Quasi-Phase Matching of Soft X-ray Light from
High-Order Harmonic Generation using
Waveguide Structures
byEmily Abbott Gibson
B.S. Engineering Physics, Colorado School of Mines, 1997

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2004

This thesis entitled:
Quasi-Phase Matching of Soft X-ray Light from High-Order Harmonic Generation using
Waveguide Structures
written by Emily Abbott Gibson
has been approved for the Department of Physics

Henry C. Kapteyn

Margaret M. Murnane

Date

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

iii
Gibson, Emily Abbott (Ph.D., Physics)
Quasi-Phase Matching of Soft X-ray Light from High-Order Harmonic Generation using Waveguide Structures
Thesis directed by Profs. Henry C. Kapteyn and Margaret M. Murnane

Ultrafast laser technology has made it possible to achieve extremely high field intensities, above 1018 W/cm2 , or alternatively, light pulses with extremely short time durations
corresponding to only a few femtoseconds (10−15 s). In this high intensity regime, the laser
field energy is comparable to the binding energy of an electron to an atom. One result of thishighly non-perturbative atom-light interaction is the process of high-order harmonic generation(HHG). In HHG, the strong laser field first ionizes the atom. The subsequent motion of the freeelectron is controlled by the oscillating laser field, and the electron can reach kinetic energies
many times that of the original binding energy to the atom. The high energy electron can thenrecollide with its parent ion, releasing a high energy photon. This process occurs for many atoms
driven coherently by the same laser field, resulting in a coherent, laser-like beam of ultrafast
light spanning the ultraviolet to soft X-ray regions of the spectrum.
In this thesis, I will present two major breakthroughs in the field of high harmonic
generation. First, I will discuss work on quasi-phase matching of high harmonic generation,
which has allowed increased conversion efficiency of high harmonic light up to the water window
region of the soft X-ray spectrum (∼ 300 eV) for the first time.[31] This spectral region issignificant because at these photon energies, water is transparent while carbon strongly absorbs,
making it a useful light source for very high resolution contrast microscopy on biological samples.
Since the resolution is on order of the wavelength of the light (∼ 4 nm for 300 eV), detailed
structures of cells and DNA can be viewed. A table-top source of light in the water window soft
X-ray region would greatly benefit biological and medical research. Second, I will present workon the generation of very high harmonic orders from ions. This work is the first to show that
harmonic emission from ions is of comparable efficiency to emission from neutral atoms thereby

iv
showing that high harmonic emission is not limited by the saturation intensity, or the intensity
at which the medium is fully ionized, but can extend to much higher photon energies.[30] Bothresults were obtained by using a waveguide geometry for HHG, allowing manipulation of thephase matching conditions and reducing the detrimental effects of ionization. The ideas from thiswork are expected to increase the number of applications of high harmonic generation as a lightsource by increasing the efficiency of the process and opening up the possibility of generating
multi-keV photon energies.

Dedication

To my parents.

vi

Acknowledgements

I was fortunate to have had the opportunity do my graduate work in Margaret Murnaneand Henry Kapteyn’s group which provided a stimulating and supportive environment for doing interesting research. I am very grateful to both Margaret and Henry for their enormous
generosity, advice, and for believing I could succeed. I would like to thank the people who contributed to my education and made my graduate experience a good one; in particular, Thomas
Weinacht, who taught me about experimental methods and whose enthusiasm was contagious,and Sterling Backus, who taught me a great deal about lasers and ultrafast optics. The work
presented here on quasi-phase matching was done with Ariel Paul, who had the ingenuity tomake the modulated waveguides. The work on pulse compression was done with Nick Wagner,
who always comes up with new physical insights and ideas in the lab. It was a great experience
working with both. Also, many others in the group, past and current: Lino Mitsoguti, Erez
Gershgoren, Chi-Fong Lei, Randy Bartels, Ra’anan Tobey, David Gaudiosi, Tenio Popmintchev,
David Samuels, Xiaoshi Zhang, Etienne Gagnon, and Daisy Raymondson were always happy tolend a hand or give scientific advice. Ivan Christov from Sofia University, Bulgaria did muchof the theory for the experiments described here and my discussions with him provided a lot of
useful insight.
I am also indebted to the many people on the JILA and Physics Instrument shops staff
who freely gave their knowledge and help with these experiments. I would like to thank the
professors who not only taught me but also believed in my abilities, especially John Cooper,
whose encouragement meant a great deal, and Stephen Leone, who continued to support me longvii
after I stopped working as an undergraduate in his group. Paul Corkum also showed me great
generosity and, even in the brief time I visited his group, influenced my understanding of the field.
I would not have made it through the tough times without the support and understanding of myfamily and without the help of some very good friends in graduate school who were responsible
for many of the good times as well, in particular, Tara Fortier and Martin Griebel.

viiiContents

Chapter

1 Introduction
1.1

1High-order Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Quasi-Phase Matching

3

6

2.1

Introduction to Quasi-Phase Matching . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Experimental Results on Quasi-Phase Matching of High Harmonic Generation . .

13

2.3

Mechanisms for Enhancement by Quasi-Phase Matching . . . . . . . . . . . . . .

28

2.3.1

Modulating the Amplitude of the Harmonics . . . . . . . . . . . . . . . .

28

2.3.2

Grating-Assisted Phase Matching . . . . . . . . . . . . . . . . . . . . . . .

33

2.3.3

Intrinsic Phase Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Generation of an Enhanced Attosecond Pulse through Phase Matching . . . . . .

43

2.4

3 High Harmonic Generation from Ions

51

3.1

Previous Experiments on HHG from Ions . . . . . . . . . . . . . . . . . . . . . .

52

3.2

Experimental Results using Hollow-core Waveguides . . . . . . . . . . . . . . . .

53

3.3

Evidence for Harmonic Emission from Ions . . . . . . . . . . . . . . . . . . . . .

56

3.4

Validity of the ADK Model in the BSI Regime . . . . . . . . . . . . . . . . . . .

59

3.5

Plasma-induced Defocusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4 High Harmonic Generation Experimental Set-up
4.1

Energy Calibration of High Harmonic Generation Spectra . . . . . . . . . . . . .

66
69

ix
4.2

Calculation of Photon Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5 Self-compression of Intense Femtosecond Pulses in Gas-filled Hollow-core Waveguides

77

5.1

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.2

A New Mechanism for Pulse Compression . . . . . . . . . . . . . . . . . . . . . .

82

6 Spectral and Temporal Characterization of High Harmonic Generation by Photoelectron
Spectroscopy

89

6.1

A Compact Magnetic Bottle Photoelectron Spectrometer . . . . . . . . . . . . .

89

6.2

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.3

Cross-correlation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Conclusion

108

Bibliography

109

Appendix

A ADK ionization rate calculation

115

B Magnetic Field Calculations

117

C Micro-Channel Plate Detector

120

x

TablesTable

5.1

Table of possible pulse shaping mechanisms that involve only 1-D propagation
effects and the propagation length over which they become significant. The conditions used in the calculation are 4 Torr of Ar in a 150 µm diameter hollow
waveguide. The gas terms were calculated for no ionization, and the plasma
terms were calculated for 200 % ionization. . . . . . . . . . . . . . . . . . . . . .

84

xi

FiguresFigure2.1

Plot of pressure for phase matching as a function of normalized ionization fraction(η/ηcr ). Beyond critical ionization, phase matching is no longer possible. . . . . .

2.2

9Diagram of quasi-phase matching plotting the second harmonic signal for theconditions of: A. Perfect phase matching, C. No phase matching, B1. First-order
QPM, and B3. Third-order QPM. From [27]. . . . . . . . . . . . . . . . . . . . .

2.3

10

a. Calculation of the signal of the 95th harmonic for a straight waveguide as afunction of propagation distance. b. Calculation for a 0.5 mm period modulated
waveguide. From [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Optical microscope image of a 0.25 mm period modulated waveguide manufactured by glass-blowing techniques. From [31]. . . . . . . . . . . . . . . . . . . . .

2.5

12

14

Experimental harmonic spectra from modulated waveguides filled with 111 Torr
He with a driving pulse of 25 fs and peak intensity ∼ 5 x 1014 W/cm2 , for different
modulation periodicities. From [56]. . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

15

Harmonic spectra from straight (black), 0.5 mm (blue), and 0.25 mm (red) periodicity modulated waveguides filled with 6 Torr neon. The spectra were takenusing the SC grating and a 10 second exposure time. . . . . . . . . . . . . . . . .

17

xii
2.7

Harmonic emission from straight (black) and 0.25 mm (red) modulated waveguides filled with 7 Torr Ar. From [31]. The spectra were taken using the SC
grating. The exposure time for the straight waveguide was 300 seconds while forthe modulated it was 200 seconds. The spectrum for the modulated waveguidehas been multiplied by 1.5 for comparison. . . . . . . . . . . . . . . . . . . . . . .

2.8

18

Harmonic spectra from a straight waveguide filled with 10 Torr Ar (blue) and 6
Torr Ar (black), and a 0.25 mm modulated waveguide with 5 Torr Ar(red). All
spectra were taken with the SC grating. Both straight waveguide spectra havea 300 s exposure time while the modulated waveguide spectrum was taken with
600 s exposure and divided by 2 for comparison. . . . . . . . . . . . . . . . . . .

2.9

19

Harmonic spectra from straight, 0.5 mm, and 0.25 mm modulated waveguides,with different pressures of neon gas. All spectra were taken with a 10 s exposure
time on the SC grating. Also plotted are the integrated counts for three different
photon energy ranges as a function of pressure. The energy ranges are indicatedwith dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.10 Experimental harmonic spectra generated in 9 Torr of neon gas for straight (black)and 0.25 mm modulated (red) waveguides. Both spectra taken with a 180 s
exposure time on the SB grating. From [31]. . . . . . . . . . . . . . . . . . . . . .

22

2.11 Experimental harmonic spectra generated in helium at 15 Torr using a 0.25 mm
modulated waveguide (black), and at 16 Torr (blue) and 10 Torr (red) using astraight waveguide. All spectra taken with 120 s exposure on SB grating. . . . .

23

2.12 Calculation of the fractional ionization of neon and helium as a function of time
during the laser pulse. The right axis plots the cutoff photon energy corresponding
to the instantaneous pulse intensity. . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.13 Comparison of harmonic spectra at the C edge using a 0.25 mm modulated waveguide filled with 9 Torr Ne (black), 42 Torr He (blue), and 60 Torr He (red). All
spectra taken with 180 s exposure time using SB grating. . . . . . . . . . . . . .

26

xiii
2.14 Comparison of harmonic spectra from 8 Torr Ne (black) and 15 Torr He (red)using a 0.25 mm modulated waveguide. Both spectra taken with 120 s exposure
time using SB grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.15 Calculations of the radial profile of the intensity (red curve) and phase (green
curve) of the laser inside the modulated waveguide. The three frames correspond
to: a. the initial mode, b. the mode giving the maximum on-axis intensity andc. the minimum on-axis intensity. From Ivan Christov. . . . . . . . . . . . . . . .

29

2.16 Diagram showing region in pulse where a given harmonic, only generated above
the threshold intensity, is turned on and off by a 10% intensity modulation. . . .

31

2.17 Plot of absorption length as a function of photon energy for 10 Torr Neon and
100 Torr Helium. Data from the Center for X-ray Optics, Berkeley Lab (wwwcxro.lbl.gov). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.18 Comparison of ionization fraction of neon as a function of time for a 22 fs duration
pulse at different peak intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.19 Harmonic intensity as a function of propagation distance for the case where thephase mismatch varies sinusoidally by 1, 3, and 5 % with a period of 2 coherence
lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.20 Plot of the harmonic intensity during the first 60 coherence lengths. . . . . . . .

37

2.21 Return phase, φf , as a function of the release phase, φ0 , determined by theclassical equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.22 Electron return kinetic energy for trajectories with different release phases, φ0 .
The long and short trajectories are indicated. . . . . . . . . . . . . . . . . . . . .

40

2.23 Intrinsic phase as a function of harmonic energy. The long trajectories acquire
more phase than the short trajectories. . . . . . . . . . . . . . . . . . . . . . . . .

41

2.24 Plot of harmonic phase in reference to the fundamental as a function of harmonicenergy. Note there is a linear dependence which is different for the long and short
trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

xiv
2.25 Plot of the intensity dependence of the harmonic phase for two different harmonicorders. The calculation was done for a fundamental wavelength of 800 nm.

. . .

42

2.26 Calculation of the phase mismatch (∆k) for the 171st harmonic in 3 Torr neon
for a 20 fs duration pulse with a peak intensity of 1015 W/cm2 . The dashed line
indicates when the QPM condition is met for a modulation period of 0.25 mm. .

44

2.27 Calculation of the harmonic signal of the 171st harmonic in 3 Torr neon for a 20
fs duration pulse with a peak intensity of 1015 W/cm2 by quasi-phase matchingwith 0.25 mm periodicity. Here, the propagation length is 1 mm. . . . . . . . . .

44

2.28 Calculation of the harmonic spectra for (a) a straight waveguide and (b) a 0.5
mm periodicity modulated waveguide for a 15 fs duration pulse with an intensityof 7 x 1014 W/cm2 in 1 Torr Ar. The calculated single atom spectrum is also
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.29 Calculation of the time dependence of the harmonic output for (a) the straightwaveguide and (b) the 0.5 mm periodicity modulated waveguide. . . . . . . . . .

47

2.30 a. Calculation of the flux of the 29th harmonic for a 20 fs laser pulse and peakintensity of 2.2 x 1014 W/cm2 as a function of argon pressure using a 3 cm
waveguide. b. Experimentally measured flux for the same conditions as part a.
From [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.31 Calculation of the fractional ionization of argon using ADK ionization rates for a
20 fs pulse with peak intensity of 2.2 x 1014 W/cm2 for a cosine (black) and sine(red) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

49

Harmonic emission from argon using a gas jet with 250 mJ pulse energy and a
250 fs pulse. From [80] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

xv
3.2

Harmonic emission from a straight 150 µm inner diameter, 2.5 cm long fiber filledwith low-pressure Ar (7 Torr), driven by an 800 nm laser with an 18 fs pulse
duration and a peak intensity of 9 x 1014 Wcm−2 . The spectrum was taken using
the SC grating and an exposure time of 300 s. . . . . . . . . . . . . . . . . . . . .

3.3

54

Harmonic emission at a higher peak laser intensity of 1.3 x 1015 Wcm−2 , using
a 0.25 mm period modulated waveguide. The spectrum was taken using the SC
grating with a 600 s exposure time. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

55

Calculated ionization levels in argon for an 18 fs laser pulse at a peak laser
intensity of 1.3 x 1015 Wcm−2 , using ADK rates. Laser pulse (black dashed); Ar(red); Ar+ (green); Ar++ (blue). The right axis shows the predicted HHG cutoff
energy for the laser intensity profile, calculated from the cutoff rule. . . . . . . .

57

3.5

Calculated ionization rates of argon species for the same conditions as in Fig. 3.4. 57

3.6

Calculated harmonic emission from neutral (red) and singly ionized Ar (blue) for
an 18 fs laser pulse centered at 800 nm, at a laser intensity of 1.3 x 1015 Wcm−2 .
The ion spectrum has not been corrected for the dynamically varying populations. 58

3.7

Static ionization rates of H and He versus electric field strength. Solid line:
numerical calculations of TDSE. Dashed line: ADK theory. From [66]. . . . . . .

3.8

60

Fraction of ionized H in the BSI regime as a function of the peak electric field
amplitude for pulses of 5, 10, and 20 fs in duration. Full line is ADK model, triangles show Krainov model, and circles show time-dependent Schroedinger equation.
The dotted line is Ebs for H. From [71]. . . . . . . . . . . . . . . . . . . . . . . .

3.9

61

Measured harmonic spectra from a straight waveguide filled with argon at 5 Torr(red) and 7 Torr (black) for a pulse peak intensity of ∼ 9 x 1014 Wcm−2 . Bothspectra taken with the SC grating and an exposure time of 300 s (using 0.4 µm
thick Zr filters). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

xvi
3.10 Harmonic spectra from a straight waveguide filled with Ne at different pressures
for a peak laser intensity of ∼ 9 x 1014 Wcm−2 . All spectra taken with the SC
grating and a 10 s exposure (using 0.4 µm thick Zr filters). . . . . . . . . . . . .

64

4.1

Schematic of Ti:Sapphire amplifier system used for high harmonic generation. . .

67

4.2

Schematic of experimental setup for the generation and measurement of highharmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Harmonic spectra generated in neon using 0.2 µm-thick B (black) and 0.18 µmthick Si (red) filters to calibrate the SC grating. . . . . . . . . . . . . . . . . . . .

