Bright Coherent Ultrafast Tabletop Light Sources Development
and the Application on EUV to Soft X-Ray Absorption Spectroscopy
byChengyuan Ding
B.S., Nankai University, 2004
M.S., Peking University, 2008
M.S., University of Colorado at Boulder, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirement for the degree ofDoctor of Philosophy
Department of Physics
2014This thesis entitled:
Bright Coherent Ultrafast Tabletop Light Sources Development
and the Application on EUV to Soft X-Ray Absorption Spectroscopy
by Chengyuan Ding
has been approved for the Department of Physics(Prof. Margaret Murnane)(Prof. Henry Kapteyn)
Date
The final copy of this thesis has been examined by the signatories, and weFind that both the content and the form meet acceptable presentation standards
Of scholarly work in the above mentioned discipline.

iii
Ding, Chengyuan (Ph.D., Physics)
Bright Coherent Ultrafast Tabletop Light Sources Development and the Application on
EUV to Soft X-Ray Absorption Spectroscopy.
Thesis directed by Prof. Margaret Murnane and Prof. Henry Kepteyn

The generation of the advanced bright coherent ultrafast light source fromultraviolet (UV) to soft X-ray have been pursued for decades. It requires the development oflight conversion technique, such as high-order harmonics generation (HHG), as well as thedriving laser. In this thesis, we first demonstrate the highest pulse energy (10 mJ), singlestage, ultrafast (45 fs) Ti:sapphire amplifier to date, with a repetition rate of 1 kHz. We
then use this laser to pump an optical parametric amplifier system and generate 1.3 µm, 30
fs pulses with sufficient energy (2 mJ) for optimally-efficient, phase matched HHG
conversion. This allows us to demonstrate the highest flux, soft X-ray HHG source to datewith > 106 photons/pulse/1% bandwidth at 1 kHz (corresponding to > 109 photons/s/1%
bandwidth) in a broadband, continuum, spectrum extending to 200 eV, through the fully
phase matched hollow waveguide geometry HHG. This photon flux represents anapproximately 3 orders-of-magnitude increase compared with past work. Meanwhile, due tothe experimental similarity, the high energy ultrashort (10 fs) UV source is implemented inparallel to the soft X-ray source by the four wave mixing (FWM) process. The pulse energy(32 uJ) of UV source is increased by more than 3 times compared with past work, with thepulse duration compressible to less than 13 fs. Finally, utilizing the unique brightsupercontinuum HHG soft X-ray source, we have demonstrated soft X-ray absorption
spectroscopy of multiple elements and transitions in molecules simultaneously, with theability to resolve near edge fine structure with high fidelity. The Xeon photon-ionization

iv
process is also resolved in the EUV transient absorption spectroscopy experiment by tuning
the soft X-ray source to the EUV region, which shows the stability, tunability, andapplicability of our tabletop extreme nonlinear light source for the time-resolved
experiments.

Dedication

To my wife, Jingjing Li, and parents.

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Acknowledgements

First of all, I would like to thank my advisors Prof. Margaret Murnane and Prof.
Henry Kapteyn for their continuous support and help in the last five years and five months.
Without them, this thesis work would not have been possible. They not only provided me
the opportunity to work in the world-class researches, especially the ultrafast coherent lightsource development, but also taught me how to succeed in the research career, which would
benefit me for life.
Next, I am very grateful to everyone in the KM molecule group who has taught me
so much. Dr. Xinbin Zhou introduced me to the KM group and the AMO project. Dr.
Robynne Lock taught me the alignment of the molecules and most of the optics lab workwhen I first started. Most importantly, I would like to thank Dr. Wei Xiong, now a new
professor in University of California, San Diego. We worked closely for a couple of years on
the soft X-ray generation and transient absorption projects. He taught me what a real
Ph.D. means. I also greatly appreciate Daniel Hickstein. He is the man I collaborated withfrom the first day in KM group till now, and he helped me in every aspect of the researchwork. I am also grateful to Tingting Fan who I have collaborated with on the transientabsorption experiment. Additionally, Dr. Ellen Keister and Dr. William Peters brought
many great ideas to our projects. Currently, Jennifer Ellis is working on the VMI project. A

vii
big thanks goes to all those in the KM molecule group who have made working here apleasant experience.
Also very important to thank is Dr. Xiaoshi Zhang. He taught me how to build themost powerful laser from scratch, and he also taught me the femto-second machining
technique, which is critical in my research on the soft X-ray generation. I am also very
grateful to Prof. Lino Misoguti. He helped me to demonstrate the brightest ultrafast UV,
which is the first achievement during my Ph.D. career. Additionally, I greatly appreciate
Dr. Tenio Popmintchev and Dr. Ming-Chang Chen for their introduction to the longwavelength driven HHG. I also owe my thanks to Matt Seaberg and Dan Adams for theircontribution on the pulse shaper, which makes the shortening of the output pulse from theregenerative amplifier possible. Moreover, I would like to thank Paul Arpin who I havecollaborated with to keep the Red Dragon working, and he started the soft X-ray transientabsorption experiment, which is the final project in this thesis.
I also would like to take this opportunity to thank other members of my thesis
committee: Prof. David Jonas, Dr. Agnieszka Jaroń-Becker, and Prof. Ralph Jimenez. It is
my great honor to have them serve on my thesis committee. I also appreciate Prof. Steven
Cundiff and Prof. Andreas Becker for serving on my Comps III exam committee.
I also would like to thank Sterling Backus and Bob Bower. The experience in
KMLabs and EUV ERC will equip me for life.
Finally, I would like to thank my friends and family for their support. My friends
have kept me sane. My parents have always believed in me. Most of all, a special thankgoes to my wife Jingjing Li, who accompanied me for five years at CU-Boulder. Without her
I would never be able to overcome the difficulties and finally accomplish my Ph.D. study.

viiiContents

Chapter

1 Introduction and thesis outline

1

1.1

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

1

1.2

Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 First 10 mJ single-stage kHz regenerative ultrafast Ti:Sapphire Amplifier

6

2.1

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

6

2.2

Regenerative amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Thermal lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Grating based optical stretcher and compressor . . . . . . . . . . . . . . . . . . 18

2.5

Performance of the 10 mJ kHz single-stage Ti:sapphire amplifier . . . . . . . . 26

2.6

Further investigation of the ultrafast amplifier with pulse shaper . . . . . . . . 29

2.7

Conclusion and future investigation . . . . . . . . . . . . . . . . . . . . . . . .

3 High flux ultrafast UV-VUV generation from 4-wave mixing process

31
33

3.1

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

33

3.2

Electromagnetic modes in the dielectric hollow waveguide . . . . . . . . . . . .

35

3.3

Mode coupling, phase matching and mode matching for the hollow waveguide . 40

3.4

Ultrafast UV generation setup and results . . . . . . . . . . . . . . . . . . . . . 47

3.5

Summary and future experiments . . . . . . . . . . . . . . . . . . . . . . . . .

55

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4 High flux soft X-ray generation by HHG

58

4.1

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

58

4.2

Single atom picture of HHG. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.3

Phase matching HHG in hollow waveguide . . . . . . . . . . . . . . . . . . . .

70

4.4

Experiment setup and results . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.5

Summary and the future investigation. . . . . . . . . . . . . . . . . . . . . . 93

5 NEXAFS from Soft X-Ray HHG and
Xe photo-ionization transient absorption spectroscopy

95

5.1

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

95

5.2

EUV- Soft X-ray (transient) absorption spectroscopy scheme . . . . . . . . . . 100

5.3

Soft X-ray NEXAFS from gas phase and solid phase samples . . . . . . . . . . 106

5.4

EUV transient absorption spectroscopy: Xe photon-ionization . . . . . . . . . . 113

5.5

Conclusion and ongoing experiment . . . . . . . . . . . . . . . . . . . . . . . . 116

Bibliography

117

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FiguresFigure2.1(a) Setup of a multipass amplifier. (b) Setup of a regenerative amplifier.
Figure captured from Ref. [43]. . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Schematic diagram of the 0.2-TW 1kHz single stage regenerative
Ti:Sapphire amplifier system.

2.3. . . . . . . . . . . . . . . . . . . . . . . . .

2.5

10

Effective focal length based on the uniform continuous pump model. The
red vertical line corresponding to real pump size used in our amplifier.

2.4

8

. .

18

diagram of the grating pair with internal lens stretcher. . . . . . . . . . . .

20(a) Schematic diagram of the grating based compressor. (b) Schematic(a) Focal spot of the single-stage laser amplifier output (KMLabs Wyvern
HETM), using a 1 m focal length lens; (b) M2 characterization of the 0.8 µm
amplifier output. Data taken at 10 mJ, 1 kHz. . . . . . . . . . . . . . . . . .

27

2.6

The stability test of the 10 mJ kHz Ti:sapphire amplifier for 8 hours.

28

2.7

FROG measurement of the amplified pulse, without the spectral filter (a),. . .

with the spectral filter before the regenerative cavity (b), with the spectral
filter inside the regenerative cavity (c).
3.130(a) Side view of the dielectric hollow waveguide. (b) End view of thedielectric hollow waveguide.

3.2. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

36(a) The far filed distribution of EH11 mode. (b) The far filed distribution of
EH12 mode. (c) The far filed distribution of LP11 mode. (d) The far filed
distribution of LP12 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

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3.3

Calculations of power coupling coefficients for the EH1m modes as a functionof the ratio of the beam waist to the guide radius, w/a. This figure is captured

3.4

from Ref. [87]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Diagram of the ultrafast UV generation in hollow waveguide.

48. . . . . . . .

3.5 (a) The throughput of 267 nm light as a function of the gas pressure inside
the hollow waveguide. (b) Phase matching pressure curve with respect tothe diameter of the hollow waveguide and the output mode. . . . . . . . . . .
3.649

The decay of light intensity inside the hollow waveguide as a function of
propagation distance. The lights of wavelength related to the FWM process
are chosen in plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

51

The spectrum comparison between the ultrafast UV light generated from theFWM process through hollow core fiber and the ultrafast UV light generated
from the 100 um thick BBO crystal. The three times broaden spectrum from
FWM implies that the pulse duration could be three times shorter than thatfrom the crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

53

SD Frog traces of the chirped 267 nm pulse (left) and compressed 267 nm
pulse (right). The SD autocorrelation is 13 fs, which implies the retrieved
pulse duration is close to 10 fs.

4.1. . . . . . . . . . . . . . . . . . . . . . . . .

54

The general feature of the HHG spectrum. The spectrum is generated fromnoble gas Ar with intensity of at 3×1013 W/cm2 at gas pressure of 15 Torr.
Figure is captured from Ref. [91]. . . . . . . . . . . . . . . . . . . . . . . . .

60

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4.2

The cartoon of the classical schematic of HHG captured from Ref. [100]. The
electric field of an intense laser extracts an electron from an atom throughtunnel ionization. The laser field then accelerates the electron, with a small
fraction of the electron returning back to the ground state of the same atom,
liberating its excess energy as a high-energy photon. . . . . . . . . . . . . .

4.3

63

The electron trajectories of different initial phase with respect to the peak of
the laser field at the tunneling time t0. The driving laser is assumed to be at
800 nm. Y axis of the plot is in the unit of the ponderomotive radius,

4.4

.

65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Physical pictures of HHG phase matching in different geometries. (a) Phasematched signal growth ensures that harmonic emission from many atoms
adds together coherently over an extended medium. (b) The tight focusing
geometry is associated curved laser and X-ray wavefronts, which limits thephase matching distance within the confocal length and has to deal with the
problem of Guoy phase shift. (c) The waveguide or cell geometry minimizes
the curvature of the laser and HHG wavefront. The nearly plane wave
propagation allows long distance phase matching HHG and has no Guoyphase or Carrier-Envelope phase (CEP) problem. The Figure is captured
from Ref. [100].

4.5(a) Critical ionization levels for the noble gas Ar, Ne and He as a function of
the driving laser wavelength. (b) Predicted laser intensity required to reachthe critical ionization level at the peak of the laser pulse on axis (solid line),
and when modal averaging is taken into account (dashed line). These figures
are captured from Ref. [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

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4.6

Photoelectron momentum spectra of the Xe ionization with respect to the
Keldysh parameter γ.

4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80(a) The ionization fraction of gas neon in a 30 fs Gaussian shaped pulse withpeak intensity of. The wavelength of the pulse is changed from

800 nm to 1300 nm to 2000 nm. Although the ionization step changes due tothe different cycle duration, the ionization rates as a function of time almost
keep the same for different driving wavelength. (b) The ionization fraction ofgas neon in a 30 fs Gaussian shaped pulse with wavelength at 1300 nm. The
peak intensity changes fromto. Ionization

fraction changes more than an order of magnitude with the intensity change.
If the intensity drops to, the ionization faction will be reduced

to 0.05%, another order of magnitude lower, which is not easily seen in thisplot. Thus, 50% of intensity change would introduce orders of magnitudechange of the ionization fraction. . . . . . . . . . . . . . . . . . . . . . . . .
4.8

81

Theoretical HHG phase matching cutoff as a function of driving laser
wavelength for three-cycle (dashed lines) and eight-cycle (solid lines) pulses.
This global phase matching picture has been validated experimentally(solid circles) using several laser wavelengths and nonlinear media. The figure
is adapted from [100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

82

Absorption-limited intensity of phase-matched HHG in a bandwidthcorresponding to a single harmonic (A) and a linewidth of 1% bandwidth (B)at the phase-matching cutoffs for laser wavelengths between 0.8 um and
10 um. The curves are normalized to the phase-matched HHG emission at
0.8 um. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

xiv

4.10 Schematic diagram of the high energy high efficiency OPA.

. . . . . . . . .

86

4.11 (a) Signal pulse evolution for a BBO type II OPA, with pump wavelength of
800 nm and signal wavelength of 1500 nm, for different lengths L of thenonlinear crystal. (b) Phase matching bandwidth for a BBO OPA at the pumpwavelength of 800 nm for type I phase matching (solid line) and type II phasematching (dashed line). Figures are captured from Ref. [30]. . . . . . . . . .

88

4.12 Pulse characterization of 1.3 um from OPA by the SHG FROG. The pulse
duration of 1.3 um light is 29.6 fs, which is shorter than the pulse duration of
the pump pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.13 Flux characterization of the optimized fully phase matched HHG emission
from a 1 cm long, 150 μm diameter with driving wavelength of 1.3 um. Noble
gas Ar and Ne are used.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.14 Flux comparison of the optimized fully phase matched HHG emission from a
1 cm long, 150 μm diameter, waveguide driven by 1.3 μm and 0.8 µm lightin various noble gases (Ar, Ne, and He). The vertical arrows indicate themaximum predicted phase matched HHG energy cutoff for each gas and laser
wavelength as predicted in Fig.4.8. The HHG flux obtained using a 1.3 μm
driving laser is comparable to that achieved using 0.8 µm. . . . . . . . . . .

92

4.15 The photo of the taper-end hollow waveguide. The taper is 1.5 mm deep, andthe outer diameter of hollow waveguide is 1.2 mm, so the angle is 43 deg. . .

94

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5.1 (a) Transitions resulting from the absorption of X-rays. (b) Variation of thelinear X-ray absorption coefficient of an atom as a function of the photon
energy. (c) Example for the X-ray absorption near edge structure (XANES)
and extended X-ray absorption fine structure (EXAFS) from K edge X-ray
absorption spectra of iron in K4FeII(CN)6 and K3FeIII(CN)6 bulk samples.
Figures are captured from Ref. [114].
5.2. . . . . . . . . . . . . . . . . . . .

96

Physics picture for X-ray absorption near edge structure (XANES) andextended X-ray absorption fine structure (EXAFS), which are originatedfrom multiple scattering and single scattering, respectively, of the out-going
photon-electron excited by the X-ray. Figure is captured from Ref. [114].

5.3. . . . . . . . . . . . . . . . . . . .

100

The sample holder designs. (a) The gas cell design. (b) The solid sample
holder design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

97

The whole soft X-ray transient absorption spectroscopy scheme enabled by
the 10 mJ Ti:saphhire pumped OPA.

5.4

. .

103

The soft X-ray transmission from gas SF6 and CS2 through the 2 mm
thick gas cell with gas pressure at 30 Torr. Data obtained from
www.CXRO.com.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

5.6 (a) Normalized High-order harmonics spectrum generated from Ar, Ne andHe gases wit1h driving laser wavelength at 1.3 μm. (b) Normalized Xetransmission spectrum observed using harmonics from Ar, and SF6transmission observed using harmonics from Ne and He. . . . . . . . . . .

106

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5.7 (a) Normalized raw data for 1.3 μm-driven HHG from Ar (blue), Ne (green),and He (red) after transmission through a 2 mm long sample cell of Xe(HHG from Ar), and SF6 (HHG from Ne and He). Note that the apparent
gap between 70 eV and 100 eV is because of Al filters used to block thefundamental light—full tuning across this range can be achieved byswitching to other filters as shown in Fig. 2. (b) NEXAFS spectrum of SF6
Sulfur L-edge 2p orbital. The high quality NEXAFS spectra were obtained in
a 300 s exposure time. However, changes in the absorption signal can bemonitored in real time, with exposures as short as 0.1 s. The modulations
between the resonance absorption peaks 6a1g, 2t2g, and 4eg arise fromindividual harmonic peaks that are separated by 1.9 eV. These modulations
can be reduced by increasing the driving pulse energy into the hollow
waveguide, implementing beam pointing stabilization, and reducing the
integration time. Figures are captured from Ref. [125].

. . . . . . . . . . .

108

5.8 (a) Schematic MO diagram for SF6. Figure is captured from [126].
(b) Schematic representation of the double-well potential (not to scale) for SF6.
The barrier divides the potential curve into two regions, the inner well andthe outer well. Valence orbitals are located mainly in the inner well. Rydberg
orbitals are located mainly in the outer well. (c) Synchrotron data for SF6
NEXAFS. Figures (b) and (c) are captured from [122]. . . . . . . . . . . . .

109

5.9 (a) NEXAFS of CS2 and SF6 obtained from our HHG Soft X-ray.
(b) Synchrotron data for CS2, H2S, SO2, (SO4)-2, and SF6 NEXAFS, adaptedfrom [126].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

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5.10 (a) NEXAS from P3HT. The red vertical line indicates where the peak X
is in (b). (b) NEXAS from thiophene (C4H4S) on Pt (111), adaptedfrom [127].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

5.11 (a) Neutral Xe 4d absorption. (b) Xe+ and Xe2+ 4d absorption. (c) Xe+ andXe2+ transient absorption signal.

. . . . . . . . . . . . . . . . . . . . . . .

