core. Step 3 is the possible recombination of the electron and ion resulting in a burst of highenergy light.

9> 1014 W cm−2 . It is therefore necessary to use ultrashort pulse lasers to prevent complete ionization of the gas before the pulse has passed through the target. To show the power of the scalingof Eq. 1.6, a laser 1070 nm wavelength and at a peak intensity of 1 × 1014 W cm−2 focused into
argon can produce photons up to an energy of about 48 eV! Assuming a pulse duration of 100 fs,<1% of the gas target will be ionized. The harmonic light is at a high enough photon energy toexcite a singly-ionized helium atom between the n = 1 → n = 2 energy levels that were mentioned
earlier. A more detailed description of HHG as it pertains to the work in this thesis will be saved
for Chapter 3.

1.2

Optical frequency combs
We will now turn our attention to the optical frequency comb before moving to how an XUVfrequency comb is produced. A review of the basic femtosecond optical frequency comb technology
can be found in Ref. [26]. The train of pulses coming from a modelocked laser looks very similar toa Dirac comb

XT (t) =∞X

δ(t − nT ).

(1.7)

n=−∞

In reality, the pulses of a modelocked laser are typically ∼ 100 fs in duration and are separated bytypically ∼ 10 ns. At 1070 nm, the period of a light wave is 3.57 fs so there are many oscillations
within a single pulse. The Fourier transform of the Dirac comb XT (t) is well known and is givenbyF

XT (t) −→ X 1 (f ) ,

(1.8)

T

so it should be of no surprise that the comb-like structure of a modelocked laser in the time domain
will lead to a comb-like structure in the frequency domain. The pulse train coming from themodelocked laser has an additional complication. The maximum of the electric field need not occur
at the maximum of the intensity envelope. Rather, it can shift from pulse to pulse. The offset of

10the electric field relative to the peak of the envelope in a single pulse is often referred to as thecarrier-envelope phase ∆φ. For a pulse train, it is more useful to describe the rate at which thephase slips and this is the carrier-envelope offset frequency f0 defined as

f0 =

∆φ
fr ,
2π

(1.9)

where ∆φ is the pulse to pulse phase slip of the electric field relative to the envelope. This is
schematically shown in Fig. 1.3. In the frequency domain, this has very important consequences.
All of the comb teeth are uniformly offset from zero by f0 . We can therefore identify any comb
tooth at optical frequencies by a combination of two radio frequencies expressed as

fn = n × fr + f0 ,(1.10)

where n is an integer. Each comb tooth fn can be used as a frequency marker to compare with CW
lasers or it can used in much the same way a CW laser is used. For example, it is possible to perform
spectroscopy with an individual comb tooth or simultaneously with multiple comb teeth [27].

1.2.1

Carrier-envelope offset frequency detection and measurement

Modelocked lasers, the basis for femtosecond optical frequency combs, have been routinely
used since the 1980’s. However, it was not until the year 2000 that an optical frequency comb wasdemonstrated [28, 29]. This is because detection of the carrier-envelope phase is quite difficult andit was not until the realization that the carrier-envelope phase slip had important consequences in
the frequency domain. It also took time to apply the tools of laser stabilization in the frequency
domain to ultrafast lasers. To stabilize the carrier-envelope phase of a modelocked oscillator, onemust be able to detect it, or at least the rate at which it slips, as previously defined in f0 . One
method, and the one originally demonstrated in 2000, is an f − 2f interferometer [28]. If we cancompare two comb teeth n and m (or sets of comb teeth) that satisfy the condition 2n = m, we will
be able to detect f0 . The problem is that this requires the optical spectrum to be at least one octave

11

Figure 1.3: Schematic drawing of an optical frequency comb in the time domain and the frequency
domain. Relevant time scales and frequency scales are identified in the schematic.

12
wide. A typical modelocked oscillator spectral width is inversely proportional to the pulse duration∆ν ∝ 1/tp and for pulses of tp ≈ 100 fs at the 1070 nm central wavelength, the spectrum only spans∆ν ≈ 4.4 THz (or ∼ 15 nm) which is nowhere near a spectral octave this optical range. To overcome
this bandwidth issue, highly nonlinear optical fibers offered an elegant solution [30]. Currently, thehighly nonlinear fiber technology is very mature, but at the time, it was still in its infancy. Highly
nonlinear fibers rely on a variety of nonlinear optical phenomena (self-phase modulation, Raman
scattering, soliton formation, etc.) to spectrally broaden the pulse to reach an octave. An exampleof this approach from our lab [31] is shown in Fig. 1.4. The spectrum is ideally matched for thestrontium optical lattice clock experiments because the octave spanning spectrum covers most of
the near infrared and provides access to telecom wavelengths near 1.5 µm. Currently, it is possible
to directly produce octave spanning spectra directly from a Ti:Sapphire-based modelocked laser,
but they are often not as robust and low maintenance as the fiber-technology counterparts [32, 33].
Once the laser spectrum spans one octave, the long wavelength side of the spectrum can befrequency doubled (by using nonlinear crystals) to spectrally overlap with the blue side of the samecomb spectrum. Fig. 1.4 contains shaded regions highlighting the relevant parts of the spectrum
used for f0 detection. By doing this, we can overlap comb teeth that satisfy 2n = m. We can thenobserve a heterodyne beat that measures f0 by

2 × (n fr + f0 ) − (m fr + f0 ) = 2n fr + 2f0 − m fr − f0 = f0 .

(1.11)

With the ability to detect f0 and fr (accomplished with a simple, fast photodetector), there is often
still a need to stabilize these quantities. Once stabilized, the potential of the frequency comb isclear; there are millions of stable and well defined frequency markers across the optical spectrum
available for use.

1.2.2

Frequency comb stabilization

There are a multitude of ways to stabilize a frequency comb. For the sake of clarity, we will
focus on a situation applicable to modern optical atomic clocks [34]. The laser to be described [31]Intensity (dB)130Oscillator
Super
Continuum

-10
-20
-30
-40
0.6

0.8 1.0 1.2 1.4
Wavelength [µm]1.6

Figure 1.4: A Yb:fiber oscillator spectrum is shown in blue. After nonlinearly broadening thespectrum in photonic crystal fiber, the red spectrum spans over one spectral octave. The red
shaded region can be frequency doubled using a nonlinear crystal to spectrally overlap with theblue shaded region to allow f0 detection.

14
was used for the strontium lattice clock [35] and precision laser comparison experiments [36]. Nevertheless, the techniques are fairly universal. Precision laser and frequency comb stabilization is an
exercise in control systems. It is often joked that you can either engineer a laser system very well,or just build a good feedback system. In reality, the best system is a combination of good engineering and good feedback system design. To avoid obfuscating the point by going into the details of
feedback loop design, it is useful to just consider the basic operating principle as it pertains to laser
stabilization.
Imagine we want to stabilize a laser frequency ΩL to a reference laser frequency ΩR . The
first thing we need is some means to measure how far away our laser frequency is from ΩR . We
typically call this a discriminator. Our discriminator will typically give us a signal in the form of
an electrical voltage V that tells us how far off we are from the goal so that V ∝ ΩL − ΩR . We cantake that signal, filter it, invert it, and apply it to the laser to try and steer the laser frequency tothe correct value. Our discriminator then reports how well we have done and the process continues
until the system settles on our desired result or V ≈ 0. The real trick involves filtering the signal
properly and re-applying with just the right strength. While this is a vast oversimplification, there
are a few important subtleties to mention. While, mathematically speaking, a phase-lock and afrequency-lock are an integral away from each other, the physical processes that can generate phasenoise or frequency noise can be quite different and have very different spectra. The actuators used
for phase-locking or frequency-locking typically typically fall into two categories: fast actuation
with low dynamic range or slow actuation with large dynamic range. Typically, both are needed.
Fast actuators are essential to remove broadband noise (or high-frequency noise) in a phase-lock or
frequency-lock. However, they tend to run out of actuation range very quickly as optical systems
drift and respond to slow (but sometimes large) environmental perturbations such as a temperaturechange. Therefore, slow but high dynamic range actuators are employed to keep the fast actuators
within optimal operating ranges.
An optical frequency comb is a complicated laser and it is schematically drawn in Fig. 1.5 withnumerous parts and stabilization components labeled. This laser system is almost identical to the15front-end of the high power system used for the HHG work in this thesis (see Chapter 2). There are
two fundamental quantities that need to be stabilized in a frequency comb, fr and f0 . The first is
possibly the easiest to understand. Two control fr , the simplest method is to change the laser cavity
length. This changes the round-trip time of the pulse in the cavity and hence the repetition rate. A
drawback to this method is that changing the cavity length usually requires moving a macroscopic
mirror. This usually introduces temporal delay in the system’s response and this is the bane of a
feedback loop. In Fig. 1.5, these cavity length changes are accomplished by piezoelectric transducer(PZT) actuators. One is attached to a saturable absorbing mirror (SAM) to provide fast changes tothe cavity length and a circular PZT stretches some fiber in the cavity to slowly change the cavity
length. However, some applications require feedback to be faster than what a PZT can accomplish.
We can then make use of an electro-optic modulator (EOM) to change the cavity length. This works
by applying a voltage to the EOM crystal to change the index of refraction of the crystal and hence
increase or decrease the optical path making the laser cavity effectively longer or shorter. This canhave feedback bandwidth almost an order of magnitude larger than common PZTs as demonstrated
in Ref. [31].
To stabilize f0 , we must refer to more sophisticated laser dynamics. Relative changes in theintracavity dispersion (relative amounts of group delay and phase delay) will lead to f0 shifts. To
accomplish this in the frequency comb of Fig. 1.5, the fiber Bragg grating (FBG) temperature ischanged. The FBG is responsible for dispersion management inside the laser cavity and by changing
the temperature, the grating’s dispersive properties change. This process is effective but very slow.
Faster control is accomplished by modulating the pump power of the laser. Modulating the pump
power will change the amount of self-phase modulation in the laser and modulate the dispersion of
the gain medium, both of which will cause f0 to change [37, 38]. However, it is often nice to use
the pump modulation actuator to reduce the amount of phase noise in the laser because fiber lasers
typically have large amplitude-to-phase noise couplings [31, 39]. With the pump laser intensity tied
up reducing phase noise in the laser by reducing the laser intensity noise, it is unavailable for directf0 control. However, this problem can be circumvented by the use of an acousto-optic modulator16(AOM) placed outside the laser cavity. An AOM uniformly shifts all the frequency comb teeth by agiven radio frequency which is typically near 40 MHz but can be up to and over 1 GHz. The AOM
can provide high feedback bandwidths and is a nearly perfect f0 actuator.
With detection of f0 described, detection of fr seems almost trivial as a photodetector witha response faster than fr is sufficient to measure it. However, it is often unwise to use this method
if the highest level of stabilization is required. As an example, stabilizing fr directly will only
fractionally preserve the stability at optical frequencies. Assuming a fr = 100 MHz signal can bestabilized to 1 Hz, that is a fractional uncertainty of ∆fr /fr = 1 Hz/100 MHz = 10−8 . Assuming
a frequency comb at 1070 nm, the central frequency is 280 THz. That means the optical linewidth∆fopt would become ∆fopt = 280 THz × 10−8 = 2.8 MHz. For some applications, this is sufficient,but for high-performing optical atomic clocks [35], this is not acceptable.
However, we are not limited to direct detection of fr . CW lasers can routinely be stabilized to∼ ∆f /f = 10−16 [40]. By making a heterodyne beat between a extremely stable laser and a single
comb tooth we can derive an excellent signal to use for stabilization. The beat frequency with thef0 dependence removed would be

fb0 = n × fr + f0 − fopt

(1.12)

fb = fb0 − f0 = n × fr − fopt

(1.13)The beat frequency fb can then be stabilized to a very stable RF reference frequency by controlling
fr . This accomplishes an important point; the fractional stability of the laser can now be transfereddirectly to fr . This has been exploited to generate extremely stable radio frequencies by subsequent
detection of fr on a simple photodetector [41]. This method also facilitates precise comparison oflasers separated by frequencies too large for direct detection [36].

17

SA

PZT
EOM

YDF

976 nm
0.6 W

FBG
WDM

PZT

ISO 1%

RIN
measurement

AOM

pump
combinerISO
double-clad YDF

fceo-beat976 nm
10 W

SHG

PCF
to optical beats

DM

Figure 1.5: A schematic of the Yb:fiber frequency comb used for optical clockwork. PZT, piezoelectric transducer. SA, saturable absorber. EOM, electro-optic modulator. YDF, ytterbium-doped
fiber. FBG, fiber Bragg grating. WDM, wavelength division multiplexer. ISO, optical isolator. RIN,
residual intensity noise. AOM, acousto-optic modulator. PCF, photonic crystal fiber. DM, dichroic
mirror. SHG, second harmonic generation.

18
1.2.3

Frequency comb performance characterization

After a frequency comb has been stabilized, it is often useful to have multiple methods tocharacterize it. Fig. 1.6 shows three possible methods. The first is to use a CW laser to comparewith an individual comb tooth. To get a true sense of how the comb is performing, it is importantthat the CW laser is “out-of-loop” (an independent component) meaning that it is not being used
as a part of the stabilization scheme. The beat between the CW laser and the comb will yield
important information about frequency stability and other possible noise sources. As an example,
this method was used to study super continuum processes and how well they preserved the frequency
comb coherence across the optical spectrum [42].
The second method and third method rely on the use of a reference frequency comb withknown stability properties. The second method uses two combs have the same fr but different
f0 . The heterodyne beat signal will contain information about f0 stability which can be useful formeasuring noise properties coupled to f0 or to measure optical linewidths across the spectrum byspectrally resolving the comb before heterodyne beat detection. We will use this method later inthe thesis when we characterize the coherence properties of the XUV comb [43]. The third method
is more complicated, it relies on the two combs having different fr and possibly different f0 . By
doing this, the relative spacing between comb teeth of each comb is growing with the comb tooth
number. The heterodyne beat spectrum will contain many signals coming from each pair of comb
teether because they are all at slightly different frequencies. This is the basis for the very powerful
technique of dual-comb spectroscopy [44, 45]. This method is often used to analyze a new class of
frequency combs based on soliton formation in torroidal resonators [46].
There are many figures of merit for frequency comb performance (or CW laser performance)
and the most meaningful often depends on the desired application [38, 47]. For the work pertaining
to this thesis, we are most concerned with the temporal coherence properties or phase stability. A
useful metric for this is the measure of the optical linewidth. Much like an atomic linewidth can giveinformation on an excited state lifetime, an optical linewidth can give information on the coherence

19

a)
Comb

Intensity

CW Laser

Frequencyb)Comb 1

Intensity

Comb 2Frequencyc)

Comb 1

Intensity

Comb 2Frequency
Figure 1.6: a) A heterodyne beat between a CW laser and a frequency comb. b) A heterodyne
beat between two frequency combs with the same repetition rate but different f0 frequencies. All
the beats from different teeth coherently add together to form a single beat. c) Heterodyne beats
between two frequency combs with different repetition rates. Each pair of teeth form individual
beats. It is therefore possible to inspect individual frequency comb teeth in the radio frequency
beat spectrum.

20time of the light. The coherence time of the light is important because it sets a timescales at whicha laser can phase coherently interact with an atomic system. These long light-matter interaction
times have the potential to reveal very subtle phenomena in quantum systems. As a quick example,
without the ability to resolve sub-Hz features on an atomic transition in strontium, many complex
many-body phenomena would have gone unobserved [48]. The resolving power is only possibleby having very phase stable light. This fact has driven much of the laser and atomic physics
development over the past 50 years – the yearning of high spectral resolution.

1.3

Extreme ultraviolet frequency combs
Creating a frequency comb in the XUV requires performing the aforementioned process of

HHG. Most HHG experiments are not performed in a manner that allows investigation of the coherence preserving properties of HHG. Certainly the spatial properties of the light were investigatedand are routinely used for coherent diffractive imaging [49]. Since we are interested in using XUVfrequency combs for high resolution spectroscopy, we are concerned primarily with the temporal
coherence properties. Earlier work has shown that adjacent harmonic orders are locked in phaserelative to each other [50]. This results in the HHG process producing a train of attosecond pulses
in the XUV. However, this does not test or guarantee absolute phase stability, especially over thetime scales of interest.
Before considering coherence properties of the HHG process, it is important to make a technical distinction between performing HHG with a frequency comb compared to a more “traditional”
laser system. Calling any laser system designed for HHG “traditional” is troublesome. High-power
laser systems are continuously evolving and the technology advances quite rapidly [51]. Nevertheless, a typical approach is to use a high pulse energy laser system. Pulse energies are typically inthe few mJ range, repetition rates are in the kHz range or lower, and average powers are in the
few W range. Central wavelengths can vary from 800 nm to 4 µm. Some sources offer the ability
to customize and shape an optical waveform and others do not even have carrier-envelope phasestability. In principle, even with a very low repetition rate, it would still be possible to observe the21

Figure 1.7: a) Single pulse HHG in the time and frequency domain. b) Multi-pulse HHG and the
XUV frequency comb. The harmonic spectrum is the same as in the single pulse case, but the highrepetition rate nature of the frequency comb creates a comb structure at each harmonic order.

22
comb structure, but in reality, when the pulses arrive so infrequently, it makes stabilization of alow repetition rate system nearly impossible. Further, it is an issue of the bandwidth of the noise
processes. If the noise-induced modulation bandwidth exceeds fr , we have no hope in uncovering a
comb structure.
The harmonic spectrum produced by a single laser pulse is shown in Fig. 1.7. The is what
is observed in low repetition rate laser systems. Near every maximum of the electric field, thethree-step process of HHG (see Fig. 1.2) can initiate. The result is a burst of XUV radiation near
the zero crossing of the laser field. Therefore, a burst of high energy photons will occur every halfcycle of the laser. This is what produces the optical harmonics spaced by 2ω0 where ω0 is thedriving laser central frequency. The harmonics are odd because the sign of the electric field of thehigh energy photon burst switches every half cycle. Also, since gas targets are typically inversion
symmetric, even harmonics are not possible. Even harmonics can be produced if the target lacks
inversion symmetry or a multi-color laser field is used to break the electric field symmetry.
The heart of the XUV frequency comb lies with the question of what happens if the HHG
process is driven with a high repetition rate pulse train (a pulse train from a frequency comb for
example). Provided the HHG process is coherent and reproducible, the XUV electric field produced
in a single HHG event will be produced every time by subsequent driving pulses. In this situation,
we are driving XUV emission at stable and well defined intervals much like a modelocked oscillator
produces optical pulses at stable and well defined intervals. We expect the overall harmonic emission
to mimic the single pulse case in terms of the overall spectral shape, but we intend to determine
whether a given harmonic order q should have frequency comb teeth separated by the driving laser
repetition rate fr . We expect the carrier envelope frequency to increase with harmonic order as
the electric field will be oscillating q times as fast. We can then re-write the frequency comb toothindexing for the different harmonic orders as

fj = j × fr + q × f0 ,(1.14)

23
where j is an integer and f0 is that of the driving laser. So, provided that the HHG process does
not lead to carrier-envelope phase noise, we hope to find a frequency comb structure in the XUVthat shares similar properties to the driving laser. For instance, it will then be possible to use thecomb structure for high resolution spectroscopy because we can treat each comb tooth similarly toa CW laser [52].

1.4

Applications
There are many examples of spectroscopic application with < 100 nm light that we will briefly

summarize below. Another feature of the XUV frequency comb has not been stressed thus far: itis actually one of the highest photon flux sources of XUV radiation available on a table-top. Not
only that, it is also nearly as spectrally bright as a synchrotron, the work horse of XUV and x-ray
science. We will also highlight some applications unrelated to high resolution spectroscopy but for
which the XUV frequency comb is well suited.

1.4.1

Spectroscopic applications

As previously mentioned, precision spectroscopy of atomic hydrogen has been very fruitful andallowed sophisticated bound-state QED calculations to be compared with experiment. The beautyof using hydrogen is that the calculations can be carried out exactly [16]. Singly ionized helium isthe next logical step after hydrogen if the goal is to compare to QED calculations [53]. Leading
order relativistic corrections to the 1S − 2S transition scale as Z 4 . Leading order correction to theLamb shift (splitting between 2S and 2P levels) also scales as Z 4 . This makes QED effects in singly
ionized helium over an order of magnitude larger than in atomic hydrogen. Precision spectroscopyof these systems will test the limits of current bound-state QED calculations and possibly reveal
new physics.
Helium-like system are also interesting because electron-electron correlations become important. There are currently no methods to perform calculations of helium properties exactly as in thecase of hydrogen. Laser spectroscopy on atomic helium has already been performed [54, 55]. At the24time of the test, the experimental uncertainties were close to the achievable theoretical uncertainties. However, more recent experimental efforts aim to beat down the experimental uncertainties tojustify improvements on the calculations [56].
Highly charged ions are a relatively untapped source of laboratory spectroscopic investigation.
It seems intimidating to trap and cool highly charged ions, but recent experiments are advancing
nicely and it will not be long before routine spectroscopic investigation is possible [57]. Highly
charged ions are a sensitive testbed to look for variations in the fine structure constant [58–60].
The sensitivity typically scales as Z 2 . Highly charged hydrogen-like ions are also postulated to be auseful test bed for looking for parity nonconservation and physics beyond the standard model [61].
In these proposed experiments, highly charged ions would be stored at relativistic velocities tobring the Doppler-shifted resonances into the VUV and XUV where lasers can be used to excite
the 1S − 2S transition and look for circular dichroism. Since the electrons of highly charged ionsare located much closer to the atomic nucleus, looking for strong nuclear affects on the electron
spectroscopy could potentially lead to new insights in nuclear physics.
Another interesting target suitable for high resolution spectroscopy is molecular hydrogenand its isotopologues (H2 , HD, D2 ). Spectra of molecular hydrogen can be used to constrain theevolution of the electron-proton mass ratio [62]. Further, basic spectroscopic investigation can beused as a test bed for QED in a molecular setting and test our fundamental understanding of themost basic molecular system [63, 64].
A final spectroscopic application of the XUV comb would be a “nuclear” clock. It would workon the same principles of an atomic clock, but rely on a nuclear isomer transition as the frequency
discriminator (rather than an electronic transition as in optical clocks). Isomer transitions are
typically at x-ray wavelengths. However, thorium is claimed to have a low lying isomer transition
near 7.5 eV. To date there is no concrete experimental evidence of this transition and the transitionenergy is still debated. If it does exist, it would likely make a very nice clock [65]. There is along-lived isomer transition in uranium with 27 minute lifetime [66, 67] at an energy just under 100
eV. It is unclear whether or not this will be useful for clock operation because the excited state

25
decay mechanisms are still unknown.

1.4.2

High-flux extreme ultraviolet source applications

The current workhorse for XUV and x-ray radiation sciences is the synchrotron. It produceshigh flux, spatially coherent, and pulsed light [68]. A small list of its applications contain microscopy,
lithography, angle-resolved photoemission spectroscopy (ARPES), crystallography, and structural
biology [69]. Most of these would benefit from a light source with similar properties but in acompact, user-friendly package that does not rely on a billion dollar user facility with limitedaccess [69]. For the sake of brevity, we will discuss application to ARPES. What follows will apply
to most photoelectron based spectroscopies that rely on XUV light for photoemission.
Many photoelectron based spectroscopies suffer from an effect known as “space charge.” For
ARPES, the ability to detect photoelectron energy and momentum at high resolution is critical.
If too many photoelectrons are simultaneously ejected from the surface of a sample, as they drift
to the detector, the momenta will change due to Coulomb repulsion and this effectively blurs out
the experimental resolution. To overcome this, scientists usually reduce the brightness of the XUVsource that is generating the photoemission, but this comes at the cost of reduced signal size
leading to worse statistics in the data (since the experimental shots occur at a fixed rate). The ideal
condition is to have a light source that produces maybe only a few photons per pulse but repeats
at a high frequency. This is exactly what the XUV frequency comb does very well. The XUV
frequency comb is of comparable brightness to a synchrotron and operates in a regime where spacecharge effects are minimal for ARPES application, or other photoelectron-based spectroscopy. An
example of how powerful a laser-based source for ARPES can be, we refer to the work of Ref. [70].
The data taken with a laser-based ARPES apparatus clearly has better energy resolution and did
not require a trip to the synchrotron. The higher energy resolution was used to observe much more
subtle physical phenomena that would have otherwise gone unnoticed in a more traditional ARPES
apparatus.

26

1.5

Thesis outline
For our work, the primary goal of the XUV frequency comb is to combine the precision laser

stabilization required to produce an optical frequency comb with the highly nonlinear process of
HHG to up convert the near infrared frequency comb into the XUV. The rest of the thesis will
describe results of the XUV frequency comb experiments.
• Chapter 2 contains the experimental apparatus including the laser system and the femtosecond enhancement cavities (fsECs).
• Chapter 3 contains a description of HHG as it pertains to power scaling XUV frequency
combs and the microscopic phenomena the XUV comb can probe.
• Chapter 4 contains the construction and operating principles of the XUV interferometerused to test the temporal coherence properties of the XUV light originating from HHG.
• Chapter 5 contains our results in performing heterodyne interferometry in the XUV to study
the XUV comb phase coherence and how we can use it to probe strong-field phenomena.
• Chapter 6 contains our efforts to use the XUV comb technology to study strong-field effects
in molecular systems.
• Chapter 7 contains conclusions and an outlook for future XUV comb work.

Chapter 2

Experimental apparatus

In this chapter, we cover the main experimental apparatus and its working principles. Details
about the XUV interferometer will be saved until Chapter 3. Here, we present the femtosecond
enhancement cavity, the linchpin of XUV frequency comb generation. We also present the laser
system and its modifications since its original construction. We will also discuss the experimental
vacuum chamber and operations unique to the XUV frequency comb.