4.4

68

70

Harmonic spectra generated in neon using 0.2 µm-thick B (black) and C (red)filters to calibrate the SB grating. . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.5

Schematic of grazing incidence grating and relevant angles. . . . . . . . . . . . .

71

4.6

Wavelength and energy calibration of SC grating using the grating equation anda two point linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

Wavelength and energy calibration of SB grating using the grating equation anda two point linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

4.9

71

72

Reflectivity of a Au mirror at 2 degree grazing incidence angle for the ideal case,and for 2 nm RMS surface roughness. . . . . . . . . . . . . . . . . . . . . . . . .

74

Measured efficiency of SB grating as a function of photon energy. . . . . . . . . .

74

4.10 Harmonic spectra at the carbon edge showing the data points used to estimate
the photon flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.11 Measured flux from the ALS beamline 6.1 and estimated flux assuming 100%efficient optics (dashed line). From [76]. . . . . . . . . . . . . . . . . . . . . . . .

5.1

76

Spectra taken after the 2.5 cm long, 150 µm diameter waveguide with different
Ar pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

xvii
5.2

Deconvolved FROG measurements of the temporal pulse shape before and after
the waveguide with different Ar pressures for an initial pulse of 32 fs. The pulse
FWHM for the different measurements are indicated. . . . . . . . . . . . . . . . .

5.3

80

Deconvolved FROG measurements of the temporal pulse shape before and after
the waveguide with different Ar pressures for an initial pulse of 29 fs. At 4 Torr,
the final pulse duration is 13.3 fs FWHM. From [78]. . . . . . . . . . . . . . . . .

5.4

Comparison of the measured spectrum with the spectrum from FROG measurement deconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

81

81

Theoretical calculation of temporal pulse shape after the waveguide with different
Ar pressures for an initial pulse duration of 30 fs using the 3-D propagation model.
The pulse FWHM are indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

86

Theoretical calculation of the pulse spectrum before and after the waveguide withdifferent Ar pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

6.1

Diagram of magnetic bottle showing direction of field lines and electron trajectories. 90

6.2

Schematic of compact magnetic bottle time-of-flight spectrometer. Dimensions
are indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

92

Measured values of the magnetic field on axis above a 1 inch dia., 0.5 inch thick,
NdFeB magnet (black dots). Theoretical prediction of magnetic field for a magnet
with a magnetization of 8000 A/m (red line). . . . . . . . . . . . . . . . . . . . .

6.4

94

Calculated z component of the magnetic field from the permanent magnet andsolenoid on axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.5

Calculated adiabatic parameter on axis, for 1 eV electrons. . . . . . . . . . . . .

96

6.6

Calculated (dashed) and measured (solid) magnetic field and adiabatic parameterfor 1 eV electrons on axis. From [43]. . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

96

High harmonic spectrum measured by the magnetic-bottle photoelectron spectrometer using different detection gases. . . . . . . . . . . . . . . . . . . . . . . .

97

xviii
6.8

Absorption at different photon energies for argon and neon gas. From CXRO
website. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9

98

Photoelectron spectrum of HHG from Ne extending to ∼ 100 eV photon energies. 99

6.10 Comparison of photon spectra measured using the magnetic bottle photoelectron
spectrometer and the Hettrick spectrometer. . . . . . . . . . . . . . . . . . . . . . 100
6.11 Diagram of resonant and non-resonant Auger processes in xenon. . . . . . . . . . 100
6.12 Photoelectron spectrum of xenon using all harmonics (blue) and harmonics passing through an Al filter blocking energies > 65 eV (red). . . . . . . . . . . . . . . 102
6.13 Absorption of xenon gas as a function of photon energy. Data from the Centerfor X-ray Optics, Berkeley Lab (www-cxro.lbl.gov). . . . . . . . . . . . . . . . . . 102
6.14 Reflectivity as a function of photon energy for the Si/Mo multilayer mirror, measured at an 85 degree incident angle. . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.15 Spectra of high harmonics generated using He gas and detected with Ne gas before(black) and after (red) the narrow-band multilayer mirror. . . . . . . . . . . . . . 104
6.16 Photoelectron spectrum of Xe using a narrow bandwidth of harmonics selectedby the multilayer mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.17 Schematic of experimental setup for measuring the pulse duration of high harmonic generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.18 Photoelectron spectra at different probe delays. The sidebands that are analyzed
are indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.19 Integrated counts for the sidebands at 22 and 25 eV as a function of probe delay. 106

C.1 MCP detector design. The micro-channel plates are in dark blue. The anode isbiased at 2 kV and is insulated from the grounded stainless steel housing. . . . . 121
C.2 Detailed schematic of the MCP detector giving dimensions for each part. All
dimensions are in inches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Chapter 1

Introduction

Ultrafast laser technology makes it possible to generate extremely high field intensities
above 1018 W/cm2 , or alternatively, to generate pulses with extremely short time durations corresponding to only a few femtoseconds (10−15 s).[8, 14] One of the most prominent applicationsof very high-power, ultrashort-pulse lasers has been the study of non-perturbative light-matterinteractions. At intensities above 1014 W/cm2 , the magnitude of the electric field of the laser
radiation is comparable to the field binding an electron to an atom. In this regime, the laser
field can easily ionize atoms. Once the atom is ionized, the subsequent motion of the free electron is controlled by the oscillating laser field, and the electron can easily reach kinetic energies
many times that of the original electron binding energy. One significant consequence of this intense light-matter interaction is the process of high-order harmonic generation (HHG).[51, 28]
In HHG, the free electron recollides with its parent ion and recombines with it, releasing a highenergy photon. Since all atoms experience the same coherent laser field, the harmonic radiationis also highly coherent. Temporally, high harmonic radiation is a series of attosecond-duration
bursts occurring every half-cycle of the driving laser. This unique property has allowed measurements of electronic processes in atoms occurring on timescales previously inaccessible.[40, 24]
The emission from HHG can extend from the ultraviolet up to the soft x-ray region of thespectrum (to ∼ 950 eV). This spectral region is interesting for studies of chemical and material
processes because these photon energies can access valence and inner shell electrons of atoms
and molecules.[11, 33, 55] Also, HHG is an extremely useful light source for applications such as2plasma interferometry[23, 60] or extreme-ultraviolet (EUV) lithography, a technology that will
be needed to implement the next generation of integrated circuits. Thus, high harmonic generation has both fundamental and practical applications. However, one major problem is that theefficiency of HHG is extremely low, typically on order of 10−5 to 10−10 conversion of laser energy
into each harmonic order. As with any harmonic generation process, the efficiency can be greatly
enhanced by phase matching. Traditional optical phase matching techniques have been applied
successfully to high harmonic generation for low harmonic orders (< 90 eV).[62, 26] However,before now there were no techniques for enhancement of HHG at higher photon energies.
In this thesis, I will present two major breakthroughs in the field of high harmonic
generation. First, I will discuss work on quasi-phase matching of high harmonic generation,
which has allowed increased conversion efficiency of high harmonic light up to the water window
region of the soft X-ray spectrum (∼ 300 eV) for the first time.[31] This spectral region issignificant because at these photon energies, water is transparent while carbon strongly absorbs,
making it a useful light source for very high resolution contrast microscopy on biological samples.
Since the resolution is on order of the wavelength of the light (∼ 4 nm for 300 eV), detailed
structures of cells and DNA can be viewed. A table-top source of light in the water window soft
X-ray region would greatly benefit biological and medical research. Second, I will present workon the generation of very high harmonic orders from ions. This work is the first to show that
harmonic emission from ions is of comparable efficiency to emission from neutral atoms thereby
showing that high harmonic emission is not limited by the saturation intensity, or the intensity
at which the medium is fully ionized, but can extend to much higher photon energies.[30] Bothresults were obtained by using a waveguide geometry for HHG, allowing manipulation of thephase matching conditions and reducing the detrimental effects of ionization. The ideas from thiswork are expected to increase the number of applications of high harmonic generation as a lightsource by increasing the efficiency of the process and opening up the possibility of generating
multi-keV photon energies.

3

1.1

High-order Harmonic Generation
High-order harmonic generation occurs when an intense linearly polarized laser field in-teracts with a gas or material. An intuitive model of HHG at the atomic level was developed
by Corkum, Kulander, and others,[22, 44] and is sometimes referred to as the three-step model.
In the first step, the strong electric field of the laser suppresses the Coulomb barrier binding
an electron to an atom, freeing the valence electron either by tunnelling or by over-the-barrier
ionization. The freed electron is then accelerated by the field. Since the laser field is oscillating, the electron can with some probability return to its parent ion and recombine, emitting ahigh-energy photon. This process occurs for many atoms driven coherently over several laser
cycles, resulting in emission of higher-order odd harmonics of the fundamental in a coherent,
low-divergence beam. The three-step model accurately predicts that the highest photon energythat can result from HHG occurs when an electron is ionized at a phase corresponding to 18
degrees after the peak of the laser cycle. This cutoff photon energy is predicted to be:

Emax = Ip + 3.2Up ,(1.1)

where Ip is the ionization potential of the atom and Up is the ponderomotive energy given by:
Up = e2 E 2 /4mω 2 = 9.33 × 10−14 Iλ2 eV,(1.2)

where e, E, m, ω, I, and λ are the electron charge, field amplitude, electron mass, fundamental
laser frequency, intensity in W/cm2 , and wavelength in microns respectively. The highest photon
energy therefore scales linearly with the laser intensity.
The recollision energies associated with the intensities required to field-ionize atoms canbe as high as hundreds of electron volts.[15, 70, 67] Thus, the emitted high-harmonic photons can correspond to the combined energy of several hundred incident photons. The linear
relationship between the cutoff and the incident intensity presents a very attractive scaling as
compared to, for example, EUV laser schemes where the power requirements scale as the photon
energy to the 3rd-5th power.[39] However, to take advantage of this favorable scaling, several

4challenges must be overcome. The HHG process necessarily ionizes the medium, generating afree-electron plasma. The dispersion of the plasma creates a mismatch in the phase velocitiesof the fundamental and harmonic light, significantly reducing the amount of harmonic signal
produced. Also, the generated plasma can defocus the laser beam, decreasing the peak intensity
and thereby limiting the maximum harmonic energy.[48] To date, the highest energy HHG emission has been achieved using short-duration laser pulses and noble gases with high ionization
potentials such as helium, which can reduce the amount of ionization for a given peak intensity. However, helium has a small cross-section giving it a lower efficiency for HHG than larger,
multi-electron atoms. In Ch. 3, I will discuss a method of generating high harmonics even in a
fully ionized gas medium, allowing HHG in argon up to photon energies of 250 eV, previously
achieved only with He gas, and demonstrating that HHG from ions is feasible. In combination
with quasi-phase matching techniques, this method should extend the efficient energy range of
HHG even further.
The simple semi-classical picture of HHG as given by the three-step model accurately
predicts the range of photon energies that can be obtained for a given incident laser intensity.
However, a quantum mechanical treatment is needed for a more complete picture of the HHG
process. In the quantum picture, as the laser field strength increases, portions of the electron
wave function tunnel through the suppressed atomic barrier and then propagate large distances relative to the atomic radius - away from the atom. When the laser field reverses, this extended
electron wavefunction re-encounters the part of the wavefunction still in the ground state of
the atom, leading to quantum interferences. As a result, rapid oscillations in the electronic
probability distribution and therefore the transient dipole moment of the atom lead to theemission of very high-order harmonics of the fundamental laser.
A fully quantum, analytical theory of HHG, derived by Lewenstein et al.[46], has been
extremely successful in describing both the general characteristics of HHG such as the photon
energy cutoff, as well as more specific characteristics such as the divergence properties of thegenerated beam[9] and the specific spectral characteristics of the emission. Qualitatively, the5spectral characteristics of HHG emission are a result of the fact that the freed electron has an
associated deBroglie wavelength corresponding to the kinetic energy acquired in the laser field,
λ = h/p, where h is Planck’s constant and p is the electron momentum. The emission also has aphase related to the total phase the electron accumulates during its free trajectory, given by:[9]Φ = qωtf − S(pst , ti , tf )/h̄,(1.3)

where q is the harmonic order, ω is the laser frequency, ti is the time the electron is releasedin the laser field, tf is the recollision time, and S(pst , ti , tf ) is the semi-classical action overthe electron’s trajectory. The phase of the emitted harmonic light is therefore not simply related to the phase of the driving laser, but also includes an intrinsic phase component thatcan vary rapidly with laser intensity. The intrinsic phase has consequences for both the spatial and spectral emission characteristics. Spatially, the intrinsic phase can result in a complex
spatial profile of the harmonic emission when generated using a beam that is converging toward a focus. This is in contrast to the gaussian spatial profile for HHG generated by a beam
diverging from a focus[9, 63, 58] or the full spatial coherence of light generated using a waveguide configuration.[49] In the spectral domain, this time-varying intrinsic phase results in large
frequency shifts and spectral broadening or narrowing of the harmonic emission.[16] The intrinsic phase also plays a role in quasi-phase matching of HHG as will be discussed in the next
chapter.

Chapter 2

Quasi-Phase Matching

Applications of HHG radiation as a table-top, coherent soft X-ray source depend upon
optimizing the total flux generated. In nonlinear optical processes, the conversion efficiency
from the fundamental to the harmonic field is enhanced by phase matching. In its simplest
manifestation, phase matching will occur when the driving laser and the harmonic signal travelwith the same phase velocity, so that signal generated throughout the conversion region adds
constructively. In HHG, since the ionization level is changing throughout the pulse, and theincreasing density of free electrons has a large effect on the phase velocity of the driving laser,
phase matching can only be obtained in limited time windows in the pulse. Experimentally,
dramatic enhancement of the flux from HHG can be achieved by pressure-tuned phase matching
using hollow-core gas-filled waveguides.[62, 26] The laser light is guided by glancing-incidence
reflection off the waveguide walls, allowing propagation over an extended interaction lengthwith a well defined intensity and phase profile. The fundamental and harmonic light propagate
through the waveguide with phase velocities determined by the dispersion of the neutral gas,
the plasma, and the waveguide at the two different frequencies. The phase mismatch is givenby ∆k = qkf − kq , where kf is the wavevector of the fundamental and kq is the wavevector of
harmonic order q. The full expression is given by:∆k = ∆kW aveguide + ∆kP lasma + ∆kN eutrals

(2.1)

7
or
∆k =

qu211 λ2π(1 − η)P q∆δ+ P ηNatm re (qλ − λ/q) −
,
2
4πa
λ

(2.2)

where λ, q, a, u11 , η, P, Natm , and re are the fundamental wavelength, harmonic order, waveguide
radius, the first zero of the Bessel function J0 , ionization fraction, gas pressure in atmospheres,
number density at 1 atm, and classical electron radius respectively, and ∆δ is the difference in
the index of refraction at the fundamental and harmonic frequencies of the generation gas at
1 atm. Here, the small contribution from the nonlinear refractive index is neglected and it isassumed that the harmonic light does not interact with the waveguide. The waveguide allows
control over the experimental parameters such as gas pressure and radius to optimize phasematching.
For low levels of ionization, the pressure can be adjusted so that the waveguide and plasma
dispersion balance the dispersion due to the neutral atoms, which is of opposite sign, and phasematching can be achieved (∆k = 0). Since phase matching is accomplished by adjusting thephase velocity of the fundamental, and the harmonic radiation travels at a phase velocity of∼c, the bandwidth of phase matching is very broad, encompassing many harmonic orders. At
higher laser intensities, the rapidly-changing ionization during the pulse means that there is alimited time-window for phase matching. However, the phase matching bandwidth is also broad
since the phase matching condition for different harmonic orders will occur at some time during
the pulse. When phase matched, the harmonic signal increases quadratically with interaction
length until it is limited by absorption of the harmonic light by the gas medium.
Unfortunately, this method of phase matching is limited to low ionization levels. Whena large enough fraction of the gas is ionized, the plasma dispersion is greater than the neutral
dispersion for any pressure so that phase matching is no longer possible. The phase velocityof the fundamental light then exceeds that of the generated harmonic signal. The value of thiscritical ionization fraction is given by:[61]ηcr = (1 +Natm re λ2 −1
) .
2π∆δ

(2.3)

8
Figure 2.1 plots the pressure required to achieve phase matching as a function of the ionization
fraction. For low laser intensities, and therefore lower harmonic orders, where the ionization
fraction is small, phase matching is primarily achieved by balancing the waveguide dispersion
with the neutral gas dispersion. At the higher laser intensities required to generate higher energy
harmonics, however, the plasma term becomes significant, and higher neutral gas pressures areneeded to compensate. Beyond the critical ionization, phase matching is not possible.[25, 21]
For the case of plane wave propagation without a waveguide in a uniform-density gas, phasematching will occur when the fractional ionization equals the critical ionization.[61] However,this requires either very high pulse energies or high gas pressure and results in spatially varying
phase matching which can create a complex spatial mode of the harmonic emission. In contrast,
the use of the hollow waveguide allows for better phase matching, resulting in buildup of an
EUV beam with a spatial mode of extremely high coherence.[10, 49] In fact, EUV light from
HHG with full spatial coherence has currently only been generated using a hollow waveguide
geometry.
Despite the many advantages of phase matching HHG in a straight waveguide, it is limited
to relatively low levels of ionization, below the critical ionization fraction (0.5%, 1% and 5% in
He, Ne and Ar). This limitation restricts the enhancement of HHG emission to photon energies< 100 eV for a 20 fs pulse since higher energy harmonics can only be generated at higher laser
intensities and therefore greater ionization fractions. At higher ionization levels, the plasma
contribution to the dispersion becomes much greater than the neutral gas contribution, making
phase matching impossible. In the absence of phase matching, the harmonic emission builds up
periodically over a coherence length and then interferes with out-of-phase light generated in thenext section of the nonlinear medium, drastically reducing the total flux.