115

1

Chapter 1

Introduction and thesis outline

1.1 Introduction
The generation of the bright advanced coherent ultrafast light sources in the
spectrum region from ultraviolet (UV) to X-ray has been pursued for decades. At the longwavelength side of the region, the excited electronic states of gas phase molecules andmaterials lie in the spectrum of UV to vacuum ultraviolet (VUV). Thus, the availability ofan ultrafast UV to VUV light would allow the excited state to be accessed. At the shortwavelength side, the ultrashort wavelength associated ultra-fine (nanometer 10-9 m) spatial
resolution and ultra-short (attosecond 10-18 s) temporal resolution make the coherent X-ray
ideal tools of the nanoworld, such as the imaging of nanometer scale small features and theattosecond time scale coherent control of the fast electrons [1]. Moreover, the chemical andelemental specificity of the coherent X-ray absorption enables the coherent X-ray source aunique probe for the complex chemical and biological samples. However, the developmentsof both the coherent ultrafast UV-VUV source and coherent X-ray source are challenging.
For the UV-VUV generation, the up-conversion from nonlinear crystals only allows thegeneration of wavelength longer than 160 nm. Meanwhile, the generated pulse duration islimited by the phase matching bandwidth of the nonlinear crystal, which is typically too
long to resolve the fast electronic transitions. The situation of coherent X-ray generation is2even worse due to complex reasons, such as the scarcity of the appropriate pumps andoptics for building an X-ray laser, and the low conversion efficiency of the extreme
nonlinear process for the up-conversion from the driving light. Therefore, special techniques
are required to develop the bright coherent X-rays source and the ultrafast coherent UVVUV source.
Specifically, to obtain the bright coherent tabletop soft X-ray source, high-orderharmonic generation (HHG) is utilized. It upconverts light from a tabletop femtosecond
laser, while retaining excellent spatial coherence when implemented in a phase matchedgeometry, as well as temporal coherence that easily supports zeptosecond to attosecond
synchronization with the driving laser [2-6]. Thus HHG provides an advanced light sourcecomplementary to facility-scale coherent X-ray sources [7-9]. By producing x-ray pulses
with durations of <100 attoseconds [10-12] and wavelengths < 1 nm [6], HHG is opening up
new research opportunities in molecular spectroscopy [13-16], materials science [17-23], as
well as nanoscale coherent diffractive imaging [24, 25]. So far, most implementations of
HHG have used Ti:sapphire driving lasers at a wavelength around ~0.8 µm. In this case,
phase matching considerations limit bright HHG to photon energies <≈ 100 eV, which lie inthe extreme ultraviolet (EUV) region of the spectrum. This limits applications of HHG tovery thin samples and a relatively small number of elements which have absorption edges
in the EUV. Fortunately, a recent breakthrough extended phase matching of HHG to muchhigher keV photon energies (i.e. shorter HHG wavelengths), by using longer wavelength
mid-infrared driving lasers at wavelengths from 1.3 to 3.9 μm [6, 26-29]. However, thesepreliminary studies were done using lasers operating at tens of Hz repetition rates, with a
total photon flux of 106 photons per second in 1% bandwidth [28], which is insufficient forbroad applications in spectroscopy or imaging. To fully harness the unique properties of soft
X-ray HHG, kHz or higher repetition-rates are required in a phase matched geometry. This3goal requires the development of stable, compact, high pulse-energy, ultrafast laser
amplifiers that can generate multi-millijoule, few cycle, mid-infrared laser pulses required
for efficient HHG.
In this thesis, we first demonstrate the highest pulse energy (10 mJ), single-stage,ultrafast (45 fs) Ti:sapphire amplifier to date, with a repetition rate of 1 kHz. This laser
then is used to pump an optical parametric amplifier system [30] and generate 1.3 µm, 30 fs
pulses with sufficient energy (2 mJ) for optimally-efficient, phase matched HHG conversion.
This allows us to demonstrate the highest flux, soft X-ray HHG source to date with > 106
photons/pulse/1% bandwidth at 1 kHz (corresponding to > 109 photons/s/1% bandwidth) in a
broadband, continuum spectrum extending to 200 eV. This photon flux represents anapproximately 3 orders-of-magnitude increase compared with past work, which used lower
repetition rates, or non-phase matched implementations [27, 31-32].
Due to the experimental similarity to the high flux soft X-ray HHG source, thegeneration of the high pulse energy ultra-short (~10 fs) UV source is investigated inparallel from the four wave mixing (FWM) nonlinear process. The similar hollow waveguide
geometry as being used in the development of the two coherent light sources makes themsharing the similar experiment setup and knowledge such as the spatial modes and nearplane-wave propagation inside the waveguide. But, the fundamental difference between
perturbative regime (UV generation) and nonperturbative regime (X-ray generation) makes
the ultrafast UV generation utilizing the completely different physics to achieve the highest
pulse energy. The UV optical parametric chirped pulse amplification (OPCPA) scheme isused to increase the output pulse energy and avoid unfavorable ionization, which is criticalin the soft X-ray HHG source. We demonstrate a more than 3 times increase of pulse energy(32 uJ) ultrafast UV source compared with past work, with the pulse duration compressible
to less than 13 fs.

4Finally, utilizing the unique bright supercontinuum HHG soft X-ray source, we
demonstrate soft X-ray absorption spectroscopy of multiple elements and transitions inmolecules simultaneously, with 0.25 eV spectral resolution, and with the ability to resolve
near edge fine structure with high fidelity. The Xeon photo-ionization process is also
resolved in the EUV transient absorption spectroscopy experiment by tuning the soft X-ray
source to the EUV region, which demonstrates the stability, tunability, and applicability ofour tabletop extreme nonlinear light source for the time-resolved experiments.

1.2 Outline of thesis
This thesis presents the development of the three ultrafast light sources, which
includes the 10 mJ, 1 kHz single-stage Ti:sapphire ultrafast amplifier (800 nm), 32 uJ, < 13
fs ultrafast UV pulse (267 nm), and the highest flux (> 109 photons/s/1% bandwidth) up to
200 eV soft X-ray HHG source (6 nm), as well as the application of the developed soft X-ray
source, which is the observation of the near edge X-ray absorption fine structure (NEXAFS)[33] and EUV transient absorption spectroscopy.
In Chapter 2, we will focus on the development of the Ti:sapphire amplifier,emphasizing on how we have achieved the highest pulse energy. The high pulse energy andthe perfect spatial mode are the two prerequisites for the efficient high quality OPA output,
which is the driving pulse of the high flux soft X-ray source. Therefore, the high energy
amplifier discussed in chapter 2 could be considered as the preparation for the soft X-ray
generation (Chapter 4).
In Chapter 3, we will discuss the ultrafast UV generation. In this chapter, we not
only explain the reason why the chirped FWM process enables us to obtain the high energy
ultrashort UV pulse, but also present the necessary physics of the EM modes inside hollow

5
waveguide, which can be applied to both the UV generation and HHG. Therefore, chapter 3
could be considered as an independent chapter, as well as the preliminary chapter for the
discussion of the HHG in chapter 4. The author do believe, the phase matching(determinant of the macroscopic emission), the mode matching (one of the determinants ofthe efficiency), and the mode coupling (the prerequisite for the other two) are the three
critical points to achieve efficient output from the hollow waveguide geometry.
In Chapter 4, we will explain the long wavelength driving HHG in detail. Bothsingle-atom response and the phase matching problem of HHG will be discussed. And boththe classical model and the quantum model will be considered. Especially, we willemphasize on the new cutoff rule in the phase matching condition, which allows us to use
the 1.3 um light to achieve soft X-ray source, and explain how the phase matching condition
would balance the low single atom yield by the increase of the pressure length product andthe transparency with long driving wavelength HHG. The high flux soft X-ray generation isa direct proof of the long wavelength driving HHG theory.
In Chapter 5, we will present the experiment results of the absorption spectroscopy
utilizing the soft X-ray source presented in Chapter 4. The high quality NEXAFS of gas SF6
and solid P3HT (polymer of thiophene) will be shown. And transient absorption experimentof Xeon photo-ionization will be presented as well. Chapter 5 could be considered as theapplication of the soft X-ray source, but it is also the start of the new tabletop X-ray
spectroscopy technique, which may bring up new physics in future.

6

Chapter 2First 10 mJ single-stage kHz regenerative ultrafast Ti:Sapphire Amplifier

2.1 Introduction
The ultrafast (ultra-short) laser pulse is an ideal tool for both scientific research andindustrial application. On one hand, the ultra-short pulse duration makes the laser pulse
the indispensable probe to the fast atomic or even electronic motion [34], as well as toaccess the strong field physics up to the relativistic nonlinear optics regime [35], since thepeak intensity is inversely proportional to the pulse duration with fixed pulse energy; on
the other hand, the ultrafast pulse enables ultra-fine machining, which can be applied toindustrial use such as semiconductor chip manufacture [36-37] and medical use such as
laser-assisted cataract surgery [38]. Therefore, people have been developing the ultrafast
laser since the invention of the laser. Actually, the first ultrafast pulse was implemented by
De Maria in 1966, only six years after the first laser was demonstrated, by using apassively mode-locked a Nd:glass laser [39], although the pulse train containing thepicosecond pulses was not regular, which remained a problem for 20 years before the use ofsaturable absorbers from semiconductors [40].
The investigation of the femtosecond laser experienced two periods generally.
During 1980s, dye amplifiers were heavily studied in conjunction with the mode-locked dye

7laser oscillator [41]. Then, because of the striking difference of saturation fluence of thegain material, for example the saturation fluence of Ti:sapphire is three order of magnitude
higher than that of dye, as well as the obvious practical difference associated with solidstate laser vs. liquid-state laser, the broadband solid-state laser has dominated femtosecond
lasers since 1990s, especially the Ti:sapphire laser. Here, we want to discriminate
femtosecond lasers by oscillators and amplifiers. The mode-locked Ti:sapphire oscillator
typically generates pulses with energies on the order of nJ, pulse duration around 10 fs, at
the repetition rate of 80 MHz to 100 MHz. This corresponds to peak intensity of order 1013
W/cm2, assuming that the beam is focused to the limit of a square wavelength. However,compared to the static Coulomb field in an atom, this electric field is still weak, so thatbound electrons remain bound in the ground state and the conventional perturbation
nonlinear optics still holds. In order to reach the strong field regime around 1015 W/cm2,
where non-perturbation theory works, an ultrafast laser amplifier is required. In this thesis,
only the amplifier, specifically Ti:sapphire amplifier, is going to be discussed. The Kerr lensmode-locking Ti:sapphire oscillator, which provide the seed for the Ti:sapphire amplifier isextensively discussed in almost all ultrafast laser textbook, so it won’t be included here.
Since the energy of the output pulse from oscillator (seed) is of nJ, and the amplified
pulse energy after the amplifier is expected to be 10 mJ, the overall gain should be around107. Due to the fundamental trade-off in amplifier design, that small signal regime is mostfavorable for achieving the high gain but is associated with low energy extraction, while thelarge-signal (saturated) regime allows efficient energy extraction but leads to low gain, inorder to achieve high overall gain and efficient energy extraction, the multipass
amplification is considered [42]. Generally, there are two types of the ultrafast amplifier,multipass amplifier and regenerative amplifier.

8

Figure 2.1 (a) Setup of a multipass amplifier. (b) Setup of a regenerative amplifier. Figure
captured from Ref. [43].

As shown in Fig 2.1 (a), the multipass amplifier utilizes a fixed geometry to
determine the number of passes the seed to be amplified in the gain medium, so thenumber is fixed; while, the regenerative amplifier combines the electro-optic element,
typically a Pockels cell, with polarization control elements, the thin film polarizer (TFP)
and quarter waveplate, to realize a polarization gating, during which the seed is trapped
inside the regenerative cavity and get amplified (Fig. 2.1 (b)). The Pockels’ “on” and “off”
switch is acting as the optical switch and the time in between is called Pockels window,
which is tunable. The advantage of the multi-pass amplifier is that it typically generates
shorter pulse compared to the regenerative amplifier, because it has less material in the
beam line as shown in the Fig 2.1, and it typically associated with higher gain therefore

9fewer passes through the gain medium is needed to achieve the same overall gain whichfurther reduces the material in the beam line. The advantage of the regenerative amplifieris that it provides a better mode spatial quality because the seed is amplified in a realcavity. Since the purpose of the newly-designed Ti:sapphire amplifier is to pump an opticalparametric amplifier (OPA) in order to obtain the high pulse energy infrared driving pulse,and couple the driving pulse into a hollow waveguide for the soft X-ray high-orderharmonics generation (HHG), a high energy regenerative amplifier is preferred as thepump laser, because both the OPA and the light coupling are very sensitive to the mode
spatial quality while OPA is not sensitive to the pump pulse duration (Section 3.3 andSection 4.4).

2.2 Regenerative amplifier
The schematic diagram of the single-stage 10 mJ kilohertz (kHz) regenerative
ultrafast Ti:sapphire amplifier system is shown in Fig. 2.2. Similar to other ultrafast
amplifiers, this regenerative amplifier utilizes the chirped pulse amplification (CPA)
technique. The seed of 800 nm light pulse is generated from the Kerr-lens passively modelocked Ti:sapphire oscillator, KMLabs Griffin Series. The typical seed power is 0.5 W withthe repetition rate of 80 MHz, which equals pulse energy of 6.25 nJ, and the bandwidth ofseed spectrum is from 730 nm to 840 nm tail to tail.

10

Figure 2.2 Schematic diagram of the 0.2-TW 1kHz single stage regenerative Ti:Sapphire
amplifier system.

In the amplifier, as shown in the big solid black block, the seed is first chirped from
10 fs to more than 100 ps by an all-reflective stretcher [44]. Then, the stretched pulse
passes through the KMLabs home-built Pockels cell, and the repetition rate of the seedpulses is picked from 80 MHz to 1 kHz before the regenerative cavity. The core part of theregenerative amplifier starts from the dashed black box, and all important elements of theregenerative amplifier are emphasized in the dashed box. The function of the thin film
polarizer (TFP) is to transmit the “p” polarized light and reflect the “s” polarized light. The
input seed is “p” polarized, so it can pass through the first TFP (on the left in Fig. 2.2) andgo into the regenerative amplifier. The function of the Faraday rotator is to rotate thepolarization 90 degree when the light travels from right to left while to keep thepolarization in the opposite direction, so the input seed can pass through Faraday rotator

11and keep “p” polarized since it’s from the left to right before it enters the regenerative
cavity. Then, the input seed passes through the second TFP (on the right in Fig. 2.2) since it
is still “p” polarized at this time. After that, if there is no Pockels cell or if the Pockels cell isoff, the “p” polarized seed will be switched to “s” polarized light once since it passes throughthe quarter waveplate twice when it bounces the end-mirror on the right-top corner, then it
will travel inside the cavity for one round trip after being reflected by the 2nd TFP since it’s“s” now, and then be switched back to “p” polarized light the second time it passes throughthe quarter waveplate twice, and finally it can pass through TFP and exit and regenerative
cavity. After that, since it passes the Faraday rotator from right to the left, it will change
the polarization again from “p” to “s”, and it will be reflected by the first TFP and exit thecore part of the regenerative amplifier. So, if the Pockels cell is off, the seed travels throughthe regenerative cavity only once, and no real amplification would happen. The key tocontrol the seed amplification, that how many times the seed is amplified inside theregenerative cavity, is by using Pockels cell. The function of the Pockels cells is like aquarter waveplate, so combining it with the quarter waveplate, when the Pockels cells is on,they work as a half waveplate, and since the seed will pass them twice every time it
bounces the end mirror, they don’t change the polarization at all, so the “s” polarized lightwill be trapped inside the regenerative cavity and get amplified until the Pockels cell turns
off, and after it is off, the amplified pulse will be changed back to “p” polarization and exit
the cavity. Meanwhile, since the pulses outside the cavity are “p” polarized, when the
Pockels cell is on, there is no chance they can change to “s” polarized light, so they will beblocked outside the cavity. (In principle, there is no pulse outside the cavity in our casesince we have used the KMLabs homebuilt Pockels cell to pick the pulse, however, thispulse-picking Pockels cell is not necessary if the “on” and “off” contrast of the Pockels cell
inside cavity is large enough and the regenerative cavity is not larger than oscillator cavity

12
so that only one seed pulse can be in the regenerative cavity, because Pockels cell inside thecavity can pick the pulse and amplify it at the same time as we explained above.) Therefore,
the Pockels cell acts like a gate, and the Pockels cell window determines the seed
amplification. Practically, the Pockels cell window should overlap the pump timing andseed timing. The time “on” should be set some time after the rising edge of the pump pulse
so the Pockels cell window overlaps with pump timing while still leaves some time forpopulation inversion to build up, and the time “off” should be set right after the gain issaturated, so the amplified pulse can have the highest energy extraction without burning.
Finally, after the seed is amplified, it’s up-collimated by a pair of convex mirror and lens,then it is compressed by the diffractive grating pair compressor and exits the laser
amplifier for use.
So far, the discussion on regenerative amplifier is quite general. Since the uniqueproperty of the newly-designed regenerative amplifier is its highest pulse energy in theworld for a single-stage amplifier, with perfect spatial mode quality, from now on, we will
focus on the key points to achieve the high pulse energy of 10 mJ. Specifically, themaximum output pulse energy obtainable from an ultrafast Ti:sapphire regenerative
amplifier is primarily limited by two factors: thermal lensing [45] and optical damage. In
order to overcome the two factors, we proposed three ideas: increasing the cavity mode sizeby enlarging the cavity size and pump mode size, cooling the crystal to the cryogenic
temperature by KMLabs patent cryogenic cell and the stretching the pulse longer by higher
groove density gratings. As to be discussed in the following sections, the first idea will
reduce the thermal lensing and the optical damage simultaneously, and the second idea will
greatly reduce the thermal lensing, and the third idea will reduce the optical damage
further. We will follow these three ideas, first discuss the thermal lensing effect in the next
section, and then derive the relationship between the grating groove density and stretched13pulse duration in the CPA process in order to demonstrate how the stretching is considered
practically in the section after that, and finally show the performance of the newly-designed
regenerative amplifier and some further investigation of the regenerative amplifier.

2.3 Thermal lensing
The term “laser” originated as an acronym for “light amplification by stimulated
emission of radiation”. Since stimulated emission requires the population inversion, which
doesn’t naturally exist because of the thermal equilibrium between the energy levels ofactive ions, atoms or molecules, a pump and an appropriate energy level system, whichenable the population inversion, are the prerequisite of a laser. The pumping process in alaser material is inevitably associated with the generation of heat for a number of reasons[46]: 1. The energy ratio of the laser emission photon energy to pump photon energy, whichis called the quantum defect efficiency. 2. Nonradiative relaxation from the upper laser
level to the ground state. 3. A certain amount of background absorption by the laser hostmaterial when the spectrum of the pump source is broad. Therefore, the thermal effects in
the gain medium, especially the solid-state material, naturally accompany the functioningof the laser, and the problem of solving the thermal issues caused by high average powerpump source has arisen almost since the invention of solid-state lasers.
Generally, the thermal effects, including thermal lensing, thermal birefringence [47]and thermal fracture [48], can be attributed to the non-uniform temperature distribution
caused by the heat dissipated inside the crystal and cooling at the boundary of the crystal
and the stress introduced by the non-uniform temperature distribution. In order to show
the causality of the thermal effects clearly, we discuss the simplest model following the14original paper, where heat is assumed to be generated inside the crystal uniformly [49]. It
presents the simple expressions of the temperature distribution and stress distribution as
well as the general physics pictures of the thermal effects. The thermal lensing effect will
be emphasized since it is one of the main obstacles to achieve high energy ultrafast
amplifier.
When the heat is generated within the laser rod uniformly by the pump laser andremoved by the coolant or just through the thermal contact with laser rod mount along thecylindrical rod surface, and when the thermal steady-state is reached which means thegeneration of heat equals the removal of heat, the radial temperature distribution in acylindrical rod can be obtained from the one-dimensional heat conduction equation [46],(2.1)( )( )

where T is the function temperature with respect to radial coordinate r, Q is the heatgenerated per volume, and K is the thermal conductivity. The solution of (2.1) is that.