2.1

Amplifying ultrafast laser pulses
Most HHG experiments use lasers that have 100 µJ – 10 mJ pulses energies while maintaining

ultrafast pulse durations (< 100 fs). With these parameters, it is very straightforward to focus thelaser beam to achieve peak intensities < 1014 W cm−2 , a sufficient intensity for HHG. These highpulse energies are achieved with low repetition rate amplifiers [71, 72]. The low repetition rateis necessitated by the amplifier pumping laser (or flashlamp) technology. The XUV frequency
comb requires similar pulse energies of 10 µJ – 100 µJ while still maintaining the full repetition
rate of the initial frequency comb of typically 100 MHz. The amount of average power required
for a 100 µJ frequency comb at 100 MHz would be 100 µJ × 100 MHz = 10 kW! Currently, no“active” amplification technology can achieve this. However, there is a “passive” approach that
relies on high finesse optical resonators. We call these special resonators femtosecond enhancement
cavities (fsECs). The details of operation will be forthcoming. However, what they achieve isquite remarkable. By coupling a frequency comb to an enhancement cavity, the pulses resonating

28
inside the cavity can have an average power potentially > 200× greater than the injected laser
while preserving the full repetition rate of the frequency comb and its ultrafast pulse duration.
Using this technology, the intracavity pulse is “passively” amplified and can be used for strong-field
experiments. Therefore, only a 50 W average power frequency comb (a power completely withincurrent technology [73]) can achieve 10 kW of average power while maintaining ultrafast pulse
durations.

2.2

The femtosecond enhancement cavity
An optical cavity is a routinely used piece of instrumentation. Here, we will outline the basic

theory of operation and highlight aspects most relevant for femtosecond pulse enhancement. For
most of the discussion, we will focus on enhancement cavities for continuous wave lasers since itsimplifies the description. However, the extension to coupling frequency combs to enhancement
cavities is quite simple because frequency combs behave as if they were composed of a series ofcontinuous wave lasers evenly spaced by the repetition rate. The enhancement cavity for a frequency
comb will be referred to as a femtosecond enhancement cavity (fsEC).
The XUV frequency comb relies on Fabry-Pérot ring resonators. Schematics of ring resonators
are shown in Fig. 2.1. Fig. 2.1a is a typical four mirror resonator. We are however not restricted
to a four optic cavity. Fig. 2.1b shows a resonator with a fifth optic placed between the curved
mirrors. This optic, the “Yost grating” [74], is used to outcouple HHG light from the cavity andwill be discussed later in Sect. 2.2.6. For the current discussion, it can be treated as an additional
high reflector in the cavity. As we will discuss, the total length of the cavities shown in Fig. 2.1 aredetermined mainly by the frequency comb repetition rate. Therefore, the separation of the curved
mirrors can be used to tune the size of the beam waist in the resonator and the transverse mode
properties.

29

a)
HR
b)
HR

CM

CM

CM

CM
HR-DG

IC

IC

Figure 2.1: a) A four mirror ring cavity is schematically shown. HR, high reflector. CM, curved
mirror. IC, input coupler. b) A five mirror ring cavity is shown. This cavity is equipped with
the Yost grating which acts as a high reflecting mirror at the driving laser wavelength. For shorter
wavelengths, it behaves as a diffraction grating. HR-DG, high reflector diffraction grating aka the
Yost grating [74].

30
2.2.1

Cavity finesse, buildup, and contrast

Referring to Fig. 2.1, we can write the intracavity laser electric field Ecav as it traverses aroundthe cavity for multiple round trips as a series given by

Ecav (ω) =√
E0 ti 1 + ri rc eiφ(ω) + (ri rc eiφ(ω) )2 + (ri rc eiφ(ω) )3 + . . . ,(2.1)

where E0 is the incident electric field, ri is the reflectivity coefficient1 of the input coupler, ti
is the transmission coefficient of the input coupler, rc is the effective reflectivity coefficient of therest of the cavity optics, and φ(ω) is the total phase shift on the electric field after one round trip
through the cavity. The phase shift term includes propagation and dispersive effects of the mirrors
and is therefore sensitive to the laser operating frequency ω. The mirror reflectivity coefficient r
and transmission coefficient t values are also in general laser frequency dependent. However, we√
will not explicitly denote this for clarity. The  term is due to imperfect transverse mode overlap
between the laser beam and the cavity mode. We refer to this term as a “mode matching factor”√
and its origin will be discussed in detail later. For now,  = 1 is for perfect overlap of the laser
beam profile and the cavity mode. Since reflectivity coefficients are less than 1, the series in Eq. 2.1
converges to

Ecav =

√

E0

ti
.
1 − ri rc eiφ(ω)(2.2)

We perform a similar treatment for the electric field reflected from the cavity. The reflected electricfield Eref is written as
1

The “reflectivity coefficient” is how much of the electric field is reflected. It is more common to refer to a mirror“reflectivity”, which is how much electric field intensity is reflected. Generally, quantities that refer to electric field
intensity will be capitalized. This is also true for mirror transmission.

31

√

√
E0 rc t2i eiφ(ω) + E0 ri rc2 t2i e2iφ(ω) + . . . ,2√ E0 t2i√= − E0 ri + 
ri rc eiφ(ω) + ri rc eiφ(ω) + . . . ,
ri
2
iφ(ω)
√
rc ti e
− ri
= E0
.
2
1 − ri rc ti eiφ(ω)

Eref = −E0 ri +

√

(2.3)
(2.4)
(2.5)

For optical cavities, we routinely measure laser intensities of the reflected and transmitted light. The
incident, reflected and intracavity intensity is defined as I0,ref,cav = |E0,ref,cav |2 . We can express
Iref and Icav as

Iref = I0(ri −rc )2 +4ri rc sin2 (φ(ω)/2)(1−ri rc )2 +4ri rc sin2 (φ(ω)/2)



,

(2.6)

Icav = I0



t2i
2
(1−ri rc ) +4ri rc sin2 (φ(ω)/2)



.

(2.7)

At this point, we can already extract useful cavity characterization parameters. By inspection, theintracavity field is largest when φ(ω) = 0 (and modulo intervals of 2π). It is useful to fully express
φ(ω) asφ(ω) = kL + φd (ω) =ωL + φd (ω),
c

(2.8)

where k is the laser wavevector, L is the round-trip cavity length, c is the speed of light, and φd (ω)
is the phase shift due to mirror dispersion. As mentioned, the intracavity power is maximized whenφ(ω) is periodic in 2π. In the absence of mirror dispersion (we will consider this later), we haveω
L = m 2π,c(2.9)

where m is an integer. The frequency spacing at which the cavity repeats itself is thus 2πc/L in
angular frequency units (it is simply c/L in Hz). This is defined as the free spectral range (FSR).
We can solve for φ(ω 0 ) to determine the half maximum intensity, again assuming the absence ofdispersion. We assume the resonant frequency is at ω0 and the frequency detuning required toachieve half of maximum intracavity intensity is ω 0 . Thus, we have32

√ i rc ) ,φ(ω 0 ) = 2 sin−1 ( 21−r
ri rc

(2.10)φ(ω 0 ) ≈(2.11)

1−r√ i rc .
ri rc(2.12)
With this manipulation, we solve for twice the frequency detuning required to achieve half transmission and define this as the full-width at half maximum (FWHM).

FWHM ≡ ∆ω1/2 ≡2|ω 0 − ω0 | ,cω0 =m π ,Lc(1 − ri rc )0
ω =− mπ ,√L
ri rc
FWHM =c 2(1 − ri rc )
,
√
L
ri rc

(2.13)(2.14)(2.15)(2.16)(2.17)

where the FWHM is given in angular frequency units (divide by 2π for units in Hz). The finesse F
is defined as the ratio of the FSR to the FWHM. We haveF=

FSR
∆ω1/2 ,

(2.18)=√
π ri rc
1−ri rc .

(2.19)

This definition of F is much more useful if expressed in terms of mirror reflectivity and notreflectivity coefficients. The mirror reflectivity is defined as R ≡ r2 . Similarly, mirror transmission
is T ≡ t2 . Mirror properties also obey

R + T + A = 1,

(2.20)

where A is the mirror loss. This is the fraction of light that is neither transmitted nor reflected. It
usually originates from scatter or absorption of the intracavity elements. If we define a mirrors total

33
loss as Lm = Am + Tm , where m is just the mirror label, we can rewrite our definition of finesse isa very useful way.

ri rc,
π 2 1−ri rc

F2 =(2.21)√= π 2 1−√RRi RRc ,(2.22)

π2,[ 12 (Li +Lc )]2(2.23)

2π
L,(2.24)i≈F=cwhere we have implicitly defined the total loss as L = Li + Lc .
Another useful quantity, and possibly the most important for our application, is a cavity’sbuildup β on resonance defined as β = Icav /I0 . When a cavity is driven on resonance, the intracavity
power will be greater than the driving laser power. Starting with Eq. 2.7 and performing similar
manipulations used to arrive at Eq. 2.24, we have

β=

Icav
F2
= Ti 2 .
I0π(2.25)

Similarly, we can determine the reflected cavity power expressed asIref = I0

F 2.
(Lcav − Li )2π

(2.26)

It can be seen that the reflected power goes to zero if the input coupler losses match the rest of thecavity losses. This is known as impedance matching. If Li > Lcav , this is an overcoupled cavity. If
Li < Lcav , this is an undercoupled cavity. The greatest buildup for a fixed finesse will occur in theovercoupled case. Looking at Eq. 2.25, for a given finesse (Lcav + Li = constant), the buildup will
be directly proportional to Ti , so bigger Ti is better.
We can also define the cavity contrast C as

34

I0 − Iref
,
I0


F 2
=1 −  (Lcav − Li )
.
2π

C=

(2.27)(2.28)

Measurement of C, F, and β constitute a full characterization of the cavity. However, C and β are
dependent on the modematching factor . Measurement of F is the best way to determine cavity loss
since it does not depend on mode matching. A straightforward way to measure finesse is to stabilize
the cavity on resonance and then rapidly turn off the laser. The cavity intensity (in transmission orreflection) will decay exponentially as a function of time. The time constant of decay can be related
to F in the following way. After a field has been stored up in the cavity and then rapidly turnedoff, we can write the energy as a function of time E(t) as

dE(t)
E0
=− .
dt
τ

(2.29)We remember that the loss of intensity per round trip is L and the energy lost per round trip is LE.
The energy lost per unit time becomes L Ec/L (This is just L E FSR/2π where the FSR is expressed
here in Hz). Plugging this in we havedE(t)= −L FSR E0
dtτ⇒(2.30)

1
L FSR .

(2.31)

c.
L(2.32)

We can now insert Eq. 2.24, rearrange and obtain

F = 2π τ

It is important to note that this is the energy (or intensity) decay time τ . The field decays twice as
slowly, so τintensity = τf ield /2.
Sometimes it is not feasible to turn the laser off rapidly or to stabilize the cavity. An alternative
method is to rapidly sweep the laser frequency across the cavity resonance at a rate much faster35
than the cavity lifetime. The cavity field that is briefly resonant will form a heterodyne beat withthe reflected sweeping laser beam. The heterodyne nature (rather than homodyne) is due to thefact that the resonant field is at a fixed frequency, but the incident beam is frequency chirped.
Therefore, when they are overlapped with the reflected beam from the cavity input coupler, they
form a beatnote. This ringdown signal can be seen in Fig. 2.2. The signal is fit to a damped,
chirped sine wave. A field decay of 1.05 µs (assuming an FSR = 154 MHz) corresponds to a finesseof about 500. The input coupler for this data was Ti = .008. This finesse measurement implies aloss of Lcav = 0.0046. Assuming perfect mode matching ( = 1), the expected cavity contrast would
be C = 0.7 and the expected buildup would be β = 200. An experimentally measured buildup
or contrast can be used to determine the mode matching factor . For this example, the cavity
loss Lcav was due to an optic with an hole drilled in the center (see Sect. 2.2.6 for information ofgeometric output coupling). When a normal optic is used, it is very routine to have a cavity with
Lcav ≤ 0.0005.

2.2.2

Gaussian beams and gaussian beam manipulation

So far, we have determined that a ring resonator has resonant frequencies evenly spaced by
the FSR, neglecting dispersion. However, we have not considered the transverse mode profile ofthe cavity. To do this, we rely on the formalism of Gaussian beam propagation and the ABCD
matrices [14]. The paraxial wave equation is given by∇2t u(x, y, z) − 2ik∂u(x, y, z)= 0,∂z(2.33)

where ∇2t is the transverse Laplacian defined ∇2t = ∂/∂x + ∂/∂y. Eq. 2.33 has well known solutions.
The solution for a Gaussian beam field is



 
r2
r2
z
−1
u(x, y, z) = p
exp −− ik z ++ i tan
,
2
2
ω(z)
2R(z)
zR
1 + (z/zR )u0(2.34)

36

150
100
Signal50

Reflection Ringdown
Data
Fit

0
50
100

Field-decay: 1.05 µs

150
0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time [µs]Figure 2.2: Cavity ringdown performed in reflection. A fit using a chirped, damped sine wave isused to extract the electric field lifetime in the cavity. This is used for determination of the cavity
finesse. A field decay of 1.05 µs is consistent with a finesse of F ≈ 500 (asssuming c/L = 154 MHz).
This should yield a buildup of β ≈ 200.

37
where r2 = x2 + y 2 . The Rayleigh range is zR = πw0 /λ, where w0 is the beam waist at the focus
and w(z) is the beam waist at a position z which can be written assw(z) = w0



1+

z
zR

2
.

(2.35).

(2.36)

The wavefront radius at a position z isR(z) = z 1 + z 2 RzAnother important feature of Gaussian beams is that there are two terms that lead to a phase shiftof the beam during propagation in z. They are−1

φ = ikz − i tanzzR= φprop + φGouy .

(2.37)

While the first term is the usual propagation phase, the second is called the Gouy phase. The Gouy
phase shift depends on how tight a Gaussian beam is focused and where the beam is relative to its
focus. While the Gouy phase usually does not have significant consequences in dealing with laser
beams, it will be very important when we consider phase matching between the driving laser andthe high-order harmonics later in this thesis.
While the above discussion is in principle all we need to know about Gaussian beams inorder to use them, it is often much more mathematically convenient to deal with the complex beam
parameter q defined as

1
1
λ
=
−i
.
q(z)
R(z)
πw(z)2(2.38)

While at the moment, this may seem arbitrary, it allows us to recast Eq. 2.34 in a slightly differentwayu0

u(x, y, z) = p
1 + (z/zR )2 
exp −i P (z) +where we have defined a new parameter P (z) as

k 2
r
2q(z)


,

(2.39)38siP (z) = ln 1 +

z
zR

2

−1

+ −i tan



z
zR


.

(2.40)The benefit of this substitution is that manipulation of the q parameter and determining it anywhere along a Gaussian beam becomes very straightforward and it contains most relevant geometric
information about the Gaussian beam. At the beam focus q is defined as q0 = i zR . At a further
distance z, q(z) becomes a very simple resultq(z) = q0 + z.

(2.41)

Therefore, with knowledge of the q parameter of a beam, the waist w and wavefront curvature R
can be determined. A couple other useful quantities are easy to extract with the q parameter−Re[q(z)]qλw0 =π Im[q(z)].

Distance to waist =(2.42)(2.43)The most powerful result of this discussion of Gaussian beams is that the q parameter can be
determined and manipulated using simple matrix operations. To go from q1 → q2 , simple rules arefollowed.
  
 
q
A
B
 2 
 q1 
 =
 
1
C D
1

(2.44)A table of useful ABCD matrices for common optics is given in Table 2.1. By careful determinationof a q parameter at a given point in a laser beam, it can be accurately predicted at a later placeeven if mode changing optics like lenses are employed.

2.2.3

Cavity geometry

We are now ready to use the briefly outlined ABCD matrix formalism to determine thetransverse mode structure of the cavity. We are able to use the transverse mode structure toOpticFree Space
Lens
Curved mirror x-axis
Curved mirror y-axis

ABCD


1 z
 0 1 1
0
−1/f 1
10−2/[Rcos(θ)] 110−2cos(θ)/R 1

39

Table 2.1: A table of common optical elements and their corresponding ABCD matrices. For thistable, f is the focal length of the lens, R is the radius of curvature of the mirror, θ is the angle ofincidence on the optic.

accurately measure important cavity parameters such as the spot size in the resonator. Assuming alaser beam with q1 is incident on the input coupler of a cavity like that of Fig. 2.1a, we can multiply
q1 by all the relevant ABCD matrices to describe the intracavity optics. When the beam returns to
the input coupler, in order for the cavity to be stable, the q parameter must return to the originalvalue. What we have just described is a powerful method to determine where the optics shouldbe placed so that a stable cavity is possible. This is equivalent to solving the eigenvalue problemof the ABCD matrix that is product of all the intracavity optics. This will need to be done forthe vertical axis of the cavity and the horizontal axis independently as they do not have identical
ABCD matrices.
Assuming a generic cavity, we have the following equationq2 =A q1 + B,
C q1 + D(2.45)

where we now require that q2 = q1 after a full round trip. We can then solve for q directly to obtain

1
D−A
1 p
=
±
(A − D)2 + 4BC.
q
2B
2B

(2.46)

Since the ABCD matrix elements are real, we can determine that the first term in the equationmust be real and the second must be imaginary because q is in general complex. Therefore, we have

40

2B
D−A ,

R=
r
w=

π

√ 2λ|B|

4−(A+D)2(2.47).

(2.48)

It is important to remember that these equations are only valid at the reference point of the cavity
determined before constructing the ABCD matrix. To determine R(z) and w(z) somewhere elsein the cavity, ABCD matrices will be needed to propagate the q parameter to that point. For
example, in order to send in a beam with the correct q parameter to the cavity, the intracavity
beam determined by the ABCD matrices can be propagated to the input coupler. If there is any
spatial mismatch between the intracavity beam and the injected beam, this imperfect overlap leads
to the mode matching factor  which was first introduce in Eq. 2.1. Any light that is not mode
matched to the cavity will be rejected and is not useful for power enhancement.
It can be seen from Eq. 2.48 that in order to have a real and physical value for the waist, wehave a constraint on A and D being,

Stability Region: |A + D| ≤ 2.

(2.49)

Assuming the overall cavity length is fixed 2 , the only way to adjust A and D for our ring resonators
is to change the distance between the curved mirrors. There is only a small range of values that will
satisfy Eq. 2.49 and hence a small range of suitable curved mirror separations and mirror radius ofcurvatures. This is referred to as the “stability region.”
The results of an ABCD matrix calculation are shown in Fig. 2.3. Assumed is a cavity withan FSR of 154 MHz and with curved mirrors with radii of curvature equal to 15 cm and 10 cm.
The angle of incidence on the curved mirrors is 3 deg. Fig. 2.3a shows the focal waist w0 of theintracavity beam changing as a function of curved mirror separation. The change is most dramaticat the edge of the stability region. Due to the angle of incident on the mirrors, the horizontal beam
2
This will be required to inject a pulsed laser into the cavity. Typically, the length of the resonator is picked so
the FSR is equal to the repetition rate

41

Figure 2.3: a) The effect on the cavity focus and the spot size on the second curved mirror as afunction of curved mirror separation. b) The beam waist as a function of position in the cavity ata fixed curved mirror separation.

42
size is not necessarily the same as the vertical axis thus leading to astigmatism. At first glance,
operating at the outer stability edge would be equivalent to the inner stability edge. However, thisis not true. Operating at the outer stability edge will lead to a weak second focus in the cavity. If
the second beam waist falls on an optic, the increased peak intensity can lead to optical damage.
For the work in this thesis, we generally operate at the inner stability edge. Operating at either
stability edge will increase the beam size on the curved mirrors, which will also help with damage
considerations. Fig. 2.3a also plots the beam waist on the curved mirror immediately after the focus
as a function of curved mirror separation. Fig. 2.3b shows the beam waist as a function of positionin the resonator. Again, the astigmatism cause by the non-normal reflection from the curved mirrors
is apparent. For Fig. 2.3b, a curved mirror separation of 12.59 cm was assumed. This leads to a
focal waist of w0 = (20.4, 22.0)µm. In practice, operating the cavity this close to the stability edge
as shown in this example is routine and often necessary for HHG work.
While it is in principle possible to determine the cavity spot size by measuring the curved
mirror separation, in practice, it is hard to make this very accurate. Instead, we can rely on
another feature of the cavity to determine the spot size. A Gaussian beam is only the lowest ordertransverse mode supported by an optical cavity. In general, the cavity supports Hermite-Gaussianmodes [14]. They are denoted by TEMmn where m, n are the transverse mode order. The Gaussian
beam we have discussed thus far is the lowest order mode denoted as TEM00 . The spatial mode
structure becomes much more complex as m, n increases. TEMmn modes behave similarly to anormal Gaussian beam. However, one important difference is that the Gouy phase is modified tobeφGouy,m,n = (m + n + 1)tan−1 (z/zR ).

(2.50)

Since a TEM00 beam will have a different round trip phase shift than a TEMmn , the resonant
frequencies of the cavity will also be different. Depending on the ABCD matrix that describes theresonator, the resonant frequencies of the q th cavity mode for a TEMmn is written as νqmn

43"
νqmn = FSR q + (m + n + 1)cos−1r!#1(2 + A + D).
4

(2.51)

It is often difficult to measure exact resonant frequencies unless a calibrated laser or a frequency
comb is employed. However, measuring the relative frequencies of high order transverse modes from
TEM00 is very straightforward. Therefore, if one can measure the relative frequency spacings ofhigher order modes, using Eq. 2.51, one can determine to a reasonable degree of accuracy what theA + D coefficients must be. If this is known, then it is straightforward to determine the spot size.
This is a very important point because it allows for very accurate peak intensity calibrations for
intracavity work. Peak intensity is one of the most important parameters for strong-field physics
experiments and is often quite tricky to measure. In the cavity, because the laser mode profile isso well-defined, cavity-based strong-field experiments do not suffer from peak intensity calibration
issues; it is very easily inferred from the measurements.

2.2.4

Cavity mirror dispersion and the femtosecond enhancement cavity

We now turn to the effect of mirror dispersion and how it affects cavity operation. This isultimately the only conceptual difference between operating an enhancement cavity for continuous
wave lasers and for frequency comb lasers (femtosecond lasers). Mirror dispersion essentially leads toa different phase shift upon reflection for different wavelengths. For a cavity driven by a continuous
wave laser, this is usually of very little consequence because you typically only excite one cavity
resonance (or a couple nearby cavity resonances) at a single time. For the work in this thesis, ourgoal is to inject a frequency comb into the cavity. Recall that the frequency spacing of the comb
teeth is determined by fr . A cavity that has zero dispersion also has perfectly evenly spaced modes
separated by the FSR. So, if one matches the frequency comb’s fr to the FSR, then all of the
comb teeth can be on resonant at the same time. However dispersion-less mirrors with reflectivity
exceeding 99.9% do not exist.
Much in the same way a Gouy phase shift will shift the cavities resonant frequency, so willa phase shift from reflection off of a mirror or if there are any dispersive materials, like a crystal,44
inside the cavity. So, the resonant frequencies of the cavity will not be evenly spaced by the FSR,
rather they will start to deviate depending on how much dispersion is present. This is a problem
for frequency combs because the spacing is always fr and it is not possible to compensate for the
non-uniformity of the FSR spacing in a dispersive resonator.
Dispersion φd (ω) (originally introduced and dropped from Eq. 2.8) can usually be expanded
in a Taylor series as

φd (ω) = φ0 + φ1 (ω − ω0 ) + φ2 (ω − ω0 )2 + φ3 (ω − ω0 )3 + . . . .

(2.52)

Each dispersion order affects a laser pulse differently. φ0 will cause the electric field to shift in phaserelative to the pulse envelope and its original value, but will not affect the position of the envelope.
φ1 will cause the pulse envelope to delay in time, but the electric field remains constant (however,the carrier phase relative to the envelope maximum may change). φ2 causes the pulse to broaden
symmetrically in time. φ3 and higher will start to distort the pulse in more complex ways. If weconsider a cavity with only a φ0 , after each round trip through the cavity, the phase of the electricfield will have shifted by some small amount relative to the carrier. The injected pulse must be
adjusted to avoid destructive interference at the input coupler. Since this happens every round trip,
it amounts to setting the carrier-envelope offset frequency f0 = φ0 fr . In the frequency domain, thisis a uniform shift of all the cavity resonances from 0. For cavity-comb coupling, φ0 and φ1 can becompensated by judicious choice of fr and f0 such thatf0 = −(φo − ω0 φ1 )fr ,(2.53)cc(1 − φ1 ).
L
L(2.54)

fr =Mirror manufacturers will usually provide group-delay dispersion (GDD) for their mirrors.
By integrating the GDD data (typically measured in fs2 ), φd (ω) can be determined. Then, φ0 and
φ1 (ω − ω0 ) can be subtracted from the data to simulate proper fr and f0 choice. The remaining

45
dispersion can then be plugged into Eq. 2.7. It is useful to express the relation of the intracavity
field to the injected cavity field as a transfer function [75, 76]. We then have

Ecav = B(ω)eiψ(ω) E0 ,(2.55)

where we have made the following definitions

ψ(ω) = arg
B(ω) =hti
1−ri (ω)rc (ω)eiφd (ω)
Ecav (ω)
ti F
E (ω)
π 0

.

i

,

(2.56)(2.57)

where F was calculated for the peak value of the mirror coating. We can interpret ψ(ω) as the
spectral phase of the circulating pulse and B(ω) as the frequency dependent coupling efficiency. The
cavity electric field can be determined by Eq. 2.2. An example of this analysis of a cavity containing
three Layertec high-power, low loss optics plus one low dispersion input coupler of R = 0.985 isshown in Fig. 2.4. The injected pulses were assumed to be 70 fs transform limited, centered at1070 nm, and a Gaussian profile in time. Fig. 2.4a shows the total GDD of all four cavity mirrors.
Fig. 2.4b,c shows the phase shift with the zeroth and linear terms removed upon a single mirror
reflection and for the whole cavity. The cavity coupling efficiency (and hence bandwidth) is shown in
Fig. 2.4d and is a plot of B(ω). For a 70 fs, transform limited pulse, the time domain and wavelengthdomain comparisons of the injected pulse and circulating pulse are shown in Fig. 2.4e,f respectively.
The bandwidth of the mirrors is quite good and very little spectral filtering is expected. However,
the residual phase shift from the cavity, even though it does not affect coupling bandwidth strongly,
is still strong enough to alter the pulse in the time domain. For this example, the effect is to increase
the pulse duration to about 75 fs. If shorter pulses are used or a cavity with higher finesse, the pulse
distortions will become much more severe because of the increased spectral filtering and sensitivity
to mirror dispersion. To date, mirrors with lower dispersion than the ones considered here arepossible. Cavities with a F = 500 that support 30 fs pulses have been developed [77]. However,their mirror technology relies on the high index material of their quarter wave stack to be Nb2 O5 .