2.1

Introduction to Quasi-Phase Matching
In recent work, we showed that the inability to phase match for relatively small amountsof ionization can be overcome for the first time by applying quasi-phase matching (QPM) tech-

9

No
Phase
Matching
Dk=0

Figure 2.1: Plot of pressure for phase matching as a function of normalized ionization fraction(η/ηcr ). Beyond critical ionization, phase matching is no longer possible.

10
niques to the HHG process. The ability to quasi-phase match HHG significantly extends therange of photon energies where it is possible to generate high-harmonic emission efficiently.
Quasi-phase matching was first proposed in 1962, shortly after the birth of nonlinear optics.
Armstrong et al.[4] proposed a phase corrective scheme whereby the phase mismatch in a nonlinear optical process is periodically corrected by introducing a periodicity to the nonlinearityof the medium corresponding to the coherence length. The coherence length is defined as the
distance it takes for the fundamental and harmonic light to become 180 degrees out of phase.
For example, quasi-phase matching of second harmonic light can be achieved by periodicallyreversing the crystal orientation and therefore the polarity of the nonlinear response so thata 180 degree phase shift is introduced every coherence length. This reverses the destructive
interference, allowing continual buildup of the harmonic signal, shown in Fig. 2.2. Practical implementation of this concept was not achieved until the development of crystal-poling techniquesin the mid-1990’s.[27]Figure 2.2: Diagram of quasi-phase matching plotting the second harmonic signal for the conditions of: A. Perfect phase matching, C. No phase matching, B1. First-order QPM, and B3.
Third-order QPM. From [27].

Unfortunately, the generation of coherent light at EUV and soft X-ray wavelengths musttake place in a gas or on a surface, making the use of standard QPM techniques impossible.
However, the extreme nonlinear nature of HHG makes alternate approaches possible. For example, quasi-phase matching of the frequency conversion process can be achieved by restricting

11
HHG to regions where the signal will be in phase and add constructively. Previous theoreticalideas for quasi-phase matching of HHG include using a modulated density gas or plasma toperiodically change the nonlinear susceptibility.[69] Another proposal suggested using counterpropagating light to modulate the intensity and phase of the driving laser.[77] Experimentally,
quasi-phase matching of HHG was first implemented using a technique proposed by Christov etal.[19] In this technique, high harmonics are generated in a hollow-core waveguide which has aperiodically changing inner diameter along the length of the guide to modulate the driving laser
intensity. The modulation of the laser intensity effectively turns on and off the generation ofharmonics near the cutoff. Figure 2.3.a shows a calculation of the signal of the 95th harmonic, inthe cutoff, as a function of propagation length in a straight waveguide using a 15 fs pulse with apeak intensity of 7 x 1014 W/cm2 and 1 Torr of argon. The calculation uses a three-dimensional
propagation model and the harmonic signal is calculated using a quasi-classical approximation
for the dipole moment. The oscillations of the harmonic signal are a result of the phase mismatchas signal builds up over a coherence length and then interferes with out-of-phase light in thenext coherence length. Figure 2.3.b shows the same calculation using a hollow-core waveguidewith a 0.5 mm period sinusoidal corrugation that changes the laser intensity on-axis by 5%.
The modulated waveguide causes a dramatic enhancement in the final signal (note the different
scales of the two plots).
We can understand the basic physics behind the enhancement of the HHG signal usingclassical nonlinear optics theory. In a simplified model of harmonic generation, the field ofharmonic order q, after propagating a distance L in a nonlinear medium, is related to the phasemismatch, ∆k, by:
Z L
Eq ∝

Eωn d(z)e−i∆kz dz,

(2.4)

0

where Eω is the fundamental field, n is the effective order of the nonlinear process, d(z) is thenonlinear coefficient, and ∆k is the phase mismatch calculated in equation 4. To model quasiphase matching, the nonlinear coefficient is expressed as a general periodic function of z with

12

Figure 2.3: a. Calculation of the signal of the 95th harmonic for a straight waveguide as afunction of propagation distance. b. Calculation for a 0.5 mm period modulated waveguide.
From [19].

13
period Λ:[27]d(z) = def f∞X

Gm eiKm z ,

(2.5)

m=−∞

where Km = 2πm/Λ is the effective wavevector of QPM and m is the order of the QPM process.
Combining Eqns. 2.4 and 2.5 shows that the harmonic signal will be enhanced when Km ≈ ∆k.
The enhancement in signal will be greatest for m = 1, but the signal will also be enhanced
for higher order QPM (i.e. m = 3,5,...). The new quasi-phase matching condition for HHG istherefore given by:∆k = ∆kW aveguide + ∆kP lasma + ∆kN eutrals − Km

(2.6)

qu211 λ2π(1 − η)P q∆δ+ P ηNatm re (qλ − λ/q) −− 2πm/Λ.
4πa2
λ

(2.7)

or
∆k =

From Eqn. 2.7, it is evident that a larger value of Km , i.e. a shorter modulation period, cancompensate for a larger ∆k, and therefore a higher level of ionization.

2.2

Experimental Results on Quasi-Phase Matching of High Harmonic Generation
The first set of experiments demonstrating quasi-phase matching of HHG showed a dra-

matic extension of the harmonic spectrum using modulated waveguides.[56] In this experiment,
25 fs duration pulses from a high repetition rate (2-5 kHz, 1 mJ/pulse) Ti:sapphire laser system
operating at 760 nm[7] were focused into 150 µm diameter modulated hollow-core waveguides
with different periodicities filled with helium gas. The modulated waveguides used in this experiment were produced using precision glass-blowing techniques starting with straight hollow-core
glass fibers, 150 µm in diameter. The modulations were approximately sinusoidal, with a radial depth on the order of 10 microns, corresponding to a 13% modulation of the fiber radius.
Figure 2.4 shows a picture taken with an optical microscope of a modulated waveguide with a
0.25 mm periodicity. The initial experiments used waveguides with modulation periods varying
from 1-0.5 mm. Figure 2.5 shows the experimentally measured HHG spectra from He for three14
different modulated-waveguide periodicities. As the modulation period is reduced from 1 mm
to 0.75 mm to 0.5 mm, the highest observable harmonic energy increases from 112 eV to 175 eV
and the signal is enhanced at higher and higher photon energies. This behavior agrees well withthe predictions from the simple QPM model that shorter modulation periods can compensate
for more ionization and thus phase match higher photon energies.

150 Microns

0.25 mmFigure 2.4: Optical microscope image of a 0.25 mm period modulated waveguide manufactured
by glass-blowing techniques. From [31].

The significance of QPM is that it enhances the higher energy harmonics that are generated at greater ionization levels, closer to the peak of the pulse. This effect becomes apparent
from a calculation of the coherence length that results from the mismatch in the phase velocity of
the fundamental and harmonic due to ionization. At high ionization levels, the phase mismatchis predominantly due to the plasma dispersion and the waveguide dispersion and is given by:∆k ≈ P ηNatm re qλ +qu211 λ
,
4πa2

(2.8)

where the assumption qλ >> λ/q is made. For example, using the same conditions as in the

15

100

L=1.0 mm
L=0.75 mm
L=0.50 mm

Intensity (Arb.)

10-1

10-2

10-3
80

100

120

140

160

180

Energy (eV)Figure 2.5: Experimental harmonic spectra from modulated waveguides filled with 111 Torr He
with a driving pulse of 25 fs and peak intensity ∼ 5 x 1014 W/cm2 , for different modulation
periodicities. From [56].

16
theory paper of Christov et al.,[19] fully ionized argon at a pressure of 1 Torr gives an electron
density of ne = 3.5 x 1016 cm−3 . From equation 2.8, the phase mismatch of the 95th harmonicorder (150 eV) is calculated to be ∆kplasma ∼ 7550 m−1 and ∆kwaveguide ∼ 6200 m−1 , givinga total ∆k ∼ 13,750 m−1 . The phase mismatch is close to the value of the effective QPM
wavevector for a modulation period of 0.5 mm of K1 ∼ 12,600 m−1 . Thus, very substantial
levels of ionization can be compensated for by using waveguides with modulation periods in therange of 1 - 0.25 mm that can be readily manufactured with glass-blowing techniques.
In the initial experiments, the ionization levels in He were still relatively low. However,in more recent experiments, we have demonstrated QPM in much higher ionization levels of> 100%. These experiments were performed using a two stage amplifier system (described inCh.4) with an output pulse energy of 1.2 mJ and a 22 fs pulse duration. Due to loss from mirror
reflections on the way to the waveguide setup, around 1.1 mJ pulse energy was focused into the
waveguide. The coupling efficiency was ∼ 60% for the straight, 0.5 mm, and 0.25 mm modulated
waveguides. The fundamental 800 nm light was blocked using a 0.4 µm-thick Zr filter with 25
nm of Ag coated on the front to prevent laser damage. Figure 2.6 shows harmonic spectra fromstraight and modulated waveguides filled with neon gas. The same gas pressure of 6 Torr wasused in the different waveguides for comparison. The decrease in harmonic signal going to lower
photon energies is due to the decreasing transmission of the zirconium filter. As expected, themodulated waveguides enhance the highest harmonic orders, and the signal from the 0.25 mm
modulated waveguide is about 8 times greater than from the 0.5 mm waveguide, showing theability of the shorter modulation periods to better compensate for the phase mismatch fromionization. The higher harmonics observed from the 0.5 mm modulated fiber could be due toa slightly better coupling efficiency. The estimated laser intensity in the waveguide is 9 x 1014
W/cm2 , producing ∼ 24% ionization in neon at the peak of the pulse. Figure 2.7 shows the
harmonic spectra from straight and 0.25 mm modulated waveguides filled with 7 Torr Ar. Here,the laser intensity is above the saturation intensity of argon so that the gas is fully ionized withsome double ionization, as verified using Ammosov-Delone-Krainov (ADK) tunnelling ionization

17
rates.[3] Even with above 100% ionization, the modulated fiber does enhance the signal by afactor of 2-3 over the entire measured spectral region. For the 109th harmonic order (170 eV),
we calculate a phase mismatch of ∆k ∼ 80,000 m−1 , therefore this signal could be enhanced by
m = 3 QPM. Where the enhancement is greatest at 110 eV (71st harmonic), ∆k ∼ 20,000 m−1 ,indicating a first order QPM process. Figure 2.8 shows harmonics from straight and modulated
waveguides filled with Ar taken on a different day. Here, the enhancement from QPM is evenmore dramatic.

In te n s ity ( a r b .)1 0 0 0 0

1 0 0 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

2 4 0

E n e rg y (e V )Figure 2.6: Harmonic spectra from straight (black), 0.5 mm (blue), and 0.25 mm (red) periodicity modulated waveguides filled with 6 Torr neon. The spectra were taken using the SC grating
and a 10 second exposure time.

The gas pressure in the waveguide is an important parameter in these experiments. Figure
2.9 shows harmonic spectra from the three different waveguides at different neon pressures.
Typically, lower gas pressures increase the highest observed harmonic orders. This effect could
be a result of reduced plasma defocusing or absorption, both of which can reduce the intensityof the driving laser. Fig. 2.9 also plots the flux for three different photon energy ranges as afunction of pressure for the different waveguides. From these plots, it is apparent that there

18

Figure 2.7: Harmonic emission from straight (black) and 0.25 mm (red) modulated waveguides
filled with 7 Torr Ar. From [31]. The spectra were taken using the SC grating. The exposure
time for the straight waveguide was 300 seconds while for the modulated it was 200 seconds.
The spectrum for the modulated waveguide has been multiplied by 1.5 for comparison.

19

Figure 2.8: Harmonic spectra from a straight waveguide filled with 10 Torr Ar (blue) and 6 Torr
Ar (black), and a 0.25 mm modulated waveguide with 5 Torr Ar(red). All spectra were takenwith the SC grating. Both straight waveguide spectra have a 300 s exposure time while themodulated waveguide spectrum was taken with 600 s exposure and divided by 2 for comparison.

20
is phase matching of higher photon energies in the modulated waveguides while in the straightwaveguide, phase matching is only achieved for the lower energies.
The next series of experiments were done at a higher laser intensity of 1.6 x 1015 W/cm2 ,accomplished by using two pump lasers on the second stage of the amplifier giving ∼3 mJ ofpulse energy at the output. The fundamental light was blocked with Ag filters of 0.45 µm
total thickness. Figure 2.10 shows the harmonic emission from 0.25 mm-modulated and straightwaveguides filled with 9 Torr Ne. At these intensities, a simple calculation of the cutoff harmonicenergy, using Eqn. 1.1, yields an expected cutoff of 330 eV. However, the observable harmonic
emission from the straight waveguide (Fig. 2.10 (black curve)) only extends to around 225
eV. In contrast, the harmonic emission from the 0.25 mm modulated waveguide (red curve)
is brighter and extends to significantly higher energies, where the carbon edge at 284 eV isclearly visible. Figure 2.11 shows the harmonic spectra when using helium gas. The modulated
waveguide enhances the harmonic signal over the entire measured spectrum by at least an orderof magnitude, and extends the observable harmonics to 284 eV.
To estimate the amount of phase mismatch in these experiments, the amount of ionization
during the time-window in the pulse that a particular harmonic is generated is calculated using
ADK (Ammosov-Delone-Krainov) tunnelling ionization rates.[3][See Appendix A for details on
the calculations] Figure 2.12 shows the fractional ionization of neon and helium as a functionof time during the driving pulse for our experimental conditions (pulse duration of 22 fs, peakintensity of 1.6 x 1015 W/cm2 ). The amount of ionization at the threshold intensity necessary
to generate 285 eV harmonics is ∼ 50% in neon and ∼ 10% in helium, while at the peak of thepulse it is ∼ 90% and ∼ 37%. The increasing amount of ionization leads to an increasing phasemismatch and therefore the dramatic decrease in the flux observed going to higher harmonicenergies. Using the expression in Eq. 2.8, which is valid at high ionization levels, the phasemismatch for the 183rd harmonic order at the C edge (284 eV) in Ne is calculated to be ∆k ∼77,000 m−1 , assuming 50 % ionization. The effective wavevector for first-order QPM for a
modulation period of 0.25 mm is K1 ∼ 25,000 m−1 . Again, the harmonic signal will be enhanced

21

6 Torr
11 Torr
20 Torr
30 Torr
40 Torr

10000

90-110 eV
110-125 eV
125-150 eV

6.0x105

Integrated Counts

Intensity (arb.)

Straight

1000

4.0x105

Divided by 2

2.0x105

0.0
100

120

140

Energy (eV)

160

5

180

10

15

20

25

30

35

40

45

Pressure (Torr)

0.5-mm Modulated
4.0x105

Integrated Counts

Intensity (arb.)

6 Torr16 Torr
21 Torr
39 Torr

110-140 eV
140-180 eV
180-210 eV

2.0x105

x10

0.0
100

120

140

160

180

200

5

220

Energy (eV)

10

15

20

25

30

35

40

Pressure (Torr)

6 Torr
11 Torr
15 Torr
20 Torr
30 Torr

10000

1000

2.0x106

Integrated Counts

Intensity (arb.)