( )

( )

( )(

)

(2.2)

where ro is the radius of the laser rod. We can see that under the assumption of the uniform
heat generation within the laser rod, the temperature profile is parabolic. Since thetemperature-dependent change of the refractive index can be expressed as

( )

[ ( )

( )] ( )

(2.3)

( )

(2.4)

by inserting (2.2) into (2.3), we can obtain

( )

( )

15
Therefore, the temperature-dependent change of the refractive index is parabolic as well.
The non-uniform temperature distribution generates stresses in the laser rod since
the hotter inside area is constrained from expansion by the cooler outer zone, as shown inequation (2.2). The expressions for radial σr, tangential σФ, and axial σz stresses under thecondition of plane strain, which means the axial thermal gradient is much smaller than theradial thermal gradient and certainly is satisfied under our assumption, and in the case of
free ends, are given by [50],

[ ∫

( )

∫

( )

[ ∫

( )

∫

( )

[ ∫

( )

( )]

]

(

)

(

( )]

(

)

(

(

)

(2.5)

)

)

(

)

(2.6)

(2.7)

where E is Young's modulus, ν is Poisson's ratio, α is the thermal coefficient of expansion.
From these expressions we can see that the maximum stress is at the surface of the laserrod. When the stress exceeds the maximum surface stress, thermal stress fracture occurs.
Since stress and strain are linked in elastic media by a stress-strain or constitutive
relationship,
(2.8)

∑ ∑

where cijkl is the elastic tensor and Skl is the strain tensor, the stress generated in the laser
rod introduces thermal strain, which, in turn, produces refractive index variations via thephotoelastic (Acousto-Optics) effect by [51],

( )

∑ ∑

(2.9)

16
where pijkl is the strain-optic tensor. From (2.8), (2.9) and the crystal symmetry the laserrod, we can obtain the refractive index change from the thermal stress. Due the complex
tensor calculation, only the final expression is shown here, and the detailed derivation is inreference [47]:(2.10)(2.11)Cr and CФ are functions of the strain-optic tensor pijkl, and typically they are not equal as
shown in the cubic crystal such as Nd:YAG [47], therefore, the thermal stress introduces
the thermal birefringence in radial and tangential direction.
Combining the refractive index change from the nonuniform thermal distribution
and thermal stress, we can obtain the total refractive index change in the cylindrical laser
rod to be,[( )(2.12)]

Therefore, the equivalent focal length of the thermal lens can be obtained from the thinlens formula

[

( )

] ⁄

(

)

(2.13)

where l is the length of the laser rod.
So far, the thermal lensing effect only considers the bulk effect inside the cylindricallaser rod. In order to obtain the full expression of the focal length of the thermal lensing,one so-called end effect, which accounts for the physical distortion of the flatness of the rod17ends or end bulging, has to be considered as well. From thermal expansion, the deviationfrom flatness of the rod ends is obtained from
[ ( )

( )

( )]

(

)

(2.14)

Then, the equivalent focal length can be obtained from the thick lens formula
⁄

(

)
)

(

[

(

(2.15))]

Therefore, the general thermal lensing formula for a laser rod has the final expression as

(

)

(

(

(

)

(

)

)

)

(2.16)

where A is the pump area, P is the pump power.
The thermal lensing comes from three terms: the temperature gradient term, thethermal stress term and the end bulging term. Typically, the first term is the dominantterm in thermal lensing [52]. If we just consider the first term in Equation (2.16), and put
the thermal parameters of the Ti:sapphire crystal at room temperature (300 K) andcryogenic temperature (77K) in, we can obtain the thermal lens effective focal length as afunction of the pump diameter, which is associated with large cavity design, and we cancompare the effective focal lengths at room temperature and at cryogenic temperature,
which is associated with cryogenic cooling to the crystal. From Fig. 2.3, we can see that:first, the focal length of the thermal length is increased by more than 200 times (30 cm to
60 meters under the pump diameter of 1.15 mm) by the cryogenic cooling. Second, since thefocal length is proportional to the pump area as in Eq. (2.16), for meters long cavity, the18pump diameter should be kept above 1 mm scale to keep the effective focal length to be longenough compared to the cavity size. These two points explain why we are building a larger
cavity and using the cryogenic cooling to the Ti:sapphire crystal.

Figure 2.3 Effective focal length based on the uniform continuous pump model. The redvertical line corresponding to real pump size used in our amplifier.

2.4 Grating based optical stretcher and compressor
For the femtosecond pulse amplification, probably the most profound concern is theultra-high peak intensity induced nonlinear phase accumulation duration the beam
propagation through any material, which could introduce self-focusing by the Kerr lenseffect [53], even just within the air if the intensity is high enough. One way to characterize

19
this nonlinear total on-axis nonlinear phase shift accumulated over the length of theamplifier is in terms of B integral, written,∫
Typically in amplifier design, one desires(2.17)( ), and forthere is a risk that self-

focusing may occur: the Kerr lens effect can become so strong that the beam collapses to avery small radius, so that the optical intensities are strongly further increased and easily
exceed the damage threshold. For a femtosecond pulse in the amplifier, for example, anbeam with pulse energy of 1 mJ, pulse duration of 50 fs, and mode diameter of 1 mm,
passing through 1 cm Ti:sapphire crystal only once without any amplification so theintensity keeps the same along the 1 cm, then the B integral is 112, which is much larger
than the desired number. In reality, the number is higher if we consider the multiple
passes to reach the gain saturation intensity and other material such as the Pockels cell
crystal included in the amplifier. Therefore, the chirped pulse amplification method is usedin the ultrafast laser amplifier, which means the seed pulse is first chirped to a much
longer pulse duration, typically tens to hundreds of ps, before being amplified through thematerial, so the accumulated nonlinear phase is 3-4 orders of magnitude smaller thenumber above, and then amplified pulse is compressed back to the fs pulse duration before
use. The chirping and compressing process are achieved by the ultrafast stretcher andcompressor.

20

Fig 2.4 (a) Schematic diagram of the grating based compressor. (b) Schematic diagram of
the grating pair with internal lens stretcher.

The schematic diagrams of the grating based ultrafast optical compressor andstretcher are shown in Fig 2.4. Some ultrafast optics terminology is needed to describe thefunction of the compressor and stretcher. From Fourier Optics, if a input pulse ein(t) has the
spectrum Ein(ω), after passing through a dispersive system that adds spectral phase ( ),the output pulse is given by(2.18)

( )

( )

∫

( ) is commonly written in terms of the Taylor series expansions as follows:

( )
where

⁄

as ( )

)

(

)

(

)

(2.19). And the propagation constant in the medium of length L is defined( ) , where

( )
so

(

⁄

(

)

(

)

(

)

. From these, the group velocity is defined as(2.20)

21(⁄(2.21))and this is equivalent to the definition of group delay(2.22)

( )
( )

since

( )

. From (2.21), if ( ) or

( ) is not linear with

, theenvelope of the pulse can’t keep invariant since light with different frequency will travel at
different group velocity, therefore the part of ( ) that is not linear inandis called nth order nonlinear phase whenis called dispersion,. The concept of the dispersion isprobably the most important one in Ultrafast Optics. From Fourier Optics, we know thatthe temporal transform limit of a pulse exits when its nonlinear spectral phase is zero, sousing dispersive devices such as compressor, stretcher and pulse shaper to balance theunwanted dispersion introduced by the materials in order to achieve the shortest pulse is ofgreat importance. Here, we will focus on the dispersion in our amplifier, specifically, thecompressor and stretcher, so that we can understand how it helps to achieve the high pulse
energy.
We first discuss the dispersion of the compressor. As shown is Fig 2.4 (a), the gratingbased compressor is composed of a pair of identical grating placed in parallel facing eachother. The distance between the two gratings is called G, the incidence angle of the input
beam is θi, then by the grating equation, the diffraction angle is determined by,
(

)

( )

⁄

(2.23)

where d is the periodicity of the grating rulings and m is the diffraction order. If we use thespeed of light c,

andis related by.
⁄ ,⁄(2.24)

22
From Eq. (2.23) and Eq. (2.24), a particular frequency component
particular diffracted angle

will travel at a

, so the compressor introduces the angular dispersion. In

general, if a beam travels through an infinite uniform dispersive medium
distance L at the reference( ) for adirection, the frequency-dependent phase difference ( )between the start and end wavefront is given by,( )(2.25)( )⁄( )where ( ) is the angle between the direction of frequency component

and the reference. This expression is true in the case of passing through a uniform slab as well [43]. From
(2.24) and (2.25),(2.26)
where ( )( ) is the optical path. Using the (2.24) and the chain rule, we can( )obtain,

[

( )

( )

If the dispersion is evaluated along the reference direction,

[

( ) ]

(
,

)]

(2.27), then,(2.28)

From the definition of group delay, we can get(2.29)Put (2.28) into (2.29)

[

( ) ]

(2.30)

23
This expression gives the group delay dispersion, coming from the material dispersion andangular dispersion, with respect to wavelength of propagating beam in the direction of thereference frequency. In the case of the grating based compressor, if we choose thereference direction to be at the diffracted angle
there is no material dispersion sinceas shown in Fig 2.4 (a), and considering

between two gratings, then, from Eq. (2.30), thedispersion from grating pairs is((2.31))

where a factor of 2 is added because an end roof mirror is used typically as show in Fig 2.4(a), so there are two passes through the grating pair following the same optical path. This
will avoid the spatial chirp in the output beam. In the case of single-pass compressor, thefactor of 2 should be omitted. We will keep the factor of 2 since it is the common case. Fromgrating equation (2.23), we can easily obtain,(2.32)

Therefore, the dispersion is(2.33)

And from (2.26), the 2nd order nonlinear phase is(2.34)

And the 3rd order nonlinear phase is

(

)

(

)

(2.35)

24
From (2.34), (2.35), we can see that the 2nd order dispersion of the grating pair based
compressor is always negative, and 3rd is always positive. Here, the way we define the signof the dispersion is following the convention of the Ultrafast Optics, where the sign of the
2nd and the 3rd dispersion of normal material are defined as positive in the visible lightregion, which means the red light travels faster than the blue light. And the reason of thecompressor is name as “compressor” is because the 2nd order dispersion of the compressor
has the opposite sign to the normal material, the total spectral phase can be balanced,
which means the pulse can be compressed by the compressor.
After obtaining the dispersion of the compressor, let’s investigate the dispersion of
the stretcher as shown in Fig 2.4 (b). Here, I want to point out that, the stretcher used in
the ultrafast amplifier is an aberration free all-reflective design [44]. It uses the concave
mirror instead of the lenses, and it uses only one concave mirror and one grating instead of
two lenses and two gratings as shown in Fig 2.4 (b). However, the optical paths areequivalent. So we’ll use this system to demonstrate the stretcher dispersion. The system
shown in Fig 2.4 (b) is also frequently used in Fourier optical systems, called a 4-f system,
since plane of where the vertical dashed line shows is the Fourier plane of the left dot on
the left grating. In Ultrafast Optics, as shown in Fig 2.4 (b), when the distance between two
gratings is are equal to 4f and a phase modulator is placed at where the dashed vertical line
is, it is a pulse shaper, and we will further discuss this in the next section; when the
distance between two gratings is shorter than 4f, which is the exact case shown in Fig 2.4(b), it is a stretcher, which is used in the ultrafast amplifier for CPA; and when the distance
between two gratings is longer than 4f, it is actually a compressor, the same as discussed
above. The reasoning is quite straightforward. Let’s first consider the case that the distance
between two gratings is exactly 4f, then phase accumulated in the two different paths
shown in Fig 2.4 (b) are equal because the lens’ law. It means when distance between two

25
gratings is exactly 4f, there is no dispersion at all. And only the distance shorter or longer
than 4f is the optical path that will introduce the angular dispersion as in the case of thegrating pair compressors. Therefore, the 2nd order nonlinear phase of the stretcher is()(2.36)In this formula, the “+” sign applies to the case when the distance is longer than 4f, and the“-” sign applies to the case when the distance is shorter. Therefore, from the sign of, we

can see that, only when distance between two gratings is shorter than 4f, the system
shown in Fig 2.4 (b), can be used to introduce the dispersion opposite to that of acompressor, which is a true stretcher.
The 2nd and 3rd order nonlinear dispersion of the stretcher and compressor are used
for the chirp pulse amplification and balancing the nonlinear dispersion introduced by thematerial in the amplifier, that is

{

(

)

(

)

(

)

(

)

(

)

(

)

Typically, only the first two orders nonlinear phase (and(2.37)) are considered, which isenough for most ultrafast amplifiers. If higher orders nonlinear phase needs to be included,
for example 4th and 5th order nonlinear phase, the prism pair stretcher and compressor canbe used together with the grating based stretcher and compressor to balance them. And, inprinciple, the pulse shaper would be able to fix nonlinear phase of any order.
The final question is how the practical parameter, the grating groove period d, andthe grating width wg is related to the maximum chirped pulse duration in the CPA, since
the pulse duration determines the pulse intensity so that the optical damage inside the26amplifier. From Fig 2.4 (a), if the input beam in x direction, then, the y coordinate of theoutput ray is given by(Then, assuming, the separation

wavelengths separated by

between two output rays at differentis|

The maximum |(2.38))||(2.39)|| is limited by the grating width wg, that |(2.39) into this relationship and use (2.33) to replace ⁄|. Put Equation, we can obtain,(2.40)

From (2.40), we can see that, for a fixed grating width, the maximum chirped pulse

duration is proportional to the grating’s groove density 1/d . In the old stretcher design,
1200 g/mm grating was used, while, in the new design 1800 g/mm is used, so the pulse ischirped 1.5 times longer, which can support pulse energy of 1.5 times more than before.

2.5 Performance of the 10 mJ kHz single-stage Ti:sapphire amplifier
First, the maximum output (compressed) pulse energy is 10.6 mJ when pumped with
50 mJ of 527 nm light from a Nd:YLF pump laser. Since the efficiency of the diffraction
grating based compressor is 70%, this represents an optical-to-optical efficiency of theregenerative amplifier of ~30% before compression. To our knowledge, this represents atleast a 50% improvement in the pulse energy obtainable from a single-stage amplifier [54].

27
Second, the 10 mJ single-stage regenerative Ti:sapphire amplifier has a nearly
perfect Gaussian mode output (see Fig. 2.5), with M2 [55] values for both x-axis and y-axisof ~1.1, which means almost all the pulse energy is in the lowest order Gaussian mode. Thisfeature ensures good conversion efficiency and output mode for the OPA [56], which in turnensures high HHG efficiency, since a good OPA mode efficiently couples into a waveguideand facilitates good phase matching (Chapter 3).

Fig. 2.5. (a) Focal spot of the single-stage laser amplifier output (KMLabs Wyvern HETM),
using a 1 m focal length lens; (b) M2 characterization of the 0.8 µm amplifier output. Datataken at 10 mJ, 1 kHz.

Third, the output pulse duration after the compressor is ≲45 fs, characterized by
SHG FROG. It is primarily limited by the bandwidth of the thin film polarizer (TFP) inside
the amplifier cavity. Unlike a conventional regenerative cavity, the dispersion introduced

28
by the material in the regenerative cavity is not the limiting factor for the pulse duration
because only 12 passes through the Ti:sapphire crystal are needed to reach the full output
power (15 mJ before compression), which is approximately the same number of passes used
in a typical multipass ti:sapphire amplifier, but with much higher efficiency. In contrast, atypical regenerative amplifier requires 20-40 passes. In the future, a shorter pulse can beachieved when broader bandwidth, higher-damage-threshold, TFPs becomes available.
Finally, the 8 hours’ stability test of the 10 mJ, 1 kHz, Ti:sapphire amplifier showsthat the fluctuation of the output power (pulse energy) is within 0.59% RMS (Fig. 2.6),
which meets the commercial specs of <1%.

Figure 2.6 The stability test of the 10 mJ kHz Ti:sapphire amplifier for 8 hours.

29
2.6 Further investigation of the ultrafast amplifier with pulse shaper
After implementing the first 10 mJ single-stage 1 kHz Ti:sapphire amplifier withperfect spatial mode, we further investigate the possibility of reducing the output pulse
duration of the regenerative amplifier, since this is the disadvantage of regenerative
amplifier compared to the multipass amplifier.
The TFP in the 10 mJ regenerative amplifier limits output bandwidth of theamplified pulse. So, we used another 5 kHz, 3 mJ, regenerative amplifier for the test, whichuses different TFP bearing broader bandwidth. Once the broader bandwidth, higherdamage-threshold, TFP becomes available for the 10 mJ amplifier, the results from the 3mJ amplifier should be implantable to the 10 mJ amplifier.
In order to shorten the amplified pulse, a spectral filter [44] is placed before or in theregenerative cavity, and a pulse shaper [57] built from the 4-f system with the liquid crystal
array placed at the Fourier plane is positioned between the oscillator and the amplifier. The
function of the spectral filter is to reduce the gain narrowing effect [44] during the pulse
amplification so that broader bandwidth can be obtained after amplification, and thefunction of the pulse shaper is to flatten the higher order phase that compressor can’t be
fixed.
The results are shown in Fig. 2.7. When no spectral filter is used, the output power
is 15 W, and the pulse duration is 44 fs. When the spectral filter is placed before theregenerative cavity, the output power is 14.5 W, and the fully compressed pulse duration is35.5 fs. When the spectral filter is placed inside the regenerative cavity, the output power is
10 W, and the fully compressed pulse duration is 29 fs, close to what the number frommultipass amplifier. Therefore, with the spectral filter and pulse shaper, the pulse durationof the regenerative amplifier can be reduced by sacrificing the output pulse energy.

30

31
Figure 2.7 FROG measurement of the amplified pulse, without the spectral filter (a), withthe spectral filter before the regenerative cavity (b), with the spectral filter inside theregenerative cavity (c).

2.7 Conclusion and future investigation
In this chapter, we first introduced the history of the development of the ultrafast
amplifier. Then, we explained the general operation principle of the regenerative amplifier.
In order to explain the three key points, the larger regenerative cavity, the cryogenic
cooling to the crystal, and the higher groove density grating in stretcher and compressor, toachieve to first 10 mJ single-stage kHz Ti:Sapphire amplifier, we extensively discussed thethermal issue and the chirping technique for the laser amplifier. Finally, we presented theperformance of the newly-designed regenerative amplifier, and investigated the possibility
to further reduce the pulse duration of the regenerative amplifier.

Future investigation:
First, test the spatially chirped pulse amplification in the ultrafast Ti:sapphire
amplifier. Gain narrowing is one of the primary limiting factors of the bandwidth of theamplified pulse. Different color competing to consume the population inversion makes thesituation even worse. Therefore, if the pulse can be spatially chirped on the crystal, and thepump beam can be shaped to amplify the side frequency, then the bandwidth of theamplified pulse could be broadened [58].
Second, further investigate the thermal lens issue. Although expression (2.16) shows
clear physics pictures of the thermal lensing and can be used to calculate the focal length of
the thermal lens quantitatively to some extent, it’s derived from oversimplified

32
assumptions which typically don’t apply to the Ti:sapphire system. On one hand, since theupper laser level lifetime of Ti:sapphire is short (3.2 μs), and the saturation power is very
high, another laser, such as Nd:YAG or Nd:YLF, is typically used as the pump source in theend pump geometry. For example, the Ti:sapphire crystal rod used in our 10 mJ system is 1
cm diameter and 1 cm long, while the typical pump beam mode diameter is typically around
1 mm in the crystal in order to make the gain high enough. Therefore the assumption of
uniform heat generation inside the crystal is not suitable radially. Meanwhile, the low
power single pass absorption at 532 nm (pump wavelength) is 92% (the absorption
coefficient γ is 2.5 cm-1), the assumption of uniform heat generation inside the crystal is notsuitable axially as well. On the other hand, the generation of the ultra-short pulse at fs
time scale is the primary reason of using Ti:sapphire crystal due to its ultra-broad lasing
bandwidth, hence, the pump source is typically also a pulsed laser with the same rep rate in
the amplifier system (1kHz in our case). Therefore, the assumption of a cw pump source, asin (2.1), is not appropriate as well. An end-pumped pulsed heating thermal lens model isgiven by R. Lausten [59]. The result differs from the simple model we have presented byorders of magnitude, which predicts a much longer focal length. And C. Durfee et al.
measured the transient thermal lens [60], which doesn’t agree with R. Lausten’s model. No
pulsed end-pumped analytical model matches the experiments so far based on the author’s
knowledge.