46
Our mirrors from Layertec use HfO2 which has a slightly lower index3 . The contrast between thehigh index layer and the low index layer (usually SiO2 ) sets the mirrors bandwidth. The maindrawback to using Nb2 O5 mirrors is that they have much lower damage thresholds compared to
HfO2 making the cavities less reliable and limit the scalability of the cavity to higher pulse energy.
So, when faced with a trade-off between high bandwidth and day-to-day robustness, we sided withrobustness.

2.2.5

Laser-cavity stabilization

We have already discussed how resonators have resonant frequencies spaced by the FSR andhave a linewidth. If a laser is incident on a cavity with a frequency close to a resonance, but still welloutside the linewidth, the cavity will reject the light. If the laser frequency is tuned into resonance, it
can be effectively injected and transmitted through the cavity. The cavity is essentially a frequency
discriminator. Lasers do not typically stay at a given frequency for very long, but they can be
disciplined to an optical cavity that can be made, relatively speaking, frequency stable. In orderto lock a laser to a cavity, there needs to be a way of determining the cavity resonant frequency
with respect to the laser. The method employed throughout this thesis is the Pound-Drever-Hall(PDH) approach [79]. It is based on a microwave Pound stabilizer but applied to lasers at opticalfrequencies. It relies on the incident laser having phase modulation sidebands and measuring thereflected cavity light, demodulated at the phase modulation sideband frequency. The approach isvery well established and used all throughout atomic physics. For example, it is used to generate themost stable lasers ever made [40, 80] and it is used as the laser stabilization method in the recently
successful gravitational wave detectors [81]. Instead of a full derivation (a useful introduction to
the matter can be found in Refs. [82, 83]), we will describe how we implement it in our system.
Our frequency modulation sidebands are generated by driving a mechanical resonance of ourlaser cavity’s PZT-driven mirror. The resonance is typically at 700 kHz or 1.3 MHz. After the3

Layertec has never confirmed their coating material with us and claim it is a secret. However, due to the damage
threshold of these mirrors and their bandwidth, we deduce HfO2 rather than Nb2 O5 or Ta2 O5 . See Ref. [78] forfurther discussion.

47

Figure 2.4: a) The group delay dispersion (GDD) of a single cavity high reflecting mirror. b) The
phase shift upon a single reflection from a high reflecting mirror with the zeroth and first orderphase terms removed. c) The phase shift of a cavity made with the mirrors in a) with the low orderphase terms removed. d) The spectral bandwidth that will fit inside a cavity made with the mirrorsin a) assuming a low dispersion input coupler of an R = 0.985. e) The top panel shows how an
injected 70 fs pulse is distorted by the mirror dispersion. The bottom panel shows how the pulse
spectrum is modified.

48

Figure 2.5: Pound-Drever-Hall error signal and the cavity transmission are shown as the laser isslowly swept across a cavity.

49
comb is injected into the cavity, the reflected light from the resonator is demodulated at the drive
frequency with the correct phase to generate the PDH error signal shown in Fig. 2.5. From there,
the error signal is filtered and then reapplied to one of two places. The correction signal can be
re-applied to the laser using the same PZT that generated the sidebands. Adding these two signals
together is accomplished with a bias-T. Alternatively, a cavity mirror equipped with a PZT canalso be used. In the former case, the laser follows the cavity fluctuations. In the latter, the cavity
follows the laser fluctuations. In either case, the laser can be maintained on resonance with a servo
bandwidth of about 50 kHz. Slow feedback is provided by a PZT stage in the fsEC or by fiber
stretching in the laser cavity. At night, when the lab is very quiet, the cavity-laser stabilization
scheme can stay locked over the course of a few hours without interruption if both fast and slow
feedback loops are active.
The frequency comb has two degrees of freedom, fr and f0 . The scheme described above
typically makes fr modifications to maintain lock. The passive stability of f0 is sufficiently goodthat it does not require active locking. Rather, f0 is set by tuning the temperature of the fiber
Bragg grating (FBG) in the laser cavity until maximum buildup is achieved. Adjustments by hand(the graduate student feedback loop), are made every 10 minutes or so if the laser is warmed up.
With both locks engaged, optimal power enhancement can be achieved. Ultrafast lasers with larger
optical bandwidth typically require active feedback on fr and f0 because it is easier for the comb
modes to walk off when the optical spectrum is very broad. However, due to our low finesse cavity
and the relative narrowness of our spectrum, only one active lock is necessary.
If there is a large amount of relative frequency noise between the laser and the cavity, feedback
loops often don’t have enough bandwidth to maintain perfect synchronization. Therefore, frequency
fluctuations of the laser (or cavity) will lead to amplitude fluctuation in the cavity. This can be very
problematic for HHG (which is very nonlinear in intensity) or if the transmitted light needs to be
used for subsequent detection. An example of the transmitted intensity noise for one of our cavities
with F = 1000 is shown in Fig. 2.6. The vertical axis is the residual intensity noise (RIN) power
spectral density (PSD) expressed in decibels below the carrier (dBc). This frequency to amplitude

50

Figure 2.6: The residual intensity noise (RIN) power spectral density is shown when a cavity
intensity servo is on and off. The actuator for the servo was an acousto-optic modulator placed
before the cavity that can steal power from the undiffracted beam. The servo bandwidth is limitedby acoustic wave propagation delay in the acousto-optic crystal.

51
noise conversion process can be combated in two ways: make the laser and cavity more frequency
stable or make an intensity servo. In Fig. 2.6, we demonstrate an intensity servo. The transmitted
light is taken and compared to a constant value to generate an error signal. The error signal isfiltered and applied to an acousto-optic modulator (AOM) that has the ability to steal power awayfrom the laser beam. The AOM is placed before the cavity. When the lock is engaged, there isa 40 dB suppression of noise out to 1 kHz with the unity gain crossing at nearly 100 kHz. Thisis extremely powerful for making the laser-cavity system more stable, especially for applications
where the transmitted light needs to be analyzed [84, 85]. We will discuss later that there is anamplitude to phase noise coupling in the HHG process as well [43, 86, 87]. This makes a quiet cavityof paramount importance if coherent XUV light is to be generated. This process will be discussed
in detail in Sect. 3.1.3.

2.2.6

Output coupling

We plan to use all of the described cavity technology to perform HHG. Since the HHG process
will be performed inside of the cavity, we need to have some method of getting the XUV light out of
the cavity. Most optical materials are completely opaque at XUV wavelengths. This is problematic
because immediately after the focus, if the XUV light is not directed out of the cavity, it will beabsorbed by the next curved mirror. There are three experimentally verified methods of outcoupling
XUV light from a fsEC. However, others such as crossing two cavities to implement noncollinear
HHG, have been proposed [88]. The first method is to use a thin piece of glass or a crystal andplace it at Brewster’s angle for the driving laser inside of the cavity. Since the index of refraction of
the material is different at XUV wavelengths, there will be a Fresnel reflection for the XUV and the
XUV light can be reflected out of the cavity. Brewster angle operation is essential to not induce any
extra loss to the cavity at the driving wavelength. A common material to use is c-plane sapphirewith a thickness around 300 µm. It is very important that the sapphire is very flat and has a very
low surface roughness, otherwise the XUV beam quality and intensity deteriorate. Another concern
with using Brewster plates is that the sapphire will be nonlinear at the high intracavity powers

52
required for HHG. Secondly, sapphire is dispersive, so it will ultimately limit how short a pulse canbe stored inside a cavity. For example, sapphire typically has a GVD of 90 fs2 /mm. So, even a 300µm piece has much worse dispersive properties than our other cavity optics.
The second method is the Yost grating and is described in detail in Refs. [74, 89]. The
method relies on placing an optic at a 70 degree angle of incidence immediately after the focus of
the resonator (See Fig. 2.1b). The optic is actually a high-reflector for the driving laser wavelength.
The topmost layer of the optic is SiO2 with a small period grating etched on it. The period of thegrating is too small for the fundamental light to cause diffraction. Thus, for the driving laser it
behaves just like a normal high-reflecting optic. However, for the shorter harmonic wavelengths, it
behaves like a diffraction grating. It causes the harmonic light to spectrally disperse, separate fromthe driving laser, and outcouple from the cavity all in one step. This is a perfect output coupler forspectroscopy applications because they usually have the requirement that the harmonic light needs
to be spectrally dispersed or filtered before shining it on a target [52]. For ultrafast applications, it
is obviously not ideal. If one were to start with a Brewster plate output coupler and then attempt
to spectrally disperse the light, there will be severe power loss as optics in the XUV are in general
terribly inefficient.
The third method is often referred as “geometric output coupling” or “holey mirror output
coupling.” The method was first proposed in our group immediately after the first demonstrations of
fsEC-based HHG [88] and was recently demonstrated a group in Garching [90]. Since the harmonic
wavelengths diverge at much slower rates than the fundamental wavelength, a small hole can be
drilled in the curved mirror immediately following the focus. The hole must be small enough to notcause large amounts of additional loss to the cavity, but large enough to allow the XUV light to leakout of the hole. Our initial attempts were not successful due to a poor effort in drilling that cause
small cracks in the optical coating near the hole. In recent years, new attempts were made inspiredby the results of Ref. [90]. The method for fabrication is to take an optical substrate, laser drill ahole with a conical shape, re-polish the substrate, and finally coat with a high-reflection coating.
Our initial attempts have been encouraging with output coupling ∼ µW/harmonic power levels,

53

Figure 2.7: Output coupling efficiency for different methods. a) The low photon energy efficiencies
are shown. All methods have similar performance. b) The high photon energy region is shown. The
improvement of using a holey mirror output coupler is clearly seen and the efficiency will increase
with wavelength. Only the Yost grating spectrally disperses and outcouples the harmonics in asingle step forgoing the need to filter the harmonics if they are to be used for spectroscopy.

54
but the performance has been far from optimal. Work is still in progress.
A summary of all three output coupling schemes is shown in Fig. 2.7. For the comparison,
the holey mirror has a radius of 50 µm in a curved mirror with radius of curvature (ROC) of 25 cm.
The focusing optic with a ROC of 7.5 cm was chosen to give a similar spot size of about 20 µm,
which is typical for our fsECs and facilitates a fair comparison to the Brewster plate and the Yost
grating. All methods shown are compatible with cavities that have a β = 200. As seen in Fig. 2.7a,the low photon energy performance from all three methods does not vary greatly. However, inFig. 2.7b, the high energy photon comparison is presented. Since the holey mirror does not rely ona reflection to output couple and since the harmonic beam diverges less at high photon energies, theoutput coupling efficiency increases with harmonic order. This is a very promising scheme because
it does not add dispersion to the cavity (good for short pulse operation) and its efficiency increases
with photon energy (great for future work requiring shorter and shorter wavelengths). This is very
enticing for possible applications in isolated attosecond pulse generation experiments or virtually
any other experiments where very high energy photons are needed.

2.3

The laser
The laser technology used for XUV frequency comb generation has rapidly progressed since

the initial demonstration [3, 4]. In the first generation of experiments, Ti:Sapphire oscillators wereused to pump fsECs. For reasons that will be outlined in Chapter 3, it is beneficial to use fsECs withrelatively low finesse which puts a stringent requirement on how much average power the frequency
comb must have. To satisfy the need for high average power frequency combs, Yb:fiber technology
was employed because it offered a platform for robust power scaling.
In the work of this thesis, we used the laser system outlined in Ref. [73]. For completeness,
we will outline the major features of the laser system and its recent modifications. The laser systemstarts with a Yb:fiber modelocked, similariton oscillator. This oscillator design is used throughout
the Yb:fiber frequency combs used in our lab [31, 36, 84, 87, 91–93]. The entire laser system,
including the oscillator details, are schematically sketched in Fig. 2.8. Modelocked operation is55
achieved by a combination of nonlinear polarization rotation and a saturable absorbing mirror witha sub-ps lifetime. The dispersion of the cavity is compensated with a chirped FBG to keep the
net dispersion near zero and in the similariton regime [94]. The FBG also doubles as a cavity end
mirror. The oscillator exhibits exceptional noise properties for a fiber laser and robust day-to-day
operation. The free running f0 linewidth is ∼ 10 kHz [92]. The oscillator produces ∼ 100 mW of
average power and pulses compressible to < 80 fs at a repetition rate of 154 MHz. The spectrum iscentered at 1050 nm and has 45 nm of bandwidth.
After the oscillator, a chirped pulse amplification (CPA) system is employed. Typically, CPA
systems rely on a grating-based stretcher and a grating-based compressor. To make our system
more robust, we replaced the grating-based stretcher with an all fiber solution making it alignment
free. Over 380 m of various fibers were required to stretch the pulses to 870 ps. The various fibers
were chosen so the higher order dispersion of the fiber is matched for the grating compressor used
after the main power amplifier. The group delay dispersion at 1070 nm was 12.4 ps2 . Since thestretching system is very lossy, the pulses were then amplified in two stages up to ∼ 3 W of average
power before being injected into the main power amplifier.
The main amplification stage relies on ∼ 9 m of polarization maintaining large mode area
(LMA) photonic crystal Yb:doped fiber commercially available from NKT Photonics. This stageof the laser was damaged and rebuilt in 2012. The fiber is pumped by a 250 W, 915 nm diode.
This was upgraded during 2014 to replace an aged diode and allow for > 80 W operation. The longfiber length and 915 nm pump wavelength were both chosen to allow the laser spectrum to redshiftduring amplification. The output of the amplifier is 100 W, but it can be briefly run up to 125
W when additional cooling methods are employed. The pulses are dechirped with a pair of 1600
l/mm transmission dielectric gratings. These were an upgrade from the original polymer gratings
that had 1590 l/mm. The small change in line spacing did not affect the performance, but thedielectric gratings offer higher damage thresholds and better power handling. After the compressor
80 W (or briefly 100 W) of average power is available for use with a pulse duration of 120 fs. The
amplifier operates with a B-integral of < 0.2 and shows no sign of nonlinearities. The laser system

56
runs daily with nearly alignment free operation. Only small tweaks to the grating separation in thecompressor is required for peak performance.

2.3.1

Nonlinear pulse compression

One of the major drawbacks of using fiber-based ultrafast laser systems is that the gainmaterials typically do not have large gain-bandwidths and this limits the achievable minimum pulse
duration. In fact, our laser system, up until recently, was unmatched in average power and pulse
duration. Nevertheless, the convenience of fiber-based systems is so compelling that there havebeen many work-arounds to the pulse duration problem. The most common way to approach this isnonlinear pulse compression after amplification [95]. However, new nonlinear amplifiers have beendeveloped, but they are still untested in XUV frequency comb applications [96, 97].
The nonlinear interaction of a laser pulse with condensed media is typically described by
the non-linear Schrödinger equation (NLSE) [98]. If we ignore higher order effects (self-steepening,
stimulated Raman scattering, etc) and only focus on self-phase modulation and dispersion, the
NLSE is expressed asi∂A
β2 ∂ 2 A=− γ|A|2 A,∂z
2 ∂T 2(2.58)

here A is the pulse envelope, β2 is the second order dispersion coefficient, and γ is the nonlinearity.
The equation is easily numerically integrated using the split-step Fourier method. For illustrative
purposes, a normalized version of Eq. 2.58 was numerically implemented and the simulation results
are plotted in Fig. 2.9. The simulation parameters are chosen to simulate injecting 80 W of our 120
fs pulses into 1 cm of commercially available LMA fiber. The injected spectrum broadens by almost
one order of magnitude and the spectral ripples are a hallmark of self-phase modulation. In thetime domain, due to the shortness of the nonlinear medium (and hence small amounts of dispersion)
hardly any perturbation is present on the pulse. However, the self-phase modulation process does
impose a large temporal chirp on the pulse that can be removed. If the pulse is dechirped, the pulse
duration shortens by almost one order of magnitude. The combination of self-phase modulation

57

Figure 2.8: Schematic of the 80 W Yb:fiber frequency comb system. PZT, piezoelectric transducer.
SA, saturable absorber. QWP, quarter waveplate. POL, polarizer. YDF, Yb-doped fiber. TEC,
thermoelectric cooler. FBG, fiber Bragg grating. WDM, wavelength division multiplexer. ISO,
optical isolator. PBS, polarizing beam splitter. FRM, Faraday rotating mirror. SC YDF, suspended
core YDF. DC YDF, double clad YDF. LMA YDF, large mode area YDF. The LMA YDF and 240
W 915 nm pump laser were replaced from the original version due to failure and age. The original
polymer grating compressor was upgraded to the dielectric grating compressor for better power
handling. Nonlinear pulse compression was implemented to reduce the pulse duration. The laser
can be operated up to 100 W for periods of time less than 1 hour limited by thermal management.

58
with temporal dechirping is a very robust way to decrease the pulse duration of fiber laser systems.

We implemented a version of this nonlinear pulse compression scheme. Instead of broadening
in an optical fiber, we chose bulk sapphire as our nonlinear medium. Nonlinearly broadening inbulk materials is often met with degradation in beam quality because of the nonuniformity of thenonlinearity (the nonlinearity is proportional to the intensity profile and hence not homogeneous
across the wavefront). Nevertheless, for our experiments we required only a modest amount ofspectral broadening due to restrictions from our enhancement cavities (See the coupling bandwidthin Fig. 2.4d). For our implementation, the laser is focused into 3 mm of bulk c-plane sapphire.
The focusing and recollimating objectives are 50 mm air-spaced doublets to ensure tight focusing
and preservation of good beam quality. The chirp imposed by the self-phase modulation process
was removed by 24 bounces on -100 fs2 /bounce chirped mirrors purchased from Altechna. After the
chirped mirrors, part of the beam is sent to diagnostics including a frequency resolved optical-gating(FROG) apparatus for pulse characterization. The rest of the beam is sent to the enhancement
cavities for use in experiments.
The results of the characterization measurements are displayed in Fig. 2.10. The increased
optical bandwidth is only slightly filtered by the optical cavity as seen in Fig. 2.10a,b. This is inagreement with the dispersive properties of the mirrors provided by Layertec and the simulation
results presented in Fig. 2.4. The pulse in the time domain (and its corresponding phase) was
measured by FROG and is shown in Fig. 2.10c. The pulse only differs from the transform limit
at the pedestal of the pulse. To verify that the cavity was not distorting the pulse in the time
domain, cavity transmission light was used for second harmonic intensity autocorrelation. Assuming
a Gaussian deconvolution factor, the intracavity pulse duration is 75 fs, consistent again with thesimulation results. The intensity autocorrelation trace is shown in Fig. 2.10d. We are able to achieve
11.5 kW of intracavity average power with 75 fs pulses. We are typically limited to similar average
powers when the Yost grating is used due to thermal heating of the optic. By reducing the pulse
duration from 120 fs to 75 fs, assuming a constant average power and constant beam waist, there59

Figure 2.9: a) The input and output laser spectrum using the Nonlinear Schrödinger equation
calculation assuming 80 W of our frequency comb power into a 1 cm piece of LMA-35 NKT Photonics
fiber. b) The input and output temporal profile are shown in blue and red. There is nearly noobservable difference between the two because the fiber is so short and does not impart dispersionof its own to the pulse. In black, the compressed pulse is shown if the chirped from the SPM
broadening is removed. A nearly 10× reduction in pulse duration is achieved.

60
will be an immediate gain of 1.6× in the peak intensity.
We also succeeded in a nonlinear pulse compression scheme based on a large mode area fiber
as a nonlinear medium. The fiber is a commercially available photonic crystal fiber from NKT
Photonics (LMA-35). We achieved 40 fs pulses at an average power of 30 W. In terms of peak
power, this is still an improvement upon our laser’s nominal operating condition of 120 fs pulses at
80 W. The average power could have in principle been increased, but we were unable to fabricate
fibers < 2 cm in house. Further, 40 fs pulses proved to be too short for our cavity mirrors (Fig. 2.4).

2.4

Typical laser-cavity operating parameters
Typical cavity-laser operating parameters are as follows. With 30 W of average power in 120

fs pulses at 154 MHz repetition rate from the laser system, a cavity with an input coupler of T =1.5% has a buildup of 200 and does not distort the 120 fs pulse duration. Therefore, the intracavity
power is 6000 W. The resonator usually operates with a 10 cm and 15 cm ROC curved mirror
combination leading to a spot size of ∼ 20 µm at the inner stability edge. The peak intensity of theGaussian beam can be calculated by

I=

Pavg
2
,
π fr τpulse w02(2.59)

where Pavg is the average power and τpulse is the pulse duration. With the aforementioned operating
parameters, the peak intensity is 0.52×1014 W cm−2 which is very suitable to make HHG in axenon gas target. We have demonstrated up to 14 kW of average power in the cavity (a four mirror
resonator not using the Yost grating) which would correspond to a peak intensity of 1.3×1014 W
cm−2 which is suitable for harder to ionize gases like argon.

2.5

The chamber
Due to the high power cavity operations, the nonlinear and dispersive properties of air will be

problematic for cavity operation. Also, XUV wavelengths (< 100 nm) will not propagate in air and61

Figure 2.10: a) The unbroadened laser spectrum, nonlinearly broadened laser spectrum, andthe cavity transmission spectrum of the nonlinearly broadened laser are shown. b) Same as a)but on a log scale. c) The time domain profile of the nonlinearly compressed pulse. The pulses
were reduced from 120 fs to 70 fs in duration. The is near the transform limit of the nonlinearlybroadened spectrum shown in a). d) The intracavity second harmonic intensity autocorrelation
trace. Assuming a Gaussian deconvolution factor, the pulse in the cavity is ∼ 75 fs in duration.
The slight increase from c) is because of the dispersion of the cavity mirrors.

62
are almost immediately absorbed. So, the entire experiment needs to be performed inside a vacuum
chamber. We do not require ultrahigh vacuum and a pressure of < 1 mTorr is generally sufficient.
This is easily accomplished with turbomolecular pumps. Due to work that involves simultaneous
operation of two enhancement cavities, a large vacuum chamber needed to be constructed and isschematically shown in Fig. 2.11. The many ports are required for allowing beam access into andout of the chamber. Despite its holey design, more holes would have been nice. Nevertheless, thechamber is mainly aluminum in construction with a steel baseplate. There are holes in the chamberbottom to allow that the breadboard in the vacuum chamber be mechanically decoupled from thechamber walls. Our cavity optics are mounted on the breadboard. This is very important so thatthe inherent vibrations of the turbomolecular pump do not disturb the fsEC. The decoupling scheme
is shown in Fig. 2.12.
We typically inject multiple atmospheres of gas pressure behind a quartz4 nozzle of ∼ 100 µm
diameter. This leads to a considerable gas load in the chamber and in order to keep the pressures
below 1 mTorr (to minimize reabsorption in the gas), pumping speeds of ∼ 1000 L/s are necessary.
However, this situation can be improved by implementing a gas dump right below the nozzle. It is
described in more detail in Ref. [89] and is shown in Fig. 2.13. A small aperture catch is placed as
close to the nozzle as possible without clipping the laser beam. The catch is hooked up to a separate
vacuum pump. When operating properly, the gas catch can reduce the background pressure in thechamber by ∼ 5×. As we will describe later, this is essential for two cavity simultaneous operationand for XUV power scaling methods. Further improvement is possible, for example, by using awider pumping tube for the gas catch thereby increasing its conductance.

2.5.1

Ozone and plasma cleaner

There is often a small amount of residual gas present in a vacuum chamber. Of particularimportance to this work are hydrocarbons. When a hydrocarbon absorbs an XUV photon, it will4

The nozzle material is important. Because of our extremely large average intracavity power, a borosilicate nozzle
will readily melt when placed close to the laser focus. Therefore, a nozzle with a high melting temperature is needed.
We have found that quartz works very nicely and can be purchased commercially.

63

2x long side
longside3p1.sldprt

Welded Aluminum C hamber

1x lid rest
lidrest3.sldprt
2x short side
shortside3.sldprt
1x Aluminum Bottom Plate
With O-ring G roove
bottom3.sldprt

1x bottom lipbottomlip3.sldprt

5x bellow
feedthroughs

Large Aluminum Vac uum C hamber:
1. 6061-T6 Aluminum plates, sealing welds on inside.
2. Does not inc lude top lids.
3. C leaned and Leak C hec ked for High Vac uum.
4. Helic oils Installed for threaded holes.

TITLE:XUVHeterodyne Beat
Chamber
REV
Tom Allison
tka3@jila.c olorado.edu D.
5

SHEET 1 OF 7
5

4

3

2

1

Figure 2.11: Drawing of the XUV comb vacuum chamber. Note the five holes in the baseplatethat are used for vibration isolation of the optical breadboard. The many ports are use to get
diagnositics in and out of the chamber.

64

Chamber Breadboard
Chamber Wall

Bellows
Optical Table

Figure 2.12: Schematic drawing of the XUV comb vacuum chamber vibration isolation scheme.
Belows are used in the baseplate to separate the optical breadboard from the vacuum chamber tomechanically decouple them. This is essential for quiet cavity operation.

65

Figure 2.13: Preventing large chamber background pressures is aided by a “gas dump.” A small
aperture hooked up to a scroll pump attempts to catch the gas coming from the injection nozzle.
We routinely observe a 5× reduction in background pressure when the dump is properly installed.