0.25-mm Modulated

1.5x106

100-130 eV
130-150 eV
150-180 eV

1.0x106
5.0x105
0.0

100

120

140

160

Energy (eV)

180

200

5

10

15

20

25

30

Pressure (Torr)

Figure 2.9: Harmonic spectra from straight, 0.5 mm, and 0.25 mm modulated waveguides, withdifferent pressures of neon gas. All spectra were taken with a 10 s exposure time on the SC
grating. Also plotted are the integrated counts for three different photon energy ranges as afunction of pressure. The energy ranges are indicated with dashed lines.

22

C EdgeFigure 2.10: Experimental harmonic spectra generated in 9 Torr of neon gas for straight (black)and 0.25 mm modulated (red) waveguides. Both spectra taken with a 180 s exposure time on
the SB grating. From [31].

23

In te n s ity ( a r b .)1 0 0 0 0

1 0 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

E n e rg y (e V )Figure 2.11: Experimental harmonic spectra generated in helium at 15 Torr using a 0.25 mm
modulated waveguide (black), and at 16 Torr (blue) and 10 Torr (red) using a straight waveguide.
All spectra taken with 120 s exposure on SB grating.

3 5 01 .0

2 8 5 e V
3 0 0
In te n s ity
N e o n Io n iz a tio n
H e liu m Io n iz a tio n

0 .8

2 5 0

0 .6

2 0 0

0 .4

1 5 0
1 0 0

0 .2

3 3 0 e V

C u to ff P h o to n E n e rg y

N o r m a liz e d In te n s ity / Io n iz a tio n F r a c tio n

24

5 0
0 .0
0
-4 0

-2 0

0

2 0

4 0

T im e ( f s )Figure 2.12: Calculation of the fractional ionization of neon and helium as a function of time
during the laser pulse. The right axis plots the cutoff photon energy corresponding to theinstantaneous pulse intensity.

25
by QPM when Km ≈ ∆k. The greatest enhancement is for m=1 QPM, but there will still beenhancement of the HHG signal for higher values of m. Thus, for the harmonics in Ne near the
C edge, it is possible to compensate for the large phase mismatch of ∼ 77,000 m−1 by thirdorder (m=3) QPM. For the harmonics above 225 eV, the phase mismatch is large enough thatthe signal can only be observed using a modulated waveguide and is below the noise level fora straight waveguide. For helium, ∆k ∼ 34,000 m−1 for the 183rd harmonic order at the Cedge, closer to first order QPM. The spectra shown in figures 2.10 and 2.11 illustrate that thehighest observed harmonic orders are not limited by the Ip + 3.17Up relation, but by ionizationinduced phase mismatch. The modulated waveguide compensates for this phase mismatch,making it possible to observe higher harmonics. Although past work observed HHG from He in
the water window, this process was not phase matched and produced significantly lower signal
levels.[15, 70]
Figure 2.13 compares harmonic spectra at the carbon edge using helium and neon as
the generation gas. All spectra were with the same experimental conditions (i.e. driving pulse
parameters and exposure time). The signal at the C edge for 9 Torr Ne is about 75% of theintensity for 60 Torr He. For the case of helium, the phase mismatch is around ∆k ∼100,000
m−1 . However, even though the phase mismatch is greater, the atom density is higher by afactor of 7, which can explain the higher flux. Further comparison between helium and neonis shown by the spectra in Fig.2.14, showing both at similar pressure. The phase mismatch at
the C edge for 15 Torr He is ∆k ∼ 34,000 m−1 compared to ∆k ∼ 70,000 m−1 for 8 Torr of
Ne. Therefore the higher signal for He compared with Ne is due to the lower order of the QPM
process.
Based on conservative estimates of detection efficiency and filter transmission, and measurements of grating efficiency, a minimum flux of between 106 − 108 photons/sec in a 10%
bandwidth at the C edge was calculated. Details of the calculation are given in Ch.4. This flux
may be sufficient for biological imaging applications and a number of improvements, such as
shorter modulation periods and longer waveguides, can lead to further enhancements.

26

In te n s ity ( a r b .)1 0 0 0 0 0

1 0 0 0 0

1 0 0 0

1 0 0
1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

E n e rg y (e V )Figure 2.13: Comparison of harmonic spectra at the C edge using a 0.25 mm modulated waveguide filled with 9 Torr Ne (black), 42 Torr He (blue), and 60 Torr He (red). All spectra takenwith 180 s exposure time using SB grating.

27

In te n s ity ( a r b .)1 0 0 0 0

1 0 0 0

1 0 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

E n e rg y (e V )Figure 2.14: Comparison of harmonic spectra from 8 Torr Ne (black) and 15 Torr He (red)using a 0.25 mm modulated waveguide. Both spectra taken with 120 s exposure time using SB
grating.

28

2.3

Mechanisms for Enhancement by Quasi-Phase Matching
In this section, I will discuss in more detail the physical processes that lead to enhance-

ment of HHG in modulated waveguide structures. From propagation models,[19] we know thatthe effect of the diameter modulation is to change the on-axis intensity in a periodic way. The
modulations of the waveguide radius occur over too short a distance for the laser mode to adiabatically follow the waveguide diameter. Instead, the change in diameter changes the boundary
conditions, exciting a transverse wave that propagates to the center of the waveguide. When thetransverse wave reaches the center of the waveguide, the interference causes oscillations in theintensity. Figure 2.15 shows calculations of the radial profile of the laser intensity and phase in
the waveguide, showing the initial laser mode and the modes giving the maximum and minimum
on-axis intensity. The calculations were performed for a 100 µm diameter waveguide with 6%
modulations and a periodicity of 0.5 mm, filled with 2 Torr argon for a 15 fs pulse with a peakintensity of 5 x 1014 W/cm2 . The amount by which the intensity is modulated can be approximately estimated by assuming the adiabatic case where the peak intensity is proportional tothe inverse of the waveguide radius squared.

2.3.1

Modulating the Amplitude of the Harmonics
The periodic modulation of the on-axis laser intensity has several different effects on thebuildup of harmonic signal over the length of the waveguide. The most dominant effect occurs
for the cutoff harmonics that are generated at the very peak of the laser pulse. These harmonics
will only be generated in the higher intensity regions (i.e. the smaller diameter sections of the
waveguide), effectively turning on and off the generation. For the lower energy harmonics, theeffect of modulating the laser intensity is more complicated. In a certain time-window during theleading edge of the pulse, the intensity modulations will also turn on and off the generation of aparticular harmonic. This effect is illustrated in Fig. 2.16 which shows the different times on theleading edge of the pulse where the intensity is great enough to generate a particular harmonic

29

(a)

(b)

(c)

Figure 2.15: Calculations of the radial profile of the intensity (red curve) and phase (green
curve) of the laser inside the modulated waveguide. The three frames correspond to: a. theinitial mode, b. the mode giving the maximum on-axis intensity and c. the minimum on-axis
intensity. From Ivan Christov.

30
when the pulse peak intensity changes by 10%. It is interesting to note that for increasing
harmonic orders, i.e. higher threshold intensities, the duration of this time-window increases.
The expected enhancement due to modulating the harmonic amplitude can be modelledusing the simple formalism described previously. Evaluating equations 2.4 and 2.5 gives anexpression for the amplitude of the harmonic field as a function of propagation length, L:[27]Eq ∝ idef f Eωn Le−i(∆k−Km )L/2 Gm

Sin[(∆k − Km )L/2],(∆k − Km )L/2

(2.9)

where the integral is dominated by the term in the sum for which ∆k ≈ Km . The quasiphase matching bandwidth is the same as the normal phase matching bandwidth shifted by theeffective wavevector, Km , so that using a longer medium with more modulated sections will
narrow the bandwidth. In the case of periodically poled materials where the phase relation
between the fundamental and harmonic is adjusted by 180 degrees every coherence length, thenonlinear coefficient can be expressed as a square wave function, oscillating between +1 and -1
every coherence length, with the coefficients given by:[13]Gm = (2/mπ)Sin(mπ/2)(2.10)

In our case, the enhancement in harmonic field is half that for the case described above since
we only enhance the signal every other coherence length. For our case with perfect quasi-phasematching (i.e. Km = ∆k), the enhancement is:Iq ∝ (def f Eωn )2 (1/mπ)2 L2(2.11)The enhancement in harmonic intensity is reduced by a factor of (1/mπ)2 compared to the caseof no phase mismatch. The greatest enhancement will be from m=1 quasi-phase matching andwill decrease for higher values of m.
The maximum enhancement of HHG that can be achieved by quasi-phase matching depends upon how strongly the harmonic light is absorbed by the generation gas. Fortunately,
going to higher photon energies, the amount of absorption by the gas decreases significantly.
Figure 2.17 plots the absorption length, Labs , as a function of photon energy for Ne and He

31

1 .2
1 .0
T h r e s h o ld In te n s ity

In te n s ity

0 .8
0 .6
H a r m o n ic is a lw a y s g e n e r a te d

0 .4
0 .2

In te n s ity m o d u la tio n s
tu r n h a r m o n ic o n a n d o ff

0 .0
-3 0

-2 0

-1 0

0

1 0

2 0

3 0

T im e ( f s )Figure 2.16: Diagram showing region in pulse where a given harmonic, only generated above
the threshold intensity, is turned on and off by a 10% intensity modulation.

32

4 0 0

A b s o r p tio n L e n g th ( c m )1 0 T o rr N e o n
1 0 0 T o r r H e liu m
3 0 0

2 0 0

1 0 0

0
0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0P h o to n E n e rg y (e V )Figure 2.17: Plot of absorption length as a function of photon energy for 10 Torr Neon and 100
Torr Helium. Data from the Center for X-ray Optics, Berkeley Lab (www-cxro.lbl.gov).

33
gases. For higher photon energies, at and above the C-edge (284 eV), the absorption length
dramatically increases reaching several meters in 100 Torr helium. For quasi-phase matching,
the optimal signal is obtained at the gas pressure where the m=1 quasi-phase matching condition is met. Previous calculations have shown that the phase matched flux is optimized whenthe medium length is > 3Labs and it saturates at around 10Labs .[21] Under our typical experimental conditions, m=1 quasi-phase matching of the harmonics at the C-edge in helium gas
is achieved at a pressure of 9 Torr (for a 0.25 mm modulated waveguide) and the absorption
length at this pressure is Labs = 1.64 m. Therefore, by simple estimates, the maximum signal enhancement possible would be ∼ (1/mπ)2 (10Labs /Lcoh )2 (1/e)10 ∼ 80, 000. With shorter
modulation periods, the same signal enhancement could be obtained at a higher pressure anda shorter waveguide. One issue with using such a long waveguide is the power loss associatedwith the fundamental mode propagation. The attenuation constant for the EH11 mode for 800
nm light and a waveguide radius of 75 µm is α = 0.323 m−1 .[50] Also, the dispersion of the
waveguide can stretch the pulse in time. The propagation length that would approximately
double the pulse duration is ∼ 14 m.[Note: This calculation is detailed in Ch. 5] Both effects
will reduce the peak intensity of the pulse as it is propagating down the waveguide, preventing
quasi-phase matching over the entire length. One elegant solution to this problem, mentioned
by Ivan Christov in Ref. [19], would be to use a tapered waveguide with a decreasing radiusalong its length, maintaining the same on-axis intensity throughout.

2.3.2

Grating-Assisted Phase Matching
Typically quasi-phase matching is implemented by a modulation of the nonlinear coeffi-

cient, however, it is also possible to achieve an enhancement from quasi-phase matching by modulating the linear optical properties. This process is termed grating-assisted phase matching.[27]Modulation of the laser intensity in the waveguide results in a modulation in the value of thephase mismatch, ∆k, along the waveguide from the changing plasma dispersion. The modulation in intensity directly modulates the ionization fraction. For a sinusoidal modulation in the34intensity, the plasma contribution to the phase mismatch can be expressed approximately as,∆kplasma (z) = P Natm re qλ(η + δη cos(2πz/Λ)),(2.12)

where δη is the change in ionization and Λ is the period of the waveguide modulations. To get
an idea of the amount by which the ionization can change, Fig. 2.18 shows the ionization as afunction of time for two different laser intensities calculated from ADK ionization rates. At thepeak of the pulse, the ionization fraction is 0.35 for an intensity of 1015 W/cm2 and 0.41 when
the intensity is increased by 5%. The value of ∆kplasma therefore changes by 16%. Dramaticenhancement of the harmonic signal can be obtained if the phase mismatch varies sinusoidallywith a period of twice the coherence length of the average value of ∆k. Mathematically, thephase mismatch is expressed by:

∆k =

π
πz
(1 + α sin( )),Lc
Lc(2.13)

where α is the fractional change. Figure 2.19 shows the calculated harmonic intensity for asinusoidally varying phase mismatch for variations of 1, 3, and 5 %. It is interesting thatthe smaller variation can give more signal enhancement, but only over a longer propagation
distance. The largest enhancement for a 1 % modulation occurs for a propagation distance of∼ 125 coherence lengths and gives a factor of 2000. Figure 2.20 shows the harmonic signal forthe first 60 coherence lengths. At the beginning, the QPM process is working until a certain
distance at which the signal buildup reverses and then oscillates.
These calculations only take into account the effect of ionization, as would be the case if
one could artificially create a longitudinally varying ionization in a straight waveguide. In themodulated waveguide, however, there is also a spatial phase due to the periodic refocusing of thefundamental light. The combination of both effects must be taken into account to understandthe amount of enhancement under real experimental conditions.

N o r m a liz e d In te n s ity /Io n iz a tio n F r a c tio n

35

1 5

1 .0 5 x 1 0
W
1 5
1 .0 0 x 1 0
W

1 .0

/c m
/c m

2
2

0 .8

0 .6

0 .4

0 .2

0 .0

-3 0

-2 0

-1 0

0

1 0

2 0

3 0

T im e ( fs )Figure 2.18: Comparison of ionization fraction of neon as a function of time for a 22 fs duration
pulse at different peak intensities.

36

1%
3%
5%

2000

Harmonic Intensity

1500

1000

500

0

0

50

100

150

200

250

300

Distance (Coherence Length)

350

400

450

500

Figure 2.19: Harmonic intensity as a function of propagation distance for the case where thephase mismatch varies sinusoidally by 1, 3, and 5 % with a period of 2 coherence lengths.

37

300
1%
3%
5%
250

Harmonic Intensity

200

150

100

50

0

0

10

20

30

40

Distance (Coherence Length)

50

60

Figure 2.20: Plot of the harmonic intensity during the first 60 coherence lengths.

38
2.3.3

Intrinsic Phase Effects
Another mechanism for quasi-phase matching arises from the finite time response of theharmonic generation process, causing an intensity-dependent phase of the emitted harmonics.
The Lewenstein model of HHG, discussed in Ch. 1, describes the phase of the emitted harmonics
as the phase the electron wavefunction acquires from when the atom is ionized to the time it
recombines. Due to this intrinsic phase, modulation of the intensity of the driving laser canperiodically adjust the phase relationship between the harmonics and the fundamental laser.
The full expression for the harmonic phase relative to the fundamental is given by:[9]Φ = qωtf − S(t0 , tf )/h̄,(2.14)

where q is the harmonic order, ω is the fundamental laser frequency and S(t0 , tf ) is the quasiclassical action of the electron from the time the atom is ionized, t0 , to when the electron
recombines, tf . The quasi-classical action is given by the expression:
Z tf
S(t0 , tf ) =

(
t0

p2
+ Ip )dt
2m(2.15)

where p is the electron momentum, m is the electron mass, and Ip is the ionization potentialof the atom. Previous theoretical work has solved for the phase of the harmonics using themethod of stationary action.[47, 29] Here, I will follow a slightly different approach, solving the
integral in Eqn. 2.15 numerically over the classical trajectory of the free electron to determine
the harmonic phase.
The classical trajectory of a free electron in the laser field is given by:mdv= −eE(t).
dt(2.16)

Making the assumption of a monochromatic laser field: E(t) = E0 Cos(ωt), the equations of
motion for the electron are:−eE0
(Sinφ − Sinφ0 )mω

(2.17)

eE0(Cosφ − Cosφ0 − Sinφ0 (φ − φ0 )),mω 2(2.18)v(φ, φ0 ) =andx(φ, φ0 ) =39where φ = ωt and φ0 = ωt0 is the phase in the laser cycle when the atom is ionized, termed therelease phase. From the equations of motion, we can solve for the phase, φf , when the electron
returns to the position of the ion. This return phase as a function of the release phase is plottedin Fig. 2.21. It is then straightforward to solve for the electron’s velocity, and therefore its
kinetic energy, when it re-encounters the ion. Fig. 2.22 plots the kinetic energy as a function of
the electron’s release phase. From Fig. 2.22 it is evident that for the harmonics at the cutoff,
there is only one classical trajectory; however, for the plateau harmonics, there are two different
trajectories (short and long) with the same return kinetic energy. For the short trajectories,
the electron is released at a phase > 18 degrees, and re-encounters the ion in less than half an
optical cycle. The duration of the long trajectories, however, is almost a full optical cycle andtherefore the electron acquires more phase.
Combining equations 2.15 and 2.17 gives the following expression for the intrinsic phaseof the harmonics due to the phase that the electron acquires in its trajectory as a function of
the release phase, φ0 , and the return phase φf :φintrinsic =Ip
2Up(φf − φ0 ) +
h̄ω
h̄ω

Z φf

(Sinφ − Sinφ0 )2 dφ,(2.19)

φ0

where the definition of the ponderomotive energy, Up = e2 E 2 /4mω 2 , is used. For the case of very
high harmonic orders where the condition Up >> Ip is met, the first term in Eqn. 2.19 can beneglected and the electron kinetic energy is approximately equal to the emitted photon energy.
Using this assumption, the intrinsic phase as a function of harmonic energy is calculated, plottedin Fig. 2.23.