33

Chapter 3

High flux ultrafast UV-VUV generation from 4-wave mixing process

3.1 Introduction
An ultrashort pulse at wavelengths in the ultraviolet (UV) and vacuum ultraviolet(VUV) is of great interest to ultrafast science and technology. On one hand, the excitedelectronic states of gas phase molecules and materials lie in the UV to VUV. Thus, theavailability of an ultrafast UV to VUV light would allow the excited state to be accessed[61-65]. On the other hand, the nonlinear crystal is limited to the generation of narrowband light at wavelengths longer than 180 nm. Although Watt-level average-power tunable
deep ultraviolet (DUV) from 185 nm to 200 nm has been generated by KBe2BO3F2 (KBBF)
crystal [66] and pulse duration is around 150 fs-200 fs by KBBF and RbBe2BO3F2 (RBBF)
crystal [66-67], the large group velocity dispersion (GVD) in the UV to VUV region for solidstate materials makes it difficult to reach broad-band phase matching to obtain the pulse
duration of < 20 fs with the practical thickness of the material. Meanwhile, thetransmission of the crystal is typically limited to 200 nm or so, for example, beta-BaB2O4(BBO) crystal is not transparent below 189 nm, and the shortest phase matched sum
frequency generation (SFG) is at 156nm by KBBF whose transmission limit is 155 nm [68].
Due to the demand of the 10 fs pulse, or even less, to match the ultrafast electronic

34
transitions in the UV-VUV region, the generation of high flux ultrafast UV-VUV pulse isdesired.
Charles G. Durfee from our group pioneered the ultrafast UV generation from noblegas as the nonlinear medium by utilizing four wave mixing (FWM) process in the hollowwaveguide geometry [69-72]. It enables an ultrashort pulse duration of 8 fs and pulse
energy of few uJ level at wavelength of 267 nm. By the cascade processes, the 4 th and 5th
harmonics of Ti:sapphire laser at 200 nm and 160 nm are generated simultaneously. Then,other schemes using gas medium were implemented as well. For example, the scheme thatcombining the UV multipass excimer amplifier and the SPM in hollow-waveguide
generated the highest pulse energy UV light pulse in the < 30 fs region, that is 200 uJ at
248 nm of 24 fs [73]; and the scheme, that using SPM to achieve the ultrashort driving
pulse at 800 nm before doing third harmonic generation (THG) through a thin gas jet,generated the shortest UV pulse so far, that is 200 nJ at 263 nm of 2.8 fs [74]. However,
although the two schemes pushed the output parameter to the limit, the first one included
another laser amplifier and didn’t obtain the ~10 fs pulse; and the second one didn’t
support enough pulse energy for applications due to the short interaction distance.
Therefore, the FWM in hollow waveguide geometry is still one of the most commonly used
schemes. And since the limitation of this scheme is the input pulse intensity, as theionization inside the waveguide would violate the phase matching process [72], Charles G.
Durfee proposed the chirped FWM () in the hollow waveguide geometry togreatly improve the output pulse energy [75], a similar idea to the CPA system asmentioned in the last chapter. The followers have generated the applicable UV-VUV pulse
by the chirped FWM process [61, 76-77]. The other commonly used scheme to obtainultrafast UV-VUV pulse is also by the FWM process but with filamentation guiding instead35of the hollow waveguide guiding [78-81]. Recently, the kagome-style hollow core photonic
crystal fiber (HC-PCF) is used to implement to ultrafast UV-VUV pulse [82-83]. Itsfabulous fiber dispersion enables many nonlinear phenomena in a super tiny core so thatthe low pulse energy high repetition rate pulse is preferred for this type of flexible fiber,
which attracts great industry interests.
In the following sections, we’ll first discuss the modes supported by the hollow
waveguide, which is also critical to both ultrafast UV-VUV generation and high orderharmonic generation, then explain the three important questions for the ultrafast UVgeneration in the hollow waveguide: the mode coupling problem, the phase matchingproblem and the mode matching problem, and finally present our experimental results to
support the theories and demonstrate that we have achieved the highest pulse energy ofultrafast UV pulse in ~ 10 fs region. The cascaded VUV generation will be discussed as oneof the further experiments since it simultaneously happens with UV generation but needs
further investigation.

3.2 Electromagnetic modes in the dielectric hollow waveguide
The diagram of the dielectric hollow waveguide is shown in Fig 3.1. It could beconsidered as a cylinder of radius a and free space dielectric constant ε0 embedded inanother cylindrical medium of dielectric (or metal) having complex dielectric constant ε.
The magnetic permeability μ0 is assumed to be that of free space for both media. The
solution of the cylindrical dielectric waveguide modes was first provided by E. Snitzer [84],while E. Marcatili and R. Schmeltzer first proposed the use of the dielectric hollow
waveguide for long distance optical transmission [85].

36

Fig 3.1 (a) Side view of the dielectric hollow waveguide. (b) End view of the dielectric
hollow waveguide.

Generally, the problem can be substantially simplified with the assumptions thatthe radius is much larger than the wavelength of the propagation light and the modes ofinterests are the low-loss modes whose propagation constants are close to that of free space:
(3.1)

| |

|

(3.2)

|

where k is the free space propagation constant;is the mth root of equation
propagation mode; and

(

)

√

is the ratio of the refractive index;, n and m are integers that characterize theis the axial propagation constant of the mode considered. The

analytical solutions under the assumptions are given by E. Marcatili and R. Schmeltzer [85].
Before we cite the formulas from E. Marcatili and R. Schmeltzer [85], we want to point out
a few things in order to pick up the necessary solutions we are interested in from thegeneral discussion of the dielectric hollow waveguide modes, and make the notations clear

37
to all readers. First, we are using the notations from E. Marcatili throughout this thesis
because they are more commonly used in the society of the UV to X-ray source generation,although in most of the text books, the notations from Snitzer are used since they are
original. For example, the important modes such as EH1m modes is labeled as HE1m in theway defined by Snitzer. Second, in general, there are three types of mode supported by thecylindrical boundary conditions with the assumptions, the circular electric mode TE0m, thecircular magnetic mode TM0m, and the hybrid modes EHnm. Considering the input lights,
800 nm light and 400 nm light, are both linearly polarized in the same direction, and thethird order susceptibility tensor only have nonzero component

( )

,

( )

the isotropic material such as the gas used in our experiment,

( )

( )

=

,

( )

, and

( )

( )

( )

(for

), thegenerated UV light has to be linearly polarized in the same direction in any third ordernonlinear problems. So, only the linearly polarized modes in the waveguide are needed tobe considered. Third, only the EH1m modes and the combination of EH-|n|.m modes andEHn+2,m modes, which can be called LP|n|,m (linearly polarized) modes [86], (), arelinearly polarized modes mathematically supported by the waveguide. When the refractiveindex, the combinations of the TE0m modes and EH2m modes are also close to linearlypolarized light [85], so, in principle, EH1m modes, LP|n|,m modes and the combinations of theTE0m modes and EH2m modes, which can be called LP1,m comprise the complete set for the
linearly polarized modes in the waveguide. Forth, the mode selection during the mode
coupling process and UV generation process has to be treated separately because of thedifferent initial condition. During the coupling process, the input light can be considered as
the lowest order Gaussian distribution TEM00. Since the asymptotical behavior of LP|n|,mmodes () (here we have combined the two kinds of combination modes mentioned

above) don’t fit the lowest order Gaussian distribution TEM00, (for example, the38
distributions of LP|n|,m modes are) [86]), the EH1m modes are considered(as the complete basis during the coupling process. However, in the UV generation process,there is no such restriction of the mode selection, so all linearly polarized modes supportedby the waveguide boundary condition should be considered.
From the discussions above, only a few modes, EH1m and LP|n|m modes, areinteresting in our experiments. The field distribution is shown in Fig 3.2 [86]. Under theassumptions (3.1), expressions of EH1m mode and LP|n|m modes are

| |

| |

( )

(

( )

(

( )

(

( )

(

(

)

)

)

)

(3.3)

( )

(3.4)

(

)

)

| |

(3.5)

(3.6)

( )

, where we have assumed the electric field is polarized in the y direction. From now on, the( ) will be abbreviated to be( ), and| |( ) will be abbreviated to be

The polarization direction is selected arbitrarily, so it becomes a 1-D problem.

( ).

39

Fig 3.2 (a) The far filed distribution of EH11 mode. (b) The far filed distribution of EH12
mode. (c) The far filed distribution of LP11 mode. (d) The far filed distribution of LP12 mode.

Another important parameter is the propagation constant

,
(3.7)

{

(

(
where

) [

)

(

( )

)]}

(3.8)

(3.9)

40

√

(3.10)√(){√
From (3.6), the phase matching problem can be discussed in the next section. From (3.7)
and (3.8), we can see that, for longer wavelength and smaller radius hollow waveguide, thebeam would attenuate faster. For example, for beams propagating in the lowest-order mode
EH11 in a waveguide with core radiusand length of 1 meter, the intensity

throughput at wavelengths 800, 400, and 267 nm are 52%, 85%, and 93%, respectively.

3.3 Mode coupling, phase matching and mode matching for the hollow waveguide
As in most of the nonlinear optical processes, the phase matching problem is themost important problem in order to obtain an efficient output of ultrafast UV. Since thenonlinear process happens in the hollow waveguide, the mode coupling and mode matching
are also determinants to the output flux. In this section, we first discuss the problem ofmode coupling since it is the first step of the whole process, affecting the other two
problems; then explain the phase matching problem from the derivation of the third ordernonlinear process; and then the importance of the mode matching will show automaticallyby comparing THG and FWM through the derivation.
As we mentioned in the last section, during the light coupling from free space into
the hollow waveguide, we assume the incoming beam is a linearly polarized lowest order
Gaussian TEM00 beam and the beam is at its waist so that the flat wave-front matches thewave-front inside the waveguide, then, the set of EH1m modes can be used as the eigen-41
basis of the hollow waveguide modes during the coupling. The TEM00 mode is decomposed
into EH1m modes as∑(3.11)

where Am is the coefficient associated with each EH1m mode. Using (3.5), (3.11) and(3.12)
where w is the radius of waist, we can obtain the power coupling coefficients for the EH 1m
modes from TEM00 mode as a function of the ratio of the beam waist to the guide radius,
w/a , as shown in Fig. 3.3 [87]. From Fig 3.3, we can see that the entrance of the hollow
waveguide is acting as a spatial filter. When w/a = 0.64, all the energy is coupled into the
waveguide, and all energy is distributed in the EH11 mode. When w/a > 0.64, some of theenergy is lost at the entrance and the energy in the waveguide is almost all distributed in
the EH11 mode. When w/a < 0.64, all energy is coupled into the waveguide and the energy
in the waveguide is distributed in the combination of the EH1m modes. In principle, we wantto achieve w/a = 0.64 by the selection of focusing lens and hollow waveguide, when the
input beam has perfect spatial profile of lowest order Gaussian distribution, M2 = 1.
Practically, it is better to choose the ratio slightly larger than 0.64 in order to clean themode by the waveguide entrance. Here, we want to point out that, from this calculation,
importance of the input mode spatial quality is presented as well. It explains one of thereasons for setting up a regenerative amplifier instead of multipass amplifier mentioned inchapter 2, because the assumptions of the TEM00 mode input mode as well as the correct
ratio of w/a chose for the coupling process are both the prerequisites that all input pulse
energy can be coupled into the EH11 mode.

42

Figure 3.3 Calculations of power coupling coefficients for the EH1m modes as a function of
the ratio of the beam waist to the guide radius, w/a. This figure is captured from Ref. [87].

After the discussion of the mode coupling, we now turn to the problems of phasematching and mode matching for generating the ultrafast UV, 3rd harmonic from
Ti:Sapphire (267 nm), in the hollow waveguide. In principle, the generation of the 3rd orderharmonic could be from THG and FWM processes. The 2nd order sum frequency generation(SFG) from fundamental (800 nm light) and 2nd order harmonics (400 nm light) is forbidden
due to the use of gas as the nonlinear medium, since the second order susceptibility
vanishes in isotopic materials. We first demonstrate that THG is not an efficient way togenerate the ultrafast UV because of the bad mode matching, although it could be phasematched by utilizing the high order modes. If the input 800 nm light and 400 nm light is43labeled as E1 and E2, and the output 267 nm light is labeled as E3, they can be represented
in the non-depletion model as:

where,( )( ) and( ) ( )[( ) ( )[( ) ( )[

) ](3.13)() ](3.14)() ](3.15)(( ) represent the mode distribution of 800 nm light, 400 nm lightand 267 nm light respectively, as shown in (3.5) and (3.7);( ) and( ) are theamplitudes of the pulses of the 800 nm and 400 nm light at the entrance of the hollow
waveguide;( ) is the slowly varying amplitude of the 267 nm light. Then the THG

equation for the slowly varying envelope of the 267 nm light in the hollow waveguide iswritten as

( )

( )

[ ( )]()(3.16)We have put the mode number nm in, since different mode has different propagation
constant and assumed the energy coupled into the waveguide is all distributed in EH11
mode by proper mode coupling. The phase mismatch is then written as(3.17)The relevant nonlinear-optical cubic susceptibility is written as [88]
( )[

( ) ∬
∬[

Integrating (3.16), we can get

( )]
( )]

(3.18)

44[( )( )] {()

From (3.19), if we neglect the absorption, thephase mismatch

[ (

) ]}

(3.19)( ) is a sinc function with respect to theas expected.

From (3.8) and (3.10), when ν is real which means the dielectric material of the
waveguide itself is not lossy, then
( )

where

(

[

) ]

( )

is the neutral atomic density, ( )contains the gas dispersion function,neutral atomic density, and(3.20)

is theis the classic electron radius. Here, we have assumed that

there is some gas with refractive index ( ) in the waveguide as the nonlinear medium. In
general, the real part of the propagation constant is the sum of four sources of dispersion:
vacuum, gas, free electrons (plasma), and waveguide. Since the THG in argon the critical
ionization fraction is only 1.3% [72], (the detailed discussion the problem of the critical
ionization for the phase matching is shown in the next chapter for high order harmonic
generation), and in UV generation the emitters are not the ionized atoms, typically, we
don’t want to introduce any ionization in the ultrafast UV generation process. The
nonlinear problem is limited, in turn, within the perturbation theory. So the third term in
the propagation constant can be neglected. Then, from (3.17), the phase mismatch is()]= 800 nm,

and

[

where

= 267 nm,respectively.

[](3.21)are gas dispersion at 267 nm and 800 nm

is positive for normal dispersion, so in order to make phase matched,

has to be positive as well, that is. Therefore, the generated UV pulse has to45
be in a mode

different fromto make the phase matching happen. For example, thelowest ordermode fulfilling the relationship,choosing the neutral atomic density
gas pressure P since, ismode, and by properlyin (3.21), which can be achieved by playing with thefor the ideal gas, THG could be phased matched in the

waveguide. If phase matching happens and we neglect the absorption terms, (3.19) changesto( )(3.22)[ ( )]

So, the 3rd harmonic efficiency is mainly determined by the relevant nonlinear-optical cubic
susceptibility.

is significantly affected by the mode matching as shown in the

integration in (3.18). A simple calculation can show that

. Therefore, unable

to generate the beam in the same spatial mode as the pump and idler spatial mode in thephase matching condition significantly reduces the efficiency of THG in the hollow
waveguide. The discussion so far is not limited only to the THG; any 3rd order SFG, say, has the similar phase mismatch relationship as in (3.21), then thenormal material dispersion has to be balanced by the mode dispersion in the way of using
the higher order mode to reach phase matching condition, so all 3rd order SFG have low
efficiency due to the bad mode matching. (This problem might be solved by using PCF, see
the section of future experiment).
FWM is proposed because it can satisfy the requirements of the phase matchingcondition and mode matching condition simultaneously. The FWM equation for the slowly
varying envelope of the 267 nm light in the hollow waveguide is written as

( )

( )

[ ( )] [ ( )][(where the relevant nonlinear-optical cubic susceptibility is written as [88]) ](3.23)

46
( )[

( ) ∬

( )]

(3.24)( )]∬[and phase mismatch can be written as

(

[

(

[

where

= 400 nm,

)]

)]

(

[

[

and

is always positive for any modedensityand](3.25)]

is gas dispersion at 400nm.

)

is positive for normal dispersion,. So, by properly choosing the neutral atomic, the phase matching condition and mode matching condition

simultaneously. By using

and equation (3.25), the phase matching condition can also

be written as()(3.26)

where Patm is the pressure of one atmosphere. Formula (3.26) means that the pressure
times the area of the waveguide is a constant for certain mode. This relationship is
important for future discussion. If phase matching happens and we neglect the absorption
terms, (3.23) can be solved to be( )[ ( )] [ ( )](3.27)

If we write (3.27) in terms of the input and output pulse energy, then,(3.28)
where,

respectively.

,, are the pulse energy of the fundamental, 2nd and 3rd harmonic,

47
3.4 Ultrafast UV generation setup and results
The schematic diagram of the ultrafast UV generation is shown is the Figure 3.4.
The 800 nm light is incident from the left. After the SHG BBO crystal, 20% of the pulse
energy of 800 nm light will be converted into 400 nm light. A beam-splitter is used after the
BBO to separate the 400 nm light from 800 nm light. There are two reasons for separating
and recombining the 800 nm and 400 nm light beams before coupling them into the hollow
waveguide. First, lenses of different focal length are used in 800 nm and 400 nm lightbeams respectively in order to optimize the mode coupling, because the waist size areproportional to the wavelength and focal length. Second, the pulse energy of the 800 nm
light has to be reduced before the waveguide, since the pulse energy of 800 nm light needed
for FWM is only a quarter of that of the 400 nm light and too much pulse energy would
ionize the gas and violate the phase matching. This is achieved by using the combination of
the half waveplate and polarizer beam in 800 nm light beam. The polarization of the 800
nm light could be controlled by them as well. After 800 nm light and 400 nm light are both
coupled into the waveguide, by controlling the gas pressure inside and the time delay
between 400 nm light and 800 nm light, pulse energy of the 267 nm light could be optimized.
The output of the 267 nm light is compressed by an UV compressor, built by a pair ofreflective gratings (300 rules/mm, 300-nm blaze, Al coating), whose throughput is 50 %.
Then the pulse duration and spectral phase is characterized by a self-diffraction FROG [70].

48

Figure 3.4 Diagram of the ultrafast UV generation in hollow waveguide.