66
typically “crack” and stick to chamber walls or optics. After long periods of XUV operation, thinfilms of carbon-containing molecules deposit on the mirrors. This typically does not hurt the fsEC
operation, however, the XUV performance of our optics degrades. The Yost grating grooves regularly
get filled up with a film and the optic will eventually stop diffracting. Similar effects have been
observed at synchrotrons for many years [99, 100]. Common cleaning methods include introducing
ozone into the vacuum chamber or running oxygen discharges [101–103]. In the early days of XUV
comb operation (before 2013), carbon contamination was a fact of life. After an optic had become
dirty, it would need to be removed from the vacuum chamber and taken into a clean-room to use
a reactive ion etcher to produce a cleaning RF oxygen discharge. This was a very time consuming
cleaning process, not to mention that a cavity would typically only last about one half hour at full
XUV flux before becoming too dirty to use.
Since this dirty little secret of XUV comb performance would clearly hinder future work,a solution needed to be reached. Since ozone was routinely used to clean optics, we decided todeliberately introduce ozone into the vacuum chamber in relatively small quantities during the HHG
work (∼1 mTorr of O2 and ∼0.1 mTorr of O3 ). This was done by purchasing a ozone generation unit
from Ozone Solutions [104] and injecting it into the vacuum chamber. After the simple modificationof introducing ozone, cavity performance can immediately recover from a contaminated state andcan sustain nearly optimal performance for month timescales. Without this simple trick, someexperiments in this thesis would have become so time consuming as to render them nearly impossible.
While the ozone trick is now a standard piece of our XUV comb operation, additional andstronger cleaning methods are sometimes needed if the ozone is ever not used or contamination isso bad that the ozone is no longer effective. The stronger cleaning method is to run an oxygen
discharge. We developed a simple method to run a small RF discharge in the vacuum chamber toisolate individual dirty optics. A small DC-AC inverter (originally intended to power laptop screen
lamps) converts a 12 V signal into a 1.5 kV signal oscillating at 50 kHz. We place this signal ona small electrode in the vacuum chamber and place it next to an optic that needs to be cleaned.
There needs to be an electric ground behind the optic in order for the RF to capacitively couple

67

Bad oxygen pressure
Too L

Good oxygen pressure

Figure 2.14: Top panel shows the plasma cleaner operation under too low oxygen flow. The bottom
panel shows ideal oxygen flow. A nice, localized plasma is formed on near the mirror to perform
cleaning. The in-situ plasma cleaner reduces the cleaning time of an optic from ∼ 1 hour to ∼ 15
minutes.

68
through the optic. A small, steady oxygen flow is introduced to the chamber and the electrode isthen switched on. If the chamber pressure is too low, the plasma becomes very unstable (for ourchamber and flow, too low means < 1 Torr). An image of non-ideal operation is shown in Fig. 2.14a.
Once higher pressures are reached, the plasma stabilizes close to the electrode and the optic tends
to be cleaned. This is shown in Fig. 2.14b. We tested a few optics and as long as good pressure
conditions were maintained for cleaning times of about 2 minutes and electrode-optic separations of
1 cm, we did not observe any additional damage and the optics would become clean. Considering
this can be done in-situ and does not require a specialized piece of equipment like a reactive ion
etcher, it is pretty convenient.

Chapter 3

High-order harmonic generation

In this chapter, we will focus on how a single atom responds to a very intense laser field andhow this interaction relates to HHG. We will focus on aspects most relevant to the XUV frequency
comb experiments. We will also describe macroscopic effects and phasematching of HHG. Special
attention is paid to the unique situation of HHG in a fsEC.

3.1

Single atom response to a strong laser field
Here, we will review strong-field ionization, a simple model for HHG, a quantum descrip-tion based on the strong-field approximation, and finally, aspects of HHG beyond the strong-fieldapproximation. We will focus on features most relevant to the XUV frequency comb experiment.

3.1.1

Ionization by an intense laser pulse

High-order harmonic generation falls under the discipline of strong-field physics and extreme
nonlinear optical phenomena. When an atom or molecule is in an intense electric field, the valence
electrons can tunnel through the severely distorted Coulomb barrier and ionize. Here, an intenseelectric field has a strength comparable to the Coulomb interaction responsible for binding the atom
or molecule together. This is typically V/Å strengths for the noble gases. In oscillating electricfields, there is also the potential for multiphoton ionization assuming the photon energy is less than
the ionization potential, a so-called “low frequency” electric field. Here, an atom or molecule will
absorb multiple photons from the laser and an electron will be excited into the continuum. In 1964,

70
Leonid V. Keldysh produced a simple means to understand what happens to atoms or solids when
they are placed in intense, low frequency electric fields [23]. The published date of 1964 (1965 inEnglish) is quite striking. Remember, the laser was still in its infancy. Nevertheless, it pre-dated
even the first observation of multiphoton ionization [105], optical tunneling [106], above-thresholdionization [107], and high-order harmonic generation [108]. The connection between Keldysh’s
work and these experimental results was not immediately apparent and it took nearly 40 years
after publication for people to realize how useful Keldysh’s physical model actually was [109]. As
expressed earlier in Eq. 1.4, Keldysh derived what we know call the Keldysh parameter γ to show
the transition of the multiphoton ionization to the tunnel ionization regimes. In some sense, thecommon interpretation of γ is problematic [23]; an interesting discourse on the meaning of γ in thecase of strong-field laser-atom interactions is givent in Ref. [110].
Nevertheless, Keldysh’s theory was the first implementation of the strong-field approximation(SFA) which stated that during the interaction of an intense laser field, bound states or resonances
can be neglected during ionization, and the effects of the Coulomb potential on the continuum
electron can be neglected. Using Keldysh’s theory, Perelomov, Popov and Terent’ev developed amethod to calculate ionization rates by intense laser fields (often called PPT rates) [112]. Later,
Ammosov, Delone and Krainov developed a simplified formulation for arbitrary atoms in intensefields (often referred to as ADK rates or ADK theory) [25]. The ADK rates are known to be poorin the multiphoton ionization regime of γ >> 1. However, due to the ease of implementation,
experimentalists often prefer ADK to PPT. A very detailed treatment can be found in Ref. [113].
Fig. 3.1 shows the ionization probability for a 120 fs, 1070 nm laser pulse using the ADK rate
modified by Tong and Lin [111] to deal with the shortcomings of the pure ADK rate when in thebarrier-suppressed regime (when over the barrier ionization plays a role). The ionization probably,or ionized fraction η, is defined as Z ∞η = 1 − exp −ωADK (t) dt ,
−∞

(3.1)

71

Ionization Fraction

100
10-2
10

Xenon
Krypton
Argon
Neon

-4

10-6
10-8

0.1

0.2

0.5

1

2

Peak Intensity [1014 W cm-2]
Figure 3.1: Ionization probabilities as a function of laser intensity for 1070 nm, 120 fs pulse. The
ionization rate was calculated using the ADK formalism with the modifications made by Tong and
Lin to include the effects of barrier-suppression [111].

72
where the integral is performed over the laser pulse duration and ωADK (t) is the ionization rate
calculated using ADK theory. The ionization probabilities are very sensitive to the ionization
potential of the atom and the laser intensity (see Table 1.1 for ionization potentials). As previously
mentioned, ADK rates are generally most applicable in the limit γ << 1, however, the Tong andLin’s modification have proven to work well when this limit is not strictly true [114]. For a 1070 nm
laser of 0.8 × 1014 W cm−2 and considering xenon’s Ip , the Keldysh parameter is γ = 0.842. As wewill discuss in Sect. 3.2, the ionized fraction of a gas target is very important when one considers
the macroscopic response of HHG and macroscopic phasematching.

3.1.2

Simple model of high-order harmonic generation

As previously mentioned, HHG was first observed as early as 1987. However, there were many
attempts to theoretically understand the phenomena and they often relied on direct integrations of
the time-dependent Schrödinger equation [115–117]. In 1993, Paul Corkum developed a simple 1
model that successfully described double-ionization, above-threshold ionization, and HHG [24]. It
is commonly referred to as the “3-step model”. The three steps for HHG are: (1) ionization with
rates determined by ADK theory or a similar theory, (2) propagation of the electron’s motion using
Newton’s equations and ignoring electron-ion interactions, (3) recombination of the electron andion to produce a dipole responsible for the generated XUV field.
The most salient features of the model are presented in Fig. 3.2. For the illustrative calculation, a single laser period is 2π. After the electron is ionized at a time ti , it propagates in theoscillating electromagnetic field according to F = ma. Since the field is oscillatory, the electron
will return to the ion at tr with the possibility of recombining and emitting a high energy photon.
Fig. 3.2a shows that there is a one to one correspondence between ionization time and return time.
However, not all trajectories will recombine. If the electron is ionized before the peak of the field,
it will simply drift away. This can be seen in Fig. 3.2c. The trajectories that have the possibilityof recombining typically ionize at the peak of the field or immediately after. Despite the one to1“Simple” meaning the calculation is at the undergraduate physics level, something experimentalists love.

73

Figure 3.2: Simple man model of HHG. All units of time have been normalized to a phase of
2π. a) The return time of the electron as a function of emission time (time of ionization). Thereis a one-to-one correspondence between the two. b) The return kinetic energy (normalized to thepondermotive energy Up ) as a function of emission time. Two emission times correspond to thesame return kinetic energy. The return kinetic energy sets the harmonic photon energy. The twotimes are referred to as the long and short trajectory referring to the amount of time the electron
spends propagating in the continuum. c) An example of electron trajectories in the laser field. If
the electron is ionized before the maximum of the wave, the electron drifts away and does not havethe chance to recombine. Electrons ionized after the peak have a chance to return to the ion andpossibly recombine and emit a high energy photon.

74
one correspondence between ti and tr , this is not true for the return kinetic energy of the electron.
Shown in Fig. 3.2b is the return kinetic energy (normalized to the pondermotive energy) as a function of ti . For a given return energy, there are two possible ti ’s tr ’s. These two paths are commonlyreferred to as the long and short trajectories indicative of the difference in the amount of time theelectron spends propagating in the laser field. The maximum return kinetic energy is 3.17Up andleads to the famous cutoff equation stated earlier in Eq. 1.6.
While Corkum’s simple model was quite successful at capturing much of the essential physics,
it is not a quantum mechanical description of the HHG process and is incapable of describing things
such as the phase of the emitted XUV light and addressing the role of quantum interferences in the
HHG process, if any. This would be addressed later in the work of Lewenstein et al. [86, 118]. While
there is nothing in Corkum’s model that would hinder future XUV comb performance, quantum
effects may shed more light on coherence properties of the XUV light and if temporally coherent
light in the XUV is even feasible.

3.1.3

The quantum description of high-order harmonic generation

The two works of Lewenstein et al. [86, 118] essentially built the quantum mechanical description of HHG within the SFA. The assumptions of the theory are similar to Corkum’s: (1) No boundstates are considered except for the ground state, (2) the ground state is not depleted, and (3) after
ionization, the electron propagates as a free particle with no influence from the ion potential. The
quantum treatment recovers much of the same physical insight as Corkum’s model with the additionof predictions on the phase of the XUV light.
Lewenstein et al. [86, 118] derived an equation for the dipole moment responsible for thegenerated XUV field. For the moment, we switch to atomic units where ~ = |e| = me = 1 [16]. The
phase term in the dipole moment is sensitive to the electron trajectory after it has been ionized andis writtenΦq = S(p, ti , t) − qωt,

(3.2)

75
where S is the quasi-classical action defined as

Z tS(p, ti , t) =


dτ

ti


[p + A(τ )]2+ Ip ,
2(3.3)

where p is the electron momentum, A(τ ) is the vector potential of the laser field, and ti is the timeof ionization. Eq. 3.3 is reminiscent of Feynman’s path integral approach. Essentially, the dipole
moment contains a sum over all possible electron paths. Eq. 3.2 is in general a rapidly varying
quantity and only conditions where the dipole contains a stationary phase will contribute to XUVgeneration. Looking for the stationary phases is often called the saddle-point approximation.
Where then can write the three saddle-point equation evaluated at the recombination time tr
as

Z tr∇p Φq |t=tr=dτ [p + A(τ )]2 = 0 ,

(3.4)

ti

∂
Φq |t=tr∂t∂
Φq |t=tr∂ti==[p + A(tr )]2+ Ip − q ω = 0 ,2[p + A(ti )]2+ Ip = 0 .
2

(3.5)
(3.6)

Since the form of the vector potential is known, it is straightforward to numerically find the rootsof this set of equations. Each of the saddle point equations gives insight into what is physicallygoing on during the strong-field process. Eq. 3.4 is simply a statement that the electron must returnto the point from which it was ionized (an assumption also made in Corkum’s model [24]). This
makes sense because the electron must return to the ion to rescatter or recombine. Eq. 3.5 is justa statement of the conservation of energy enforcing that the emitted photon energy is a sum of
the kinetic energy of the electron plus the ionization potential. It can readily be seen Eq. 3.6 does
not have a solution for real values of ti (because Ip is positive). The imaginary part of ti has beeninterpreted as the “tunneling time” which many claim exists and even try to measure. This assertion
may be fanciful and can even be shown to be gauge dependent [110]. One of the cornerstones ofphysical theories is that measurable observables should not be gauge dependent. A discourse on the

76
idea of “tunneling times” can be found in Refs. [119, 120]. Nevertheless, if complex time values are
assumed, roots to the saddle-point equations can easily be found.
By numerically finding the roots of Eq. 3.4–Eq. 3.6, assuming a xenon target and a 1070
nm driving laser, it is possible to calculate the dipole phase from Eq. 3.2 using the return times
determined from the saddle point equations. Fig. 3.3a shows the return time (emission time) as
a function of harmonic order. Two roots per harmonic order are found, corresponding to the longand short trajectories. Two sets of solutions are shown corresponding to different laser intensities.
Fig. 3.3b shows the dipole phase (and hence phase of the emitted XUV light) as a function of laser
intensity for the short and long trajectories of the 15th harmonic. The short trajectories have nearlyno dependence on intensity while the long trajectories have a quite strong dependence. The coupling
between intensity and phase is parametrized in atomic units as

φ = −αjUp (I)
I= −αj 3 ,
ω
4ω

(3.7)

where j is the index of the short or long trajectory. αj is plotted in Fig. 3.3b in units of π similar
to Refs. [43, 87, 121]. We will use this convention later in our experimental investigation of theintensity-dependent phase presented in Chapter 4.

3.1.4

Beyond the strong-field approximation

The SFA is not without its shortcomings, but it has nevertheless been a very useful tool. For
the work in this thesis, we are typically in a regime where the SFA is certainly a bad approximation.
Below- or near-threshold harmonics are the harmonics with photon energy less than or comparable
to the ionization potential, respectively. These low order harmonics violate two out of the threeassumptions of the SFA, namely that bound states are not important [122] and that the Coulomb
potential is safe to ignore allowing the electron to be treated as a free particle when propagating
the in the electromagnetic field [121, 123].
The theoretical treatments of below- or near-threshold harmonics are beyond the scope of
this thesis. However, some of the outcomes of the analysis are relevant to our work. First, for77

Figure 3.3: a) The return time (time of XUV photon emission) is plotted versus harmonic orderfor two laser intensities using the strong-field approximation. Two trajectories contribute to eachharmonic order. b) The intensity dependent dipole phase for the short and long trajectory as afunction of laser intensity for the 15th harmonic. The inset contains the coupling parameter between
intensity and phase. For all presented here, the laser wavelength was 1070 nm and the target was
xenon.

78

b)

a)
H9

0.7
0.6

3

0.5
0.4
0.3

2

0.2
0.1

H9

0.9

4

0.8
0.7
0.6

3

0.5
0.4
0.3

2

0.2
0.1
0.0

1

H11

H13

2

2

Intensity (10 W/cm )

H13

1.0

2

0.0

H11
4

13

13

13

13

0.8

Intensity (10 W/cm )0.9

1

Intensity (10 W/cm )

H7

1.0

2

Intensity (10 W/cm )

H7
4

3
2

4
3
2
1

1
-4

-2

0

2

4

6

alpha units of π)

8

10 -4

-2

0

2

4

6

alpha units of π)

8

10

-4

-2

0

2

4

6

alpha units of π)

8

10 -4

-2

0

2

4

6

alpha units of π)

810Figure 3.4: The effects of considering effects beyond the SFA. The plots were reproduced from
Ref. [121]. a) The calculation shows the density of quantum paths when only tunnel ionizationis considered. There is only one quantum path present. b) The calculation shows the density ofquantum paths when tunnel ionization and over-the-barrier ionization (including uphill trajectories)
are included. The multiple quantum paths for below-threshold harmonics were initially surprisingand they do not originate from tunnel ionization like their above-threshold counterparts.

79
below-threshold harmonics, if only the SFA is considered, there is only a single quantum path
relevant for HHG. This is shown in Fig. 3.4a which has been reproduced from Ref. [121]. The plotshows the distribution of possible αj as a function of laser intensity assuming a xenon target anda 1070 nm driving laser. What is most apparent is that the single trajectory actually has a verystrong intensity to phase coupling. Work from our group showed that below-threshold harmonics do
indeed have multiple quantum paths leading to harmonic generation [87]. This surprising result has
allowed our collaborators from the Schaeffer/Gaarde group to work out the origin of these harmonic
paths. Fig. 3.4b shows the calculation where instead of relying only harmonics to be initiated bytunnel ionization (as in the SFA), we included over-the-barrier ionization with uphill trajectories
and downhill trajectories [121]. Now, multiple quantum paths are present, which are consistent with
our initial experimental observation [87]. Subsequent work reported in this thesis (see Chapter 4)
was able to isolate the short quantum path component of these trajectories and make a precision
measurement of the intensity-dependent dipole phase [43], thus confirming the weak intensity tophase coupling present in below- and near-threshold harmonics. These findings further emphasize
their non-tunnel ionization-based origin. The often ignored below- and near-threshold harmonics
are a rich system for studying strong-field light-matter interactions. Further, the presence of theshort trajectory with a weak intensity to phase coupling in the below- and near-threshold harmonics
is rather fortuitous for future work using the harmonic light for precision spectroscopy applications.
Since the HHG process will be less sensitive to intensity fluctuations, the coherence properties of
the XUV light will be much less sensitive to laser intensity noise.

3.2

Macroscopic response and phasematching
Like most nonlinear optical phenomena, the nonlinearly converted light rarely (if ever) per-fectly mimics the single-atom response discussed thus far. The macroscopic emission of harmonics
is dominated by phasematching. Phasematching is well understood in typical HHG experiments
and we will review it within the framework of Ref. [124]. The analysis was originally formulatedfor HHG experiments performed by very intense, low-repetition rate laser systems. However, the80
XUV frequency comb operates in a very different regime employing a much tighter laser focus and arepetition rate nearly six orders of magnitude higher than more traditional systems. The fsEC also
complicates the phasematching situation. We will cover how these effects modify phase matchingconsiderations and, despite the added difficulty, show that the XUV frequency comb is the single
most powerful source of XUV radiation originating from HHG in terms of average power in its
operating spectral region.

3.2.1

Single-pass case

One of the key differences between HHG and other nonlinear harmonic generation (for example, in crystals) is that the nonlinear medium is also strongly absorptive at the wavelengths being
generated. Conversion efficiency can then be limited by absorption and not other phase matchingconditions.
Reaching the absorption limit in phase-matching HHG requires a delicate balance of many
macroscopic parameters. Ref. [124] introduced a simple formulation of the phase-matching criterion
that well describes high conversion efficiency results with ∼800 nm [124] and ∼1030 nm [125] laser
systems operating in a regime where the laser is focused only loosely. It was originally anticipatedthat it would not be experimentally feasible to reach the absorption limit in cases where the laser
beam is tightly focused (a condition applicable to high repetition rate laser systems like the XUV
frequency comb), but analysis and experiment have shown that it is indeed possible to reach thislimit albeit in low repetition rate laser systems thus far [126, 127].
The primary distinguishing operating characteristics of the XUV frequency comb are the highrepetition rate and tight laser focus needed for performing HHG. Performing HHG inside a fsEC
has led to record high XUV flux from HHG systems, but has come with many additional operating
constraints pertinent to proper phase matching [114, 128].
The one-dimensional model used for analysis follows the work of Ref. [124]. The harmonic
flux of the q th harmonic is Pq (t) given by

81





L
L
4 ρ2 (t) A2q Labs (t)2
Lmed
− L med− 2L med(t)
(t)
abs
abs
1+e
,
Pq (t) ∝− 2 cos πeLcoh (t)1 + 4π 2 Labs (t)2 /L2coh (t)(3.8)

where ρ(t) is the neutral density that varies with time as ρ(t) = ρ0 (1 − η(t)), with η(t) being theionization fraction and ρ0 being the original target gas density. Aq is the atomic dipole of theq th harmonic obtained quantum mechanically [118] or empirically [129], Lmed is the length of thegas target, the absorption length is Labs (t) = 1/σρ(t) where σ is the absorption cross-section of
the harmonic of interest. The coherence length Lcoh (t) measures the phase mismatch between thedriving laser and harmonic q as Lcoh (t) = π/∆k with ∆k = |kq − q k0 |, k0 is the wave-vector ofthe driving laser, and kq is the wave-vector of the q th harmonic. Details of all the relevant phasematching terms have been covered extensively elsewhere [72] and can be summarized by∆k = ∆katomic + ∆kplasma + ∆kGouy + ∆kdipole ,(3.9)

where ∆katomic is from the un-ionized gas dispersion, ∆kplasma is from the free electron dispersion,∆kGouy is from the differential Gouy phase shift (Eq. 2.37), and ∆kdipole is from the quantum pathcontribution of the HHG process [86] and depends on the laser intensity gradient.
Phase-matching occurs when ∆k ≈ 0 and only transiently over ultrashort laser pulses as
Lcoh (t) evolves with the ionization fraction η(t). Absorption limited phase matching is generallyachieved when

Lmed > 3 Labs ,(3.10)

Lcoh(3.11)> 5 Labs ,ensuring the macroscopic yield is more than half of the maximum yield [124]. An illustration of
how the harmonic yield is affected by relative sizes of Lmed , Lcoh , and Labs is in Fig. 3.5. If Lcoh
and Labs are comparable in size, oscillations in the harmonic yield as a function of target length

82
will be present. These are often called “Maker fringes”. If no absorption is present, a quadraticallyincreasing harmonic yield is expected. However, even a slight amount of absorption will lead to asaturation of the harmonic yield as a function of target length. The reason is that for every emitter
added to the target, an absorber is also added and the net harmonic flux does not increase.
While the one dimensional model was originally intended to apply in the case where the laser
beam is only loosely focused, Refs. [126, 127] showed that it is still applicable to the case where
the laser beam is tightly focused. A balance of the Gouy phase shift, ionization fraction, and gas
density can result in ∆k ≈ 0. Fig. 3.6a shows the required density to achieve phase matching as afunction of beam radius for different ionization conditions. The critical ionization fraction ηcrit is
the ionization fraction where the neutral atom dispersion cancels the free electron dispersion in theplasma. An important feature of Fig. 3.6a is that a tight laser focus requires commensurately hightarget densities (and hence nozzle backing pressure). Current fsEC technology limits the feasible
spot size to the shaded region and shows that multi-atmosphere backing pressures will be requiredfor phasematching.

3.2.2

Enhancement cavity case

One would hope that these previously discussed conditions could readily be applied to thecase of HHG in fsECs. However, the cavity is very sensitive to nonlinear phase shifts present inside
the resonator because the cavity effectively increases the nonlinear interaction length [114, 128, 130].
Since the repetition rate of the laser is high, the plasma does not have time to clear the focus of thelaser beam, leading to a steady-state plasma during the generation process that can be ∼ 5× larger
than the single-pass case. The steady-state plasma leads to an intensity-dependent phase shift in
the resonator and the resonator become bi-stable. The phase shift can be quantified by

φN L = 2 r0 ρηLmed F/k0 ,(3.12)

where F is the cavity finesse and r0 is the classical electron radius. While there are methods to deal
with this [114, 130], they require detuning the laser from the cavity and reducing the enhancement

83

med

Figure 3.5: The harmonic yield plotted as a function of medium length for various ratios of thecoherence and absorption lengths. In the limit that the coherence length is much larger than theabsorption length, harmonic flux does not increase with increasing target length. This is in stark
contrast to other nonlinear optical phenomena like second harmonic generation where absorption
usually does not play a role. A typical quadratic scaling is also predicted in the absence of absorption
which is what is expected when the emitters add coherently.

84of the cavity. To illustrate the immediate impact of the fsEC on phase matching, Fig. 3.6b shows thenonlinear phase shift under phase matched conditions. Operating too close to the critical ionization
and with too small of beams results in a cavity that will not be stable and makes reaching thephase-matched condition nearly impossible unless the thickness of the target is greatly reduced.
The process of HHG has been shown to be scale invariant suggesting our operating conditions can,in principle, reach absorption limited phasematching [131]. Nevertheless, it is important to keep
the nonlinear phase shift φN L ≤ π to avoid this. We impose an additional optimization conditions
and require

φN L ≤ π.

(3.13)

Fig. 3.6b contains two shaded regions corresponding to feasible spot sizes obtainable in the resonatorand the region where the nonlinear phase shift is small enough to allow stable, reliable operation.
The overlap of the two shaded regions is clearly very small highlighting how important effects ofthe cavity are for achieving phasematched operation.
To further complicate the matter, when femtosecond pulses propagate through field-ionizing
media, the laser pulse spectrum shifts to higher frequencies [114, 130, 132]. Usually this effect issmall for optimal HHG conditions, but in the cavity-enhanced approach, small spectral shifts cause
notable effects due to the high cavity finesse [114, 130]. When the blue-shift becomes severe, theintracavity spectrum no longer matches the injected laser spectrum and the enhancement is reduced.
Until now, there was no suitable method to reduce the spectral blue-shifting besides reducing thecavity finesse and requiring larger driving lasers. We speculate about a potential method at the endof Chapter 6.
To apply the one dimensional model (Eq. 3.8) to HHG in fsECs, extra considerations are
required. First, the ionization fraction includes a steady-state value determined by the intracavity
intensity [114, 128]. Second, the plasma ramp across the laser pulse leads to an energy loss andspectral blue shifting that reduce the intracavity intensity. The direct energy loss has little effect

atm

85

Figure 3.6: a) Phase matching conditions. For a given ionization fraction, the beam radius andtarget gas density can be tuned to achieve phase matching conditions when ∆k ≈ 0. b) Nonlinear
phase shift in the cavity. The phase shift from the neutral/plasma target must be kept less than πin order to keep the fsEC stable. For the phase matching conditions in a), the nonlinear phase shift
is shown as a function of beam radius and ionization fraction. The gray shaded regions correspondto regions where the beam waist in the fsEC is feasible to operate for HHG. The purple shaded
region in b) is the region where the nonlinear phase shift is small enough for operation.