To find the harmonic phase relative to the fundamental, it is necessary to include

the first term in Eqn. 2.14. The result for the harmonic phase relative to the fundamental isshown in Fig. 2.24.
For quasi-phase matching, it is important to know how the harmonic phase changes withlaser intensity. Fig. 2.25 plots the result for the intensity dependence of the harmonic phase fortwo different harmonic orders, assuming an 800 nm driving laser and the condition Ip = 0. Thelong trajectories have a linear intensity dependence equal to ∼ -23 x 10−14 Radians·cm2 /W. The

40

3 5 0

R e tu rn P h a s e (d e g .)

3 0 0

2 5 0

2 0 0

1 5 0

1 0 0

5 0
1 8

3 6

5 4

7 2

9 0

R e le a s e P h a s e ( d e g .)Figure 2.21: Return phase, φf , as a function of the release phase, φ0 , determined by the classicalequations of motion.

3 .5

3 .2 U p

R e tu r n K in e tic E n e r g y ( U p )

3 .0

2 .5

2 .0

1 .5

L o n gS h o rtT r a je c to r ie s

1 .0

0 .5

0 .0
0

2 0

4 0

6 0

8 0

1 0 0

R e le a s e P h a s e ( d e g r e e s )Figure 2.22: Electron return kinetic energy for trajectories with different release phases, φ0 .
The long and short trajectories are indicated.

41

6

L o n g T r a je c to r ie sp

/ h v )

5

In tr in s ic P h a s e ( U

4

3

2

S h o r t T r a je c to r ie s
1

0
0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

H a r m o n ic E n e r g y ( U p )Figure 2.23: Intrinsic phase as a function of harmonic energy. The long trajectories acquire
more phase than the short trajectories.

1 2
1 0

R e la tiv e P h a s e ( U p /h v )86

S h o r t T r a je c to r ie s
4
2
0

-2

L o n g T r a je c to r ie s
-4
-6
0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

H a r m o n ic E n e r g y ( U p )Figure 2.24: Plot of harmonic phase in reference to the fundamental as a function of harmonicenergy. Note there is a linear dependence which is different for the long and short trajectories.

42

R e la tiv e P h a s e ( R a d ia n s )

2 0 0

4 5

th

2 5

th

H a r m o n ic
H a r m o n ic

1 0 0

0

-1 0 0L o n g T r a je c to r ie s :-1 4s lo p e = - 2 3 x 1 0
R a d ia n s * c m-2 0 02/W-3 0 0
0

2

4

6

8

1 0

In te n s ity ( x 1 0

1 2
-1 4

W

1 4

/c m

1 6
2

1 8

2 0

)

Figure 2.25: Plot of the intensity dependence of the harmonic phase for two different harmonicorders. The calculation was done for a fundamental wavelength of 800 nm.

43
short trajectories have a more complicated dependence on intensity which is not exactly linear
and is significantly smaller than the long trajectories. For the condition Ip = 0, the intensity
dependence of the phase is not a function of harmonic order. It is interesting to compare
these results with previous work done using the method of stationary action. Lewenstein et al.
calculated that for Ip = 0, the harmonic phase varies with intensity as -4.08 Up /h̄ω at the cutoff
and as -6.28 Up /h̄ω in the plateau region (accounting only for the long trajectories).[47] Using
the definition for the ponderomotive energy, these values correspond to: -15.7 x 10−14 cm2 /W
and -24.2 x 10−14 cm2 /W for 800 nm light.
For the long trajectories, an intensity change of 1.4 x 1013 W/cm2 would produce aphase shift of π, while the same phase shift for the short trajectories at the cutoff would requirea change of 3.4 x 1013 W/cm2 . This corresponds to an intensity change of 1.4 and 3.4 % ifusing a driving laser intensity of 1015 W/cm2 . Therefore, the intensity-dependent harmonic
phase does play a role in the quasi-phase matching of the emission of harmonics generated at
the cutoff for intensities around 1015 W/cm2 and above.

2.4

Generation of an Enhanced Attosecond Pulse through Phase
Matching
An important aspect of phase matching and quasi-phase matching of high-order harmonic

generation is the changing amount of ionization during the pulse which creates a narrow time
window over which the phase matching condition is met. Only the harmonics generated in thistime window are enhanced, making it possible to generate a single enhanced attosecond-duration
pulse. To illustrate this, Fig. 2.26 shows a calculation using ADK theory of the phase mismatchas a function of time during the laser pulse. The dashed line in the figure shows when the phasematching condition is met for quasi-phase matching with a periodicity of 0.25 mm. Fig. 2.27
shows a simple calculation of the expected intensity of the 171st harmonic as a function of time
during the laser pulse from quasi-phase matching using Eqn. 2.9. Note that longer propagation
lengths will both increase the signal and shorten the duration of the time window.

3 5 0 0 0

1 .0

3 0 0 0 0

0 .8

0 .2 5 m m

m o d u la tio n

0 .6

2 5 0 0 0

0 .4

2 0 0 0 0

In te n s ity ( N o r m .)P h a s e M is m a tc h ( m

-1

)

44

0 .2
1 5 0 0 0
0 .0
1 0 0 0 0
-4 0

-3 0

-2 0

-1 0

0

1 0

2 0

3 0

4 0

T im e ( fs )Figure 2.26: Calculation of the phase mismatch (∆k) for the 171st harmonic in 3 Torr neon for
a 20 fs duration pulse with a peak intensity of 1015 W/cm2 . The dashed line indicates when the
QPM condition is met for a modulation period of 0.25 mm.

1 .2 x 1 0-6

1 .0 x 1 0-6

8 .0 x 1 0-7

6 .0 x 1 0-7

4 .0 x 1 0

-7

0 .4

2 .0 x 1 0

-7

0 .2

H a r m o n ic S ig n a l

0 .8

0 .6

N o r m a liz e d In te n s ity

1 .0

0 .0

0 .0
-3 0

-2 0

-1 0

0

1 0

2 0

3 0

T im e ( f s )Figure 2.27: Calculation of the harmonic signal of the 171st harmonic in 3 Torr neon for a 20
fs duration pulse with a peak intensity of 1015 W/cm2 by quasi-phase matching with 0.25 mm
periodicity. Here, the propagation length is 1 mm.

45
Enhancement of a single attosecond-duration pulse by quasi-phase matching was firstpredicted by Ivan Christov in Ref. [19]. Propagation of the laser in the hollow-core waveguide ismodelled by numerical solution of the three-dimensional scalar wave equations. The harmonic
emission is calculated from a quasi-classical approximation of the dipole moment. Fig. 2.28
shows the harmonic spectrum after propagation through a 1 cm long straight and modulated
waveguide. Fig. 2.29 shows the temporal properties of the harmonic output for the straight andmodulated waveguides. Because the coherence lengths for neighboring harmonics are similar, themodulated waveguide enhances several harmonics in the cutoff region. Temporally, the emission
from these harmonics is phase matched over a single cycle of the driving laser, creating a single
attosecond-duration pulse.
Experimentally, the effect of phase matching over a single cycle of the driving laser wasobserved by Durfee et al. in Ref.[26]. For phase matching in a straight waveguide, they calculated
that tuning the pressure would allow phase matching at different half cycles of the pulse (i.e.
different ionization levels). Figure 2.30.a shows their calculation of the pressure dependenceof the flux of the 29th harmonic for a 20 fs pulse in Ar gas. The different peaks correspondto optimal phase matching at different ionization ”steps”. Experimentally, they observed thatin the pressure region corresponding to phase matching, the flux was rapidly oscillating, Fig.
2.30.b, attributed to the sensitivity of the amount of ionization on the carrier-envelope offset(CEO) of the pulse, illustrated in Fig. 2.31. However, in the lower pressure region where theprocess is not phase matched, the signal does not fluctuate since it is not sensitive to the CEOof the pulse.
The rapidly increasing ionization from the HHG process, together with a stable carrierenvelope offset, may make it possible to phase match only the emission from a particular single
half cycle of the driving laser, giving a single, enhanced attosecond-duration pulse. Withoutphase matching, it is only possible to generate a single attosecond pulse by using an extremely
short driving laser (i.e. 6 fs in duration). Generating such a short pulse at the fundamental
involves complicated pulse compression techniques which greatly decrease the pulse energy and46

Figure 2.28: Calculation of the harmonic spectra for (a) a straight waveguide and (b) a 0.5 mm
periodicity modulated waveguide for a 15 fs duration pulse with an intensity of 7 x 1014 W/cm2
in 1 Torr Ar. The calculated single atom spectrum is also shown.

47

Figure 2.29: Calculation of the time dependence of the harmonic output for (a) the straightwaveguide and (b) the 0.5 mm periodicity modulated waveguide.

48

Figure 2.30: a. Calculation of the flux of the 29th harmonic for a 20 fs laser pulse and peakintensity of 2.2 x 1014 W/cm2 as a function of argon pressure using a 3 cm waveguide. b.
Experimentally measured flux for the same conditions as part a. From [26].

49

1 .0
0 .1 00 .6
0 .0 60 .4

0 .0 40 .0 20 .2

0 .0 0

0 .0
-2 0

-1 0

0

1 0

N o r m a liz e d In te n s ity

Io n iz a tio n F r a c tio n

0 .8
0 .0 8

2 0

T im e ( fs )Figure 2.31: Calculation of the fractional ionization of argon using ADK ionization rates for a
20 fs pulse with peak intensity of 2.2 x 1014 W/cm2 for a cosine (black) and sine (red) pulse.

50
stability (since they rely on nonlinear processes to broaden the spectrum). Phase matchingin hollow waveguides thus presents a more elegant, stable, and efficient method for attosecond
pulse generation.

Chapter 3

High Harmonic Generation from Ions

The highest photon energies obtained from high harmonic generation (HHG) have generally been limited not by the cutoff rule, but by the detrimental effects of ionization when using
high laser intensities. The resulting plasma can defocus the laser beam,[59] limiting the peakintensity and therefore Up . It also causes a significant phase velocity mismatch between the driving laser and the harmonic light, greatly reducing the harmonic signal. Ionization can be kept
to a minimum by using shorter-duration laser pulses or atoms with a large ionization potential,
thus allowing the atoms survive to high laser intensities before ionizing. The highest harmonics
observed to date, at around 950 eV, have been generated using helium,[15, 70, 67] which has
the largest ionization potential of the noble gases. However, helium also has an exceptionallysmall effective nonlinearity, which limits the harmonic flux. Larger, multi-electron atoms such asargon typically generate greater signals, but the highest observable harmonic orders have been
comparatively lower. For example in Ar, photons of greater than 100 eV energy have never been
observed using an 800 nm driving laser,[83, 80] even using very short 7 fs duration pulses[65] orlarge pulse energies. In other work, a longer-wavelength driving laser was used to increase theharmonic emission in Ar to ∼ 150 eV.[68] Using longer wavelengths increases the value of Up forthe same laser intensity, however, this approach also reduces the efficiency of the process since
the electron wavefunction is more delocalized, lowering the probability of recombination.

52

3.1

Previous Experiments on HHG from Ions
Harmonic emission from neutral atoms is limited by the saturation intensity, or theintensity at which ∼ 98% of the atoms are ionized. However, emission from ions can in theory
extend to very high energies, since the saturation intensity for each successive stage of ionization
is progressively higher. Krause, Kulander, and Schaffer calculated that emission from neutrals
and ions is of comparable strength with emission from ions extending to higher energies.[42] In
previous experiments, Wahlstrom et al. observed harmonic emission from Ar ions up to ∼ 100
eV using an 800 nm driving laser with 250 fs pulse width and 250 mJ of pulse energy focused to
an intensity of ∼ 1015 W/cm2 . Their results are shown in figure 3.1. The emission attributed
to ions is several orders of magnitude weaker in intensity than the emission from neutral atoms
and is still short of the predicted Ip + 3.2Up rule.[79, 80]

Figure 3.1: Harmonic emission from argon using a gas jet with 250 mJ pulse energy and a 250
fs pulse. From [80]

53

3.2

Experimental Results using Hollow-core Waveguides
In our experiment, high harmonics are generated by focusing intense laser light into ahollow-core waveguide filled with low pressure gas. The waveguide has several advantages overa gas jet. It counteracts the effect of plasma-induced defocusing, allowing high laser intensities
to be achieved in a fully ionized gas medium. Also, using a modulated-diameter waveguide, thelarge phase mismatch associated with ionization can be partially compensated by quasi-phasematching. As a result, we observed harmonic generation from argon up to 250 eV, an extensionof 100 eV over previous results using a gas jet setup for HHG. These results demonstrate thatemission from ions can greatly increase the photon energies obtainable from HHG and that largeions such as Ar, with high nonlinear susceptibilities compared to He, can be used to generateharmonic emission at energies above 200 eV, in the soft x-ray region of the spectrum.
In our experiment, we use a 1 kHz Ti:sapphire laser system[8] producing 22 fs-duration
pulses, with pulse energies of 3 mJ. The laser light is focused into 150 µm inner-diameter, 2.5
cm long, hollow-core waveguides filled with argon. The harmonic emission emerging from thewaveguide passes through a zirconium or silver filter to block the fundamental light. Frequencyresolved-optical-gating (FROG) measurements of the laser pulse before and after the fiber filledwith low-pressure Ar show some shortening of the pulse from the 22 fs input to 18 fs at the exit
due to ionization effects to be discussed in Chapter 5. Care was taken to prevent contaminationof the gas by evacuating the waveguide first to ∼10−2 Torr, and then flowing Ar continuously
through it. Figure 3.2 shows the harmonic emission from a straight waveguide filled with 7 Torrof Ar at a peak laser intensity of ∼ 9 x 1014 Wcm−2 . Under these conditions, the harmonic
emission extends up to 180 eV. The falloff in the spectrum toward low energies is a result of thetransmission characteristics of the Zr filter used to reject the laser light. Here, we use a 0.4 µm
thick Zr filter with 25 nm of Ag coated on one side. When a 0.25 mm-modulated waveguide wasused under the same conditions, the highest photon energy observed was the same; however, thesignal was 2-3 times stronger.

54
The laser intensity was determined from the experimentally measured pulse energy andpulsewidth, and is accurate to within 10%. This intensity was also compared with that required
to generate the experimentally observed harmonic cutoff, which gives a lower limit for theintensity. Figure 3.3 shows the harmonic emission from Ar at a slightly higher laser intensityof 1.3 x 1015 Wcm−2 using a waveguide with a periodically modulated diameter.[56] We use a0.45 µm thick Ag filter that is 5-15% transmissive from around 120 to 385 eV to reject the laser
light. In Fig. 3.3, the emission from Ar at this higher intensity now extends up to 250 eV. Notethat the increase in observable harmonic order scales linearly with intensity, as expected fromthe cutoff rule provided that saturation effects do not limit the emission. The signal is clearly
emission from HHG since the signal disappears if circularly polarized driving pulses are used.

5 0 0 04 0 0 0In te n s ity ( a r b .)3 0 0 02 0 0 01 0 0 0

1 0 0

1 2 5

1 5 0

1 7 5

2 0 0

2 2 5

E n e rg y (e V )Figure 3.2: Harmonic emission from a straight 150 µm inner diameter, 2.5 cm long fiber filledwith low-pressure Ar (7 Torr), driven by an 800 nm laser with an 18 fs pulse duration and a peakintensity of 9 x 1014 Wcm−2 . The spectrum was taken using the SC grating and an exposure
time of 300 s.

55Intensity (arb.)

10000

1000

100

10
100

150

200

250

300

350

Energy (eV)Figure 3.3: Harmonic emission at a higher peak laser intensity of 1.3 x 1015 Wcm−2 , using a
0.25 mm period modulated waveguide. The spectrum was taken using the SC grating with a
600 s exposure time.