In experiments, we have tested the hollow waveguides with diameters of 100 um,
150 um and 250 um, and with lengths of 20 cm and 100 cm in order to prove the theories inlast section and find the optimized parameters for ultrafast UV generation. Noble gas argon(Ar) is supplied into the hollow waveguide as the nonlinear medium. Most features of theultrafast UV generation are shown in the pressure scan of the throughput of UV (267 nm)
as in Fig 3.5 (a), which is taken by using the hollow waveguide of 150 um diameter and 100
cm long.

49

Figure 3.5 (a) The throughput of 267 nm light as a function of the gas pressure inside thehollow waveguide. (b) Phase matching pressure curve with respect to the diameter of thehollow waveguide and the output mode.

50First, the efficient output of the 267 nm is only shown when the input light iscoupled into hollow waveguide properly, that all energy of both 400 nm light and 800 nm
light are distributed in the EH11 mode, and when the gas pressure is tuned to the phasematching pressure. This is a manifestation of the optimization of the mode coupling, phasematching and mode matching simultaneously. The results of the output pulse energy andpulse duration of the 267 nm light mentioned later is both taken under this condition.
Second, when the mode coupling is clean, as shown in the blue curve, the 267 nm light is“only” phase matched at one mode, EH11. This is because the mode matching between thephase-matched higher order signal modes and the pump and idler mode, EH11, is too little,
so that they don’t show clear output in the pressure scan, although they could be phasematched. When the mode coupling is not clean as shown in the red curve, the 267 nm light
can be phase matched at multiple modes in the pressure scan. This is because the coupledhigher order modes of the pump light could help the higher order modes of 267 nm light tomode match with them. Therefore, some higher order modes of the 267 nm light can begenerated. Third, the phase matching pressure of each mode can be directly obtained fromthe pressure scan. Combing the phase matching pressure, the mode number and thediameter of the hollow waveguide, the phase matching equation (3.26) can be proven, whichis shown in Fig 3.5 (b). The red curve is calculated from (3.26), and the black dot is obtained
from the pressure scan of the 267 nm throughput signal. The experimental results agreewith theoretical prediction perfectly. This is a direct proof of the theories of mode coupling
and phase matching mentioned in the last section.

51

Figure 3.6 The decay of light intensity inside the hollow waveguide as a function of
propagation distance. The lights of wavelength related to the FWM process are chosen inplots.

Having proven the theories, we are aiming for the high energy ultra-short UV pulse,which are coupled problems. We first compare the efficiency of UV generation for different
lengths of the hollow waveguides. When we are using the 20 cm long waveguide, thecoupling efficiencies of both 800 nm light and 400 nm light are both close to 60%. (Thecoupling efficiency is characterized as the throughput pulse energy after the waveguide
divided by the input pulse energy before the fiber). The short interaction length limits theconversion efficiency from 400 nm light to 267 nm below 10 %. When we are using 1 m longfiber, the coupling efficiency of 400 nm light drops to 40% and the coupling efficiency of 800

52
nm light is even lower due large decay of the light at long wavelength (Fig. 3.6). However,the long interaction length improves the conversion efficiency to above 30% when the
coupled energy of 400 nm light is below 20 uJ, and close to 20 %, when the coupled energyof 400 nm light is around the saturation energy, above which the ionization of the gas
would greatly reduce the coupling efficiency and conversion efficiency so that more coupled
400 nm light won’t generate more 267 nm light. Therefore, all the results below are basedon the 1 m long waveguide. Then, we compare different diameter waveguides, and expect
larger diameter waveguide would help to improve 267 nm output pulse energy. In principle,
if we keep the coupled intensity the same (below the saturation intensity), we can get more
pulse energy coupled into the bigger waveguide, and if the conversion efficiency is the same,then we can get higher 267 nm output pulse energy. However, since phase matchingpressure is inversely proportional to the waveguide core area, when bigger waveguide isused, the phase matching pressure drops simultaneously, and the conversion efficiency
drops as well. A brief explanation of this phenomena: If we assume the gas density is
proportional to the pressure, then the number of gas atoms inside the waveguide is, where n, A, L, P are gas density, core area, waveguide lengthand phase matching pressure respectively. So the number of UV emitters will not change
under the phase matching condition by using the bigger waveguide of the same length.
Therefore, using the bigger waveguide won’t increase the output pulse energy efficiently,
which is consistent with our experiment result. All the results below are based on the 150
um diameter hollow waveguide.
When we use input 800 nm light and 400 nm light both with pulse duration around30 fs, which are un-chirped input beams, the maximum pulse energy of 267 nm we canobtain is 12 uJ, through a 150 um diameter, 1 m long, waveguide with noble gas Ar. In

53order to obtain more pulse energy, we chirped both 800 nm light and 400 nm light to 100 fs
by the fused silica rods (1 inch long for 800 nm light and 1 cm long for 400 nm light due tothe different GVD), and the maximum output pulse energy of 267 nm increases to 32 uJ,
which is more than 3 times of what people have got before [3.10]. Then, the questions are if
the chirped 267 nm pulse compressible and how short it can be compressed (Here, we meanby using the conventional compressors such as grating pairs or prism pairs. The pulse
shaper is not included).

Figure 3.7 The spectrum comparison between the ultrafast UV light generated from theFWM process through hollow core fiber and the ultrafast UV light generated from the 100
um thick BBO crystal. The three times broaden spectrum from FWM implies that the pulse
duration could be three times shorter than that from the crystal.

54
We first checked the spectrum of the output 267 nm light, since the width of thespectrum determines the transform limited pulse duration. As we can see from Figure 3.7,
the bandwidth of spectrum of the 267 nm light from FWM through the hollow waveguide isabout three times of that from a thin BBO crystal, which means the pulse duration of the
267 nm through hollow waveguide could be 3 times shorter than that from the crystal.
(Here, we only showed the bandwidth at the high output pulse energy; actually we cansacrifice the output pulse energy to obtain broader bandwidth through self-phasemodulation and cross-phase modulation, which have been demonstrated in our
experiments.) Then, we characterized the pulse duration of the 267 nm light after thecompressor built from a pair of aluminum gratings. By properly choosing the incident andangle and distance between the two gratings, the pulse could be compressed to around 10 fs(Fig 3.8), with an efficiency of 50 %. Therefore, we have generated ultra-short high energy
UV pulse, achieving at least three times more pulse energy than before.

Figure 3.8 SD Frog traces of the chirped 267 nm pulse (left) and compressed 267 nm pulse(right). The SD autocorrelation is 13 fs, which implies the retrieved pulse duration is close
to 10 fs.

55
3.5 Summary and future experiments
In this chapter, we first introduced the history and state-of-art techniques of theultrafast UV generation. Then, we investigated the electromagnetic modes allowed in thehollow waveguide, since it is the basis of the mode coupling, mode matching and phasematching problems in the hollow waveguide. Then, we extensively discussed the model
coupling, phase matching and mode matching problems in the ultrafast UV generation. It’s
worth noting that the knowledge of the mode coupling and mode matching apply to thehigh-order harmonics generation as well, which will be discussed in the next chapter.
Finally, we verified the theory of the FWM phase matching in hollow waveguide anddemonstrated the ultra-short (10 fs) high energy (32 uJ) UV pulse in experiments,
achieving at least three times more pulse energy than before.
The future experiments are proposed:First, keep improving the ultra-short UV pulse energy through FWM, andinvestigate the phase transfer problem. We have shown that, in order to increase theoutput pulse energy of UV light, we have to chirp the input pulse of 800 nm and 400 nm
light. In principle, they should be chirped to the same pulse duration to optimize thetemporal overlap. However, this would inevitably introduce the high order spectral phase.
The reason is: from the mathematics of the Fourier transform, we know that the Fouriertransform of a product is a convolution. Therefore, from equation (3.27), the spectral phaseof the 3rd harmonic is the convolution of the spectral phase of the second harmonic and thecomplex conjugate of the spectral phase of the fundamental light. This means that, if one of
the input lights has the flat phase (400 nm or 800 nm) and the other input light has thelinear chirp, the 3rd harmonic would be linear chirped as well, the same sign as 400 nm

56light if 400 nm light is the chirped one and the opposite sign to 800 nm light if 800 nm lightis the chirped one. However, if 400 nm light and 800 nm light are originally having similarpulse durations, this will not be favored by the conversion efficiency; otherwise the pulse
durations would differ a lot after one of them is chirped. So, in principle, the spectral phaseof the 3rd harmonic is the convolution of three linearly chirped spectral phases, which has
high order spectral phase. Therefore, at some point of chirping input pulses, the chirped 3rd
harmonic won’t be able compressed just by a pair of gratings or prisms. It may require bothor even pulse shaper to compress it, although I believe the bandwidth of the spectrum isalways there. Moreover, if we could control the phase transfer to obtain the negatively
chirped UV output, we may just use materials to compress it.
Second, a more interesting experiment is to use the PCF to generate the ultrafast
UV. The reason is that: Although the phase matching equation still holds due to the same
boundary condition, the PCF would change the material dispersion from normal toabnormal [83]. Recall that, the reason that THG in hollow waveguide doesn’t work is thematerial dispersion and the waveguide dispersion are in the opposite sign in equation (3.21)
when the material dispersion is normal and all light have the same mode. If the PCF canchange the sign of material dispersion, then THG can be phase matched with perfect mode
matching. Therefore, the ultrafast UV generation scheme can be simplified from two input
lights scheme to one input light scheme. Furthermore, PCF typically has very tiny core < 50
um, which would bring up the coupled intensity orders of magnitude higher if the input
pulse energy keeps the same. On the other hand, if we keep the coupled intensity the same
which is preferred by UV generation, we can use a lower pulse energy pump, such as a fiber
laser. Then, we can obtain a high rep-rate ultrafast tabletop UV source.
Third, in order to obtain the VUV light, the cascaded processes in hollow waveguide
could be utilized. Actually, the cascaded process happens simultaneously with the FWM

57
process, and efficiency is on the order of 10% [71-72]. We have generated the brightest
ultrafast UV light which means the VUV light generated from the cascaded process isprobably also the much brighter than before. However, the phase transfer problem as wellas the conversion efficiency problem still needs to be investigated to optimize the VUVgeneration. Moreover, VUV generation could be the bridge between UV generation and highorder harmonic generation. What’s the physics there could be investigated as well.

58

Chapter 4High flux soft X-ray generation by HHG

4.1 Introduction
High-order harmonics generation (HHG) describes the phenomena that harmonics
result from a strong nonlinear interaction between the electric field of the incident laser
pulse and the individual atoms, in which the perturbative nonlinear picture doesn’t apply.
The incident laser field simultaneously drives the nonlinear generation of harmonics in alarge number of atoms or molecules all at the phase coherent with the incident laser field.
Thus, resultant harmonics add in phase in the direction of the driving pulse, enabling theefficient high-order harmonics emission in a relatively narrow forward radiation cone [89].
The first observations of HHG date back to the late 1980s. In 1987, the 17th harmonic (14.6nm) was observed when the intense picosecond 248 nm light (3-10 mJ) was focused into
noble gas neon (Ne) [90]. In 1989, the 33rd harmonic (32.2 nm) was observed when thestrong 1064 nm light was focused into noble gas argon (Ar) [91]. In 1990s, due to thedevelopment of the femtosecond laser, especially the Ti:sapphire amplifier, the harmonics
photon energy cutoff had been greatly increased [92]. When high ionization potential gas
helium (He) was used, the cutoff reached the “water window” region [93]. The “new”
ultrafast technique, self-phase modulation (SPM) [94], also provided the way to obtain the

59ultra-short 800 nm driving pulse of 5 fs pulse duration, which enabled the water window
harmonics as well with low driving pulse energy [95]. All works above were based on thetight focusing geometry using the gas jet. The short interaction lengths limited the HHG
efficiency and coherence. In 1998, Andy Rundquist demonstrated the first fully phasematched HHG by using the hollow waveguide geometry [2]. The harmonic flux wasincreased by 100-1000 times compared to the previous tight focusing geometry, because theinteraction distance was greatly increased while the fully phase matching HHG ismaintained in the hollow waveguide. In the waveguide geometry, the fully spatial
coherence of HHG was demonstrated as well, so the high harmonic (HH) beam became a“laser” like beam [3]. In order to increase the photon energy cutoff of HHG in the phasematching geometry, the quasi-phase matching scheme was proposed first [96-97]. Then, in2008 Tenio Popmintchev proposed the new way to increase the cutoff by using driving pulseof the long wavelength [27]. Thanks to the recent development of the ultrafast opticalparametric amplifier (OPA) and optical parametric chirped pulse amplification (OPCPA)
systems, multi-mJ level infrared (IR) pulses are available [29-30]; and the fully phasematched water window harmonics has been demonstrated by using driving pulse of
wavelength at 2 um [28], and the fully phase matched harmonics of photon energy up to 1.6
keV was implemented by using driving pulse of wavelength at 3.9 um [4].
The HHG spectrum is generally consisted of three regions: a steep decrease for the
first few order harmonics, a plateau, and a sudden cutoff for highest few order harmonics,
as shown in Fig. 4.1 [91]. Specifically (Fig. 4.1), in region I, the yield of the 3rd, the 5th andthe 7th harmonic drops fast at a rate as the perturbative nonlinear theory predicts. In
region II, for harmonic orders from the 9th to the 29th, the harmonic yield keeps most a
constant level, called the plateau which can’t be explained by the perturbative nonlinear
theory. In region III, harmonic yield drops dramatically from harmonic order of the 31th to

60
the 35th, and almost disappears at the 35th harmonic order, which is called the harmonic
cutoff. The corresponding photon energy is called the cutoff photon energy, which is 40 eV
in this case.

Figure 4.1 The general feature of the HHG spectrum. The spectrum is generated from noble
13 W/cm2 at gas pressure of 15 Torr. Figure is capturedgas Ar with intensity of atfrom Ref. [91].

61
Simply speaking, the discussion of the HHG is focused on region II and III, and theresearches of HHG source development could be considered as two groups: 1. increase thephoton energy of the HHG cutoff so that broader spectrum could be achieved for
applications; 2. demonstrate the shortest pulse from HHG since it is so far the only way toachieve the attosecond pulse which is the shortest event human beings can capture. Since
the goal of our HHG development is to obtain high flux in soft X-ray region for realapplications (chapter 5), in this chapter we will focus on the discussion of increasing HHG
cutoff photon energy and the flux, which is implemented by using driving pulse of long
driving wavelength and the fully phase matching geometry. We will discuss the physics in
it in the next two sections and present our experiment results after that. The summary andfuture investigation is given finally.

4.2 Single atom picture of HHG
The physics process of HHG is classified into two categories. One is the single atom
response of the noble gas in the intense laser filed. The other one is the macroscopic effectduring the propagation of the harmonic fields in the noble gas as a dispersive andabsorptive medium. We will discuss the single atom picture in this section first, because, it
determines the theoretical limit of the photon energy cutoff of the HHG, which suggests to
use longer wavelength driving pulse and higher driving intensity to obtain higher cutoff
energy. However, it also brings up the problem that single atom yield of HHG scales toin a certain bandwidth and constant laser intensity, which needs to be overcomeby the macroscopic effect, the phase matching of HHG, that is to be discussed in the next
section.

62
The most straight forward way to derive the photon energy cutoff of HHG is from a
simple classical model called “three-step model” proposed by K. C. Kulander and P. B.
Corkum in 1993 [98-99]. In this model, the whole HHG process is divided into three steps
as shown in Fig. 4.2. First, when a linearly polarized strong laser field is focused into thegas atoms. It bends the coulomb potential, and the electron in the ground state may tunnel
out to the free continuum state with initial velocity of zero. Second, the free electron isaccelerated by the driving laser field and gains kinetic energy from it. Third, after theelectric field reverses, the electron is accelerated in the opposite direction. It might returnand recombine with the parent ion, back to the ground state of the atom. The excess kinetic
energy it has gained during the travel in the laser field will be emitted as high energy
photons, which are called high order harmonics. Despite the simplistic approach, the“three-step model” successfully explains the maximum photon energy, called single-atom
HHG cutoff, which could be obtained in the HHG,
(4.1)
where Ip is the ionization potential and Up represents the average oscillation energy of afree electron in the field, which is commonly called the ponderomotive potential,
are driving laser’s intensity and wavelength.

and

63

Figure 4.2 The cartoon of the classical schematic of HHG captured from Ref. [100]. The
electric field of an intense laser extracts an electron from an atom through tunnel ionization.
The laser field then accelerates the electron, with a small fraction of the electron returning
back to the ground state of the same atom, liberating its excess energy as a high-energy
photon.

We will use the classic model to obtain equation (4.1), and then use the quantum
mechanics model from M. Lewenstein [101] to confirm this equation and explain the reason
why the long wavelength driving HHG was not expected to have efficient throughput at theearly age of HHG.
The first step of the “three-step model” is called the tunneling ionization. The
ionization rate in tunneling ionization regime can be approximated by the AmmosovDelone-Krainov (ADK) model [102]. Since the ionization rate is a critical factor in the phasematching of HHG, we will leave the detailed discussion of the ionization to the next section.
In the quantum mechanics model from M. Lewenstein, it is described by a transition dipole
moment from ground state to the continuum state as shown laser.

64
The second step of the “three-step model” directly determines the kinetic energy afree electron can gain from the laser field. Consider an electron tunneling out from groundstate to continuum state at t0 in the electric field), where the tunneling could(

be considered as instantaneous compared to the duration of the electron traveling in thelaser field before recombining, the equation of the motion of the free electron follows the
Newton’s second law:
̈

(

(4.2)

)

where e is the electron charge, m is the electron mass, x is the displacement of the electronwith respect to the parent iron,is the initial phase with respect to the peak of the laser

filed when the electron tunnels out at t0. The laser field is assumed to be linearly polarized,
so this is just a one dimensional second order derivative equation. With the initial condition
that ( )and ̇ ( )( )[[101, 103], the solution of equation (4.2) is easily obtained

(

)

(

)

(

)

(

)]

(4.3)

Then, the kinetic energy is written as,

( )

[

where

set

(

)

(

)

(

(

)]

)]

(4.4)

is the ponderomotive potential,is the intensity of theis the wavelength of the driving laser. The chosen ofis arbitrary, so we caṅlaser, and[̇

Then, the relationship of between the trajectory of the free electron ( ) and theinitial phaseis given by

65
( )

[

(

)

(

)

(

(4.5)

)]

Figure 4.3 The electron trajectories of different initial phase with respect to the peak of thelaser field at the tunneling time . The driving laser is assumed to be at 800 nm. Y axis of
the plot is in the unit of the ponderomotive radius,.