86
under our operating conditions. Most of the reductin in laser intensity comes from the mismatch
between the blue-shifted intracavity field and the unshifted field we inject into the resonator [114].
Third, the nonlinear phase shift due to the steady state plasma is maintained to be small enoughfor practical implementation. To achieve reliable operation, we impose Eq. 3.13.

3.3Power scaling
Up until recently, armed with the just described phasematching knowledge, it was possible togenerate ∼ 200 µW/harmonic. This already makes the XUV frequency comb about 1/10 as brightas the ALS synchrotron [69]. This amount of power was sufficient to perform the first spectroscopy
experiments [52]. However, as is often true in any laser-based experiment, having more power isbeneficial. The greatest hindrance to further power scaling the system is the troublesome persistentplasma that remains in the laser focus for many cavity round trips. Reducing the steady state
plasma will improve conversion efficiency and make the system more reliable and stable.

3.3.1

Methods

Our method for power scaling beyond the 200 µW/harmonic level is straightforward. For ourinvestigations, we use our fsEC equipped with a diffraction grating output coupler [74], a system well
suited for spectroscopy [52]. The cavity and HHG detection is schematically sketched in Fig. 3.7.
To improve performance, we modify the properties of our gas target using tricks often employed insupersonic beam experiments [133]. A simple estimate and demonstration of our scheme is shownin Fig. 3.8a. For a beam waist of w0 = 20 µm and a peak intensity of 0.5 × 1014 W cm−2 , theradial profile of the beam and the ionized fraction of the gas (calculated using ADK theory) are
shown in Fig. 3.8a. We can estimate the speed of the beam using Ref. [133] and we find that thetime it would take to replenish the previously ionized region is 35 ns, or about 5 cavity round trips.
This is consistent with previous observations and modeling [114]. Seeding a heavy gas (like xenon)
into helium molecular beam is a common method to increase the jet velocity of the heavier gas.
Following Ref. [133], we can estimate the speed increase of a xenon beam mixed with helium and this

87

DET1

Au
SS
CM
DG

HR

Intensity(arb.)

Au

DET2

7

11

17

pixelsCM
IC

Figure 3.7: Schematic of the apparatus. The harmonics are out-coupled from the cavity using ahigh reflector/diffraction grating optic. The 11th and the 17th harmonic can optionally pass throughthe sodium salicylate detection plate by apertures. They can then be sent to independent detectors.
The inset shows a typical harmonic spectrum with all orders incident on the sodium salicylate
plate with various harmonic orders identified. The relevant harmonic orders can be identified bytheir horizontal spacing on the plate since this is determined by the line spacing of the diffraction
grating. The image was taken with a simple CCD located outside of the vacuum chamber. CM,
curved mirror. HR, high reflector. DG, diffraction grating output coupler. IC, input coupler.
DET1, aluminum coated silicon photo-diode. DET2, NIST calibrated vacuum photo-diode. SS,
sodium salicylate plate. Au, gold mirrors.

88
is shown in Fig. 3.8b. For large helium fractions, we can greatly speed up the beam and replenish
the previously ionized region much more quickly. This will in turn lead to a reduced steady stateionization fraction and hence better phase matching and stability. Since we operate at intensities
much too low to effectively ionize helium, we will only maintain harmonic yield from xenon. The
potential cost is a reduced xenon density or requiring unfeasible backing pressures for the gas nozzle.

3.3.2

Results

In a sweeping cavity mode, a cavity mirror with a PZT sweeps the cavity resonance acrossthe comb teeth. The cavity operates with an enhancement factor of ∼ 200. The cavity is equippedwith a 10 cm and a 15 cm ROC mirror and the cavity is set to near the middle of the stability range
to achieve a focal beam waist of w0 = 25 µm. The beam waist is determined by measuring thefrequency spacings of higher order transverse modes of the cavity (see Sect. 2.2.3). Gas is injected
into the focus of the cavity via a quartz nozzle. The harmonics are outcoupled from the cavity using
the Yost grating (See Sect. 2.2.6 and Ref. [74]). We detect the 11th harmonic and the 17th harmonic
independently and simultaneously. The 11th harmonic is detected on a NIST calibrated vacuum
photodiode [134, 135]. The photodiode was biased to -100 V, but the photocurrent was insensitive tothe bias voltage between -60 and -120 V. The photocurrent was dumped over a resistor of 10 kOhm
and the voltage was amplified using a low noise voltage amplifier. The NIST calibrated diode wasused to calibrate the 17th harmonic detector which is an aluminum-coated silicon photodiode [136].
The calibration is performed by comparing photo-currents on both diodes and comparing the relative
yields on a sodium salicylate plate. The sodium salicylate acts as a uniform scintillator [13] and thefluorescence is easily measured with a camera, as seen in the inset of Fig. 3.7. The reflectivity of thegold mirrors is calculated with knowledge of the photon energy, angle of incidence, polarization andthe optical constants of gold [137]. While the optical properties of gold are well known, the specificcharacterization of the gold mirrors employed in our measurements are unknown. It is likely thatthe reflectivity is lower than the theoretical value. Similarly, the performance of the NIST vacuum
photodiode is assumed to be perfect. We have no way of independently verifying this, so all power

50

0.0125

25

0.000
40

-

b)

20
- 20 Beam Radius
0
[μm]

0
40

Speed [m/ s]

0.025

I ntensity [TW/ cm2]

I onization Fraction

a)

1800

6

1000

3

200

0

0.5
Helium Fraction

Cavity Round Trips

89

10

Figure 3.8: a) Beam radial profile and the ionization fraction radial profile. The ionization fraction
is estimated by using ADK theory covered in Sect. 3.1.1. b) The speed of the xenon gas jet as afunction of helium seeding and the number of round trips the ionization profile shown in a) willtake to replenish the interaction region. The jet speeds were estimated following Ref. [133] andeffects like velocity lag were ignored. Replenishment was defined by when the ionized fraction moveat least one ionized fraction width away. This figure should only be interpreted as an estimate to
the replenishment. More sophisticated fluid flow simulations could in principle be performed. The
round trips assumed a laser with a repetition rate of 154 MHz. The round trip estimate can beeasily scaled to compared with other systems by the ratio of the repetition rates under consideration.
Simply dividing the repetition rate by two should lead to nice improvements on the steady stateionization problem.

90
levels reported here should be interpreted as lower bounds and the power levels may very likely
be higher. While we have taken precaution to ensure the detector was never contaminated, thesedetectors are known to degrade due to air exposure or other surface contamination [135]. All
detectors were verified to be blind to the driving laser.
To determine if we are reducing the steady state ionization fraction by using a faster atomic
beam, we can sweep the driving laser across the cavity resonances and observe the self-locking
phenomena described throughly in Ref. [114]. Simply, the more steady state ionization, the more
the cavity will follow the laser as it is sweeping across the resonance, leading to a lineshape thatlooks similar to a saw-tooth shape and not a Lorrentzian as in the case of a normal, linear cavity(for an empty cavity, see Fig. 2.5). Fig. 3.9 illustrates this mechanism. Fig.3.9a shows the resultof injecting pure xenon into the cavity through a 100 µm nozzle. The intracavity power, the 11th
and the 17th harmonic power are monitored as a function of time while the resonance is swept. The
departure from the Lorrentzian lineshape of the cavity resonance is a manifestation of self-locking.
By simply switching to a 11:1 helium to xenon mixture and increasing the nozzle backing pressure
until approximately the same partial pressure of xenon is achieved, the distortion becomes lesssevere and the harmonic yield is greatly enhanced, as seen in Fig. 3.9b.
Fig. 3.10 documents the maximum achievable harmonic yield for various mixing conditions
and nozzle diameters. The laser peak intensity was kept approximately the same in all cases (∼0.67 × 1014 W cm−2 , or 12.2 kW of average power) and gas pressure was chosen empirically tomaximize yield. We also observe a systematic increase of harmonic yield with decreasing nozzle
diameter. Decreasing the nozzle diameter requires much larger backing pressures on the nozzle,
making it difficult to experimentally realize with the large mixing ratios. We also verified that no
measurable harmonic yield (< 0.05 µW outcoupled) was observed when using pure helium as ourpeak intensities were too low to effectively ionize helium.
As a final demonstration of power scaling, we moved to conditions that we have experimentally
determined to optimize yield; the data is also presented in Fig. 3.10. By using a 100 µm quartz
nozzle, backed by 100 psi of a 4:1 He:Xe gas mixture, we are able to outcouple 60 µW in the 11th

91

11250

56.25

7500

37.50

3750

18.75

0

40

20
0
20
Sweep Time [µs]

40

0.00

15000

75.00

11
17

11250

56.25

XUV Power [µ W]

11
17

b)
IR power [W]

75.00
XUV Power [µ W]

IR power [W]a) 15000

7500

37.50

3750

18.75

0

40

20
0
20
Sweep Time [µs]

40

0.00

Figure 3.9: The data in both panels was aquired by sweeping the laser across the cavity resonance.
A piezo on a cavity high reflector used for the sweep. All the data was recorded on photodetectors
simultaneously and averaging with the oscilloscope was performed. a) Data for a pure xenon target.
b) Data for an 11:1 He:Xe mixture. The harmonic yield is greatly enhanced and the self-locking
phenomena is reduced compared to a). The xenon densities in these two cases is approximately thesame. The nozzle diameter was ∼ 100 µm. The beam waist was w0 = 25 µm. The driving laser
power was also approximately the same. However, due to the reduction in steady state plasma,b) sweeps to a slightly higher intracavity power. However, harmonic power levels at identicalintracavity power levels show a clear improvement for the case of the 11:1 He:Xe mixture so theboost in harmonic yield is not an artifact of sweeping to a larger intracavity power and is due toimproving phasematching conditions by reducing the steady state plasma.

92
and the 17th harmonic order simultaneously. Assuming the diffraction grating output coupler isperforming as well as theory predicts [74], this would correspond to a generated harmonic yieldof 0.7 mW and 0.63 mW at the 11th and the 17th harmonic order respectively. By switching to
a 50 µm quartz nozzle and applying 400 psi with a 4:1 gas mixture, we are able to outcouple 71µW at the 11th harmonic, corresponding to 0.84 mW generated at 12.7 eV photon energy (4 × 1014
photons/s). Further improvements are achieved with 275 PSI applied with the 9:1 mix resulting
in 78 µW at the 11th harmonic, corresponding to 0.93 mW generated at 12.7 eV photon energy(4.8 × 1014 photons/s).
We also performed HHG with nonlinearly compressed pulses (See Sect. 2.3.1). By broadening
in bulk sapphire and compressing with chirped mirrors, we delivered 70 fs pulses at up to 53 W of
average power to the fsEC. This method yields an optical spectrum and beam profile suitable tomaintain intracavity buildups of 200. The intracavity pulse duration was 75 fs, measured by second
harmonic intensity autocorrelation of the intracavity field. We are able to outcoupled harmonic
flux at the 50 µW/harmonic level and we observed a 25% boost in conversion efficiency (average
driving laser power to average XUV power). By using nonlinearly compressed pulses, we observed
a decrease in the self-locking phenomena at similar target densities and peak intensities confirmingthat the shorter pulses are ionizing less of the gas at equivalent peak intensities.
It is useful to compare our results with high-flux results obtained with titanium-sapphire based
XUV frequency comb generation. Ref. [138] reports outcoupled power levels of 45 -77 µW over asimilar wavelength range. When one considers the inherent ∼ λ−6 wavelength scaling of HHG [139],
it becomes clear that the improvement obtained in this work is very encouraging. If we scaled our
60 µW at the 17th harmonic result (1070 nm driver) to directly compare with the titanium-sapphire
results (800 nm driver), this would amount to a scaled power of 342 µW, implying generation levelsof ∼3.4 mW. For applications that do not require very high photon energies (lower harmonic cutoff),
it is worth reconsidering driving HHG at 800 nm or even possibly the second harmonic of Yb:fiberbased laser systems at ∼ 532 nm. It is important to note that the outcoupled power of fsEC-based
XUV sources is in principle totally deliverable on target for experiments, without the need for extra

93

Figure 3.10: System scaling. Output coupled harmonic yields are shown as a function of nozzle
diameter and mixing ratio. The laser intensity was approximately the same in all cases and thepressure was choses to maximize yield.

94filtering to remove parasitic driving laser light, which is typically very lossy for the XUV light [137].
This is in sharp contrast to more traditional approaches to HHG. Therefore, for direct comparison
with lower repetition rate systems, it is more useful and informative to compare generated power
levels for both cases. The fsEC-based approach is much higher [125]. For implementation of the
XUV light in experiments, filtering out the driving laser is almost always necessary. In the fsEC
system described here, the driving laser light is effectively removed by output coupling of the XUVlight from the cavity. A broad comparison of HHG sources is plotted in Fig. 3.11. The figurecontains the generated XUV power as a function of repetition rate. It should be obvious that for anominally fixed conversion efficiency, the generated power increases with repetition rate (although,pulse energy commensurately goes down). For linear spectroscopy, average power is the relevant
quantity and the XUV comb clearly leads the pack.
The only drawback to our demonstrated method is the increased gas load to the vacuum
chamber. However, by having sufficient pumping speed (∼700 L/s) and proper gas dump design(see Sect. 2.5 and Ref. [130]), we are able to operate under safe conditions for the turbomolecular
pumps and at low enough background pressures that re-absorption of the harmonics is minimal.
The improvements outlined in this chapter are simple to implement and can easily be extended toother collinear output coupling techniques [88, 90] and possibly non-collinear geometries [88, 150].
A simple estimate suggests that by using the geometric output coupling technique [88, 90], we canincrease output coupling efficiency by 3× and deliver ∼ 180 µW at 19.7 eV. This will greatly aid
application of XUV frequency combs for precision spectroscopy, strong-field physics, and any otherapplication where a high photon flux source of XUV radiation is needed.

95

Wavelength
1060 nm
800 nm532 nmöäMethod
fsEC

ä

Trad.
C.C.
Pulse EnergypFigure 3.11:
Generated power levels are plotted as a function of laser repetition rate. The
XUV frequency combs clearly has the advantage in terms of photon flux. The very low XUVpulse energies delivered by the XUV frequency comb are inconsequential for linear spectroscopy.
Benko [140], Cabasse [141], Cingöz [52, 130],Constant [124], Hädrich1 [125], Hädrich2 [142], Hergott [143], Lee [138], Lorek [144], Pupeza [90], Rothhardt [126], Rudawksi [145], Takashi1 [146],
Takahashi2 [147], Wang [148], Willner [149].

Chapter 4Interferometry in the extreme ultraviolet

In this chapter, we will focus on aspects related to performing interferometry in the XUV.
Various components of the XUV interferometer will be discussed along with some design considerations.

4.1

Motivation and brief history
The goal of building an interferometer in the XUV is to test the temporal coherence propertiesof the XUV light originating from HHG. Often, we will refer to the temporal coherence level byquoting the inverse of the temporal coherence time and refer to it as the frequency resolution. 1
Prior to this work, temporal coherence was tested at the 10 MHz frequency resolution levelby multiple experiments. The first attempt by Bellini et al. [151] only tested coherence at the
few fs level, but the experiment nevertheless showed the temporal coherence was indeed there (atleast within a single pulse) [151]. Future applications in high resolution spectroscopy will require
coherence at a much higher level that will potentially extend over billions of pulses.
After the pioneering work by Bellini et al. [151], it was not until much later that pulse-to-pulse
coherence was demonstrated. Two methods were employed to test the temporal coherence. Ourgroup employed an interferometric approach by constructing a time-delayed homodyne interferometer to measure the 7th harmonic of our 1070 nm fiber laser [87]. The group of Eikema developed a1This is not a rigorous definition of frequency resolution, but nevertheless useful for our discussion. In Chapt. 5,
we will discuss how optical phase noise and temporal coherence are related to laser linewidth and hence frequency
resolution.

97
spectroscopic approach akin to Ramsey spectroscopy. There, the 15th harmonic of an 800 nm laser
was used to perform spectroscopy in helium [56, 152]. The spectroscopic signature is evidence ofpulse to pulse coherence. Both groups established frequency resolution at the ∼10 MHz level.
In a heroic experimental effort, our group also established the ability to use the frequency comb
to perform absolute frequency measurements deep into the XUV by performing direct frequency
comb spectroscopy on atomic argon and neon [52, 89]. This, however, still only demonstrated
coherence at the ∼10 MHz level but did establish the efficacy of using XUV frequency combs forhigh resolution spectroscopy. Ultimately, the fractional frequency uncertainty of the transition inargon was measured to about 10−10 (∼ 500 kHz) which is about four orders of magnitude better
than a typical spectrograph operating in the XUV can achieve [13].
The method we employ to directly measure the fundamental limits of temporal coherence in
the XUV was essentially established in our group with the first XUV comb demonstration [3] with aconcurrent demonstration in Garching [4]. Our original method relied on generating two UV combs,
one in a gas target and one using traditional methods by tripling in crystals. The two sources had
different f0 ’s which upon recombining would form a heterodyne beat (see Fig. 1.6c). By measuring
the linewidth of the heterodyne beat, coherence properties could be determined. While this result
may not be surprising since it was applied to only the 3rd harmonic, it demonstrates the principle
we will rely on to test coherence at harmonics much higher than the third. For our work, we will
construct a heterodyne interferometer capable of operating between 1070 nm and 56 nm.

4.2

The extreme ultraviolet interferometer
Our interferometer is an extension of a traditional interferometer design known as a Mach-

Zehnder interferometer shown schematically in Fig. 4.1a. A laser beam is split with a 50:50 beam
splitter and then recombined, after traversing independent paths, on an additional 50:50 beam
splitter. It is possible for these beam splitters to be polarizing, but it is not necessary. There are
two output ports where detectors can be placed. The output ports will change from bright to dark
depending on the relative phase between the two paths. In general, the two ports are anticorrelated:

98
meaning that if one port is bright, the other is dark such that the total optical power out of theinterferometer is conserved.
The XUV version of the Mach-Zehnder interferometer is shown schematically in Fig. 4.1b.
The first beam splitter will be an AOM. This will spatially split the driving laser beam and put afrequency shift on one arm relative to the other. A cavity-based HHG apparatus will be placed ineach arm of the interferometer and the resulting HHG beams will be recombined in the XUV. One
detection port will be available in our scheme and the interference pattern will oscillate from brightto dark at a rate proportional to the frequency set by the AOM. This is the beat frequency, or thebeatnote, that we aim to detect. While what we are trying to achieve is conceptually very simple,
the practical implementation is far from it.
The full schematic of the interferometer is shown in Fig. 4.2. Our driving laser is the previously described Yb:fiber comb (See Chapter 2). There are optional modulators (amplitude orphase) placed in one arm of the interferometer. Inside the vacuum chamber, each arm contains
an independently stabilized fsEC. The harmonics are outcoupled using sapphire Brewster plates.
The harmonic beams are then recombined on a beam combiner in space and time. The combined
beam is then sent to a spatial filter and finally an XUV photodetector. There is also an optional
monochromator to analyze the performance of the HHG process and XUV optics. Since this is avery non-trivial interferometer, each of the components will be described in detail.

4.2.1

The acousto-optic modulator

The first component of the XUV interferometer is the AOM. When used in conjunction witha frequency comb, the AOM acts as an f0 -shifter. The AOM can be used for high-bandwidthstabilization of f0 with servo bandwidths of up to 250 kHz, typically limited by the propagation
delay of the acoustic wave in the AOM crystal [31]. For this implementation, it will just produce astatic, relative f0 shift between the driving laser of the two arms. We would like to have the AOM
put a small frequency shift between the arms since the frequency shift of the harmonics will growas fq,0 = q × f0 where q is the harmonic order. Therefore f0 needs to be kept low so it will be

99

AOM

Figure 4.1: Schematics of a Mach-Zehnder interferometer and an XUV Mach-Zehnder interferometer shown in a) and b) respectively.

100

Figure 4.2: Detailed schematic of the XUV interferometer. The Yb:fiber comb is split in two with an
acousto-optic modulation (AOM). One patch contains an additional AOM to act as an amplitude
modulator. Each cavity is independently stabilized using the PDH method. The XUV beams
are outcoupled with sapphire Brewster plates. The XUV beams are recombined (see Sect. 4.2.2)
and detected after being spatially filtered (see Sect. 4.2.3 and Sect. 4.2.4). The optional sphericalgrating is for operation of the Seya-Namioka monochromator (see Sect. 4.2.6). Temporal overlap of
the XUV pulses is achieved by temporal delay of the driving laser using a simple delay stage. PMT,
photomultiplier tube.

101
within a possible XUV photodetector bandwidth. We settle on a 1 MHz relative detuning between
the arms. However, AOMs do not operate at such low frequencies.
To overcome this, we use an AOM that operates at frequencies near fr such that fr - faom = 1
MHz. When a frequency comb is split by an AOM with a frequency of fAOM and interferometrically
recombined on a photodetector, there will actually be two beat frequencies below fr . The possible
beat frequency fb can be fb = fAOM , fr -fAOM . This can be seen by referring back to Fig. 1.6. The
latter option is typically referred to as the conjugate beatnote. If we simply detect the conjugate
beatnote fb = fr -fAOM , any fluctuations in fr will couple into the signal. Since we do not want this,
it is removed by a phase-lock trick shown schematically in Fig. 4.3 for the black signal pathway. To
remove the fr dependence of the low frequency beatnote, we enforce fr -fAOM = 1 MHz by phaselocking it to an ultrastable Wentzel crystal oscillator. By doing this, the fr dependence is transferredto the higher frequency beatnote which is left undetected in our case.
As is true of any interferometer, any acoustic noise, mechanical vibrations, or air current
fluctuations that are not common to both interferometer arms will cause the relative phase of theinterferometer arms to fluctuate. This can be very problematic if the goal is to test the phasestability of the optical signal; it can easily be corrupted by a poorly constructed interferometer.
Since it is very difficult to construct an interferometer like ours that is passively stable to less thanan optical fringe over many seconds, we must resort to actively stabilizing the interferometer.
To accomplish this, we need a method to detect fluctuations between each arm of the interferometer. When we ultimately detect XUV beatnotes in the interferometer, a small amount ofour driving laser light will also make it to the detection plane. For the driving laser, there is abeatnote at 1 MHz that will be sensitive to any phase fluctuations due to the interferometer not
being perfect. We use our driving laser beatnote at 1 MHz and compare it to another stable 1 MHz
Wentzel source. This comparison signal is referred to as the interferometer error signal. The error
signal is sensitive to any phase fluctuations and they are actively removed by slight adjustment to
the AOM drive frequency phase. To accomplish this, the additional phase lock is implemented as
schematically shown in Fig. 4.3 with the blue signal pathway added. With this additional phase

102

Figure 4.3: The radiofrequency drive scheme for the AOM and the optional stabilization method.
An fr signal is derived from a photodetector and compared with a voltage-controlled oscillator(VCO). The difference is then compared to 1 MHz and that is used to produce an error signal
for the phase lock that sets the AOM frequency to be fr -1 MHz exactly. When the interferometerstabilization is required, the blue signal path is added. A second error signal at 1 MHz is produced
from the interferometer that is proportional to phase fluctuation in the interferometer. This is error
signal is filtered with a loop filter (LF) and applied to the AOM stabilization scheme to cancel out
the fluctuations.

103
lock, we can be free of technical noise of the interferometer and focus on the ultimate stability of
the XUV light.
The choice of AOM optical material is also critical for the experiment. Initially, an AOM
constructed of fused silica was chosen. Most AOMs are not designed to operate with up to 80 W of
average laser power. Therefore, large beam diameters are needed as well as large RF powers in orderto maintain a minimum of 50% diffraction efficiency. A fused silica AOM worked, however, at highlaser power, the optical mode strongly distorts due to thermal lensing. This is a common problem
with high average power laser systems like ours. To overcome thermal lensing, the laser beam waist
can be made larger or additional telescopes can be used to try and manipulate the beam at highpower. As it turns out, fused silica just did not have good enough thermal properties. Therefore,
we had to switch to a tellurium oxide (TeO2 ) AOM. It had much better thermal properties atthe expense of more optical dispersion. The extra dispersion is easily compensated with the laser
compressor and the residual thermal lensing was manageable with telescopes. An additional TeO2
AOM is optionally used as an amplitude modulator in one arm of the interferometer.

4.2.2

Construction of the beam combiner

As previously discussed, there simply is not suitable optical material in the XUV that is transparent. This poses a great challenge for building an interferometer than can operate in the XUV.
Beam combination is usually achieved with partially reflecting/transmitting optics or polarization
beam splitters. These do not exist in the XUV. The aforementioned beam splitters are typicallyreferred to as amplitude division beam splitters. However, there is another class of beam splitters,
wavefront division beam splitters, that are suitable for application in the XUV as they only requirereflections.
To combine two XUV beams, we rely on a mirror with a hole in it. This is a common trickin the XUV world. By using a mirror with a hole in it, one XUV beam is focused and transmitted
through the hole while a second XUV beam reflects off the surface. The two beams will have partial
overlap in the far field. To construct such a mirror, we first started with a silicon wafer with a104

reﬂectedtransmitted

combined

Silicon beam combiner

Figure 4.4: Images of the XUV beam combiner. The transmitted (XUV2), reflected (XUV1),and combined beam images were all taken with the driving laser at 1070 nm (see Fig. 4.2 for the
relevant beam paths). In order to observe fringes like these with a simple CCD, the frequency
detuning between XUV1 and XUV2 must be set to zero. This can be achieved by driving the AOM
with the repetition rate (obtained from a simple photodetector).