56

3.3

Evidence for Harmonic Emission from Ions
The laser intensities required to generate the highest harmonics shown in Figs. 3.2

and 3.3 are above the saturation intensity for neutral Ar atoms of 7.5 x 1014 Wcm−2 , for an
18 fs laser pulse duration. Thus, we attribute the highest observed harmonics, in the energy
range from ∼ 160 eV to 250 eV, to emission from Ar ions. To illustrate this, Fig. 3.4 shows
the calculated population fractions of Ar, Ar+, and Ar++ as a function of time during thelaser pulse at a peak laser intensity of 1.3 x 1015 Wcm−2 , and for a laser pulsewidth of 18
fs. These parameters correspond to the experimental conditions of Fig. 3.3. This calculation
was done using the well-established Ammosov-Delone-Krainov (ADK) ionization rates, given in
more detail in appendix A.[3] The left axis gives the population fractions for the various species.
The right axis plots the HHG cutoff calculated from the cutoff rule for the laser intensity profile.
In this calculation, hνmax = Ip + 3.17Up , where Ip is the ionization potential Ar of 15.8 eV,
and where the instantaneous laser intensity is used to calculate Up . (Note that Ip for Ar+ is
27.6 eV, which would shift the harmonic energies slightly higher for the emission from ions).
At a laser intensity of ∼ 7.7 x 1014 Wcm−2 , Ar is fully ionized, and the highest observableharmonics from the neutral atoms should therefore be approximately 160 eV. At the same laser
intensities where the population of Ar neutrals is zero, the population of singly ionized Ar+ ions
is significant. Figure 3.5 shows the ionization rates multiplied by population at a given time
during the driving pulse for Ar and Ar+. Near the peak of the pulse, the rate for Ar+ dominates
and therefore is the source of the HHG emission. Note that in this analysis, the assumed 18 fs
pulse width is conservative; if the laser pulse is longer, full ionization of neutrals occurs at evenlower laser intensities in the pulse and therefore strengthens the argument for emission fromions. The laser pulse cannot be shorter than 18 fs because of spectral bandwidth limits. Finally,
the same arguments can be used for the lower intensity data corresponding to Fig. 3.2 - thehigher harmonics must be emitted from ions, because Ar is fully ionized well before the peak of
the pulse.

57

1 .0
2 5 0

Io n iz a tio n F r a c tio n

A r+
A r+ +

0 .6

2 0 0

1 5 0
0 .4
1 0 0
0 .2
5 0

C u to ff E n e rg y (e V )

A r

0 .8

0 .0
0
-3 0

-2 0

-1 0

0

1 0

2 0

3 0

T im e ( fs )Figure 3.4: Calculated ionization levels in argon for an 18 fs laser pulse at a peak laser intensityof 1.3 x 1015 Wcm−2 , using ADK rates. Laser pulse (black dashed); Ar (red); Ar+ (green);Ar++ (blue). The right axis shows the predicted HHG cutoff energy for the laser intensity
profile, calculated from the cutoff rule.

x 1 0

1 5

1 .4
0 .0 2 0
1 .2
1 .0

0 .0 1 50 .8
0 .0 1 0
0 .6
0 .4
0 .0 0 5

2

)

A r+
In te n s ity ( W /c m

Io n iz a tio n R a te ( a .u .)

A r

0 .2
0 .0

0 .0 0 0

-3 0

-2 0

-1 0

0

1 0

2 0

3 0

T im e ( fs )Figure 3.5: Calculated ionization rates of argon species for the same conditions as in Fig. 3.4.

58

-6

Log Intensity (arb.)

-7

-8

-9

-10

-11
0

50

100

150

200

250

300

350

Energy (eV)Figure 3.6: Calculated harmonic emission from neutral (red) and singly ionized Ar (blue) for an
18 fs laser pulse centered at 800 nm, at a laser intensity of 1.3 x 1015 Wcm−2 . The ion spectrum
has not been corrected for the dynamically varying populations.

59
As further evidence for HHG from ions, we can compare the observed spectra with thepredictions of a fully quantum single-active electron approximation model that uses a soft corepotential to fit the ionization potential of the atom or ion under consideration.[37, 20] Only
qualitative comparisons are possible since the theory does not account for propagation effects orthe changing fraction of ionized species. Figure 3.6 shows predictions of the quantum model forthe harmonic spectrum emitted by Ar and Ar+ for an 18 fs driving-laser pulse at a peak intensityof 1.3 x 1015 Wcm−2 . The model predicts strong emission from neutrals up to photon energiesof 125 eV. For higher harmonics, the intensity drops rapidly by many orders of magnitudewith increasing harmonic order due to the complete ionization of Ar neutrals. In contrast,
emission from Ar+ is much stronger up to 260 eV. These theoretical predictions are in very
good qualitative agreement with our experimental observations. We also note that the quantum
calculations predict comparable emission intensities (i.e. effective nonlinear susceptibilities)
for neutrals and ions. This is consistent with our observations, which do not show a large
discontinuity in emission intensity with higher harmonic orders.

3.4

Validity of the ADK Model in the BSI Regime
One important question to address is whether the ADK model, which is valid in thetunnelling ionization regime, is applicable in the barrier suppression ionization (BSI) regime.
Barrier suppression occurs when the electric field strength is comparable or greater than thestrength of the binding potential of the atom, allowing the electron to escape directly from thepotential well without tunnelling. This would lead to inaccuracies in the predictions of Figs.
3.4 and 3.5, although the quantum rate calculations and their predictions shown in Fig. 3.6 are
valid in this regime. Barrier suppression of an atom occurs for a field strength of Ebs = Ip2 /4, or,in the case of Ar, an intensity of 2.5 x 1014 Wcm−2 .[41, 5] Previous theoretical work has shownthat in this regime, ADK theory may overestimate the ionization rate and thus underestimate
the cutoff harmonic order from neutrals.[66, 71] Figure 3.7 shows the static field ionization rates
for H and He from a numerical solution of the time-dependent Schroedinger equation and from60
ADK theory. Above the barrier supression field strength, ADK begins to overestimate the rate.
However, when calculating the total ionization over the pulse, the error is less because only
the peak of the pulse has an intensity in the BSI regime. In Figure 3.8, the total amount ofionization of H is calculated as a function of peak electric field amplitude and pulse duration.
As expected, for a shorter duration pulse, the error in the total amount of ionization is greater
since a higher proportion of the pulse is at the BSI intensity.

Figure 3.7: Static ionization rates of H and He versus electric field strength. Solid line: numericalcalculations of TDSE. Dashed line: ADK theory. From [66].

Another source of error can result from the fact that the ADK theory describes a DC
tunnelling rate. In the quasi-static approximation (QSA), these rates can be applied to oscillating
electric fields if the tunnelling time is less than the oscillation period, or when the Keldysh
parameter, γ ≤ 1/2, whereγ=(

Ip 1/2 p)= 2mIp ω0 /E.
2Up

(3.1)

However, A. Scrinzi et al. [66] point out that in the BSI regime, γbs = 16ω0 /(2Ip )3/2 , which
for 800 nm light and noble gas atoms (Kr, Ar, Ne, He) ranges between γ ≈ 0.5 − 2, so thatthe quasi-static approximation may not be valid. By comparing numerical calculations withcalculations using the QSA of the total amount of ionization produced from a pulse, they find61

Figure 3.8: Fraction of ionized H in the BSI regime as a function of the peak electric field
amplitude for pulses of 5, 10, and 20 fs in duration. Full line is ADK model, triangles show
Krainov model, and circles show time-dependent Schroedinger equation. The dotted line is Ebs
for H. From [71].

62that the deviation is less than 5% when the Keldysh parameter is between 0.5 < γ < 2 for 800
nm light. They conclude that the Keldysh condition which is calculated assuming tunnelling
ionization is not the same as for the BSI regime.
In our experiments, we use an 18 fs duration pulse with a peak intensity of 1.3 x 1015
Wcm−2 . The peak electric field amplitude is Ep = 2.3Ebs for argon; however, the peak fieldthat the neutral argon atoms survive to will be somewhat less. From the calculations for He,
the actual rate is about 80% what ADK theory predicts when the field is twice the barrier
suppression field. Therefore the error in the ADK calculation of the total ionization fraction
during the pulse should not be significant. For Ar+, the calculated ionization rate using the
ADK model should be accurate, since we are below the barrier suppression intensity of the ion(2.4 x 1015 Wcm−2 ) for our peak intensities. Previous work has shown very good agreement
between experiment and theory using ADK ionization rates for ions.[81, 17] Therefore, it is valid
to conclude that the significant extension of HHG we observe in Ar is due to emission from ions.

3.5

Plasma-induced Defocusing
We attribute our ability to observe these very high harmonic orders in argon primarily

to the use of a waveguide for high harmonic generation.[50] The waveguide minimizes the effectof laser beam defocusing, resulting in high driving laser intensities even in the presence ofa substantial plasma. Theoretical calculations[59] and experimental observation[2] show thatplasma-induced defocusing effects limit the maximum intensity that can be achieved when alaser is focused into an ionizing gas. These effects become significant when the defocusing
length, defined in Ref.[59] as lD = λnc /2ne , where ne is the electron density and nc is thecritical density, is smaller than the confocal parameter. In effect, the spatially varying plasma
creates a negative lens and the defocusing length is the length at which the phase advance at thecenter of the beam is π/2. Under the conditions in our experiment, 100% ionized gas at 7 Torr,the defocusing length is around 1 mm. This is an order of magnitude shorter than the laser
beam confocal parameter of ∼ 1 cm, and indicates that without the waveguide, the ionizing

63
gas would defocus the pulse before reaching its peak intensity. Experimentally, we observe thatthe transverse profile of the beam emerging from a 2.5 cm long waveguide is unaffected by thepresence of the ionized gas for argon pressures up to 7 Torr, and that the energy decreases by
10-15%. This energy loss corresponds to a deposited energy of ∼30 eV for each atom in the
waveguide, and will lead to double-ionization of atoms in the center of the waveguide. We also
observe a significant spectral broadening, blueshifting of around 20-50 nm, and a shorter pulse
duration (from 22 fs to 18 fs) of the driving laser pulse when the gas is present, all of which
are indicative of a strong laser-plasma interaction that would be enhanced with guiding. At thehigh laser intensities and ionization levels present in the waveguide, spectral broadening of theharmonic peaks can also occur, as seen in Fig. 3.3.
When the argon pressure is increased above 9 Torr, the output mode breaks up andharmonics are no longer observed. Therefore, there exists a limit to the amount of ionizationinduced defocusing for which the waveguide can compensate. Experimentally, we observe that
reducing the gas pressure in the waveguide will lead to an increase in the highest observed photon
energy. Figure 3.9 shows HHG spectra for argon at 5 and 7 Torr in a straight waveguide. At
the lower pressure, the flux at lower photon energies is reduced, but the higher photon energies
are enhanced. Figure 3.10 shows a more dramatic result for harmonics from neon at the same
intensity. There are several possible explanations for these results. It is possible that there is
still some defocusing effect which increases with higher pressures, reducing the intensity. Also, at
higher gas pressures, the laser energy at the output of the fiber is reduced, either from ionization
or from other loss mechanisms in the waveguide. The reduction in laser energy and therefore
intensity could also explain the decrease in harmonic energy.
The ability to generate harmonics in highly ionized media and from atomic ions holds
great promise to advance the energy range and flux of soft x-ray sources from HHG. Previously,
it was believed by many in the field that the limit to HHG energy was the saturation intensity.
In this work, we demonstrate that the true limit is given by the cutoff rule. In Chapter 2, it was
shown that the flux can be enhanced by using modulated waveguides even in 100 % ionization.

64

In te n s ity ( a r b .)3 0 0 02 0 0 01 0 0 0

0
8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

E n e rg y (e V )In te n s ity ( a r b .)Figure 3.9: Measured harmonic spectra from a straight waveguide filled with argon at 5 Torr(red) and 7 Torr (black) for a pulse peak intensity of ∼ 9 x 1014 Wcm−2 . Both spectra takenwith the SC grating and an exposure time of 300 s (using 0.4 µm thick Zr filters).

6 T o rr
1 1 T o rr
2 0 T o rr
3 0 T o rr
4 0 T o rr

1 0 0 0 0

1 0 0 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

E n e rg y (e V )Figure 3.10: Harmonic spectra from a straight waveguide filled with Ne at different pressures
for a peak laser intensity of ∼ 9 x 1014 Wcm−2 . All spectra taken with the SC grating and a 10s exposure (using 0.4 µm thick Zr filters).

65
For the results shown in Fig. 3.3, the falloff in the emission at higher energies is likely due tothe rapidly decreasing coherence length at higher levels of ionization, from ∼0.5 mm at 100 eV
to ∼50 µm at 250 eV. Although a 0.25 mm modulated waveguide was used, the process was
not optimally phase matched and further enhancement of the flux is expected by using shorter
modulation periods and longer waveguide lengths.

Chapter 4

High Harmonic Generation Experimental Set-up

In this chapter, I discuss the experimental apparatus design and calibration used for the
results of Chapters 2 and 3. The laser system was designed by Sterling Backus and was originallybuilt by Sterling Backus and Randy Bartels. It is comprised of a 1 kHz two stage Ti:Sapphire
amplifier system, shown schematically in Figure 4.1.[8] The output of a Ti:Sapphire oscillator is
stretched to 200 ps using a grating stretcher, then passes through a zero length stretcher witha deformable mirror in the focal plane where the pulse is spectrally dispersed. The low energy
pulse is then amplified in two stages to a final output pulse energy of 5 mJ, using 2 pump lasers
on the 2nd stage. The amplified output is compressed in time with a grating compressor. The
deformable mirror is used in conjunction with an Evolutionary Strategy algorithm to optimize
the second harmonic signal, giving a final compressed pulse width of ∼22 fs +/- 2 fs. The loss
from the grating compressor reduces the compressed signal to around 3 mJ.
The laser is focused with a 50 cm focal length mirror into a 150 µm inner-diameterhollow-core waveguide with a coupling efficiency of ∼ 60%.
Figure 4.2 shows the experimental setup for high harmonic generation. The laser isfocused through a 250 µm-thick sapphire window and into the waveguide setup. The central,
2.5 cm fiber is held in a glass capillary with holes drilled in it to allow input of the gas, and there
are two small fiber sections at each end which allow differential pumping to vacuum on eitherside. The harmonic signal and residual fundamental 800 nm light exit the waveguide into the restof the vacuum chamber. A valve after the waveguide setup makes it possible to vent and change

67

10-9 Torr
77 Kelvin

Figure 4.1: Schematic of Ti:Sapphire amplifier system used for high harmonic generation.

Gas Input Bellows

Flexible

3 Section
Glass WaveguideVacuum10-3 Torr

1-3 mJ
22 fs
1 kHz Ti:S

Removable Mirror
and Viewport

Valve

Varied-Groove
Density Grating

Fixed Curvature Vertical
Focusing Mirror

Entire Set-up under Vacuum
10-6 Torr

Adjustable Curvature
Horizontal
Focusing Mirror

Filter to Block
Fundamental

Hettrick Scientific
Soft X-ray Spectrometer

Removable Filters
for Calibration

Baffle Filter
Assembly

Andor CCD camera
Location of Flat Field Focus

Flexible
Bellows

68

Figure 4.2: Schematic of experimental setup for the generation and measurement of high harmonics.

69
waveguides without venting the entire chamber. The harmonic light is focused and spectrally
resolved with a grazing incidence soft X-ray spectrometer made by Hettrick Scientific, Inc. The
spectrometer has vertical and horizontal focusing mirrors. The curvature of the horizontal mirror
can be adjusted to remove any astigmatism at the focus on the CCD camera. The gratings havea varied groove density, allowing a flat field focus at the CCD.[35] The angle of the grating canbe adjusted in vacuum to scan different wavelength ranges. A thin metal filter (Zr or Ag) is
placed inside the spectrometer after the first curved mirror to block most of the fundamental
light and pass the harmonic light. There is also a light-tight baffle with another thin metal filter
after the spectrometer to eliminate any remaining fundamental light. A movable filter holder
after the spectrometer allows insertion of different thin filters to calibrate the spectrum. The
spectrum is measured using an Andor vacuum CCD camera.