A few trajectories of different initial phases

are plotted in Fig. 4.3 by equation(4.5). From the trajectories, we can see that: First, whencome back to its parent iron, which equals to ( )has solution after, the electron could; when, the electron cannot come back. Second, the highest kinetic energy to be gained by anelectron accelerated in the continuum can be determined by solving ( )phases of the laser electric field. The slopes of the trajectories in Fig. 4.3 at ( )for various, labeledby the dashed lines, are indicatives of the velocity of the electron at the point ofrecombining with the ion core and therefore represents the kinetic energy available. From

66
numerical calculation, it is found that 3.17Up is the maximum kinetic energy to be acquired,
which occurs when the electron has been ionized at a phase of the laser electric field of 17º.
And this is the origin of the photon energy cutoff of HHG from the classical model. Third,from 0º to 17º, the gained kinetic energy is increasing and then falls off beyond an electricfield phase of 17º. It means that there are two trajectories that may impart the same kinetic
energy to the free electron ionized at the initial phase in opposite sides of the phase of 17º.
Trajectories of the free electron acquiring the same kinetic energy are the so-called long andshort trajectories, which are defined as

the long trajectory andshort trajectory that can be easily understood by their duration before the recombination.
The third step of the “three-step model” says the electron recombines with theparent iron, and the transition from continuum state to the ground state happens.
Therefore, besides the gained kinetic energy from the laser field, the combining energy
which is called the ionization potential

will also be emitted in the third step. So the

maximum energy, cutoff, of the HHG is given by equation (4.1).
The classical model explains (4.1) in an elegant way, however, it doesn’t show thequantum nature of the ionization, the quantum diffusion when the electron travels in thelaser field and the recombination. Therefore, it can’t explain the questions such as HHG
single atom yield and HHG intrinsic phase. So we will follow quantum mechanics modelfrom Lewenstein to recheck the whole process.
The Lewenstein model calculates the time-dependent dipole moment which is
responsible for the HHG. The electric field is calculated from the second derivative of thedipole. Finally, a Fourier transform reveals the spectrum. Derivation of this model begins
with the Schrödinger equation in the length gauge| ( ⃗ )⟩[( ⃗)( ) ]| ( ⃗ )⟩

(4.6)

67
where, ( ⃗) is the molecular potential and( ) is the laser field. Next, in writing the

wavefunction, several approximations are made. First, only the ground bound state | ⟩ isconsidered. Second, depletion of the ground state is neglected. Up should be large enough, sothat, but still below the saturation level, Usat, when all atoms ionize during theinteraction time. So the ionization is in the tunneling regime. Third, in the continuum, theelectron can be treated as a free particle moving in the electric field with no effect of ( ⃗).
Mathematically, it is expressed as⟨ ⃗| ⃗| ⃗⟩

(⃗

(4.7)

⃗⃗⃗⃗)

where the continuum-continuum transitions are rewritten to reflect only the most singularpart. | ⃗⟩ represents the continuum state of velocity ⃗ . Then, the wavefunction can be
written as| ( ⃗ )⟩
where ( )[ ( )| ⟩∫is the ground state amplitude and(4.8)⃗ ( ⃗ )| ⃗⟩]( ⃗ ) is the amplitude of the continuum

state. Insert (4.8) into (4.6) and consider (4.7), ⟨ | ⃗⟩as well as

in themomentum representation. The Schrödinger equation for ( ⃗ ) reads as(⃗ )where ⃗( ⃗)) (⃗ )

(

( )

(⃗ )

( )

( ⃗)

(4.9)

⟨ ⃗| | ⟩ denotes the atomic dipole matrix element for the bound to freetransition and( ⃗) is the component parallel to the polarization axis. The solution toequation (4.9) is

(⃗ )

∫

( )

[⃗

⃗( )

⃗( )]

{

∫

[ (⃗

⃗( )

⃗( ))

]} (4.10)

68
where ⃗( )() is the vector potential of the laser field. Using equations (4.8)( )and (4.10), the x-component of the dipole moment is( )⟨ ( )| | ( )⟩Using the canonical momentum ⃗

( )

∫

∫

⃗

∫

(4.11)( ⃗) ( ⃗ )⃗⃗⃗( ) , dipole moment becomes

( )

(⃗

⃗( ))

(⃗

⃗( ))

∫

{ [⃗

⃗(

)]

}

[

(⃗

)]

(4.12)

where

(⃗

)

(4.13)

Equation (4.12) shows the quantum mechanics version of the three-step model.
( )(⃗⃗( )) describes ionization at timeas the step 1. The electronic wave

function is then propagated until the time t and acquires a phase factor equal to[(⃗)], where ( ⃗) is the quasiclassical action, as the step 2. This phase iscalled the intrinsic phase of HHG. ( The effects of the atomic potential are assumed to besmall between t' and t based on the assumption 3, so ( ⃗) actually describes the motionof an electron freely moving in the laser field with a conserved canonical momentum ⃗,although it does incorporate some effects of the binding potential through its dependence onIp.) Finally, the recombination at time t is described by
Since

(⃗

⃗( )), as the step 3.

is of the order of one period of the laser field which as in Fig 4.3, thequasiclassical action varies much faster than the other factors entering Eq. (4.12).
Therefore, the major contribution to the integral over ⃗ in Eq. (4.12) comes from thestationary points of the classical action

⃗

(⃗

)

(4.14)

69
Considering

⃗

(⃗

)

⃗( )

⃗( ), equation (4.14) just means the most the dominant

contribution to the harmonic emission comes from the electrons which tunnel away fromthe nucleus but then reencounter it while oscillating in the laser field, which is actually theassumption of the three-step model. By (4.14), the integral over ⃗ in Eq. (4.12) might beperformed using a saddle-point method. The result is

( )

(

∫

)

(

(

)

( ))

(

(

)

(

))

(

)

[

(

)]
(4.15)

where

(

)

[

( )

(

)]

is the stationary value of the component of the

momentum in the direction the of the polarization of the laser field. The value of thequasiclassical action in Eq. (4.15) is given by

(

)

∫

(

(

))

(

)

[

( )]

( )

(

(4.16)

)

where

( )

( )

(4.17)( )The Fourier transform of the dipole moment ( ) gives the spectrum of the HHG. We don’t
go further than this, since (4.16) and (4.17) already show the cutoff energy and single-atom
HHG field for our interests. More detailed derivation could be found [101].
First, in equation (4.16), we can see that ( ) determines the variation ofa function of t in the unit of Up. Since the action

(

(

) as

) is the integral of the kinetic energy

over t", the maxima of ( ) correspond to the maxima of the kinetic energy gain of theelectron at t. The detailed derivation shows that the maximum of 2 ( ) in the unit of Up is

70
the maximum kinetic energy, and maximum of 2 ( ) is 3.17. So from the quantum model,
we can obtain the same HHG cutoff as the result obtaining from the classical model.
Second, from the first factor in equation (4.15), we can see that the dipole moment( ), so the single atom yield of HHG is proportional tosingle atom yield of HHG is proportional to. Since, it means theas well. However, this is just count fromsomething called from quantum diffusion from free electron. An additional

factor arises

from the conversion factor from photon number to energy since HHG cutoff is scaled by
Totally, the single atom yield of HHG is scaling around.

.

More precise theoretical solutions of the time dependent Schrodinger equation(TDSE) showed slightly worse scaling factor of[104-106], and Shiner et al.

experimentally obtained an even worse scaling factor of[107]. These results mean that,when the driving wavelength is doubled, the single-atom harmonic yield is expected to be
decreased by 50-100 times. Although the longer wavelength driving pulse is expected to
produce more energetic harmonics, the lower single-atom yield “seems” to be the problem.
However, all things considered so far are based on the single-atom response. In the next
section, we’ll show that the phase matching could compensate low single-atom yield by theincrease of the gas pressure-length product and transparency in the case of longer driving
wavelength.

4.3 Phase matching HHG in hollow waveguide
The macroscopic effect of HHG could be considered as the generation and the
propagation of the high-order harmonics from the fundamental light in the noble gas as adispersive and absorptive medium. Assuming the harmonics are all propagating in the z

71
direction, under the slowly varying envelope approximation (SVEA), the wave equations of
the fundamental light and qth order harmonic can be written as,

(

(

where

(

)and

)

(

)

(

(

)

(

)

(

)

)

(4.18)()((4.19))) represent the slowly varying amplitude of the fundamental lightand qth order harmonic propagating in the gas medium with wave vectors ofrespectively.

and) is the nonlinear polarization of the gas medium, which should(

calculated by the quantum mechanics instead of assuming to be proportional to [ ()] asin the perturbation theory. If we confine the discussion in the z-direction and neglect the
spatial variation in the x-y plane, (4.19) can be written as
(

where

)

[(

) ]

[

]

(4.20)

is the phase mismatch between the fundamental light and qth order harmonic.

From (4.20), we can see that as the fundamental light propagates long z direction, it ionizes
the atoms at position z which generate the qth order harmonic of spatial phase of,coherent with the fundamental light at z. After being generated, the qth order harmonichas its own propagation speed, associating with the spatial phase ofdifferent from the spatial phase of the fundamental lightwhich might be. Therefore, the de-phasing

between the fundamental light and qth order harmonic in propagation limits the nonlinear
macroscopic HHG efficiency. The maximum conversion efficiency from the fundamental
light to the macroscopic high-order harmonic yield happens when the phase matching HHG,
, could be maintained over long enough distance before touching the limit set by the72reabsorption of the high-order harmonic field from the medium. Equivalently, it meansthat the high-order harmonic light should travel at the same speed as the fundamental
light in the dispersive medium over enough distance.
Just based on the discussions of HHG phase matching so far, we can compare thedifferent geometries that have been used for HHG. Typically, there are two types ofgeometries being used for HHG: tight focusing geometry and plane wave geometry. As
shown in Fig. 4.4 (b), the tight focusing geometry causes a large Gouy phase shift throughthe focus, which is particularly problematic for HHG because of the large difference inwavelength and divergence of the harmonics compared with those of the driving laser [108].
Although, by considering the intrinsic phase as an effective additional wave vector added tothe overall phase matching of the HHG process, the interplay between Gouy phase andintrinsic phase could enable the phase matching happen at some point, for example, when
the gas jet is placed at the position where the laser is slowly diverging from the focus, theGouy phase induced phase mismatch could be compensated by the intrinsic phase [109-110],the short confocal length would limit the interaction distance since the laser intensity drops
fast beyond confocal length, which in turn would change the ionization rate or plasma
generation, an critical factor in phase matching problem as shown later. Therefore, theconversion efficiency in the tight focusing geometry is low although it is relatively easy to
be implemented.

73

Figure 4.4 Physical pictures of HHG phase matching in different geometries. (a) Phasematched signal growth ensures that harmonic emission from many atoms adds together
coherently over an extended medium. (b) The tight focusing geometry is associated curved
laser and X-ray wavefronts, which limits the phase matching distance within the confocal
length and has to deal with the problem of Guoy phase shift. (c) The waveguide or cellgeometry minimizes the curvature of the laser and HHG wavefront. The nearly plane wave
propagation allows long distance phase matching HHG and has no Guoy phase or CarrierEnvelope phase (CEP) problem. The Figure is captured from Ref. [100]74Contrarily, the plane wave geometry maintains the phase matching condition overan extended distance, making it possible to build up a fully spatially coherent and fully
phase-matched high-harmonic beam (Fig. 4.4 (c)). Especially, by using the hollow
waveguide geometry, the guided mode ensures the near plane-wave propagation of thedriving beam, maintaining a flat uniform phase and high peak intensity over a large
interaction distance, avoiding any variation in the intrinsic quantum phase. The waveguidealso makes it straightforward to create a region of controlled, confined and wellcharacterized gas pressure, with optimal differential pumping at the entrance and exit of
the waveguide.

This feature is more critical for achieving absorption-limited phasematched HHG flux for longer driving wavelengths, since the high-pressure gas medium
together with an abrupt transition to vacuum becomes increasingly important as the HHG
driving laser wavelength is increased. Therefore, we will focus on the phase matching HHG
in the hollow waveguide geometry in the following discussion, and this is consistent withthe discussion in the ultrafast UV generation.
The near-plane-wave geometry can also be realized using a large focus and highlaser energy when the confocal parameter of the beam is much longer than the gas medium.
Such a high-energy HHG geometry has been effectively used in experiments that benefit
from high peak power rather than high repetition rate. However, considering theapplication end, the high repetition rate is always preferred, so we will not discuss it in thefollowing paragraph.
We have extensively discussed the modes inside the hollow waveguide in Chapter 3.
The wave vector (real part of the propagation constant) of mode EHnm is given by (3.20) as
( )

[

(

) ]

( )

(4.21)

75
whereis the neutral atomic density, ( ) contains the gas dispersion function,neutral atomic density, andis theis the classic electron radius. So, the wave vector is the sumof four sources of dispersion: vacuum, gas, free electrons (plasma), and waveguide. In
contrary to the UV generation, HHG has to be associated with the ionization as describedby the three-step model, so the third term can’t be neglected in HHG. If we assume thedriving laser is coupled properly into the hollow waveguide mode EH11 and the ionization
fraction of the gas is ( ), then, the wave vector of the driving laser is

[

( )]

(4.22)( )where P is the pressure of gas in the waveguide,

is the difference between the refractive

index of the gas at specific driving wavelength at one atmosphere and the refractive index
of vacuum, which is one.

is the gas density at one atmosphere. Obviously, we haveused the same relationship that P~N as in Chapter 3 to obtain (4.22). For the wave vectorof the qth order harmonic, considering the phase velocity of the high harmonics is close to
the speed of light in vacuum because the refractive index for high photon frequencies (farabove most atomic resonances) is close to unity, it could be written as,( )(4.23)

Here, we neglect the waveguide term for two reasons: first, the HHG is very sensitive to theionization, the ionization is stronger at the center of the hollow waveguide than close to theboundary, so it hard to describe it in a single mode; second, when we consider it asis only, itof the sum of wave-vectors of the fundamental waveguide terms. If the mode

number is not huge (which doesn’t make sure because of the mode matching), this term

76should be very small. So, the phase mismatch of HHG in qth order harmonic in the hollow
waveguide geometry is

( )

{

[

( )]

) ( )

(

}

(4.24)The physics explanations of equations (4.22), (4.23) and (4.24) are: from (4.23), thehigh harmonic travels at a speed close to the speed of light in vacuum, although it isslightly off by the plasma term and waveguide term; from (4.22), the speed of thefundamental light is reduced by the normal dispersion from the neutral gas and increasedby the waveguide dispersion and plasma dispersion. The change of speed is quite dynamic,
depending on the ionization induced by the driving pulse; from (4.24), in order to makephase matching happen, the modified speed of the fundamental light should be the same as
the speed of the high harmonic, which is close to the speed of light in vacuum. And since thespeed of the fundamental light dynamically depends on the ionization ( ), from (4.24),
phase matching actually requires the term in the braces to be positive which sets the upper
limit of ( ), called critical ionization level. Below the critical ionization level, phasematching is possible, and higher ionization fraction is preferred since the ionized atoms are
the real emitters of high harmonic, while, above the critical ionization level, phasematching is impossible and no macroscopic harmonic emission would happen.
From the discussion above, we can see the critical ionization levelplay animportance role in the phase matching. From now on, we will show it actually sets thephoton energy cutoff of HHG in the phase matching condition. From (4.24), the critical
ionization level is given by

[(

)(

)

]

(4.25)

77
when, equation (4.25) can be approximated to be((4.26))

From (4.26), the critical ionization level monotonically decreases as the driving laser
wavelength increases. Put the parameters
as well as the numbers ofandof specific gas species and driving wavelengthin Eq. (4.26), we can see that the values forare onthe order of a few percent in the near-IR region, approximately 4% for Ar, 1% for Ne, and
0.5% for He, respectively, at driving laser wavelengths of 0.8 um and decreased to about
0.01% in the mid-IR region. Sincechanges by only less than 1% when the wavelengthchanges from near-IR region to mid-IR region, equation (4.26) can be approximately written
as(4.27)The plot ofwith respect to the driving laser wavelength

for noble gases Ar, Ne, and He

are shown in Fig 4.5. (a) [27].
After obtaining the values of the critical ionization level in phase matching HHG,
the further and more important question is what’s the maximum intensity we can use in
HHG? On one hand, since the critical ionization level sets the upper limit of the ionization
fraction in phase matching condition, if the driving intensity is too high, it will over-ionize
the atoms and exceed the critical ionization level, so the high harmonics can’t coherently
build up; on the other hand, we always want to have as much ionization as allowed by thephase matching, since the ionized particles are real emitters. Although the low driving
intensity allows phase matching (see chapter 3), the no-emitters’ phase matching means nohigh harmonic, which actually changes back to the theory in Chapter 3. Therefore, in78
principle, to obtain efficient high harmonics, we want to apply the driving intensity right
below what would introduce the ionization associated with the critical ionization level.

Figure 4.5 (a) Critical ionization levels for the noble gas Ar, Ne and He as a function of thedriving laser wavelength. (b) Predicted laser intensity required to reach the critical
ionization level at the peak of the laser pulse on axis (solid line), and when modal averaging
is taken into account (dashed line). These figures are captured from Ref. [27].

79
Moreover, this intensity will determine the phase matching cutoff of HHG. In the sing-atom
response, we have seen that the, which means the single atom HHG

cutoff can be scaled by both the laser intensity and the laser wavelength; they are both freeparameters. However, in the phase matching HHG, at specific wavelength, the critical
ionization level determines the maximum intensity we can use, so the intensity is no longer
a free parameter. In order to get the matching cutoff of HHG, we have to find therelationship between

and, then put it into the single-atom formula to obtain the new

cutoff formula, which should have only one free parameter.
To obtain the maximum intensity the critical ionization level allows at specificwavelength, we need to treat the ionization more carefully. Generally, the ionization could
be categorized as multiphoton ionization and tunneling ionization by Keldysh parameter ,defined as√

ionization, and When

[4.30]. When, the ionization is described by the multiphoton, the ionization is described by the tunneling ionization. The

detailed discussion on the photon-ionization is beyond the content of this thesis. V. S. Popovreviewed the photon-ionization in a strong laser field in Ref. [111]. Here, from ourexperiment data as shown in Fig. 4.6, we can see that the photoelectron momentum
distribution in the multiphoton ionization is more isotropic, and photoelectron momentum
distribution in the tunneling ionization is more linearly parallel to the polarization of thelaser field. Obviously, HHG favors the tunneling ionization since the ionized electrons haveto recombine with the parent iron. So, we will restrict the ionization process in thetunneling region.

80

Figure 4.6 Photoelectron momentum spectra of the Xe ionization with respect to the
Keldysh parameter .

In the tunneling region, the ionization of an atom is well described by the ADK
tunneling ionization model, particularly for the longer wavelength driving lasers. The ADK
ionization rate is given by

) | |

((

√

|
(

|

(

) (

(

)

√

)

)

(

(

)

)

(4.28))

where Z is the ion charge,

is the E-field of driving laser, and l and m represent theangular momentum and magnetic quantum number, respectively. Factors

and (

)

are given by

|

|

(

where

(

)

and

(

)

.

(
| | (|) ()( | |)|)( | |))

(4.29)(4.30)

81

Figure 4.7 (a) The ionization fraction of gas neon in a 30 fs Gaussian shaped pulse withpeak intensity of. The wavelength of the pulse is changed from 800 nm to1300 nm to 2000 nm. Although the ionization step changes due to the different cycle
duration, the ionization rates as a function of time almost keep the same for different
driving wavelength. (b) The ionization fraction of gas neon in a 30 fs Gaussian shaped pulse
with wavelength at 1300 nm. The peak intensity changes fromto. Ionization fraction changes more than an order of magnitude with theintensity change. If the intensity drops to, the ionization faction will be
reduced to 0.05%, another order of magnitude lower, which is not easily seen in this plot.
Thus, 50% of intensity change would introduce orders of magnitude change of the ionization
fraction.

82
From ADK model, we can demonstrate that the tunneling ionization rate in HHG isalmost wavelength independent, while strongly depends on the laser intensity (Fig. 4.7).
Considering these two points, we can find the driving laser intensity required to reach thecritical ionization level decreases slightly as the driving wavelength increases (Fig. 4.5 (b)).
As a result, the phase matching HHG cutoff scales approximately as(4.31)The relationship is shown in Fig. 4.8 and has been demonstrated by the experiments [100].