105
100 µm square, pyramidal aperture shown in Fig. 4.4. The pyramid structure is important because
it allows a larger acceptance angle for the beam that will be focused through the hole. Such asilicon wafer is an “off the shelf” item and is produced by KOH etching, but it needs to be altered
to perform well as an XUV mirror. First, the silicon wafer is only 3 mm x 3 mm. To make thismore robust and to facilitate polishing it later, we anodically bond the silicon wafer to a pyrex
substrate that also has a large hole in it of about 1 mm diameter. It is important the substrate ispyrex or else the anodic bonding process will not work well. To anodically bond the two pieces,they are sandwiched together with a large voltage (∼ 1.5 kV) applied across them and under a hightemperature (∼ 300 C). There is a small current that flows between the electrodes which can bemonitored. Once the current has almost stopped (after a couple hours), the two optics should bebonded together after the optic cools.
After the anodic bond is formed, the silicon wafer needs to be polished so it can later becoated with a material that reflects well in the XUV. The anodic bond is strong enough to survive
polishing. After the silicon is polished to an ∼Å rms roughness level and the surface is determined
to be optically flat, it is ready for coating. We chose to coat the optics with boron carbide (B4 C)
because it has a high reflectivity of about R = 0.3 out to 55 nm at normal incidence. This is a “highreflector” at normal incidence in the XUV. The coating was performed by Rigaku [153] and further
discussion of XUV optics is in Sect. 4.2.5.
The nominal performance at 1070 nm wavelength is shown in Fig. 4.4. The transmitted beamretains its normal Gaussian structure. However, the reflected beam does not look quite as nice.
Despite the structure on the beam, the wavefront phase is still relatively flat. When the beams arerecombined, a bull’s-eye fringe is present in the far-field profile. The bull’s-eye structure is due tothe fact that the reflected beam and transmitted beam have different wavefront curvatures. If thisbeam was incident on a detector, the total interference pattern would average to zero. To overcome
this, the central fringe is selected out with another aperture before detection. This throws away asmall amount of optical power, but it ensures a high contrast interference fringe for detection.

106
4.2.3

Beam combiner theory and scaling

To begin to describe the beam combiner operation, we rely on Babinet’s principle (principleof complementary apertures) to describe the reflected beam. The far-field reflected beam uR can be
written as combination of a Gaussian beam and the beam generated by the inverse of the aperture.
For our consideration, we will assume a circular aperture of radius a. The schematic used for
calculation is in Fig. 4.5a. For the reflected beam amplitude, we haveuR = uG − uH

(4.1)

where uG is just a Gaussian beam with the focus of ω1 at a distance L from the hole mirror. The
functional form of the Gaussian beam can be found in Eq. 2.34. The contribution of the apertureis [14]−iπN (r/a)2uH (r, z)/u0 = iπN e
Z 1  00r2πrr00 −iπN (r0 /a)2 r
duG (r )e
J0
,
a
a
a2
0

(4.2)

where N is the Fresnel number defined as N = a2 /zλ. While this equation is quite cumbersome, itcan easily be treated numerically. The full far field combination is a sum of all three contributions

utotal = uR + uT ,(4.3)

where uT is the transmitted beam, which is also a Gaussian of form Eq. 2.34 with a focus at z = 0
and a beam waist of ω2 .
As an illustrative example, the results of a numerical simulation are shown in Fig. 4.5b,c with
L = 10 cm, z = 40 cm, λ = 83 nm, ω1 = ω2 = 25 µm and the beam combiner aperture a was left
to vary. The spatial filter aperture was chosen to isolate the central fringe. The signal to noise wasestimated by considering the contrast of the signal illustrated in Fig. 4.5c with the in and out ofphase beam profile of the combined beams. The contrast is C = (Imax − Imin )/(Imax + Imin ) and√ p
assuming a shot-noise-limited detection, the signal to noise is 2 (Imax + Imin )C.

107

Figure 4.5: a) Schematic of the beam combination geometry relevant for calculation. A roughly
collimated beam is used in reflection and a tightly focused beam is used for transmission. The beamcombiner is assumed to have an aperture of width a. The detection plane will contain a spatial filter
to select the central interference fringe. The degree of beam collimation, beam combiner aperturesize, and spatial filter aperture size were all choses to maximize interference contrast and signalto-noise. For b,c) the following parameters were chosen. z = 40 cm, L = 10 cm, λ = 83 nm, andω1 = ω2 = 25 µm. The parameter a was varied and the spatial filter size was chosen to isolate
the central fringe. b) Shown is the compromise of SNR, power used and contrast as a function ofaperture size. The bottom panel shows the optimal spatial filter size. c) The electric field is shownin for the transmitted and reflected beam for a ≈ 50 µm. The in and out of phase profile of the
combined beam is also shown.

108
If briefly we ignore the effect of the aperture on the reflected beam and analyze how thedifferent wavefront curvatures of the two combined beams affect the spatial fringe, we find that√the distance from the center of the beam to the first interference minimum actually scales as λ.
This can be seen by looking at Eq. 2.34 and realizing the wavefront phase scales inversely to λbut proportionally to the radial distance squared. This rather favorable scaling implies that theinterferometer should work over a wide range of wavelength ranges without having to constantly
modify experimental parameters such as z, L, ω1 or ω2 . By setting the aforementioned parameters, asingle spatial filter at the interference plane can also be chosen to maintain near optimal performancefor harmonics 7–17. As will be presented later, we observed the heterodyne beats between 56 nm
and 1070 nm without changing anything but the detector.

4.2.4

Extreme ultraviolet detectors

To detect heterodyne beats in the XUV, we need a detector that is both sensitive at XUVwavelengths and has a fast response since the beatnotes will be at multi-MHz frequencies. The mostcommon approach to detecting XUV photons is to use a scintillating material. Sodium salycilate
is a good material because it absorbs light below 355 nm and fluoresces around 420 nm. Detecting
420 nm light is straightforward with a photomultiplier tube (PMT) or even a CCD camera. The
upper state lifetime of sodium salycilate is also ∼ 10 ns allowing for high bandwidth detection. An
additional benefit is that the quantum efficiency is nearly perfectly flat between 5 eV and 15 eV
making comparisons between harmonic orders very straightforward [13].
Another option is to use an electron multiplier or an multi-channel plate (MCP). Electron
multipliers work in a similar way to PMTs. They can have very large internal gains (up to 107 ) and
can operate directly in a vacuum chamber (unlike PMTs). The work function of common electron
multipliers is typically 9 eV or higher, making them “solar blind”. For us, that means they do not see
the driving laser or the 3rd harmonic, which is a huge benefit, and can thus allow for very sensitive
detection without saturating the detector on the abundant 1070 nm or the 357 nm light. Electron
multipliers can also have ∼ 10 ns response times again allowing for high bandwidth detection.

109
In the power scaling effort reported in Chapter 3, two other detection methods are employed.
A windowless silicon diode with a thin metallic coating can be used [136]. The metallic coating
filters out low photon energy light and the silicon behaves in much the same way as it does withvisible light. The second was a NIST calibrated vacuum photodiode [134]. The photodiode was apolished piece of aluminum oxide that serves as a photocathode. Photoelectrons emitted from thealuminum oxide are collected on a biased anode ring. The device was calibrated by NIST. These
two detection methods are often the most accurate. However, they are also very difficult to operatewith high gain and high bandwidth. They are therefore not ideal for the heterodyne interferometry
experiment.

4.2.5

Extreme ultraviolet optics

One of the most common optical coating materials for XUV radiation is gold. At grazing
incidence, gold can have excellent reflectivity from the XUV out to hard x-ray wavelengths [137].
However, for our interferometer, it is impossible to build it with only grazing incidence optics. The
actual optical layout in reality is close to what is shown in Fig. 4.2. For the path of XUV2 in the
figure, a 70 deg grazing incidence reflection and one normal incidence reflection will be required to
get the beam to the combiner. Fig. 4.6 show the expected reflectivity if gold optics are used. The
net reflectivity of both optics combined is only ∼ 2% for the wavelengths of interest.
Another, less commonly used optical material is B4 C. At normal incidence, B4 C has a reflectivity 2×-3× that of gold [154–156]. Fig. 4.6 also shows the expected reflectivity for XUV2 if
all B4 C mirrors are used. The advantage is clear. When one considers that the heterodyne beat
between XUV1 and XUV2 will be proportional to the product of their fields, enhancing the netreflectivity by 5× for each arm of the interferometer will give 5× more heterodyne beat power. The
reflectivity reported in Fig. 4.6 was calculated by the known optical constants of both materials.
One drawback to using B4 C or other boron-based optical coatings is that they tend to decay overtime and primarily limit the reflectivity at short wavelengths at normal incidence. We have indeed
observed this decay and a mirror that was used over the course of a year experienced almost a factor

110

Figure 4.6: The reflectivities of boron carbide and gold. a) The reflectivity of both optics in thepath of XUV2. b) The net reflectivity of both optics for XUV2 assumes both gold mirrors or both
B4 C mirrors. The improvement by choosing B4 C is very clear. In all cases, the reflectivity wascalculated with knowledge of the optical constants of the materials.

111
of 3 drop in reflectivity at 200 nm. We were unable to determine the loss at shorter wavelengths,
but suspect it was even more severe.

4.2.6

The Seya-Namioka monochromator

The final piece of the experiment is not actually required for interferometer operation. However, it is a very useful tool for diagnosing the HHG process and confirming proper operation of our
XUV optics. The Seya-Namioka monochromator is a simple and robust method of detecting HHG
radiation [13]. We added a Seya-Namioka monochromator to our apparatus by having a removablespherical grating placed near the detection plane as seen in Fig. 4.2. The monochromator only works
for the XUV2 beam path. The beam combiner acts as an entrance slit to the monochromator. The
spherical grating diffracts and re-focuses the light to an exit slit. After the exit slit is a sodium
salycilate coated light pipe fed to a PMT. A DC stepper motor drives a ferrofluidic feedthrough thatis connected to a rotation stage that holds the grating. A key feature of the Seya-Namioka design isthat the angle between the entrance slit and exit slit is fixed (∼ 70 deg) and only the grating needs
to rotate. This makes for easy implementation and easy interpretation. The achievable resolution
is in principle a few Å.
The results of a monochromator scan are shown in Fig. 4.7. The monochromator is operated
in low resolution mode (∼ 1 mm exit slit) but all the harmonics orders are clearly resolved. The
scan takes about 30 seconds. Harmonics 7–21 are clearly visible (and possibly the 23). The relative
heights of all the harmonic orders are strongly influenced by our choice of optics. For the XUV2
path, Fig. 4.8 contains the reflectivity of all the optics. The trace labeled B4 C accounts for thenormal incidence reflection and the grazing reflection in the beam path. The grating efficiency was
given by the grating manufacturer. The total throughput of the system is not that great, being only
0.2% at maximum. Nevertheless, we can use this information to reconstruct the harmonic spectrum
that is generated and how much will be present at the beatnote detection plane (with the grating
removed). The scaled harmonic spectrum is shown in Fig. 4.9. The raw spectrum is taken fromthe data in Fig. 4.7. The exit plane data represents the light that will be available for heterodyne

112

Figure 4.7: The harmonic spectrum measured with a Seya-Namioka monochromator. The spherical
diffraction grating is rotated about its center while the photocurrent from a photomultiplier tube
is measured. Signals that do not fall on harmonic orders are due to second order diffraction of thegrating. The data is the result of a single scan that took about 30 seconds to acquire.

113
detection. The generated data is the inferred harmonic spectra at the focus of the cavity. Clearly,
we make an enormous amount of 7th harmonic, but this is consistent with previous observations
and is relatively unexplained [89]. Our harmonic spectra exhibits a “traditional” plateau shape, oneof the early and intriguing features of the HHG spectrum.

4.3

Putting it all together
With all of the key operating components discussed, the final operation mode can be estab-lished. The output of the Yb:fiber comb is split 50:50 by the AOM operating at fr -1 MHz. Eachpath, XUV1 and XUV2, has an independent set of mode matching optics. The two arms are slightly
different due to the fact that the 1070 nm beam traverses different directions in the ring resonators.
The light is coupled into independent fsECs. The input coupler is R = 0.985 so the finesseis ∼ 400, and the buildup is ∼ 200. Since the Yb:fiber comb only has two degrees of freedom andboth cavities require a total of four degrees of freedom for simultaneous stabilization, the Yb:fibercomb is left free running and all the servo actuation to maintain stabilization is performed with thefsECs. The PDH lock is actuated on by two PZTs in each fsEC. One PZT is a fast, low voltage
PZT attached to the cavity HR mirror. A second, slow but long travel PZT pushes a translation
stage located underneath the same HR mirror. The servo loops for each cavity are independent.
Fortunately, since both cavities have identical optics, the required f0 was the same for both cavities(including the 1 MHz offset by the AOM) and could be adjusted by manipulating the temperatureof the fiber Bragg grating inside the laser oscillator. Generally, no active feedback was necessary
for f0 .
The focal spot size is 19 µm × 22 µm in both cavities, so average powers between 4 kW
and 6 kW are necessary for good HHG conversion. The focal spot is determined by measuring thefrequency spacing of higher order transverse modes as discussed in Chapter 2. Xenon is injected
through quartz nozzles with 150 µm diameter. The light in XUV1 and XUV2 is outcoupled using
a 330 µm thick sapphire Brewster plates (see Fig. 2.7 or Fig. 4.8 for expected performance). Eachof the XUV beams are steered inside the vacuum chamber by motorized B4 C coated mirrors. After114

Figure 4.8: The reflectivities of various optics in the XUV interferometer and monochromator.
This data is used to back out the relative amount of harmoinc power at the generation location.

115

Figure 4.9: The relative harmonic powers at different points in the interferometer. The data inboth figures is the same but shown in linear and log scales for a) and b) respectively. The large
amount of 7th harmonic is consistent with previous observation and its origin is not completely
clear.

116
recombination in space and time, the beams are directed to a spatial filter of a 100 µm square
aperture before detection. The time overlap between the XUV pulses is achieved by delaying thetime of the Yb:fiber comb pulses using a delay stage in the XUV2 beam path.
Because fsECs XUV1 and XUV2 are both pumped by a common NIR laser, the noise in theradio frequency (RF) beatnotes is immune to the common-mode frequency noise of the Yb:fiber laser.
Thus, the apparatus directly measures noise from the HHG process or the cavity-plasma dynamics.
However, because the interferometer is not actively stabilized, there are small amounts of relative
noise induced by vibrations in the optics, giving the two sources a non-zero relative linewidth thatis technical in origin. This tecnical contribution can be removed by the active stabilization scheme
previously discussed.
Harmonics 3–7 could be detected using a sodium salycilate screen with a PMT. Harmonics 9–
19 were detected directly with an electron multiplier. The photocurrent coming from the detectors
is expected to have the form

I(t) = I1 (t) + I2 (t) +p
I1 I2 cos(qfAOM t + q(fr − fAOM )t + φ(t)),(4.4)

where φ(t) are any relative phase fluctuations. However, we have enforced ∆f = fr − fAOM = 1
MHz and qfAOM is far too fast for our detector’s response time, Eq. 4.4 reduces toI(t) = I1 (t) + I2 (t) +p

I1 I2 cos(q∆f t + φ(t)).

(4.5)

The output of the detectors is amplified and sent to an RF spectrum analyzer. Fig. 4.10
contains the RF heterodyne beats for harmonics 3–17. The harmonic beatnotes were heterodyne
translated from their 3 MHz, 5 MHz, 7 MHz, etc. tones to lower frequencies for display on an FFTanalyzer. There is also a beatnote from the fundamental laser at 1 MHz, but this is not shown.
It is separately detected by picking off a small fraction of the edge of the beam that would beotherwise wasted by the spatial filter. The B4 C optics and the Brewster plate still have a very smallreflectivity at 1070 nm facilitating this detection. There iss also a beatnote at 19 MHz, however,117

Figure 4.10: Heterodyne beats from harmonics 3–17. The harmonics were originally detected at 3
MHz – 17 MHz but were frequency shifted to much lower frequencies using RF mixing to help highresolution display on an FFT analyzer. The data is presented on a 30 Hz resolution bandwidth.

118
the signal-to-noise is very poor. We suspect this is due to a reduction in the B4 C reflectivity, whichis hard to accurately characterize in our system [154–156]. The relative heights of the beatnotes inFig. 4.10 also resembles the expected relative heights from Fig. 4.9.
The data in Fig. 4.10 is already great news for the XUV comb. The f0 scaling of eachharmonic order is exactly fq,0 = q × f0 . Also, the beatnotes are clearly sub-kHz. Already this is ahuge improvement over previous work and shows that nothing drastic is happening to destroy thecoherence in the XUV. With each harmonic order individually accessible in the RF domain, it is
now possible to analyze them independently and use them to investigate strong-field physics withphase-sensitive techniques.

Chapter 5

Phase noise and phase sensitive detection in the extreme ultraviolet

In this chapter, we will discuss and answer the basic question of “How good is the XUV
frequency comb?” Here, “goodness” is referred to as a long coherence time or a narrow opticallinewidth. Further, in this chapter we will describe how phase noise scales in the harmonic generation
process and test the limit of XUV coherence by measuring sub-Hz linewidths at various harmonicorders. We will also demonstrate how we can use phase-sensitive detection in the XUV to probe
strong-field physics which occurs at attosecond timescales.

5.1

Optical phase noise
Phase noise in periodic signals is a very well studied problem. Much of the work well pre-dates laser development and finds most of its development in the context of RF signals and RF
communications. We will focus mainly on how phase noise affects oscillators or waves (specifically,
their power spectrum), but we refer to Ref. [157] for thorough investigation of amplitude, phase andfrequency noise processes. For discussion related to RF oscillators, Ref. [158] is quite useful. In thecontext of laser stabilization, a thorough discussion is in Refs. [47, 159]. For the purposes here, wewill review the most important aspects as they relate to the XUV comb experiments.
Before establishing the characteristics of optical phase noise, it is useful to consider how acoherently phase modulated wave behaves and then move to a wave modulated by random phasefluctuations. A phase modulated wave is written as

120

E = E0 e−iω0 t−iφ(t) .

(5.1)

the instantaneous frequency is ω = ω0 + dφ(t)/dt. If we assume that φ(t) = β sin(Ωt), where β isthe phase modulation depth 1 and Ω is the modulation frequency, we can expand Eq. 5.1 in a seriesof Bessel functions as

E = E0 J0 (β)e

−iωt

+

∞
X



E0 Jn (β) e−i(ω+nΩ)tn −i(ω−nΩ)t+ (−1) e



,

(5.2)

n=1

where Jn is a Bessel function of order n. An interesting property that is preserved and is not
immediately obvious from the expansion is that EE ∗ = E02 , meaning a phase modulation process
does not add to or remove from the total power of an oscillating wave. The expansion in Eq. 5.2
tells us that the sinusoidal modulation creates symmetric “sidebands” about the carrier wave at
ω0 . However, the sign of the electric field amplitude at frequencies below the carrier changes withorder n. If the modulation depth β is large, there will be multiple sidebands generated at integer
multiples of the modulation frequency Ω. We can represent the amount of power in the first order
sideband PSB relative to the power in the carrier PC byPSB
J1 (β) 2=.
PC
J0 (β)(5.3)

2 /2 where β
In the limit that β is small, PSB /PC ≈ βrms
rms is the root-mean-square of β. Thisis an instructive result because it shows that the amount of power moved from the carrier to the
sidebands is nonlinear in the size of the phase modulation. To illustrate why this can be problematicin the context of optical harmonic generation, consider the second harmonic gerneration process ina crystal. The second harmonic field intensity will be proportional to the square of the driving laser
electric field 2 E2ω ∝ Eω2 . Explicity, we have
1

Previously, we used β to refer to enhancement cavity buildup. In this chapter, we use β to mean phase modulation
depth. The choice of using β is based on convention. The meaning of β will be clear from context.
2
Note that this is E 2 and not |E|2 , a subtle but extremely important difference.

121

E2ω ∝ Eω2 ,(5.4)

Eω2 = E02 e−i2ω0 t−i2β sin(Ωt) .

(5.5)

There are two important results here. First, the modulation frequency did not change by the
harmonic generation process. Second, the phase modulation depth changed and β → 2β meaning
the amount of power taken from the carrier also changed to PSB /PC → 4 PSB /PC . This can begeneralized to higher order harmonics (assuming they are proportional to powers of the electric field)
to be (PSB /PC )qω = q 2 (PSB /PC )ω where q is the harmonic order. If the harmonic order becomes
too high, eventually all of the power can be taken away from the carrier. If we imagine that theφ(t) is no longer a pure sinusoid, but a random process, we can still decompose it into its Fouriercomponents and we can have a similar expansion as in Eq. 5.2 but for each Fourier component(we will make this claim more rigorous shortly). What we realize is that with increasing harmonicorder, the amount of power in the noise relative to the carrier will grow quadratically for frequency
doubling.
Instead of dealing with discrete phase modulation frequencies, we can generalize to the casewhere φ(t) is a normal, random process. If we want to know how the random phase modulation
affects the pure wave, we often look at the frequency power spectrum to analyze how much power
is at frequencies other than the carrier. The Wiener-Khintchine theorem states that the frequency
power spectrum is just the Fourier transform of the temporal autocorrelation. Explicitly, the electricfield autocorrelation isR(τ ) ≡ hE(t)E ∗ (t + τ )it .

(5.6)

Invoking the Wiener-Khintchine theorem, the power spectrum is

1
PE (ω) = √2π
We can insert Eq. 5.1 into R(τ ) to getZdτ e−iωτ R(τ ).

(5.7)

122

R(τ ) = E02 eiω0 τ hei(φ(τ +t)−φ(t)) it ,(5.8)

and using the Gaussian moment theorem [158, 159], we can make the final manipulation

R(τ ) = E02 eiω0 τ eRφ (τ )−Rφ (0) ,(5.9)

Rφ (τ ) = hφ(t)φ(t + τ )it .

(5.10)

where

Before moving forward, we can identify important characteristics of Rφ (τ ). First, if the noise
is random and uncorrelated, Rφ (τ ) has an important limiting form of

lim Rφ (τ ) = 0.

(5.11)Rφ (0) = hφ(t)φ(t)it = hφ(t)2 it = ∆φ2rms .

(5.12)

τ →∞Secondly, for τ = 0, we have

These conclusions allow us to rewrite R(τ ) in the limit of large τ in a very suggestive way. We have2

lim R(τ ) = E02 e−∆φrms eiω0 τ .

τ →∞(5.13)This shows that the amount of power in the carrier is just exp −∆φ2rms .
We can also define the phase noise power spectral density of the electric field Sφ (ω) using the
Wiener-Khintchine theorem. It is

1
Sφ (ω) = √2πZdτ e−iωτ Rφ (τ ) .

(5.14)The phase noise power spectral density is a very useful quantity. Note that is has units of [rad2 /Hz].
To illustrate its utility, we assume that our wave has small phase fluctuations. This means that

123
both Rφ (τ ) and Rφ (0) are small. Starting with Eq. 5.7 and expanding the exponent in R(τ ), we
have

Z
E02
PE (ω) = √dτ e−i(ω−ω0 )τ (1 + Rφ (τ ) − Rφ (0)),
2π
Z

Z
E02
−i(ω−ω0 )τ
−i(ω−ω0 )τ
=√dτ e(1 − Rφ (0)) + dτ eRφ (τ ) ,2π

(5.15)= E02 [(1 − Rφ (0))δ(ω − ω0 ) + Sφ (ω − ω0 )] ,where δ(ω − ω0 ) is a Dirac delta functional for ω = ω0 . Therefore, we see that the power spectrum
has a coherent carrier at ω = ω0 that is reduced by an amounts ∆φ2rms . The second part of thepower spectrum is just proportional to the phase noise power spectral density centered at ω = ω0 .
The coherence properties of a laser are often parametrize by its linewidth (inverse of thecoherence time). The linewidth is usually defined as the FWHM of the power spectrum. However,
the meaning of FWHM is usually restricted to cases where the function form of the power spectrum
is known (for example, Gaussian or Lorrentzian. See Ref. [158] for limiting cases). A more generaldefinition of the laser linewidth is therefore desirable. One proposed metric [47, 160] is that the
linewidth is the frequency from the carrier at which half of the power spectrum is located in thenoise and half is in the carrier. We can make this more explicit by using the phase noise power
spectral density

Z ∞dωSφ (ω) ≈ 1 rad2 .

(5.16)

δωwhen δω is chosen so the integral totals to 1 rad2 , approximately half of the optical power is in thenoise and half in the carrier frequency. We can then define the FWMH as being FWHM= 2δω.
Returning to our earlier example of how the phase modulation depth scaled with harmonicorder, if we start with the premise that φqω0 = qφω0 , that is that the phase of the q th harmonic
advances at q× the phase of the fundamental frequency, our analysis results in that the phase noise
power spectral density of the harmonic tone Sqω0 ,φ will grow quadratically with harmonic order.
That is124

Sqω0 ,φ (ω) = q 2 Sω0 ,φ (ω).

(5.17)

It therefore follows that the linewidth will also grow quadratically with harmonic order. Further,
if Sω,φ (ω) is not sufficiently small before the harmonic generation process, the coherent carrier will
eventually disappear for large q. This is known as carrier collapse [160] and is discussed in detail inthe context for RF oscillators in Ref. [161].