4.1

Energy Calibration of High Harmonic Generation Spectra
Energy calibrations of the harmonic spectra were done using removable thin filters. The

lower energy spectra were taken with grating SC which has 65 lines/mm. Calibrations weremade with silicon and boron filters, which have sharp absorption edges at 99.9 eV and 188.35
eV respectively. Figure 4.3 shows the harmonic spectra taken with the two different filters. The
higher energy spectra were taken with grating SB, which has 160 lines/mm, and calibrated using
the boron and carbon edges at 188.35 eV and 284 eV, shown in figure 4.4. Both calibrations
were done by a linear fit of the wavelength of the filter edges and the pixel numbers. However,this is only an approximation since the wavelength is not exactly linear with pixel spacing on
the CCD camera. The grating equation describing the angle at which a given wavelength isdispersed is given by,
nλ = d(cos α − cos β),(4.1)

where n is the order, λ is the wavelength, d is the grating line spacing, and the angles α and β
are the incident and reflected angles defined in figure 4.5. The angle β is related to the positionon the CCD, x, by x = r tan β, where r ∼ 80 cm is the distance between the center of the grating

70
and the CCD array. The dispersion of the grating is then given by,dλd=
sin β cos2 β.
dx
nr

(4.2)

Because the reflection angle β off the grating changes across the CCD array, the dispersion is
not linear with x. Calibrations using the grating equation (Eqn. 4.1) and a two point linear
fit are calculated to find out if the amount of error is significant. The Hettrick spectrometer isdesigned so that when zero order is centered at the output, the incident and reflected angle areboth 2 degrees. As the grating angle is then adjusted so that the first order is centered on the
CCD, α+β=4 degrees still holds. It is necessary to know the wavelength at the center of the
CCD to calculate α. The number of CCD pixels is 1024 along the dispersed axis, so that, in thecase of the SC grating, the silicon edge is close to the center pixel, giving a value of α = 1.34
degrees. Figure 4.6 plots wavelength and energy as a function of position x from the center of
the CCD, using the grating equation and a two point linear fit. From these plots, it is evident
that the linear calibration is off at the edges of the CCD array; however, for the wavelength
region of interest in our experiments, the error is small.

For the SB grating calibration, the1 0 0 0 0 0

In te n s ity ( a r b .)1 0 0 0 0

1 0 0 0 0

1 0 0 01 0 0 0B e d g e
S i e d g e
4 0 0

4 5 0

5 0 0

5 5 0

6 0 0

6 5 0

7 0 0

7 5 0

8 0 0

8 5 0

9 0 0

P ix e l

Figure 4.3: Harmonic spectra generated in neon using 0.2 µm-thick B (black) and 0.18 µm-thickSi (red) filters to calibrate the SC grating.

71
1 0 0 0 0 0

In te n s ity ( a r b .)1 0 0 0 0

1 0 0 01 0 0 0B e d g e

C e d g e

1 0 0

0

2 0 0

4 0 0

6 0 0

8 0 0

P ix e l

Figure 4.4: Harmonic spectra generated in neon using 0.2 µm-thick B (black) and C (red) filters
to calibrate the SB grating.

baIncreasing pixel #and photon energy

Figure 4.5: Schematic of grazing incidence grating and relevant angles.

30

400

Wavelength (nm)20
15

Si Edge

10

B EdgePhoton Energy (eV)

Grating Equation
Linear Fit

25

Grating equation
Linear fit

300

200

100

5
0

-1.5

-1.0

-0.5

0.0

0.5

X Displacement (cm)

1.0

1.5

0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X Displacement (cm)Figure 4.6: Wavelength and energy calibration of SC grating using the grating equation and a
two point linear fit.

72

14

500

Grating Equation
Linear Fit

10
8

B Edge6

C Edge420Grating Equation
Linear Fit

400

Photon Energy (eV)Wavelength (nm)

12

300

200

100
-1.5

-1.0

-0.5

0.0

0.5

X Displacement (cm)

1.0

1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X Displacement (cm)Figure 4.7: Wavelength and energy calibration of SB grating using the grating equation and a
two point linear fit.

73
wavelength at the center is approximately 6.2 nm, so α = 1.19 degrees. Figure 4.7 compares the
wavelength and energy calibration using the grating equation and linear fit for the SB grating.
Again, error in the linear fit is negligible for the energy range in our experiments.

4.2

Calculation of Photon Flux
An ultrafast soft X-ray source at the carbon edge is interesting for many applications

such as biological microscopy or studying the dynamics of organic molecules. Many applications
need a minimum amount of flux for reasonable data rates. To estimate the flux of our source,
we consider the possible losses through the spectrometer and the detection. The harmonic lightis reflected by two Au coated mirrors at an incident grazing angle of approximately 2 degrees.
Figure 4.8 shows the reflectivity as a function of photon energy for one mirror. Surface roughness
has a significant effect on the reflection efficiency at shorter wavelengths but in the best case, thereflectivity at 284 eV is 66%. The harmonic light is diffracted by the SB grating. The grating
diffraction efficiency, measured at the Berkeley Advanced Light Source, is shown in Fig. 4.9
and is around 2% at 284 eV. The harmonic light passes through 0.4 µm of Ag to filter out thefundamental 800 nm light. The transmission of this thickness of Ag is 6% at 284 eV. Finally, theCCD detector has a quantum efficiency of roughly 40% at the carbon edge. The total efficiency
is therefore at most 0.02%.
Figure 4.10 shows the baseline-subtracted harmonic spectra at the C edge. The highlighted data points are used to calculate the flux. There are 1108 counts in a 1% bandwidth
at 283.3 eV for a 180 second exposure. When corrected for the efficiency of the spectrometerand filters, this gives 30,000 photons/s in a 1% bandwidth. This value is the absolute minimum since contamination of the filters and/or mirrors is likely and would reduce the measured
signal. More precise calibration will be performed in the future using a soft X-ray mirror witha known reflectivity and bandwidth and detection either with a calibrated vacuum photodiodeor CCD camera. It is useful to compare the flux at these extremely high orders with flux atlower harmonic orders. Previously, the flux of a broad range of harmonic orders centered at

74

0 .9 5

N o S u rfa c e R o u g h n e s s
2 n m R M S S u rfa c e R o u g h n e s s

0 .9 00 .8 5R e fle c tiv ity

0 .8 00 .7 5
0 .7 00 .6 5
0 .6 00 .5 5
0 .5 0
1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

P h o to n E n e rg y (e V )Figure 4.8: Reflectivity of a Au mirror at 2 degree grazing incidence angle for the ideal case,and for 2 nm RMS surface roughness.

0 .2 0E ffic ie n c y

0 .1 50 .1 00 .0 50 .0 0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

P h o to n E n e rg y (e V )Figure 4.9: Measured efficiency of SB grating as a function of photon energy.

75
31 eV (harmonic orders 17-23) was measured using a vacuum photodiode.[10] The harmonics
were generated with a 5 kHz repetition rate laser in argon using a 10 cm long hollow-core 150µm inner-diameter waveguide. The measured flux was ∼ 2 x 1012 photons/second. For further
comparison, Fig. 4.11 shows the flux from the soft X-ray beamline at the Advanced Light Source
at Berkeley. At the carbon edge, an estimated flux of 1013 photons/s in a 0.1% bandwidth isproduced before the monochromator. Despite the higher photon flux of the Advanced Light
Source, the pulse duration much greater than that of the soft x-ray light produced from HHG.

76
2 0 0 06 0 T o r r H e liu m
4 2 T o r r H e liu m
9 T o rr N e o n

In te n s ity ( a r b .)1 5 0 01 0 0 0D a ta p o in ts u s e d
fo r flu x c a lc u la tio n

5 0 0

0
2 5 0

2 6 0

2 7 0

2 8 0

2 9 0

3 0 0

E n e rg y (e V )Figure 4.10: Harmonic spectra at the carbon edge showing the data points used to estimate thephoton flux.

Figure 4.11: Measured flux from the ALS beamline 6.1 and estimated flux assuming 100%efficient optics (dashed line). From [76].

Chapter 5

Self-compression of Intense Femtosecond Pulses in Gas-filled
Hollow-core Waveguides

One surprising experimental result that came out of my research on high harmonic generation was the observation of temporal compression of the 800 nm driving pulse after propagating
in a hollow-core waveguide filled with low-pressure argon at ionizing intensities. Unlike otherstandard pulse compression techniques, no subsequent dispersion compensation was used. Typically, pulse compression is accomplished by broadening of the spectrum of the pulse throughnonlinear self-phase modulation. This broadened pulse bandwidth can then be compressed to ashorter pulse by appropriate dispersion compensation. Self-phase modulation of the pulse canbe done by propagation through an optical fiber[53, 73] or a gas-filled hollow waveguide[54] at
non-ionizing intensities and dispersion compensation is provided by prisms, gratings, or chirped
mirrors to create a negative group velocity dispersion. These schemes have been employed tocompress pulses to under 4 fs pulse duration[82]; however, such techniques are lossy and can bequite complicated. Furthermore, all previous pulse compression schemes have been limited to
sub-millijoule pulse energies due to the damage threshold of the fibers or ionization of the gas.
The use of ionization to broaden the spectrum of a pulse for compression has been previously
considered theoretically.[72] However, the compression also depended on negative group velocity
dispersion compensation.

78

5.1

Experimental Results
In the experiment, light from a Ti:sapphire amplifier producing ∼ 30 fs near-transform

limited pulses with 2.5 mJ pulse energies at 1 kHz was focused into a 2.5 cm long, 150 micron
inner-diameter hollow waveguide filled with low pressure argon. The spot diameter of the lasermode at the entrance of the waveguide was ∼ 85 µm FWHM and the energy throughput of
the waveguide was ∼ 60% with no gas. The focusing region before and after the waveguide
was held under vacuum to prevent any additional nonlinear effects. The pulse was characterized before and after the waveguide using second-harmonic frequency-resolved-optical-gating(SHG-FROG).[74] All FROG measurements were deconvolved using commercial software from
Femtosoft Technologies (FROG Version 3.03) to an error of < 1% for a 256x256 sampling grid.
The spectral marginals for each measurement agree with the experimentally measured spectra
to within 5%. After exiting the waveguide, the pulse passes through a 250 µm thick sapphire
window and a 1 mm thick fused silica beam splitter in the FROG setup. The total dispersion ofthese elements corresponds to a group velocity dispersion of 61 fs2 and a third-order dispersionof 40 fs3 at a center wavelength of 760 nm, which would change the pulse duration at most by 3
fs, not enough to significantly affect the experimental results. Also, the material gives a positive
group velocity dispersion, which would not explain pulse compression by the typical schemes.
The pulse shape after the fiber without gas is the same as the input to within our measurement
resolution.
The intensity in the waveguide, approximately 1015 W/cm2 , is high enough to completely
doubly ionize the argon gas by the peak of the pulse (for a 30 fs duration pulse). This rapidlychanging ionization during the pulse results in an index of refraction that decreases with time,causing a blue shift of the overall laser spectrum. The spectra at the exit of the waveguide withvarying pressures of argon is shown in figure 5.1. Pressures as low as 1 Torr result in significant
blue shift and spectral broadening and blue shifts of up to ∼ 50 nm are observed.

Figure 5.2

shows the deconvolved FROG measurements of the temporal pulse shape after the waveguide

79

N o
1 .1
2 .0
2 .8
3 .8

2 0 0 0G a s
T o rrT o rrT o rrT o rrIn te n s ity

1 5 0 01 0 0 0

5 0 0

0
7 0 0

7 5 0

8 0 0

8 5 0

9 0 0

W a v e le n g th ( n m )Figure 5.1: Spectra taken after the 2.5 cm long, 150 µm diameter waveguide with different Ar
pressures.

80

In p u t
1 .1 T o rr
2 .0 T o rr
2 .8 T o rr
3 .8 T o rr

1 .0

In te n s ity ( N o r m .)0 .8

3 2 .02 2 .3
1 9 .2
1 7 .7
1 5 .5

fs
fs
fs
fs
fs

0 .6

0 .4

0 .2

0 .0

-6 0

-4 0

-2 0

0

2 0

4 0

6 0

T im e ( fs )Figure 5.2: Deconvolved FROG measurements of the temporal pulse shape before and after the
waveguide with different Ar pressures for an initial pulse of 32 fs. The pulse FWHM for thedifferent measurements are indicated.

81

1 .0

In p u t 2 8 .9 fs
4 T o rr 1 3 .3 fs
6 T o rr
8 T o rr
9 T o rr

In te n s ity ( N o r m .)

0 .8

0 .6

0 .4

0 .2

0 .0

-5 0

0

5 0

1 0 0

1 5 0

T im e ( fs )Figure 5.3: Deconvolved FROG measurements of the temporal pulse shape before and after the
waveguide with different Ar pressures for an initial pulse of 29 fs. At 4 Torr, the final pulse
duration is 13.3 fs FWHM. From [78].

1.0OutputInput

13.3fs28.9fsMeas ured
R econs tructed

3000
2500

0.6
13.3 fs
0.4

Intens ity

Intens ity

0.8

2000
1500
1000

0.2

500
0

0.0
-40

-20

0
T ime (fs )

20

40

650

700

750
800
Wavelength (nm)

850

900

Figure 5.4: Comparison of the measured spectrum with the spectrum from FROG measurement
deconvolution.

82
for different argon pressures. With only 1 Torr of argon, the initial pulse width of 32 fs has
been drastically reduced. As the pressure is increased, the pulse width decreases to 15.5 fs at
4 Torr. Figure 5.3 shows the results for an input pulse of 29 fs duration. At 4 Torr, the final
measured pulse duration is 13.3 fs. As the pressure is increased, the pulse duration remains
relatively constant until 9 Torr, at which point the pulse begins to break up. The deconvolved
spectrum from the FROG and the experimentally measured spectrum are shown in figure 5.4 forthe 13.3 fs pulse. The excellent agreement between the two indicates that the measurement has
high accuracy and is self-consistent. The spectral bandwidth can support an 11 fs transformlimited pulse. Both the input and compressed pulses are near-transform limited, within 20%of the time-bandwidth product. The power exiting the waveguide is reduced by ∼ 5 % with 4
Torr of Ar compared with no gas in the waveguide, while at 8 Torr, the loss is 10%. Also, thespatial mode after the waveguide remains good at low gas pressures. Using a larger diameter,
230 µm waveguide, the pulse is only compressed to 24 fs from a 30 fs duration input. When the
waveguide is replaced with a 1 cm diameter gas-filled cell, there is negligible compression of thepulse.

5.2

A New Mechanism for Pulse Compression
Conventional 1-D pulse propagation models[1, 13] such as those used to calculate self-phase modulation in optical fibers, microstructured fibers, or gas-filled hollow waveguides, failto explain the experimental self-compression observed. The blue-shift and broadening of thespectrum can be explained by self-phase modulation from the changing plasma index; however,this will not alter the temporal envelope. The dispersion of the low-pressure gas, the plasma, or
the waveguide is too small to affect the temporal shape as well. Higher-order nonlinear effects
can cause self-steepening of the pulse which can reshape the trailing edge, but will not shorten
the pulse in time.
Table 5.2 summarizes the potential 1-D pulse shaping mechanisms and the characteristic
propagation length over which they become significant for our conditions.[13] The dispersion

83
length is given by:
Ldis =

T2

(5.1)

2

d k
| dω
2 |,

where T is the pulse duration (1/e intensity) and k = ωn/c. For a 30 fs pulse FWHM, T =18 fs. The dispersion length is the length over which the pulse duration would approximately
double due to group velocity dispersion (GVD). Self-phase modulation becomes important when
the spectral broadening is on the order of the spectral width of the initial pulse. The amount of
SPM broadening is given approximately by dφ(ω)/dt or ∼ ∆φN L /T where ∆φN L ∼ n2 IωL/c.
The bandwidth of the input pulse is approximately 1/T . Therefore, SPM is significant when
∆φN L ∼ 1 and the effective length is given by:
LSP M =

c
.
ωn2 I

(5.2)

Another nonlinear mechanism is self-steepening of the pulse. The self-steepening process(g)depends on the nonlinear group index which is defined as n2(g)assumption n2

2= n2 + ω dn
dω . In our case, the≈ n2 is valid. The effective length over which self steepening is significant isgiven by:
LSS =

cT
(g)

.

(5.3)

n2 I

Lastly, the time-varying index due to ionization during the pulse also affects the pulse
spectrum. The plasma index is given by:
r
np =

1−

ωp2
ωp2
'
1
−
ω2
2ω 2 ,(5.4)

where ωp is the plasma frequency given by: ωp2 = Ne e2 /m0 , e and m are the electron charge andmass, and Ne is the electron density. Because the electron density is increasing in time during
the leading part of the pulse, the plasma index is also changing during the pulse. In this way, theplasma index acts like the nonlinear index, n2 I, the difference being that it only decreases during
the pulse, instead of increasing and decreasing over the duration of the pulse. Therefore, thespectrum predominately broadens to shorter wavelengths. The length over which the effective
nonlinear index from the plasma will affect the spectrum from self-phase modulation is given by
LP lasmaSP M = c/ωnp and the effective self-steepening length given by LP lasmaSS = cT /np .