Figure 4.8 Theoretical HHG phase matching cutoff as a function of driving laser
wavelength for three-cycle (dashed lines) and eight-cycle (solid lines) pulses. This global
phase matching picture has been validated experimentally (solid circles) using several laser
wavelengths and nonlinear media. The figure is adapted from [100].

83
From equation (4.31), although the freedom of scaling up the HHG cutoff by driving
laser intensity is restricted by the requirement of the phase matching condition, the phasematching HHG cutoff still could be scaled up by the increase of the driving laser
wavelength. People have known that the HHG cutoff from 800 nm driving laser is limited
less than 150 eV. Therefore, in order to obtain bright soft X-ray light (> 100 eV), using
longer driving wavelength is proposed.
The last question is whether the phase matching process could help the HHG tobalance the low single-atom yield at long driving wave length, since the single atom
response is scaled as

or even worse. From E. Constant’s semi-analytical model, for theqth harmonic, the number Nout of photons emitted on axis per unit time per area is [112]:

(

where

is the gas density,

harmonic frequency

)

[

(

)

(

)

(

)]

(4.32)

is the is the amplitude of the single atom response at the. The absorption length iscross-section, and coherent length is, whereis the ionization. From this model, we can have a few

insights to the HHG flux from long wavelength driving pulse.
First, the asymptotic behavior ofmatching is achieved so

can be obtained by assuming the phaseand the gas medium length (hollow waveguide length). Under the assumptions, the asymptotic behavior of (4.32) is(4.33)
From equation (4.33), we can find that, although

would decrease due to the single

atom response at long driving wavelength, the absorption cross-section tends to decrease atlong wavelength as well. So, from asymptotic behavior, if we can extend the harmonic

84
emission to the photon energy region that the absorption is much lower than the EUV
region. The decrease from single atom response might be balanced by the decrease of thereabsorption.
Second, combining Eq. (4.24) and (4.26), and considering the critical ionization levelat long wavelength is typically less than 1%, the phase matching pressure around thecritical ionization level is(4.34)It means the phase matching pressure is much higher when we use long driving
wavelength, typically from 1 atm to 80 atm when the wavelength change from 1300 nm to4000 nm.
Third, from Eq. (4.31), there is another important result: in order to approach theasymptotic behavior,andare required. Since the absorption

cross-section greatly reduces at the high energy region, theincreases significantly. For

example, close to keV region, for gas He, it requires a medium length as large as tens ofcentimeters to approach the asymptotic value [27].
In summary, by using the long driving wavelength, it requires a long interaction
distance with high pressure. It means the hollow waveguide geometry is the best choice inexperiment to achieve the high photon flux. The large pressure length product in hollow
waveguide geometry would make the photon emission from HHG to the asymptotic value in
Eq. (4.33), where the reduction of absorption might balance the low single atom yield.
Finally, we cite Tenio Popmintchev’s prediction of the HHG flux from Ref. [27] to thank his
contribution to the long wavelength driving HHG.

85

Figure 4.9 Absorption-limited intensity of phase-matched HHG in a bandwidthcorresponding to a single harmonic (A) and a linewidth of 1% bandwidth (B) at the phasematching cutoffs for laser wavelengths between 0.8 um and 10 um. The curves arenormalized to the phase-matched HHG emission at 0.8 um.

86
4.4 Experiment setup and results

Figure 4.10 Schematic diagram of the high energy high efficiency OPA.

As discussed in the last two sections, the scheme of high flux soft X-ray generation
has two steps: first, using OPA to obtain the longer driving wavelength at 1.3 um; second,using the hollow waveguide geometry to generate the fully phase matching high-orderharmonic. Then, the characterization of the soft X-ray flux is done by using the soft X-ray
spectrometer and CCD Camera. The details are showing bellow.
A homebuilt 3-stage OPA that achieves greater than 40% conversion of the 0.8 µm
light into 1.3 μm (signal) and 2 μm (idler) beams to are used to obtain the high-quality 1.3
μm pulses to drive soft X-ray HHG (Fig 4.10). Type II BBO crystals are used as thenonlinear crystals in the OPA to amplify the seed from the white light generation process.
The reasons for using Type II BBO are: First, compared to the type I phase matching, type87
II phase matching reduces the effect from the group velocity mismatch, which enable amuch longer interaction distant compare to the splitting length defined by the groupvelocity mismatch, so it gives a higher conversion efficiency (Fig. 4.11 (a)). Second, although
the type I phase matching allows broader phase matching bandwidth, which supports
shorter pulse duration, the tunability of the output wavelength is much worse than type IIphase matching (Fig. 4.11 (b)). Since the HHG cutoff is determined by the driving
wavelength, the tunable driving wavelength would make the generated soft X-ray from
HHG also tunable in photon energy, which has great practical importance. So we decided
choose the Type II BBO, aiming for high conversion efficiency and high tunability otherthan the ultra-short pulse duration. Crystal thicknesses of 2.5 mm, 2.0 mm and 1.5 mm
were used in the first, second and the third OPA stage respectively.
The high energy OPA design is the standard one used by many groups worldwide,similar for example to the HE-TOPAS from Light Conversion. Due to the excellent beam
quality of the high-energy regenerative amplifier, very high conversion efficiency of 40% is
possible in the OPA. For these measurements, the total pulse energy used to drive the OPA
was 8.5 mJ. In the first stage, 1% of the pulse energy is used for white light generation andthe first stage pump. The second stage uses 9% of the pulse energy as the pre-amplification
pump, while 90% of the pulse energy is used to pump the final stage. The output OPA
signal pulse energy is 2.25 mJ, with a spectral full-width of greater than 200 nm centeredat 1.3 μm. The pulse energy of the idler is 1.4 mJ, giving a total conversion efficiency in the
OPA from pump to signal plus idler of approximately 43%. The good coupling of the OPA
mode into our soft x-ray waveguide and resultant record soft x-ray flux (discussed in thefollowing section) demonstrates that the OPA mode is of high quality, and indeed was the
motivation for the new regenerative amplifier design. Additionally, the broad spectrum of
the 1.3 μm light originates from white light generation, and supports an ultrashort pulse

88
duration of 30 fs (characterized by a second harmonic FROG measurement) (Fig.4.12). As
expected in parametric down-conversion of multi-cycle ultrafast pulses, this signal pulse
duration is considerably shorter than the 45 fs 0.8 µm pump pulse.

Figure 4.11 (a) Signal pulse evolution for a BBO type II OPA, with pump wavelength of 800
nm and signal wavelength of 1500 nm, for different lengths L of the nonlinear crystal. (b)Phase matching bandwidth for a BBO OPA at the pump wavelength of 800 nm for type I
phase matching (solid line) and type II phase matching (dashed line). Figures are captured
from Ref. [30].

89

Figure 4.12 Pulse characterization of 1.3 um from OPA by the SHG FROG. The pulse
duration of 1.3 um light is 29.6 fs, which is shorter than the pulse duration of the pumppulse.

Bright soft X-ray harmonics are generated by focusing the 2 mJ pulse energy, 1.3
μm, OPA signal output into a gas-filled hollow waveguide (length 1 cm and diameter 150μm) using an f = 25 cm lens. The coupling efficiency is more than 50% - even with highpressure gas in the waveguide, so that the driving pulse peak intensity inside of the
waveguide can reach 5 × 1014 W/cm2, with the energy well coupled into an EH11 mode. The
waveguide is filled with Argon (Ar) or Neon (Ne) to generate the soft X-ray. Since each
noble gas has a different refractive index and ionization potential, each gas optimizes at adifferent phase matching pressures and laser intensity, and also extends efficient HHG tosome maximum phase matched photon energy cutoff. By controlling the gas pressure inside
the waveguide and the 1.3 μm pulse energy before the waveguide, we can optimize the90
HHG flux at the phase matching pressure, as shown in Fig. 4.13. For a 1 cm long, 150 µm
diameter waveguide and 1.3 μm driving lasers, the optimized pulse energies, phasematching pressures and phase matching cutoff photon energies are 0.95 mJ, 700 torr and
110 eV for Ar, and 1.6 mJ, 1300 torr, and 200 eV for Ne. By accounting for the absorption of
the metal filters, the efficiencies of the spectrometer (Hettrick Scientific), and the quantum
efficiency of the CCD camera (Andor, Inc.), the photon flux of the harmonics is estimated atmore than 106 photons/pulse in 1% bandwidth up to 200 eV (Fig. 4.13), corresponding to 109
photons/s in 1% bandwidth at kHz repetition rates. This photon flux represents anapproximately 3 orders of magnitude increase compared with past work (which used lower
repetition rates or non-phase matching geometry). Additionally, a long term stability test
demonstrates that the HHG flux is stable over many hours, limited primarily by beam
pointing drift (when no beam pointing stabilization is used).

Figure 4.13 Flux characterization of the optimized fully phase matched HHG emission froma 1 cm long, 150 μm diameter with driving wavelength of 1.3 um. Noble gas Ar and Ne are
used.

91
Since numerous experiments have been implemented utilizing 0.8 µm-driven EUV
harmonics, we compared the optimized fully phase matched HHG flux driven by 0.8 µm
light with the HHG flux driven by 1.3 μm light. We used the same waveguide forcomparison, since the geometry and quality of the waveguide might affect the harmonic
yield. The pulse energy used for the 0.8 µm-driven HHG from Ar, Ne, and He is 0.40 mJ, 1.5
mJ and 2.2 mJ, respectively. We find that at 100 eV photon energy, the 1.3-μm-driven HHG
flux is within a factor of two of the flux of 0.8 µm-driven HHG, while above 150 eV, the 1.3µm HHG flux becomes much more efficient. This shows that many applications based on
EUV harmonics can now be extended to the soft X-ray region. Although our laser/OPA
system did not provide sufficient pulse energy to generate fully optimized phase-matched
HHG flux from He, we were in-fact able to generate coherent light up to ~300 eV from He.
However, the HHG flux obtained was significantly lower (by ~100x) than the data of Fig.
4.14, which still represents a record HHG flux of 107 photons/s in 1% bandwidth at 300 eV,
at kHz repetition rates. We predict that a further modest increase in pulse energy (by ≲2x)
will allow us to generate flux comparable to that of Fig. 4.14 up to the water window using
He as the nonlinear medium.
Finally, it is worth noting that high harmonics peak to peak modulation is muchshallower when driven by 1.3 um than driven by 800 nm. The spectroscopic study (nextchapter) also demonstrates that harmonic spectrum using Ne gas is a true coherent
broadband supercontinuum - an observation which is consistent with other recent work
demonstrating that phase matched soft X-ray HHG driven by mid-infrared lasers naturally
emerges as an isolated, attosecond-duration, burst [113].

92

Figure 4.14 Flux comparison of the optimized fully phase matched HHG emission from a 1
cm long, 150 μm diameter, waveguide driven by 1.3 μm and 0.8 µm light in various noble
gases (Ar, Ne, and He). The vertical arrows indicate the maximum predicted phase matched
HHG energy cutoff for each gas and laser wavelength as predicted in Fig.4.8. The HHG flux
obtained using a 1.3 μm driving laser is comparable to that achieved using 0.8 µm.

93
4.5 Summary and the future investigation
In this chapter, we introduced the HHG history first. Then, to answer the question
how to generate the high flux high energy harmonics, we started from the theoreticalmodels for single atom response of HHG in both classical and quantum pictures. Although
they both agree that HHG photon energy cutoff could be increased by increasing the
wavelength and the intensity of the driving pulse, the low single atom response hits the
HHG from long driving wavelength would have low flux. This problem is overcome by thephase matching HHG in hollow waveguide. The decrease of the absorption cross-section at
higher photon-energy, associated with the increase of the pressure length product requiredby the phase matching condition, might balance the low single atom yield in a fully phasematching hollow waveguide HHG. Promoted by this idea, we have achieved a photon flux ofmore than 109 photons/s/1% bandwidth up to 200 eV at kHz repetition rates by using thehigh quality 1.3 um driving light from a high efficiency OPA. The flux is 1000 times higher
than past work. The stability and brightness of this tabletop soft X-ray source indicate
great potential for future time resolved molecular and material dynamics studies.
Future investigations:
Preliminary studies from our research have shown that, by tapering the entrance ofthe hollow waveguide (Fig. 4.15), the HHG flux from Ne with 1.3 um driving light can beincreased by 3 times. The taped entrance doesn’t help to increase to the coupling efficiency
but increases the phase matching pressure. This suggests that the increase of the HHG flux
comes from the higher phase matching pressure which is associated with more emittersinside the waveguide. From the phase matching theory, the increase of the phase matchingpressure also means the change of the modes inside the waveguide as well. A new field

94
distribution (mode matching) with high phase matching pressure (phase matching) may
help the HHG flux, which is to be studied.

Figure 4.15 The photo of the taper-end hollow waveguide. The taper is 1.5 mm deep, andthe outer diameter of hollow waveguide is 1.2 mm, so the angle is 43 deg.

Second, a more interesting experiment is to use the PCF to generate the HHG. PCF
typically has very tiny core < 50 um, which would bring up the coupled intensity orders ofmagnitude higher if the input pulse energy keeps the same. Therefore, this is the way toobtain high rep rate high order harmonic which is the ideal tool to applications, including
lithography, imaging and spectroscopy. More details can be seen in Chapter 3.

95

Chapter 5

NEXAFS from Soft X-Ray HHG and Xe photo-ionization transient absorption
spectroscopy

5.1 Introduction

X-ray absorption spectroscopy measures the absorption of X-rays as a function of theincident X-ray photon energy. The general features are (see Fig. 5.1(b)): 1. With theincrease of the photon energy, the overall absorption decreases. 2. A few saw-tooth-likefeatures with a sharp rise at discrete energies, called absorption edges, are present in theoverall decrease of the X-ray absorption. These absorption edges are corresponding to thephoton-electron transition originated from the innermost electron orbitals to the freecontinuum (ionization of core orbitals) or to unoccupied bound levels just below theionization limit, for example, originated fromelectron orbital is called K-edge, fromelectron orbital called L-edge (LI refers to 2s, LII to 2p1/2, and LIII to 2p3/2 orbitals), andM, N, etc refers to the corresponding higher lying bound core shells. 3. Fine structures (Fig.
5.1 (c)) are shown around the absorption edges, which could be divided into the X-ray
absorption near edge structure (XANES) and the extended X-ray absorption fine structure(EXAFS). Different references name these fine structures differently, so from now on, the
X-ray absorption near edge structure (XANES) is called the near edge X-ray absorption fine96
structure (NEXAFS) in this thesis, which is more consistent with EXAFS and is used in thebook [33]. Simply speaking, the general feature 1 tells us the overall absorption behavior of
all elements, which could be derived from quantum mechanics. And, it also explains thepossibility and one of the motivations of the long wavelength driving HHG (chapter 4). The
general feature 2 provides us the information of the energies of the core energy levels,
which is useful but relatively old. And the general feature 3 contains much more
information we are interested in, such as the molecular structures, therefore, we arefocused in this region. If it is combined with pump-probe technique, a movie of a specificmolecular or material dynamics could be recorded by observing the fine structures change.

Figure 5.1 (a) Transitions resulting from the absorption of X-rays. (b) Variation of the linear
X-ray absorption coefficient of an atom as a function of the photon energy. (c) Example for

97
the X-ray absorption near edge structure (XANES) and extended X-ray absorption finestructure (EXAFS) from K edge X-ray absorption spectra of iron in K4FeII(CN)6 andK3FeIII(CN)6 bulk samples. Figures are captured from Ref. [114].

Figure 5.2 Physics picture for X-ray absorption near edge structure (XANES) and extended
X-ray absorption fine structure (EXAFS), which are originated from multiple scattering andsingle scattering, respectively, of the out-going photon-electron excited by the X-ray. Figure
is captured from Ref. [114].

Specifically, the EXAFS region covers the high-energy continuum above theionization threshold, whose energy region is from about 50 eV to even keV above theabsorption edge. In this regime, the scattering of outgoing electrons is such that the only
significant contributions to the final state wave function in the vicinity of the absorbing
atom comes from paths in which the electron is scattered only once (see Fig. 5.2) [115]. An
oscillatory structure in the spectrum (Fig. 5.1 (c)) is then present due to the single

98
scattering. In the energy domain, the oscillatory structure can be obtained directly from theabsorption spectrum as,
( )

( )

( )
( )

(5.1)

where, ( ) oscillatory absorption coefficient from the spectrum,
varying atomic-like background absorption, and( ) is the smoothly( ) is the normalization factor. From

the relationship between energy and momentum of a free particle,√
whereis the X-ray photon energy,

(

(5.2)

)

is the binding energy. The oscillatory absorption

can be expressed as a function of the wave-vector. Then, by the standard EXAFS formula:
( )
where

|

∑

( )|

(

is the interatomic distances,

)

(

( ))

(

)

(5.3)

is the number of the equivalent scatters (thesubscript j refers to the group of Nj atoms with identical properties, e.g., bond distance andchemical species),
length,( )is the temperature-dependent root mean square fluctuation in bond

| ( )|(( )) is the backscattering amplitude,

is the central-atom

partial-wave phase shift of the final state, ( ) is the energy dependent photon-electronmean free path, andsuch as the bond length

is the overall amplitude reduction factor, the structure information
could be obtained from EXAFS.

NEXAFS covers an energy range from slightly below the absorption edge to about 50
eV above the edge. The region that lies below the ionization threshold is dominated byexcitations such as transitions to bound valence states or quasi-bound states (so-calledshape resonances in molecules), core excitons in ionic crystals, unoccupied local electronic
states in metals and insulators, etc. Therefore, structures in this region differ by different
classes of materials. The region above ionization threshold comes from the multiple

99
scattering of the outgoing electrons by neighboring atoms (Fig 5.2). As a result, NEXAFS
contains information of both the electronic and the molecular structure, different fromEXAFS which mainly contains the structure information. Much valuable electronic andstructural information can be unraveled by a careful analysis of the NEXAFS regions.
Furthermore, the signal strength of the NEXAFS is about one order of magnitude higher
than that of EXAFS, since it is either from the multiple scattering or the transitions tobound states other than the single scattering. So, in synchrotron, measuring EXAFS isconsidered as a tool to demonstrate to feasibility and stability of the X-ray fine structure
measurement since its signal is weaker than NEXAFS (If EXAFS could be obtained,
NEXAFS should be easily obtained as well).

The disadvantage of the NEXAFS is its

information is not as clean as EXAFS, which could be directly used by the simple standard
formula (5.3). Complimentary knowledge might be required to analyze the NEXAFS data.
In this thesis, we are focusing on using the soft X-ray to measure the NEXAFS fromboth solid sample and gas sample, which means the soft X-ray from HHG is the probe lightother than the pump light as in most other HHG experiment before [116]. The pioneer
work of using HHG source as the probe light in the transient absorption spectroscopy is
Leone’s group [117-121]. However, they are limited by the 800 nm HHG source, therefore
the photon-energy range of the probe light is limited below 100 eV.
Finally, we end up the introduction with definition of the absorbance ( ), which isfrequently being used in the later sections:
( )
where

( )

( )

( ) is the X-ray intensity after the sample, andwithout the sample.

(5.4)( ) is the intensity before or

100
5.2 EUV- Soft X-ray (transient) absorption spectroscopy scheme

Figure 5.3 The whole soft X-ray transient absorption spectroscopy scheme enabled by the
10 mJ Ti:saphhire pumped OPA.