5.2

Noise scaling of high-order harmonic generation
As we have just discussed, in harmonic generation of oscillating signals, the phase noise power

spectral density (and hence the linewidth) is expected to grow quadratically with harmonic orderfrom the phase noise of the driving laser, even if the harmonic generation process is noiseless [161].
With the XUV interferometer described in Chapter 4, we use the resulting heterodyne beatnotes
presented in Fig. 4.10 to test this assertion for the case of optical HHG. The relative linewidth
between the two XUV sources (refer to Fig. 4.2 for the schematic of the apparatus) is presented inFig. 5.1. The linewidth as a function of harmonic order is fit to a quadratic function and excellent
agreement is obtained. For this data, the interferometer is not phase-stabilized and all the beatnotes
are shifted from the MHz frequencies to audio frequencies and analyzed on a FFT machine so ahigh resolution signal could be obtained allowing numerical fitting for the linewidth.
For the data in Fig. 5.1, it is worth mentioning that we approximated the linewidth using a
Lorentzian function. However, nearly identical scaling results were obtained if a Gaussian was used.
In reality, neither function perfectly matches our observed power spectrum. This is not surprisingsince the relative noise in the interferometer can arise from a wide variety of sources. We refer toRef. [158] for examples of when a Gaussian versus a Lorentzian power spectrum is expected. In
general, the power spectrum will be a convolution of these or even more complicated [159, 160].
The phase noise scaling was also tested in an independent way utilizing coherent phase modulation instead of noise. By placing phase modulation sidebands on the driving laser, we observe how

125

Figure 5.1: Beat linewidth versus harmonic order to show phase noise scaling. The data is thesame as presented in Fig. 4.10. The interferometer was not actively stablized for this measurment.
The linewidth was approximated with a Lorentzian function, but a similar results was obtained
assuming a Gaussian. The shown fit is to a simple quadratic equation.

126
the power in the sidebands changes at various harmonic orders. By phase modulating one arm of
the interferometer and using the heterodyne beat of the driving laser, the phase modulation depth
is determined to be βP1 = 0.049. At the 17th harmonic, we expect this to be β17 = 17 × P1 . The
heterodyne beat at the 17th harmonic with the modulation on and off is displayed in Fig. 5.2. A
clear reduction in the carrier power is observed and large phase modulation sidebands are present.
We use Eq. 5.3 to determine that the sideband should be at -6.8 dBc which is in good agreementwith the measured -6.5 dBc. This sort of measurement was repeated at harmonics between 7–17
with good agreement.
The observations with the linewidth scaling and the modulation sidebands indicates that the
HHG process is not adding any additional noise to the laser beyond what is expected by frequency
multiplication. With this, we already have very strong predictive power on what sort of levels ofcoherence can be expected in the XUV. For example, if we start with a comb that has a 1 Hz
absolute linewidth, we can expect to have a XUV comb with a linewidth of 441 Hz at the 21st
harmonic or a linewidth of 10.2 kHz at the 101st harmonic.
With our interferometer not actively stabilized, it is equivalent to performing a relative phasenoise characterization between two lasers that have a relative linewidth of about 1 Hz. The 1 Hz
relative linewidth was not deliberate but just the natural manifestation of our apparatus. Whatremains to be tested is the following. Imagine our lasers had a relative linewidth of 1 mHz, whatis the linewidth in the XUV? What about 1 µHz? At what level does the HHG process start toadd phase noise beyond what pure frequency multiplication dictates? To answer these questions,
we need the interferometer to be completely phase stable so that the relative linewidth between thetwo sources at the driving laser wavelength is non-existent (at least to the degree that we can makea good phase lock). Fortunately, we have the ability to do exactly that.

5.3

Testing the limits of coherence in the extreme ultraviolet
As previously described in Sect. 4.2.1, we use the heterodyne beat from the driving laser at

1 MHz to actively phase stabilize the relative path lengths of the two arms in the interferometer.

127

Figure 5.2: The 17th harmonic with and without the presence of phase modulation on the driving
laser. A clear reduction in the carrier is present. The size of the phase modulation sidebands at
the 17th harmonic is consistent with the measured sidebands of the driving laser. The modulation
sidebands were places on the driving laser by the AOM used as a beam splitter for the interferometer.
The data is shown on a 30 Hz resolution bandwidth.

128
The feedback loop parameters were chosen to have a deliberately low feedback bandwidth as to
not interfere with any subsequent and deliberate laser modulation (amplitude or phase). Since thenoise in the interferometer was already made small at low frequencies due to its passive stability,
a high bandwidth feedback loop was not necessary to remove most of the noise. Fig. 5.3a,b show
the heterodyne beat between the driving laser with the interferometer unstabilized and stabilized,
respectively. Only a subtle change is observed due to the action of the phase lock, but it is clearthat the fluctuations have been removed. Fig. 5.3c,d show the results of the heterodyne beats forthe 17th harmonic with the interferometer unstabilized and stabilized. The change is much more
dramatic and note the difference in the x-axis of the plot. The beat in Fig. 5.3d appears to be a
δ-function and its narrowness is limited by the observation time. This is often referred to as being
limited by the resolution bandwidth. For this data, the resolution bandwidth was 1 Hz and thislevel of coherence was confirmed at all harmonics from the 3rd to the 19th . We can further increase
the observation time of the heterodyne beat to 16 seconds. Fig. 5.3f shows a near resolution limited
beat of 62.5 mHz. This linewidth implies a coherence time of 16 seconds. This also demonstrates
that we were able to maintain perfect phase stability for over 109 optical pulses at our repetition
rate of 154 MHz. At this level of investigation, we do not observe any deleterious effects to thephase coherence from the HHG process, a testament to its coherence maintaining properties despite
its enormous nonlinearity. We also confirmed that our observation was not a coincidence of picking
the same nonlinear medium for both cavities (in this case xenon). By injecting xenon into XUV1
and krypton into XUV2, we observed a 1 Hz resolution limited beat at the 15th harmonic, again
limited by the resolution bandwidth of this measurement.
Coherence at the 62.5 mHz level is great news for future prospects of performing highresolution spectroscopy in the XUV. The best available lasers in the near IR have linewidths down
to 30 mHz [80]. Assuming we discipline our driving laser to this, we can expect to start with afrequency comb with comparable linewidth. Due to the quadratic scaling of the phase noise, we
expect to have a linewidth of 8.7 Hz at the 17th harmonic. The data presented in Fig. 5.3 showsthat the XUV frequency comb can indeed support this level of coherence. Current laser technology

129

Figure 5.3: Ultra narrow heterodyne beats test XUV comb coherence at long time scales. a) andb) show the beatnote between the driving laser with the interferometer unstabilized and stabilized
respectively. A subtle change to the IR heterodyne beat power spectrum is observed. c) and d)show the beatnote between the 17th harmonic with the interferometer unstabilized and stabilized
respectively. A dramatic difference is observed. Note the difference in the x-axis of the plots. We
can further zoom in to d) and uncover an ultranarrow beat, resolution limited at 62.5 mHz inf ). To show that the observed heterodyne beats were possible if different gases were used, the15th harmonic beat between XUV1 making harmonics in xenon and XUV2 making harmonics inkrypton is shown in e) with a 1 Hz resolution bandwidth limited heterodyne beat. At 1 Hz, theXUV frequency comb already exceeds the capabilities of current laser stabilization techniques inthe visible and near-infrared. RBW, resolution bandwidth.

130(in terms of linewidth) would need to improve by nearly two orders of magnitude to justify testing
the XUV comb linewidth beyond the level presented here. Our result of no additional noise at the62.5 mHz level is eight orders of magnitude better than previous demonstrations [52, 56, 87].

5.4

Measurement of intensity dependent dipole phase
Access to phase-stable light in the XUV opens up many exciting possibilities. For example,

two-dimensional spectroscopy in the XUV, XUV holography, and of course, high-resolution spectroscopy. To demonstrate the power of having access to optical phase in the XUV, we decided
to investigate one of the most salient features of HHG presented in Sect. 3.1.3. The quantum
description of HHG predicts that the phase of the XUV light will be linked to the driving laser
intensity. The relationship is parametrized as φ = −αj I/4ω 3 , where αj is the coupling constantand the index j refers to the index of the different possible quantum paths (typically, just the shortand long trajectories). Ref. [87] established that below threshold harmonics contained both longand short trajectory contributions to the overall HHG yield. The investigation relied heavily on
theory to extract values for αj but was nevertheless successful. Most investigations of the intensity
dependent dipole phase of above threshold harmonics rely on the interference of the long and short
trajectories with each other, and rarely direct, independent investigation of an isolated trajectory
was performed [87, 162]. Our heterodyne interferometry scheme has the ability to isolate the short
trajectory by observing the heterodyne beats directly on axis where the contribution from the long
trajectory is expected to be very negligible due to its much larger diffraction angle in the far field
and its reduction due to macroscopic phasematching. With systematic variation of the laser intensity, phase shifts on the heterodyne beats can be linked directly to this sub-laser cycle (and hence
attosecond timescale) physics of HHG.
To illustrate the principle of this measurement, Fig. 5.4 shows the heterodyne beats at the
15th harmonic and at the fundamental at two different intracavity laser powers. The 15th harmonicin Fig. 5.4a shows a clear phase shift (and slight reduction in beat power) as a laser intensity ischanged while there is hardly a difference in fundamental shown in Fig. 5.4b. The phase shift in the131

Figure 5.4: Heterodyne beats at 15th harmonic and fundamental in the time domain for two
differrent laser powers shown in a) and b) respectively. The phase shift in the XUV is clearlypresent while there is virtually no detectable phase shift at the fundamental beat (due to thephaselock).

132
15th harmonic originates from the intensity dependent phase. However, one must be very careful ina measurement like this. For example, by changing the laser intensity, we may inadvertently change
the driving laser phase. Since we have already established that small phase shifts at the fundamental
scale with harmonic order in the XUV, small perturbations can be very important. Therefore, wewill rely on modulation techniques and lock-in detection methods to make measurements of theintensity dependent phase. Simultaneous monitoring of the XUV beats and the fundamental beatwill be necessary to correct the signal for any parasitic phase modulation on the fundamental. The
schematic for the modulation/demodulation scheme is shown in Fig. 5.5. As describe earlier, an
additional AOM is placed in the beam path to act as an amplitude modulator. The following section
will describe the signal extraction and analysis scheme.

5.4.1

Theory of measurement

By trying to measure the intensity dependent phase that results from HHG, we are effectively
measuring the AM-PM coupling with the AM (amplitude modulation) being on the driving laser
and the PM (phase modulation) being on the XUV light. We can mathematically describe a beatsignal as

S(t) = (1 + A Sin(Ω t + φm )) Cos(ω t + P Sin(Ω t + φm )).

(5.18)

ω is the frequency of the beat and Ω is the frequency of the applied modulation and φm is its phase.
A is the amplitude modulation depth and P is the phase modulation depth. The phase response
is directly proportional to the amplitude response, when we assume that these two response arein phase with each other. Each beatnote is characterized by its own values for A and P. To avoid
confusion, the subscripts will refer to the signal it represents. For example, AIR is for the amplitude
modulation of the driving (fundamental) laser and Pq is the phase modulation of the q th harmonic.
Our task is to determine the values for A and P at the fundamental and the harmonics. By taking theratio of Pq to AIR and using proper units, we can extract the intensity dependent phase coefficient133

Figure 5.5: Detailed schematic of the measurement scheme for the intensity dependent dipole phase.
AOM, acousto-optic modulation. AM, amplitude modulation. Ω, the modulation frequency. DET,
photodetector. φ, a phase shifter in the RF signal chain. The dashed lines represent the signal chain
for the fundamental detection and the solid lines represent the signal chain for harmonic detection.
It is very important that the interferometer is phase stabilized during the data detection process lest
the double demodulation scheme will not work. It is also important that all RF sources involved in
the heterodyne beats and interferometer stabilization are mutually phase coherent.

134
αj .
We need to extract the relevant parameters of Eq. 5.18 at both the harmonic of interest andthe driving laser simultaneously. To do this, we use a two step demodulation process. By taking
Eq. 5.18 and mixing it with a stable RF signal (LO1) at a frequency of ω and relative phase offset
φ, we getS(t) ⊗ V1 Cos(ω t + φ) = S1 (t)S1 (t) ≈ V1 (Cos(φ) −(5.19)AP
Sin(φ) + A Cos(φ)Sin(Ω t + φm ) − P Sin(φ) Sin(Ω t + φm )).
2(5.20)

Where we have ignored terms at 2ω. We can further low pass the signals at Ω and obtain a “DC"signalS1 (t)

Low Pass→

SDC = V1 (Cos(φ) −APSin(φ)).
2(5.21)

Eq. 5.21 will be one of our primary signals. Note that the phase is set by the phase of LO1. This
also assumes that the phase of the XUV beat is stable. This is true if we phase stabilize ourinterferometer which we do for this measurement.
The signal S1 (t) contains terms that oscillate at the applied modulation frequency Ω. We candemodulate our signal once more at the correct phase φm and ignore terms at 2Ω to obtain

S1 (t) ⊗ V2 Sin(Ωt + φm ) = SLIA ≈ V1 V2 A (Cos(φ) −P
Sin(φ))
A(5.22)SLIA is our second signal. With Eq. 5.21, Eq. 5.22 and some independently measured parameters,
we can extract our parameters of interest.
By applying AM to the pump laser on one arm of the interferometer, we can control AIR very
well. It is also easily measured with a photodetector. By varying the phase of LO1, we can measure
SDC and SLIA (Eq. 5.21 and Eq. 5.22) simultaneously. With the modulation (A,P) turned off, SDC

135
tells us the phase of the beat. With the modulation on, the relative phase between SDC and SLIA
can tell us the ratio of A/P. Since A can be measured independently, we can extract the amountof phase modulation, P. This procedure needs to be done with the IR signal and the XUV signal
simultaneously recorded to prevent any systematic errors. Due to the nonlinear nature of HHG,
simply measuring AIR and PIR does not give us enough information about the XUV light. This isbecause AIR 6= Aq . However, as we discovered in our linewidth studies Pq = q × PIR . This will be
an important point for calibrating our signal.
To measure the AM-AM coupling, we can use our beat signals. Our XUV beatnotes are
directly proportional to the amount of XUV power in each beam. The amount of beat power canalso be easily measured on an RF spectrum analyzer. By changing the amount of power in one of
the fsECs and observing how the beat power changes, we can determine how much the XUV power
must have changed for a given laser intensity change. Since Aq is no longer a fit parameter, we canuse the measurements of SDC and SLIA with the XUV signal to extract Pq . Once we have both Pq
and AIR , we need to correct the measurement of Pq to account for any small amount of PIR that
may have been in the system. We do this by simply Pq → Pq − q × PIR . This is to remove any
residual phase modulation that does not originate from the dipole response in HHG. From here, we
can take the ratio of Pq /AIR and scale by appropriate units to determine αj .

5.4.2

Measurement systematics

In order to determine αj , we need to determine AIR , PIR , φIR , Aq , Pq , and φq . It is easy
to directly control and measure AIR , but due to various imperfections in the system, attempts
to put pure amplitude modulation on the IR light results in small amounts of PIR . However, bymeasurements of SDC and SLIA with the IR signal, we can extract the amount of PIR and use it tocorrect our system. This is important to do because Pq = q × PIR and this Pq does not originate
from the atomic dipole phase of interest. This is just a manifestation of how the phase of the XUVlight is tied to the phase of the IR light via the harmonic process. Modulating the intensity has
notable effects on the neutral/plasma density ratio inside the fsEC. However, the modulation is slow

136
(at 2 kHz) and the PDH lock can easily follow it to maintain resonance. Due to the neutral/plasma
density changes, there is a small amount of PM induced by AM on the driving laser. This was verified
by measuring the effect of AM with gas present and absent. This can easily corrupt the intensitydependent phase measurement and necessitates the correction described previously. Furthermore,
any cavity oscillation due to bistability [114] can render the signals too noisy. Further, to avoid
the aforementioned bistability, we deliberately lock the laser off the peak of the cavity resonance byadding a small DC offset to the PDH error signal. This is problematic because as the amplitudeof the driving laser is modulated, so will the size of the PDH error signal be modulated. The
effective detuning from the laser to the cavity resonance is set by the ratio of the DC offset value
to the peak-peak value of the PDH signal. This is now dynamically modulated by our imposed
AM and thus the cavity/feedback loop is now converting this to PM by modulating the cavity-laser
relative detuning. This was systematically measured on both the fundamental and harmonic light toensure proper correction and no nonlinearity. A final systematic check was performed with “dummy
signals” originating from RF synthesizers. The dummy signals were injected into the electronics at
the location of the photodetectors. The signals were then analyzed by the electronics to ensure no
artificial AM-PM couplings due to nonlinearities in either the lock-ins or the RF mixers. Similarly,
AM-PM conversion of the photodetectors was also verified to be of negligible sizes by studying their
response to controlled light amplitude variations.

5.4.3

Results and data analysis

The results of the intensity dependent phase measurement for harmonics 3–17 are shown in
Fig. 5.6. This is the main result of our attempt to measure the intensity dependent dipole phase.
The laser peak intensity was 0.34 × 1014 W cm−2 with an intensity modulation of 15%. Each data
point is determined by fitting ∼ 10 runs of beat measurement. By combing multiple runs of data,
we can construct a weighted average and determine a final value for αj at a given harmonic. The
case of the 15th harmonic is shown in Fig. 5.7. The error bars seem to be larger than the actual
scatter of the data. This just means there is some “over-scatter” indicating we did not make a large

137

Figure 5.6: Data summary of the intensity dependent dipole phase as a function of harmonicorder. The purple regions is where the SFA predicts the values to lie for above-threshold harmonics.
The gray regions is where the near-threshold harmonics are expected to lie. The sign depends oncontributions from uphill or downhill over-the-barrier ionization trajectories.

138
enough statistical average for each run, but clearly the systematics were small. Normalizing thefinal error bar by the χ2red will account for this. An example of an individual run of data for the
IR and the 15th harmonic data is shown in Fig. 5.8. The data is displayed as signal voltages as afunction of φ (the phase of the LO1). From these two traces and the measured value of AIR , we canmeasure any unintended PIR . Each run yields a value of αj with an error bar determined by thestatistical noise in the data. All the data is then fit to a standard mean, weighted by the standard
uncertainty of the individual fits to αj , and an error bar is given by the standard error of the pooleddata normalized to the χ2red of the fit to a constant.
5.4.4

Conclusion

Our data, summarized in Fig. 5.6, shows the intensity dependent dipole phase as a functionof harmonic order from below-threshold to above-threshold. Our method of measurement is unique
because it directly determines the contribution of the short trajectory by sampling of optical phaseand does not rely on spatial interference between multiple pathways. It is complementary to RABBITT and related methods based on photoelectron spectroscopy [72, 163], which seek to determine
the time delay (or equivalently phase shift) between adjacent harmonic orders at a given intensity.
Our data also has the ability to help distinguish important aspects of near-threshold harmonic
generation. For example, in the theory work of Ref. [121], the treatment of the atomic potential andionization dynamics is very important for determining the intensity dependent phase of the nearthreshold harmonics. By using the semi-classical model (SCM) or using over-the-barrier corrections(OTB), the sign of the intensity dependent phase parameter αj can change and the trajectory
contributions are modified. Our data is more consistent with theory when the OTB corrections
are included. This confirms the claim that short trajectories of the below-threshold case do not
originate from tunnel ionization, unlike short trajectories of the above-threshold harmonics which
do originate from tunnel ionization. We previously presented Fig. 3.4 that shows the difference
between the SCM and OTB case.
Our phase measurement technique is able to resolve phase shifts with uncertainties at the 10−2

139

Figure 5.7: A dataset for the intensity dependent dipole phase measurement at the 15th harmonic.
a) The raw phase modulation depth of the 15th harmonics. b) The scaled phase modulation depth.
The scaling factor is determined from the amount of phase modulation present on the fundamental
light which is shown in c). d) The total dataset for the 15th harmonic with a fit shown in red. The
mean values and the χ2red normalized standard error is reported in the inset.

140

LO1 Phase [rad]LO1 Phase [rad]LO1 Phase [rad]LO1 Phase [rad]Figure 5.8: An example of a single data run measuring the intensity dependent dipole phase at
the 15th harmonic. The phase of LO1 is scanned as the voltages from the DC detector and the LIA
are measured. a,b) contains the DC signal for the fundamental and the 15th harmonic respectively.
c,d) contains the LIA signal for the fundamental and the 15th harmonic respectively.

141
rad level, which corresponds to a time uncertainty of 6 as. In contrast to typical experiments thatutilize direct attosecond timing resolution, we measure the attosecond electron dynamics imprinted
on the phase of the emitted XUV light originating from HHG. With system improvements, it isfeasible to extend this into the ≤1 as regime, rivaling the highest achievable temporal resolutionof attosecond electron dynamics provided that a direct link between XUV phase and time can beestablished.
To conclude this detour into attosecond physics, our apparatus provides a unique ability todirectly sample optical phase in the XUV. We performed a proof of principle experiment where
we measure the intensity dependent dipole phase from below-threshold to above-threshold. Our
technique can be further extended to probe attosecond physics in other atomic samples or aligned
molecules.

Chapter 6

Strong-field physics with molecules

The motivation for studying strong-field physics of molecules is based on the success of Chapter 5 where we were able to extract the intensity dependent dipole phase, a signature of the strongfield light-matter interaction responsible for HHG. The hope is that we can do similar work withmolecules and investigate HHG physics as a function of molecular alignment and laser intensity,for example. The work of Ref. [164] essentially started the field of high-harmonic spectroscopy
for molecular orbitals. The current status of the HHG spectroscopy/attosecond science is wellsummarized by a series of review articles [165–168]. An interesting perspective of high harmonic
spectroscopy from someone at the frontline of the high harmonic spectroscopy battleground is inRef. [169]. We seek to offer a clean, highly sensitive measurement of HHG phenomena and without
the need to invoke too complicated theories.

6.1

Impulsive stimulated Raman scattering
Raman scattering and stimulated Raman scattering (SRS) [170] are ubiquitous processes in

optical science. Some notable examples include manipulation of ultracold molecules [171], nonlinear spectroscopy [172] and microscopy [173], and ultrafast molecular phase modulators [174]. Of
particular interest for ultrafast science is the use of SRS as an ultrafast phase modulator capableof generating multi-octave spanning optical spectra [174, 175]. The resulting spectrum is capableof supporting sub-fs optical pulses [176] or generating complex optical waveforms [177]. In theseexperiments, intense, short laser pulses are ideally suited to investigate SRS and use SRS as a tool

143
to manipulate optical fields. SRS generates a coherent vibration or rotation in a molecular system (typically H2 or isotopologues). The molecular vibration or rotation modulates the index ofrefraction in the gas, producing sidebands at the corresponding frequencies.
In some SRS experiments, two separate pulses of ∼ns−ps durations at two different frequencies
are used to stimulate the Raman responses from the molecules. These long pulses operate in the
adiabatic regime where the modulation process persists during the entire laser pulses. However,with a broad spectrum being generated from the modulation process, trains of ultrashort (possibly
sub-fs) pulses may emerge within the long parent pulses [174]. It is also possible to perform SRS withpulses of ∼fs durations. In this regime, known as impulsive stimulated Raman scattering (ISRS), a
single pulse has a sufficiently wide spectral bandwidth to provide Raman coherence. Unlike the casewith longer pulses, the coherent vibration or rotation can persist long after the pulse has passed
through the molecular sample [178]. On a rotational excitation, this is field-free molecular alignment(FFMA) in the ISRS limit [179, 180].
Impulsive, field-free molecular alignment has greatly aided the study of molecular systems
because it is a simple and robust way to create an aligned molecular sample in the absence ofany electromagnetic fields [179, 181]. The impulsive molecular phase modulation has also been
demonstrated to produce ultrashort laser pulses [182]. More importantly, this technique has led tomajor advancements in strong-field physics with high-order harmonic generation (HHG) experiments
performed in molecules [163, 164, 183, 184]. In the impulsive regime, the pulse duration is much
shorter than the characteristic vibrational or rotational period of the system. Therefore, assumptions
made in describing adiabatic SRS are no longer valid. The impulsive nature has important effects
on the driving laser [178, 185, 186]. The laser pulse experiences a self-phase modulation-like process,which leads to a red-shifting of the original optical spectrum. This features is distinct from SRS
in the adiabatic regime [178, 185, 186]. Impulsive FFMA also has important consequences forstrong-field physics [163, 164, 179, 183] since the HHG-producing laser pulse will experience spectraland phase shifts related to ISRS during the HHG process. Indeed, we can now provide a clean
measurement of this effect with our fsEC approach.