The whole EUV-soft X-ray transient absorption spectroscopy scheme is shown in
Figure 5.3. It starts from the 10 mJ, 45 fs, 1 kHz, Ti:saphhire amplifier, which typicallyruns around 8.8 mJ for experiments, by reducing the pump power to 42 W and adding one
more round trip to the amplification of the seed in the regenerative cavity to reach the gain
saturation. The main reason of not running at full power is the pump laser mode (532 nm)
becomes better when the pump power is lower, so the amplified cavity mode is better.
Although the mode matching between the pump mode, seed mode and cavity mode would
help the amplified mode quality compared to that of the multipass amplifier, a bad pumpmode would still affect the output mode quality. Some oxygen is filled into the cryogenic cell101
to help to reduce the asymmetry of the amplified mode. An elliptical pump (800 nm) mode
will not affect the performance (efficiency and pulse duration) of the OPA, but will reducethe coupling efficiency of 1.3 um light into hollow waveguide, since the mode of 1.3 um lightfrom the OPA has very similar shape to the pump mode and the hollow waveguide has
cylindrical symmetry. The asymmetry in the pump mode would also affect the overlap
between the pump and the probe in the dynamics experiment. (It’s important to note thatthe asymmetry does not have to be shown as a bad M2 number as mode is still clean.)The high pulse energy 800 nm light then pumps the efficient three stages OPA (seeChapter 4). The output pulse energy of 1.3 um is around 2 mJ with the pulse duration of 30
fs. The spatial and temporal qualities of the 1.3 um are typically checked by a tight focusinglens. Round, bright, symmetric plasma should be obtained at the exit of the OPA with thelens. The high quality 1.3 um light is then coupled into a 5 mm-1 cm long hollow waveguideby a lens of 25 cm focal length. The gas is supplied through two side holes in the hollow
waveguide and pumped efficiently from the entrance and exit of the waveguide by tworoughing pumps. Without gas supplied in, the front and back end of the waveguide shouldbe pumped below 100 mTorr, ideally 30 mTorr with the roughing pump we have, todemonstrate no leak. The pumping capacity, especially at the front entrance, is ofparticular importance, since any residual gas before the waveguide would reduce thecoupling efficiency and violate the coupled mode and the residue gas after the waveguide
will reabsorb the high harmonic. After the waveguide a differential pumping block is used
to keep the vacuum at the experimental side not affected by the gas inside the waveguide,
as high phase matching pressure is required for long driving wavelength HHG; thepressure after the different pumping block is typically below 10-4 Torr. A plunger is placed
after the differential pumping block for checking the coupled mode quality and measuring
the coupled 1.3 um pulse energy. Even with high gas pressure supplied into the waveguide,102
the coupling efficiency should be 50 % or more for the 1.3 um light, which indicates the good
spatial quality of 1.3 um, proper mode coupling (chapter 3) and no leak.
A high vacuum valve is used after the plunger to separate the soft X-ray generation
setup from the sample chamber, and one more high vacuum valve is used between thesample chamber and the soft X-ray characterization system, which includes the Hettrick Xray spectrometer, metal filters and the EUV-X-ray camera. Therefore, the soft X-ray
generation setup, the sample chamber and the detection system are well separated, and we
can work on each specific part of the whole X-ray beamline without venting the wholesystem.
In the sample chamber, a 0.2 um thick silver (Ag) filter is placed right after theentrance to block the residue 1.3 um light. Then, a nickel coated toroidal mirror is aligned
at a 4 degree grazing angle with respect to the X-ray beam to focus the X-ray down to 100
um diameter at the focus. This incidence angle is critical since it not only affects the focal
size but also the focal shape of the X-ray beam (the probe), which determines the overlap
between the pump and probe light. The flatness of the Ni coating is critical as well, since
the X-ray optics requires a finer surface than the EUV optics due to the wavelength of the
X-ray is shorter. The reflectivity of a layer of Ni at grazing angle of 4 degree should bearound 60 %, but the “coarse” surface may easily reduce it to 30 %. After the toroidal mirror,
the X-ray passes through a mirror with a hole at the center. The “with hole mirror” is used
to combine the pump and probe beam before the sample, where the probe beam transmits
through the hole and the pump beam is reflected on the surface, so the pump and the probe
beams are nearly parallel after the mirror into the sample. The sample is set at the focus of
the X-ray on a 3 dimensional controlled translational stage to optimize the transmission of
the X-ray through the sample. The sample holder is also mounted on a rotational stage withthe rotation axis perpendicular to the X-ray beam, so that the sample surface (in the case of

103
the solid sample) could be set perpendicular to the X-ray precisely. The detailed designs forthe gas sample and the solid sample are shown in Fig. 5. 4.

Figure 5.4 The sample holder designs. (a) The gas cell design. (b) The solid sample holder
design.

A 2 mm thick gas cell is used for the gas phase experiment. The gas is supplied froma side hole by a 1/8 inch diameter hose. The entrance and exit for the X-ray beam on the gas
cell are 150 um or 200 um diameter holes at the center of the cell. The gas density inside
the cell is controlled by the supplied gas pressure. Typically, the pressure inside the gas cellis from 20 Torr to 50 Torr to introduce enough absorption depth (Fig 5.5). The sample
chamber is pumped by a 300 liters/s turbo pump, therefore, although the sample gas isleaking from the gas cell through the entrance and exit holes for the X-ray beam, thebackground gas pressure inside the sample chamber is being kept around 10 mTorr.
Considering the sample chamber length is 40 cm, the absorption from the gas density

104
outside the gas cell to the total absorption signal is less than 10 % inside the sample
chamber.

Figure 5.5 The soft X-ray transmission from gas SF6 and CS2 through the 2 mm thick gas
cell with gas pressure at 30 Torr. Data obtained from www.CXRO.com.

So far, this number is limited by the relatively small hole on the chamber wallconnecting the turbo pump, which reduces the pumping capacity. If we can increase the sizeof the hole to the required dimension, then the background pressure should be able toreduce to less than 1 mTorr, and the absorption from the gas density outside the gas cell to
the total absorption signal will be reduced to less than 1%, which could be neglected. For
the static absorption experiment, the absorption of the gas outside the gas cell is not a
problem, but for the dynamics experiment, it will affect the signal to noise (s/n) ratio.

105
The solid sample film is coated on a thin (30 nm) Silicon Nitride membrane window
from solution either by the spin coating method or the simple drip and evaporation method.
The 30 nm thick Silicon Nitride window will absorb the soft X-ray beam by 35 %-50 % at
photon energy range from 100 eV to 200 eV. And the thickness of the sample is typicallyaround tens to hundreds of nanometer depending on the solution density. The Silicon
Nitride window is 500 um by 500 um big, which is sitting at the center of a 7.5 mm by 7.5mm big frame. Since the focus of the X-ray is around 100 um diameter, the X-ray beam canthrough the solid sample and window without any clipping.
The last element in the sample chamber is another metallic filter, which is used toblock the residue pump light after the sample. After the sample chamber is the EUV-X-ray
spectrometer (Hettrick scientific). It is consisted of a pair of Kirkpatrick-Baez (KB) mirrors
and a set of EUV-X-ray gratings. The X-ray beam is focused and spectrally dispersed bythem. Then after another two or three filters, which thoroughly block any residue pump orstray light, the spectrum is recorded on the EUV-X-ray CCD camera at the imaging point.
Finally, plenty of choices of the pump wavelength and sufficient pump power areavailable for the dynamics experiment due our high energy amplifier and the light
conversion OPA. For example, the residue 800 nm light (3 mJ to 4 mJ ) after the OPA could
be used for the strong field ionization and dissociation experiment; small amount of 800 nm
light could be split before the OPA (dash line in Fig 5.3) when the pump of high mode
quality is needed; and with the nonlinear crystal or crystal combinations, wavelengths of
200 nm, 267 nm, 400 nm, 500 nm, 1 um, and 2 um have all been achieved for the pumplight already. At the meantime, a shutter is placed in the pump beam line, which issynchronized with the CCD camera. 0.1 second exposure time of the camera (5 Hz shutter
speed) is chosen to real-time subtract pump on and pump off signals in order to reduce theeffect from HHG fluctuation.

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5.3 Soft X-ray NEXAFS from gas phase and solid phase samples

Figure 5.6 (a) Normalized High-order harmonics spectrum generated from Ar, Ne and He
gases with driving laser wavelength at 1.3 μm. (b) Normalized Xe transmission spectrum
observed using harmonics from Ar, and SF6 transmission observed using harmonics fromNe and He.

By passing the high harmonics through a 2 mm long, 200 μm diameter gas cell, theabsorption signal from the sample can be directly observed from the transmitted HHG
spectrum (Fig 5.6). For example, for SF6 gas, using harmonics generated from Ne, we
simultaneously observe three Sulfur 2p inner well resonance absorption peaks [122] 6a1g,

107
2t2g, 4eg, as well as 2t2g and 4eg shape resonances. When the SF6 gas transimssion ismeasured using He HHG, one additional Sulfur 2s Fano-type resonance absorption peak[123] 2s-1 6t1u appears in the spectrum. The absorption peak 6a1g, which is not obvious in
the spectrum of the harmonics from He, is much sharper in the Ne HHG spectrum. Notethat in the He data, an absorption at a photon energy of 285eV originates from the carbon(C) K-edge, due to C contamination on the spectrometer optics. Thus, we can measure theabsorption spectrum of multiple elements simultaneously. Figure 5.6 (b) also shows the 4d
giant resonance absorption in Xe [124] at high spectral resolution using harmonics from Ar
gas, which clearly resolves absorption from different spin states. (We are going to discuss
the Xe absorption in details in the photon-ionization transient absorption experiment.) This
demonstrates element specificity and sensitivity to electronic structure over a very broad~250 eV spectral range in a tabletop setup.
High quality absorption spectra can be obtained by taking the logarithm of the ratio
between the transmitted harmonic spectrum without (Figure 5.6 (a)) and with the samplegas (Figure 5.6 (b)), for the same exposure time, as the definition from absorbance asmentioned in the introduction. Figure 5.7 (b) shows the SF6 Sulfur L-edge 2p orbital
NEXAFS structure. Different transitions and spin states are clearly resolved and assigned
using synchrotron data [122], and the 1.2 eV energy splitting due to the Sulfur 2p orbital
spin states splitting in the absorption peaks 6a1g and 2t2g agrees well with synchrotron data.
Although the energy resolution is currently 0.25 eV, limited by 160 g/mm groove density of
the grating used in our setup, further increases in spectral resolution will be possible using
a higher groove-density grating.

108

Figure 5.7 (a) Normalized raw data for 1.3 μm-driven HHG from Ar (blue), Ne (green), andHe (red) after transmission through a 2 mm long sample cell of Xe (HHG from Ar), and SF 6
(HHG from Ne and He). Note that the apparent gap between 70 eV and 100 eV is because of
Al filters used to block the fundamental light—full tuning across this range can be achievedby switching to other filters as shown in Fig. 2. (b) NEXAFS spectrum of SF6 Sulfur L-edge
2p orbital. The high quality NEXAFS spectra were obtained in a 300 s exposure time.
However, changes in the absorption signal can be monitored in real time, with exposures as
short as 0.1 s. The modulations between the resonance absorption peaks 6a1g, 2t2g, and 4eg
arise from individual harmonic peaks that are separated by 1.9 eV. These modulations canbe reduced by increasing the driving pulse energy into the hollow waveguide, implementing
beam pointing stabilization, and reducing the integration time. Figures are captured from
Ref. [125].

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Figure 5.8 (a) Schematic MO diagram for SF6. Figure is captured from [126]. (b) Schematic
representation of the double-well potential (not to scale) for SF6. The barrier divides thepotential curve into two regions, the inner well and the outer well. Valence orbitals arelocated mainly in the inner well. Rydberg orbitals are located mainly in the outer well. (c)Synchrotron data for SF6 NEXAFS. Figures (b) and (c) are captured from [122].

The observed SF6 NEXAFS could be explained by the molecular orbital (MO) theory.
Generally, the resonances in NEXAFS spectra here arise from the excitation of a coreelectron into a virtual orbital. The final state of the system is commonly described byreferring to the character of the newly occupied orbital. In particular, the transition may be

110
to an inner-well or outer-well state, which is similar to the atomic partial-wave potential
energy diagram for an electron of angular momentum l > 1, where centrifugal contribution
to the potential energy may combine with the electrostatic contributions to produce a
potential barrier as shown in Fig 5.8 (b). In molecules the electrons around the outer atoms
produce a similar barrier along the bond directions. Virtual orbitals with a spatial
distribution primarily inside the potential barrier are known as inner-well orbitals. If anelectron is promoted into such a valence orbital, the system is said to be in an inner-wellstate. Allowed transitions to these states are normally quite intense in NEXAFS spectra. If
the virtual orbital is located mainly in the outer well, the transition is to a Rydberg state.
These transitions are normally weak in NEXAFS spectra as a result of poor spatial overlap
between the initial-state and final-state orbitals. Transitions to bound inner-well states lie
at energies below transitions to Rydberg states. The Rydberg resonances have a series ofenergies which converge on the absorption edge from below. Transitions to quasiboundinner-well states are referred to as shape resonances and occur at energies above theabsorption edge. Specifically, the 6a1g resonance (blue line in Fig. 5.8 (a)) is corresponding to
the excitation from 2t1u orbital, originated from Sulfur 2p orbital (LII, LIII edge depending on
the angular momentum), to the 6a1g bound inner-well state; 2t2g, 4eg resonance (blue line in
Fig. 5.8 (a)) are excitation from 2t1u orbital to 2t2g, 4eg quasibound inner-well states, shaperesonances; and the 6t1u resonance (green line in Fig. 5.8 (a)) is excitation from 3a1g orbital,
originated from Sulfur 2s orbital (LI edge), to 6t1u inner-well state, referred as Fano-type
resonance. In the synchrotron data (Fig 5.8 (c)), the Rydberg resonances (red line in Fig. 5.8(a)) were also observed. In high harmonic spectra, the harmonic peak fluctuation is muchlarger than the fluctuation of the overall counts; therefore, the Rydberg resonances haven’t
been captured by our HHG soft X-ray source so far. However, comparing our data with thesynchrotron data, the inner-well resonances including the spin information are all captured.

111
Another sample gas CS2 is also measured by our soft X-ray source, and the NEXAFS
is shown in Fig 5.9 (a). Comparing with the synchrotron data (Fig 5.9 (b)), the absorption
fine structure above 166 eV is resolved. However, the fine structure below the 166 eV (thered shadow in Fig 5.9 (a)) is overwhelmed by the harmonic peak fluctuation. Therefore, thepeak fluctuation is the main limitation in our absorption measurement. Since the harmonicpeak is an indication of the pulse train of the generated high harmonic, and the longer
driving pulse, such as the 2 um wavelength, would generate single atto-burst under thephase matching condition [113], the better absorption data would be obtained by using
longer driving pulse HHG. For the soft X-ray source from 1.3 um, over-ionize the noble gasfor HHG is a way to reduce the harmonic peak fluctuation.

Figure 5.9 (a) NEXAFS of CS2 and SF6 obtained from our HHG Soft X-ray. (b) Synchrotron
data for CS2, H2S, SO2, (SO4)-2, and SF6 NEXAFS, adapted from [126].

112

Figure 5.10 (a) NEXAS from P3HT. The red vertical line indicates where the peak X is in(b). (b) NEXAS from thiophene (C4H4S) on Pt (111), adapted from [127].

After investigating the gas sample, the solid sample P3HT, Poly (3-hexylthiophene-2,
5-diyl), is measured because of its popularity in the solar cell material research. An
absorption peak splitting around 166 eV is shown in the NEXAFS (see Fig. 5.10 (a)). Since
the peak is spitted by 1.2 eV, they cannot be both from the harmonic fluctuation. In order tounderstand the origin of the absorption, we searched in the synchrotron data, and only thedata of thiophene (C4H4S) is found. (P3HT could be considered as the polymer of thiophene)[127]. In the thiophene NEXAFS, two absorption peaks X and Y are shown in the photon
energy range we are discussing now. Peak X corresponds to transitions of S 2p electrons toclosely spaced unfilledandmolecular orbitals associated with the S-C bond in

thiophene. And Peak Y is a substrate-enhanced atomic resonance in the continuum due to113
excitations to d-like final states, because the thiophene is on the Pt (111) substrate in thatexperiment. Therefore, the absorption peak shown in Fig. 5.10 (a) is probably thetransitions of S 2p electrons to unfilledandmolecular orbitals.

5.4 EUV transient absorption spectroscopy: Xe photon-ionization

Besides the static absorption, the high repetition rate enabled high flux broadband
high-order harmonics also allows us to follow the dynamics, such as the charge transfer,using femtosecond-resolved transient absorption spectroscopy. To demonstrate this, westudied the ionization process of Xe gas using harmonics from Ar. The residual 800 nm lightleft after pumping the OPA is focused into the cylindrical gas cell to photon-ionize the Xe
gas as the pump beam by a lens of 25 cm focal length. The ionization induces additional
absorption peaks in Xe and these peaks can be detected using harmonics from Ar. The Xegas pressure inside of the gas cell is 30 Torr. The residual pump pulse is blocked by a 0.2
um thick Al filter after the cell.
The neutral Xe 4d absorption peaks (Fig. 5.11 (a)), match the data from synchrotron[124]. When the pump pulse is incident before the probe beam, four introduced absorption
peaks appear in the spectrum (Fig. 5.11 (b)). The photon energy corresponding to the fourpeaks are 55.4 eV, 56.1 eV, 57.1 eV and 57.6 eV. According to previous studies of the Xe ion
4d giant resonance [128], peak 1 and 2 corresponding to the Xe+ (4d105p5 2P →4d95p6 2D)
resonances, and the transitions are 2P3/2 →2D5/2 and 2P1/2 →2D3/2 respectively. Peak 3 and 4
are corresponding to the Xe2+ (4d105p4 →4d95p5) resonances, which include four transitions
theoretically. Additionally, the intensity ratios between peaks 1 and 2 and peaks 3 and 4
are close to the published oscillator strength ratios. Thus, the ability to resolving a set of

114
congested absorption peaks demonstrates the fine frequency resolution of transient spectra
taken using a HHG soft X-ray source. Furthermore, even without further compression, thehigh harmonic beam can be used to probe ultrafast dynamics on the order of tens offemtosecond. With a 10 fs step size all four absorption peaks build up within 75 fs after thetime zero, as shown in the difference data from the subtraction of the pump on and pumpoff counts (See Fig. 5.11 (c)).

115

Figure 5.11 (a) Neutral Xe 4d absorption. (b) Xe+ and Xe2+ 4d absorption. (c) Xe+ and Xe2+
transient absorption signal.

116
5.5 Conclusion and ongoing experiment

In this chapter, we first introduced the X-ray absorption features, especially, the finestructures of the absorption edge. Then, we introduced the Soft X-ray transient absorption
spectroscopy setup we have implemented. After that, we presented the NEXAFS from thegas sample SF6 and CS2, and solid sample P3HT using our soft X-ray source. Finally, wedemonstrated the transient absorption spectroscopy by resolving the Xe photon-ionization
process in the EUV region.
Ongoing experiment
The samples for static NEXAFS measurements are all the proposed samples for soft
X-ray region transient absorption spectroscopy experiments we are doing. For example, SF6
shape resonance could be used for detecting the vibrational dynamics pumped by impulsive
stimulated Raman scattering; P3HT could be used for studying the charge transfer in thesolar cell material.

117

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