144

6.2

Field-free molecular alignment
Field-free molecular alignment is an elegant way to use ultrashort laser pulses to coherently

excite molecules such that at later times after the initial excitation the molecular axis will become
periodically aligned with respect to the laser polarization. This method is not restricted to linear
molecules, but for this work we will focus on those solely. Here, we briefly outline the basic formalismof FFMA for linear molecules as it pertains to our experiments. A more detailed treatment canbe found in Ref. [179, 180]. During the laser pulse interaction with the molecules, the effective
Hamiltonian is,

Hef f (t) =B 2 1
J − ∆αE 2 (t)cos2 θ.
~2
4

(6.1)

Here, B is the rotational constant, J is the angular momentum operator, E(t) is the electric field,and θ is the angle between the primary molecular axis and the laser polarization. The anisotropic
polarizability ∆α is defined as ∆α = α|| − α⊥ , where α|| − α⊥ are the polarizabilities of the molecule
parallel and perpendicular to the molecular axis, respectively. The time-dependent Schrödinger
equation can be solved by expanding the wave function in the rigid rotor basis, and the degree ofalignment can be quantified by the expectation value of hcos2 θi = hψ(t)|cos2 θ|ψ(t)i. To comparewith experiments, gas samples usually begin in thermal equilibrium, so the expectation value ofhcos2 θi needs to be thermally averaged, denoted as hhcos2 θii (this will be described shortly). In
the rigid rotor basis, the cos2 θ operator can connect only J → J 0 = J, J ± 2 in a two-photon,
Raman-like process. The molecules receive an impulsive kick from the field, and many rotational
states are coupled owing to the large bandwidth associated with the short pulse. After the pulse
has passed through the gas target, the molecular wave function then evolves freely as

ψ(t) =

X
J,M

B

a0J,M e−i ~ J(J+1)t |J, M i,

(6.2)

145
where a0J,M is the amplitude of the |J, M i state right after the interaction with the pulse has ended.
Eq. 6.2 shows that the laser pulse has created a coherent superposition of rotational states thatevolve coherently after the pulse. It is evident that at time t = 2π ~/B and its integer multiples,
the wave function will repeat itself. At these wave-packet revival times, the hhcos2 θii expectation
value will be maximized. The thermal average to obtain hhcos2 θii is performed in the usual way by2

hhcos θii = Z

−1

X
J



−BJ(J + 1)
exp
kB T

 M
=J
X

hcos2 θiJ,M ,

(6.3)

M =−J

where Z is the rotational partition function, kB is the Boltzmann constant, and T is the rotationaltemperature. The rotational partition function is given by

Z=

X
J


−BJ(J + 1),
wJ (2J + 1)exp
kB T


(6.4)

where wJ is the weight of the J th energy level determined by nuclear statistics. Despite the thermal
nature of the gas jet we are attempting to align, the coherences are still present as manifested by the
dynamic evolution of hhcos2 θii. There may be times of strong alignment at fractions of t = 2π ~/Bthat depend on the molecule under investigation [179, 180].
To illustrate the alignment phenomena, a numerical implementation is shown in Fig. 6.1. For
Fig 6.1a,b a 5 TW cm−2 laser pulse of 130 fs duration and a rotational temperature of 70 K was
assumed for N2 and N2 O respectively. The field-free evolution is shown after the laser pulse at t = 0.
A list of common molecular parameters is shown in Table 6.2 for reference. It is clear that the N2
and N2 O have quite different behaviors. The obvious features are the different timescales for the
dynamics which is due to the difference in rotational constants. Further, the degree of alignment isstronger in N2 O because of the larger anisotropic polarizability. A more subtle differences is that
N2 O only has full and half revivals while N2 has full, half, and quarter revivals. This difference isdue totally to nuclear statistics.
Impulsive FFMA can also have important consequences for precision studies of strong-field
physics [163, 164, 179, 181, 183, 184, 187] in aligned molecules since the HHG-driving laser pulse

146

N2

N2 O

Figure 6.1: Rotational revivals shown for N2 and N2 O in a) and b) respectively at 70 K interactingwith a 5 TW cm−2 , 130 fs pulse. The expectation value hhcos2 θii is plotted as a function of field-free
evolution time.

147

a)

~ 10 ps~ 120 fsb)

1/154 MHz = 6.5 ns
HR

XUV

QWP

PR

IR

PBS

Figure 6.2: a) Schematic of the experiment. A high repetition-rate train of dual pulses, one
for alignment and the second for HHG, are coherently coupled into a high-finesse, femtosecond
enhancement cavity. At the focus of the cavity, the molecules are aligned by the first pulse andhigh-order harmonic generation is performed by the second pulse. The XUV light can be extracted
from the cavity using a sapphire plate placed at Brewster’s angle for the driving laser. b) The dual
pulse feature of femotsecond pulse train is produced from a Gires-Tournois interferometer, with atunable delay between the two pulses. HR, high reflector. PR, partial reflector. QWP, quarter
waveplate. PBS, polarizing beamsplitter.

Molecule
N2
O2
N2 O
CO2

B(cm−1 )1.98961.42970.41900.3902

Trot (ps)
8.383
11.67
39.8
42.74

∆α(Å3 )
0.93
1.099
2.8
2.11

148
Ip (eV)
15.58
12.07
12.89
13.78

wJ (even J)
201
1wJ (odd J)11
10

Table 6.1: A table containing the rotational constants, rotational periods, anisotropic polarizabilities of common molecules used in field-free molecular alignment.

will experience spectral and phase shifts related to ISRS during the HHG process. Despite FFMA
being a well understood phenomenon [179, 181], the effects on the driving laser originating fromISRS are often ignored in HHG and FFMA experiments due to their small effects when thin,freely expanding gas targets are used at low densities. However, in the cavity-enhanced approachdescribed here, we are more sensitive to the effects of ISRS on the driving laser due to the cavity
effectively increasing the interaction length of our sample by a factor proportional to the cavity
finesse. Indeed, we can now provide a clean measurement of the effects of ISRS with our fsEC
approach while maintaining otherwise similar experimental conditions to conventional molecular
HHG experiments. Our measurements indicate that modulations (amplitude and phase) of thedriving laser cannot be ignored in future experiments utilizing fsECs and must be characterized toproperly describe the XUV frequency comb spectral amplitude and phase.

6.3

Field-free molecular alignment experiments
The apparatus is schematically shown in Fig. 6.2. The intracavity pulse duration is veri-

fied with second-harmonic intensity autocorrelation. The intracavity spectrum is measured with
an optical spectrum analyzer using the light transmitted through a cavity high-reflecting mirror.
Performing HHG with aligned molecules requires two successive pulses, the first to rotationallyexcite the molecules and the second to drive HHG (henceforth, pump and probe respectively). To
accomplish this goal, we use a mirror combination similar to a Gires-Tournois interferometer (GTI)
to convert the original laser pulse train into two with a tunable timing delay, schematically shown in
Fig. 6.2 b). The combination of a partial reflector (R ≈ 0.1) and a high reflector (R ≈ 1) along with

149
a quarter (λ/4) waveplate and a polarizing beamsplitter cube generates the dual-pulse train. This
combination of optics also generates weaker pulses after the probe pulse, but these do not affect therotational dynamics generated by the pump and observed by the much larger probe. Additionally,
the GTI combination of optics is more efficient in generating the pulse train than a more traditional
Michelson-type interferometer that relies on polarization optics. By tuning the value of the partialreflector, the relative heights of the pump and probe can be adjusted. By simply changing the distance between the partial and high reflector, the timing delay between the pulses can be tuned. The
pulse train after the GTI combination was analyzed with second-harmonic intensity autocorrelation
to ensure the pulse separation and ultrashort pulse durations were maintained. For our experiment,
the intensity ratio of alignment pulse (pump) to HHG-driving pulse (probe) is 1:8; however, thisvalue can easily be varied from 1:3 to 1:30 limited only by the available PR mirrors. The pulse train
is coherently coupled into the fsEC, and at the cavity focus we inject N2 O gas through a quartz
nozzle with a ∼ 120 µm diameter with up to ∼ 3 atmospheres (atm) of constant backing pressure.
XUV light produced with HHG is outcoupled from the cavity using a 250 µm thick sapphire plate
placed at Brewster’s angle for the fundamental driving laser.

6.3.1

Single-pulse caseWe begin with investigating the effects of a single pulse propagating and interacting with an
N2 O gas jet inside the fsEC. N2 O was chosen because of the relatively large anisotropic polarizability
to facilitate alignment, the small rotational constant to create a large delay between revivals, andthe low ionization potential to facilitate observation of above-threshold harmonics. We estimate
the rotational temperature of the N2 O gas to be 30-60 K [133]. The pulse has a peak intensityof 0.7 × 1014 W cm−2 in the absence of molecules. We monitor both the intracavity spectrum andpower as a function of the applied pressure to the gas nozzle. As shown in Fig. 6.3, we observe a
clear red shift of the spectrum, accompanied with a systematic decrease of the power. The power
decrease results from spectral red-shifting decreasing the enhancement of the cavity and not fromlight scattering out of the cavity. The red shift by ISRS is contrary to what is usually observed

150

Figure 6.3: a) Pressure dependence of the buildup power when a single pulse is interacting withmolecules inside the cavity. As the N2 O pressure is increased, a decrease in the intracavity power isobserved due to loss of power enhancement in the cavity. A trend line is shown with the data. The
loss is due to the spectral red-shift from ISRS. b) The intracavity spectrum exhibits a clear shift tothe red when gas is present, a signature of impulsive stimulated Raman scattering. The red shift
is dominant even in the presence of strong ionization, which would shift the spectrum to the blue.
The red shifted spectra corresponds to the data in a) at 1.7 atm of applied pressure. An interesting
application of the spectral red-shift could be to counteract the spectral blue-shift of ionization toaid in maintaining high buildup during HHG in atomic systems.

151– spectral blue shift – when a femtosecond pulse propagates through field-ionizing media [114]. This
demonstrates that ISRS has a more dominant effect on the pulse when interacting with N2 O at these
intensities and densities. The red shift is a manifestation of energy transfer from the field to themolecular rotations.

6.3.2

Multi-pulse caseWe proceed to inject two pulse trains into the fsEC. The first pulse (pump) excites a rotational
wave-packet. The second pulse (probe) is of greater intensity and generates high-order harmonics.
We observe two important features as a function of delay between the two pulses. First, the HHG
yield of the probe pulse is modulated at the revivals of N2 O (more on this in Sect. 6.4). Second,
thanks to the cavity-based measurement and the multi-pass effect, we observe clear effects of thepump pulse on the probe pulse that occur at the rotational revivals of N2 O. We observe these effects
by monitoring the intracavity spectrum and power relying on the fact that the intracavity spectrum
is dominated by the much more intense probe pulse and that the delay-dependent effects can only
affect the probe. The effects on the probe pulse persist even if the probe intensity is too weak forappreciable HHG ( < 0.2 × 1014 W cm−2 ) further showing that the observed modulations are notpurely an ionization related phenomena.
The effects on intracavity power are presented in Fig. 6.4. At ∼20 ps and ∼40 ps of delay
between the pump and probe, the half- and full-revival of N2 O alignment, respectively, modulate
the intracavity power of the probe, as shown in Fig. 6.4a,b. The intensity uncertainty is ∼ 1%. Thispower modulation is related to spectral shifts of the intracavity spectrum, shown in Fig. 6.5a,b as afunction of pump-probe delay for the half- and full-revival respectively. The dominant contribution
to the modulations in the power arises from the reduced overlap between the intracavity comb andthe incident probe. The center of the intracavity spectrum shifts as the delay is scanned, along withthe corresponding intracavity power, as shown in Fig. 6.5a,b. The conditions for the half-revival
and full-revival data were the same as Fig. 6.4a,b at 2 atm of nozzle backing pressure. The increase
in intracavity power occurs with a slight blue-shift in the spectrum and the decrease with a slight

152

Figure 6.4: a,b) The effects of the half-revival (a) and full-revival (b) on the probe pulse as afunction of pressure on the intracavity power. The signal is normalized to the baseline (∼ 16 ps and∼ 36 ps, respectively) and offset vertically for clarity. The pressure reflects what is applied to the120 µm diameter quartz nozzle. c,d) The size of the half (c) and full (d) revival signals, determinedwith |max(signal)-min(signal)| of the revival structure. The dependence of the signal size on sample
density is fit to a linear function.

153
red-shift. When the laser spectrum shifts to the blue, the spectrum becomes closer to the empty
cavity case. The improved overlap leads to an increase of the intracavity power. Conversely, a
red-shift reduces the spectral overlap further and the intracavity power decreases. It is important
to note that in the single pulse case (e.g., Fig. 6.3), only red shift of the spectrum is observed
because only one laser pulse interacts with the target. However, with the combination of pump andprobe pulses, we measure the effects on the probe field. The molecular alignment/coherent rotation
can transfer energy back and forth between the molecule and the probe field, depending on thepump-probe delay time [178].
To provide a systematic investigation of the intracavity response, we have measured its dependence on a number of important parameters including the target gas density and pump/probe
intensities. As the molecular density is increased, we observe a linear increase in the size of therevival effect on the intracavity power (measured by |max(signal)-min(signal)| of the revival structure), as shown in Fig. 6.4c,d. This dependence is consistent with ISRS [178]. We also see a linearly
increasing response as either the pump or probe is increased independently. This effect is also consistent with ISRS and the degree of alignment hhcos2 θii increasing linearly in the small intensity
limit. These observations have been verified by numerically simulating the alignment process.
To complete our investigation of the field-molecule interaction, we use the observed spectral
shifts to estimate the effects of ISRS on the phase of the probe pulses. We determine the phase shifts
by analyzing the measured central wavelength of the intracavity spectrum as a function of pumpprobe delay. The delay dependent spectral shifts are smaller than the red-shift observed when thegas is introduced (see Fig. 6.3b.). The spectral shifts and their corresponding intracavity power are
shown in Fig. 6.5a,b for the half- and full-revival respectively. Using the relation ω(t) = ω0 + dφ/dt,
we extract the phase dependence as a function of delay [188]. The phase shift results are shown in
Fig. 6.5c,d for half- and full-revival respectively. Large phase shifts are introduced to the driving
laser. This will be important to understand and control for experiments with HHG because small
phase shifts on the pump will be transferred to the harmonic light and scale with harmonic order(as discussed in Chapter 5).

154

Figure 6.5: a,b) The intracavity central wavelength and the intracavity power are shown as afunction of pump-probe delay for the half-revival (a) and full-revival (b) of N2 O. c,d) The phase as
a function of delay for the half-revival (c) and full-revival (d) of N2 O respectively using the relation
ω(t) = ω0 + dφ/dt. The conditions for the data were the same as Fig. 6.4a,b with 2 atm of pressure
for the half-revival and full-revival data.

155

6.4

High-order harmonic generation with molecules.
With this clear determination of the molecular alignment effect on the probe pulse that isused to drive HHG, we now turn our attention to the measurement of the HHG yield as a function ofmolecular alignment. A pump pulse of 7.5 × 1012 W cm−2 is used to prepare a rotational wavepacket
before a probe pulse of 0.6 × 1014 W cm−2 is used to perform HHG. The total yield of harmonics
15 - 19 is detected simultaneously with an electron multiplier as the delay between the pump andprobe pulses is scanned. These harmonics are isolated with an aluminum filter and the bandwidthof our B4 C optics. The data is documented in Fig. 6.6. The XUV yield is normalized to the baseline
at t ∼ 5 ps. The delay is scanned continuously at a rate of at 1 ps/s and no averaging of the datais performed besides low-pass filtering at 300 Hz. Further averaging could be performed to improvesignal to noise and is not presented here. In Fig. 6.6b, we report the observation of a nearly 50%
modulation in the XUV yield. The XUV yield is shown with the hhcos2 θii expectation value forreference. The similarity between the data and hhcos2 θii is expected when the harmonics are far
from an interference region [189, 190]. This XUV yield does not mimic the effect on the driving
laser, meaning that the modulation in the XUV yield are not driven by modulation in the driving
probe power but is an intrinsic signal from the molecular dynamics. With our fsEC apparatus weare easily able to achieve probe intensities of 1 × 1014 W cm−2 and we have also performed HHG
experiments with molecules of higher ionization threshold such as CO2 or harder to ionize molecules
like O2 and observed similar modulations in the HHG yield arising from rotational revivals. We
also observed a weaker harmonic modulation from O2 , but O2 proved to be much harder to ionize
than the Ip suggests resulting in lower harmonic yields and a much noisier signal.

6.5

Future potential of cavity-based field-free molecular alignment
Observing HHG from aligned molecules at high repetition rates opens exciting avenues forprobing molecular rotational dynamics [184, 191] with more rapid data acquisition and increased
measurement precision. It may also be possible to enhance the degree of alignment with repetitive

156

Figure 6.6: a) High harmonic signal generated from aligned N2 O molecules. Harmonics 15-19 are
measured as a function of pump-probe delay. The half and full revival of N2 O is shown at ∼19 and∼ 39 ps of delay, respectively. Revival signals on the harmonic yield are observable out to ∼ 80
ps. b) The amplitude modulation on the XUV yield is shown in detail. The hhcos2 θii expectation
value is shown for reference. It is assumed that the harmonic yield will approximately follow thehhcos2 θii value if the harmonic order is far from a resonant feature. Strong harmonic modulations
were also present down to the 3rd harmonic. The harmonic modulations are not identical to the IR
modulations shown in Fig. 6.4.

157
impulsive kicks [192]. With the demonstration of heterodyne interferometry with XUV combs [43]we expect to measure both the amplitude and phase of XUV light from aligned molecules [164].
There are also a few open questions about the system. First, we observed the higher fractional
revivals in N2 O than what is predicted by hhcos2 θii. These are shown in the highlighted sectionof Fig. 6.7a. The higher order fractional revivals could be used to probe additional light-moleculeinteraction dynamics [179]. Second, the coherence decays much faster than we expected. The decayof the driving laser revival signal is shown in Fig. 6.7b. It is possible that the presence of theplasma is affecting the rotational coherence at these fast timescales. The collisional dynamics ofthe molecules and the plasma are potentially interesting to study. Third, there is a possibility
to use the modulations from the molecule to tune the nonlinearities in the cavity. This can beused to compensate for plasma blue-shifting or to tune phase shifts for possible attosecond pulse
generation experiments. This effect is shown in Fig. 6.7c. We tested the hypothesis by measuring
the intracavity spectrum using pure xenon, pure N2 O, and a mix of xenon and N2 O such that thexenon density was the same as in the pure xenon case. The inset to the figure shows the differential
spectra relative to the empty cavity case.
Another possibility is to use the cavity-enhanced FFMA signals to non-resonantly detectmolecules. An example of where this could be useful is detecting OH in the presence of H2 O. OH
and H2 O have very close and congested absorption spectra in the mid-IR, however they have very
different rotational constants. Therefore, the revival features of the two molecules would be very
different. It should then be possible to pick a pump-probe detuning to focus on OH rather than H2 O
and use highly-sensitive detection techniques (such as lock-in detection on intensity modulations orpolarization modulations) to look for the revival signal. Using a much higher finesse cavity thanthe one used for this work can also increase the sensitivity. An added benefit of a method like thisis that it can also be very flexible with the specific molecule to be detected since the wavelengthof the laser is essentially irrelevant and only the pump-probe delay is selected in order to isolate acertain molecule. It may also be possible to use multiple probe pulses at different times to multiplex
the detection. It is unclear whether absolute concentrations can be determined, but if polarization

158
modulation is used for detection, it should be possible.

159

Figure 6.7: a) The data is the IR intensity modulations from N2 O. The highlighted regions show
revival struction at the quarter of the full rotational period. These do not originate from the hhcos2 θii
expectation value. b) The decay of the revivals as a function of time. The lifetime is much shorter
than is expected. This could be a method to determine if there is collisional decoherence in thejet. c) Use of molecules and a traditional HHG target, xenon. The xenon causes an ionization
blue-shift. The N2 O causes a ISRS-induced red-shift. Together, they can cancel each other. The
insert is the difference spectra of each case relative to no gas. The small reduction in blue-shift canbe seen in the cyan trace which corresponds to the mixed gases. Further optimization could yield
more dramatic results.

Chapter 7

Conclusion and Future Prospects

The riddles of God are more satisfying than the solutions of man.
–G. K. Chesterton

7.1

Conclusion
We have shown that the XUV frequency comb is not only one of the brightest sources of XUVlight originating from HHG, it is also by far the most temporally coherent of all available XUVlight sources. Certainly, it is now in an excellent position to leverage both of these properties for
future spectroscopy application. Picking the perfect spectroscopy target will be left up to future
generations of the experiment. We have also shown that there is a good amount of strong-field
physics that can be investigated with the XUV comb apparatus. Certainly there will be more
opportunities in the future. For instance, measuring harmonic phases as a function of molecularalignment is completely feasible and the XUV comb promises a level of sensitivity not reached inany other effort to date. Further, the possibility of turning the XUV comb into an attosecond pulse
machine is intriguing (if not for the amazingly short pulse duration but for the continuous XUVspectrum these pulses require). We have helped show a few tricks that may help in future efforts,
but progress is being made in other groups around the world. It seems only a matter of time before
it becomes a reality. Another interesting avenue is using an optical parametric oscillator (OPO)
to pump the enhancement cavity [93]. The OPO is a tunable source outputing at wavelengths of
3-5 µm. Due to the favorable scaling of HHG with driving laser wavelength, building a soft X-ray

161
frequency comb seems within reach. A modest power scaling of the OPO technology would be
required from the current 1 W of power to about 10 W of power. However, the OPO pumping laser
technology is certainly available. The only remaining question would be if an enhancement cavity
that can support 10 kW of average power in the mid IR is possible [193].
The fact that there are femtosecond pulses circulating inside the fsEC is definitely underutilized. We made an effort to exploit this with the field-free molecular alignment work, but beyond
using them for just making harmonics, little work has been done in the time domain. This is set tochange with recent experiments in transient-absorption spectroscopy and should start a whole newwave of cavity-based ultrafast experiments [194]. There are a whole host of possible pump-probestyle experiments that can now be envisioned with increased power, sensitivity, and statistics. This
will definitely be a very exciting direction if not completely orthogonal to our groups efforts focusing
on high-resolution spectroscopy.

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173

Appendix A

Numerical Code Repository

GithubDuring the course of this work, a decent amount of code was generated to numer-

ically simulate various things relevant to the XUV frequency comb project. The numerical codeis stored on Github at https://github.com/c-benko. A very brief description of the repositories
relevant to this work are below. It is written almost entirely in Python.
CavityThis contains an implementation of ABCD matrices for ring resonators. It also contains
simulations of how dispersion affects the fsEC intracavity spectrum and pulse duration. This code
was used in Chapter 2.
crankyc
This contains a small library of functions to load data from various experimental equipmentused in lab. It also contains simple functions for commonly used mathematical manipulations suchas Allan deviations and FFTs.
HHG
Contains a numerical implementation of Corkum’s three step model [24] and the quantum
version of Lewenstein [86, 118]. It also contains an implementation of and ionization rate calculation
using ADK theory. This code was used in Chapter 3.
HHG_Phasematching
A general HHG phasematching simulation in a single-pass geometry. It is based on theabsorption-limited phase matching work of Constant [124] and can be applied to a wide variety of174experimental conditions. This code was used in Chapter 3.
HHG_phasematching_fsEC
A general HHG phasematching simulation in a single-pass geometry. It is based on theabsorption-limited phase matching work of Constant [124] and can be applied to a wide variety ofexperimental conditions. This code attempts to include the effects of the fsEC.
Molecular_Alignment
A variety of numerical implementations of FFMA using various methods within the Python
programming language. The code highlights methods to integrate C/C++ with Python for fastercomputation. This code was used in Chapter 6.
NLSE
A simple implementation of the normalized nonlinear Schrödinger equation to simulate nonlinear broadening of ultrashort pulses in optical fibers or materials. This code was used in Chapter 2.
XUV
A set of scripts to calculate mirror reflectivity in the XUV. This was used in Chapter 4

175

Appendix B

Publication list

Since I spent almost two years of my graduate career on the strontium lattice clock experiments, I have added my full list of publications which includes a handful of papers relevant to thestrontium experiments in context of many-body physics, clock operation, frequency comb development, and laser stabilization.
Publications[B.1] J. R. Jameson, D. Ngo, C. Benko, J. P. McVittie, Y. Nishi, and B. A.Young, “Dielectricrelaxation study of hydrogen exposure as a source of two-level systems in Al2 O3 ,” Journal of NonCrystalline Solids 357, 2148–2151 (2011).
[B.2] C. P. Weber, C. Benko, and S. C. Hiew, “Measurement of spin diffusion in semi-insulating
GaAs,” Journal of Applied Physics 109, 106101 (2011).
[B.3] A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong,
J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum
frequency comb,” Physical Review A 84, 011806 (2011).
[B.4] M. Bishof, M. J. Martin, M. D. Swallows, C. Benko, Y. Lin, G. Quéméner, A. M. Rey, andJ. Ye, “Inelastic collisions and density-dependent excitation suppression in a 87 Sr optical latticeclock,” Physical Review A 84, 052716 (2011).
[B.5] C. Benko, A. Ruehl, M. Martin, K. S. E. Eikema, M. E. Fermann, I. Hartl, and J. Ye, “Fullphase stabilization of a Yb:fiber femtosecond frequency comb via high-bandwidth transducers,”
Optics Letters 37, 2196–2198 (2012).

176[B.6] A. Ruehl, C. Benko, M. J. Martin, K. S. E. Eikema, M. E. Fermann, I. Hartl, and J. Ye, “Phasestabilization of a Yb:fiber frequency comb via high-bandwidth transducers,” in “CLEO: Science andInnovations,” (Optical Society of America, 2012), pp. CTh1J–1.
[B.7] M. D. Swallows, M. J. Martin, M. Bishof, C. Benko, Y. Lin, S. Blatt, A. M. Rey, and J. Ye,“Operating a 87 Sr optical lattice clock with high precision and at high density,” IEEE Transactions
on Ultrasonics, Ferroelectrics and Frequency Control. 59, 416–425 (2012).
[B.8] M. J. Martin, M. Bishof, M. D. Swallows, X. Zhang, C. Benko, J. von Stretcher, A. M. Rey,and J. Ye, “A quantum many-body spin system in an optical lattice clock,” Science 341, 632–636(2013).
[B.9] T. K. Allison, A. Cingöz, C. Benko, D. C. Yost, A. Ruehl, M. Fermann, I. Hartl, and J. Ye,“High brightness XUV frequency combs via intracavity high harmonic generation,” EPJ Web ofConferences 41, 11006 (2013).
[B.10] A. M. Rey, A. V. Gorshkov, C. V. Kraus, M. J. Martin, M. Bishof, M. D. Swallows, X. Zhang,
C. Benko, J. Ye, N. D. Lemke, and A. Ludlow, “Probing many-body interactions in an optical latticeclock,” Annals of Physics 340, 311–351 (2014).
[B.11] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr,
F. Riehle, G. D. Cole, and M. Aspelmeyer, “Reduction of residual amplitude modulation to 1× 10−6
for frequency modulation and laser stabilization,” Optics Letters 39, 1980–1983 (2014).
[B.12] C. Benko, T. K. Allison, A. Cingöz, L. Hua, F. Labaye, D. C. Yost, and J. Ye, “Extreme
ultraviolet radiation with coherence time greater than 1 s,” Nature Photonics 8, 530–536 (2014).
[B.13] C. Benko, L. Hua, T. K. Allison, F. Labaye, and J. Ye, “Cavity-enhanced field-free molecularalignment at a high repetition rate,” Physical Review Letters 114, 153001 (2015).
[B.14] C. Benko, N. Dörre, and J. Ye, “Power scaling extreme ultraviolet frequency combs to themW level,” in preparation. (2016).