84

Mechanism
Waveguide GVD

Length scale
2 2 3
L = T ua2 cω

Gas GVD

L = T 32πcd2 nPlasma GVD
Gas SPMGas self-steepening
Plasma SPM
Plasma self-steepening
Plasma defocusing

2

11

2

λ dλ2
2
3
L = T ωcω2p
L = ωnc2 I
L = cT(g)n2 I
L = 2ωcωp2
2T cω 2L = ω2p
L = 12 kω 2 = 2ωcωp2Distance
14 m
1.5 km
1.4 m
10 cm
4m
2 mm
1 cm
1.5 mmTable 5.1: Table of possible pulse shaping mechanisms that involve only 1-D propagation effects
and the propagation length over which they become significant. The conditions used in thecalculation are 4 Torr of Ar in a 150 µm diameter hollow waveguide. The gas terms were
calculated for no ionization, and the plasma terms were calculated for 200 % ionization.

85
A 1-D propagation model was performed by Tempea et al.[72] for similar conditions:pulse intensities of 7 x 1014 to 1.5 x 1015 W/cm2 , 125 µm inner-diameter hollow waveguide, 20
fs input pulse, and argon pressure of ∼ 0.5 Torr. They found significant spectral broadening toshorter wavelengths for long propagation lengths (90 cm) with a positive chirp on the output
pulse, but did not observe any temporal compression from their model.
Since 1-D nonlinear wave propagation cannot explain our experimental observations andbecause the waveguide is necessary to achieve pulse compression, a 3-D propagation model was
employed. Propagation of the laser pulse in the waveguide was modelled by numerical solutionof the 3-D scalar wave equation. This model was developed by Ivan Christov and is described in
more detail in Ref. [18]. The model takes into account the dispersion from the transient plasma,
power dissipation from ionization, and dispersion due to the neutral gas at each radial positionin the waveguide as a function of time. The 3-D diffraction by the waveguide is included andautomatically accounts for the waveguide dispersion. The waveguide loss was also measured
experimentally and incorporated into the model.
The theoretically predicted pulse durations and spectra as a function of pressure areshown in Figs. 5.5 and 5.6. The model is in excellent agreement with experimental data,
predicting that the pulse undergoes temporal reshaping that reduces the pulse width from 30
fs to approximately 13 fs with increasing pressure, and that the minimum pulse duration is
obtained around 4 Torr. The leading edge of the pulse also steepens in time, accompanied
by a longer trailing edge that develops a shoulder as the pressure is increased - again in very
good agreement with experiment. The model also predicts less compression for larger diameterwaveguides and little or none for the case of no waveguide, as is the case experimentally. Finally,
the corresponding pulse spectra blue shift and increase in bandwidth due to the combined effectsof rapidly changing plasma index and self-phase modulation are also in good agreement withexperiment. At high pressures above 8 Torr, the pulse splits and the spectrum develops alarge long-wavelength wing, also observed experimentally. The only adjusted parameter in thecalculations to obtain agreement was the input diameter of the focused laser mode into the86

In p u t - 3 0 .0
2 T o rr - 2 5 .4
4 T o rr - 1 7 .8
6 T o rr - 1 5 .0
8 T o rr

1 .0

In te n s ity ( N o r m .)

0 .8

fs
fs
fs
fs

0 .6

0 .4

0 .2

0 .0

-6 0

-4 0

-2 0

0

2 0

4 0

6 0

8 0

T im e ( fs )Figure 5.5: Theoretical calculation of temporal pulse shape after the waveguide with different
Ar pressures for an initial pulse duration of 30 fs using the 3-D propagation model. The pulse
FWHM are indicated.

87

N o G a s
2 T o rr
6 T o rr
8 T o rr

1 .0

In te n s ity ( N o r m .)

0 .8

0 .6

0 .4

0 .2

0 .0

0 .7 00 .7 5

0 .8 00 .8 50 .9 0W a v e le n g th ( m ic r o n s )Figure 5.6: Theoretical calculation of the pulse spectrum before and after the waveguide withdifferent Ar pressures.

88
waveguide.
The self-compression is a result of the combined effects of plasma-induced refraction andblue shifting, diffraction of the laser beam, phase modulation due to the rapidly-changing plasma
index, and guiding by the hollow waveguide. The role of plasma defocusing combined with the
waveguide refocusing governs the spatial propagation and is very important in the compression
process. The model shows that as the pulse begins to ionize the gas, the trailing edge of thepulse refracts from the center of the waveguide due to plasma-induced defocusing. However,this portion of the beam is then reflected back into the guide by the waveguide walls. This
leads to a temporally compressed pulse with a sharp leading edge and flat temporal phase. The
reshaping also leads to a small periodic oscillation in the beam diameter and pulsewidth as thepulse propagates down the waveguide. This high-field pulse reshaping process has similarities
with passive mode locking of lasers using saturable absorbers - the trailing edge of the pulse isslowly depleted periodically along the waveguide due to ionization-induced defocusing, where
the role of a laser resonator is replaced by the waveguide. Both experimentally and theoretically,
the pulse compression process is sensitive to the initial mode coupled into the waveguide. Also,
theoretically, the laser beam must be focused to a tighter focal spot diameter than the ideal
100 µm for coupling into the lowest-order mode of 150 µm waveguide, in agreement with ourexperimental observations.
A new mechanism for pulse compression that operates at laser intensities above the ionization threshold has been demonstrated both theoretically and experimentally. This techniqueallows compression of pulses with millijoule energies and has the advantage of minimal distortionof the laser mode, low power loss, and no need for additional dispersion compensation. It is
likely to scale to higher intensities and pulse energies, making it useful for high-field science andapplications.

Chapter 6

Spectral and Temporal Characterization of High Harmonic
Generation by Photoelectron Spectroscopy

In this chapter, I discuss the design and development of a simple in-line apparatus tomeasure the spectral and temporal characteristics of high harmonic generation. The apparatus can be used in front of an experiment without disrupting the HHG beam. The spectrum
is measured by detecting photoelectrons emitted as a result of ionization of a noble gas by
the extreme-ultraviolet (EUV) light. The photoelectrons are energy analyzed using a compact
magnetic-bottle time-of-flight spectrometer. The spectrum can be reconstructed from the known
ionization potential and absorption of the detection gas. The time duration of the EUV pulse
is measured by cross-correlation with a portion of the driving laser pulse.

6.1

A Compact Magnetic Bottle Photoelectron Spectrometer
The magnetic bottle time-of-flight (TOF) apparatus is designed to collect and energy

analyze photoelectrons emitted over initial trajectories within a 2π solid angle. The increased
collection efficiency is necessary to measure the photoelectron spectrum from ionization of a gas
by high harmonics where the signal level is typically 2-3 electrons per laser shot. The detection
region has a strong magnetic field aligned along the TOF axis. The field is then reduced toa lower, constant value in the time-of-flight region. In this way, electrons generated in thedetection region undergo cyclotron motion and follow the magnetic field lines to the detector.
As they pass from the high field to the low field region, their velocities are parallelized so that

90

Micro-channel Plate Detector

Low B - Fieldqf

qi
Detection Region

e-

N
S

EUV

High B- Field

Figure 6.1: Diagram of magnetic bottle showing direction of field lines and electron trajectories.

91
even electrons initially travelling in a direction away from the flight tube end up with most of
their velocity redirected along axis. A schematic of the apparatus indicating the magnetic fieldis given in Figure 6.1.
The theory of operation of the spectrometer is described in detail by P. Kruit and F. H.
Read.[43] The initial angular momentum of the electron is given by` = m2 v 2 sin2 θi /eBi ,(6.1)

where m, v, and e are the mass, velocity, and charge of the electron, θi is the initial angle ofthe velocity with respect to the time-of-flight axis, and Bi is the initial, strong magnetic field.
From conservation of angular momentum, the final velocity angle of the electron, θf , in thelower magnetic field, Bf , is given by the relationsin θfBf 1/2=() .
sin θiBi(6.2)

If the difference between the initial and final fields is large enough, the final velocity direction
will be along the TOF axis for a range of different initial angles. For example, if Bf /Bi =10−3 , then for the maximum initial angle, θi = 90 degrees, the final angle is given by, θf =sin−1 (Bf /Bi )1/2 = 1.8 degrees. It is important to have a large difference in the initial and final
field so that electrons with the same initial velocity will have the same drift time in the TOF
tube. Also, for the best energy resolution, it is important for the changing-field region to be as
short as possible while remaining in the adiabatic limit. The adiabaticity parameter is definedbyχ=

2πmv dBz||,
eBz2 dz(6.3)

which is the fractional change in the field that the electron experiences in one cyclotron orbit.
The design of the spectrometer is shown in Figure 6.2. The spectrometer consists of a
stack of permanent magnets to generate the strong field in the detection region and a flighttube with a solenoid constructed around it to create a uniform lower field. A cone with a 2 mmdiameter aperture is located before the entrance to the time-of-flight tube to allow differential

92

Turbo Pump

12.00"

Micro-channel Plate
Detector

2 mm
Aperture

3.00"

2.00"

Gas Nozzle

2.00"

1.5”
Dia.

1.00"

Dia.

NdFeB Magnets

Figure 6.2: Schematic of compact magnetic bottle time-of-flight spectrometer. Dimensions areindicated.

93
pumping and to limit the detection region. At the top of the TOF is a micro-channel plate(MCP) detector and a 60 L/s turbo pump which evacuates the TOF region to a pressure of
10−6 Torr. A 3 cm long, 150 µm inner-diameter hollow-core glass fiber is used as a gas nozzle
to continuously leak gas into the detection region. Another 60 L/s turbo pump is attachedadjacent to the detection chamber so that the pressure in the detection region does not reachabove ∼ 10−4 Torr with the nozzle on. All of the surfaces in the detection region and TOF tube
are coated with conductive carbon powder to eliminate any charge buildup effects. Also, care
was taken to ensure that all parts were constructed with aluminum or non-magnetic stainlesssteel. The MCP detector design is given in Appendix C. The current signal from the detector
is amplified by a 1 GHz Amplifier and Timing Discriminator (EG&G Ortec Model 9327) andthen recorded by a timing board on a PC (ACAM Time Measuring Device Model AM-F1,
manufactured by Acam-messelectronic gmbh Karlsruhe, Germany). The start signal is takenfrom an SRS Delay Generator which also provides the trigger for the Pockels cell of the amplifier.
The true start time is obtained by direct detection of scattered photons from the driving laser,
producing a current signal from the MCP detector.
To aid in the design of the spectrometer, the total magnetic field was modelled. The
magnetic field of the permanent magnets are calculated using the equation for the magnetic
scalar potential,[36]1ΦM (x) =4π

0n0 · M(x )da0.
|x − x0 |
S

I

(6.4)

The surface integral is evaluated over the top and bottom faces of the magnet. The value ofthe magnetization, M, was found by fitting the experimentally measured value of the field on
axis at several distances from the magnet face, shown in figure 6.3. The magnetic field of thesolenoid was calculated using the equation for the magnetic field of a current loop, given inreference [45], p.112. The solenoid field is then the sum of all the loop fields. The solenoid
parameters are loop radius of 2.05 cm, current of 1 A, and 9 loops/cm with a total length of 30
cm. The solenoid starts 5 cm above the top face of the permanent magnet. The code used forthe calculation is given in Appendix B. Figure 6.4 shows the total magnetic field on axis due to94

M a g n e tic F ie ld ( G a u s s )4 0 0 03 0 0 02 0 0 01 0 0 0

0
0 .0

0 .5

1 .0

D is ta n c e fr o m

1 .5

2 .0

2 .5

M a g n e t F a c e (c m )Figure 6.3: Measured values of the magnetic field on axis above a 1 inch dia., 0.5 inch thick,
NdFeB magnet (black dots). Theoretical prediction of magnetic field for a magnet with amagnetization of 8000 A/m (red line).

95
both the permanent magnet and the solenoid. The decrease at around 32 cm is due to fringing
at the end of the solenoid.

B z (G a u s s )1 0 0 0

1 0 0

1 0

0

5

1 0

1 5

D is ta n c e fr o m

2 0

2 5

3 0

3 5

M a g n e t F a c e (c m )Figure 6.4: Calculated z component of the magnetic field from the permanent magnet andsolenoid on axis.

Figure 6.5 plots the adiabatic parameter, calculated using Eqn. 6.3, as a function oflength on axis for 1 eV electrons. For higher energy electrons, the parameter scales as√

E.

The adiabatic parameter is still less than one, even for electrons with energies as high as 30
eV. For comparison, Figure 6.6 shows the magnetic field on axis and the adiabatic parameterfor the magnetic bottle spectrometer design by Kruit and Read. In their design, the changingfield region is shorter, which can give an improved energy resolution. This is achieved by amore complicated water-cooled electromagnet to generate the strong field. Other magnetic
bottle designs use specially designed pole pieces with permanent magnets to achieve similarfield profiles.[75]96

A d ia b a tic ity P a r a m e te r

0 .2 00 .1 50 .1 00 .0 50 .0 0
0

5

1 0

1 5

2 0

D is ta n c e ( c m )Figure 6.5: Calculated adiabatic parameter on axis, for 1 eV electrons.

Figure 6.6: Calculated (dashed) and measured (solid) magnetic field and adiabatic parameterfor 1 eV electrons on axis. From [43].

97

6.2

Experimental Results
Despite its simple design, the compact magnetic bottle spectrometer has good resolution

over a broad range of electron energies. Figure 6.7 shows a typical spectrum from ionizationof Ar and Ne gas by high harmonic EUV emission. The high harmonics are generated in a 2.5
cm-long, 150 µm inner-diameter hollow waveguide filled with Ar. The driving laser pulse energy
is ∼ 0.7 mJ and pulse duration is 22 fs. A typical spectrum can be acquired in 60 seconds witha data rate of 2000-3000 counts/s. The photon energy is calculated by taking the photoelectronenergy and adding the ionization potential of the detection gas. The difference in the spectrausing Ar and Ne as the detection gas is due to the difference in their absorption spectra, shown in
Figure 6.8. Therefore, to get an idea of the actual photon spectrum, the photoelectron spectrum
must also be corrected for the energy dependent absorption. From figure 6.7, it is apparent thatthe magnetic bottle is able to resolve harmonic peaks over a broad energy region. However,there is a background for the lower energy photoelectrons ranging from 0 to ∼8 eV.

200

Argon
Neon

Counts

150

100

50

0

20

25

30

35

40

45

50

55

60

65

70

Photon Energy (eV)Figure 6.7: High harmonic spectrum measured by the magnetic-bottle photoelectron spectrometer using different detection gases.

98

0.8
0.7
Neon
Argon

Absorption

0.6
0.5
0.4
0.3
0.2
0.1
0

10

20

30

40

50

60

70

80

90

100

Photon Energy (eV)Figure 6.8: Absorption at different photon energies for argon and neon gas. From CXRO website.

99
At higher photon energies, the magnetic bottle spectrometer also performs well. Figure
6.9 shows high harmonic generation from a hollow-core waveguide filled with Ne at higher driving
laser intensities (∼ 1 mJ and 22 fs) measured by detecting photoelectrons from Ne gas. The
photon energy extends up to ∼ 100 eV. In addition, the photon spectrum was measured at thesame time by a Hettrick EUV spectrometer, which uses a grating and CCD camera. Figure 6.10
compares the spectra taken with the two different spectrometers. The resolution of the Hettrick
spectrometer should be almost constant over the region of the spectrum. The magnetic bottle
resolution appears to decrease at lower photoelectron energies. Because the spectral width of
the harmonics is unknown, it is difficult to give an exact value for the resolution.

40
35

Counts

30
25
20
15
10
5
0
10

20

30

40

50

60

70

80

Photoelectron Energy (eV)Figure 6.9: Photoelectron spectrum of HHG from Ne extending to ∼ 100 eV photon energies.

With harmonic photon energies up to 100 eV, it is possible to look at inner-shell ionization
processes in xenon. The inner 4d-shell of xenon has two energy levels with ionization potentials
of 67.5 and 69.5 eV. When the inner shell is ionized, Auger processes, where the excited ion
rearranges to a lower energy state by ejecting a second electron, are observed. A diagram of theAuger process in xenon is shown in Figure 6.11. It is also possible to observe Auger electrons

100

Intensity (arb)1

Magnetic Bottle
Hettrick

0.8

0.6

0.4

0.2

0
35

40

45

50

55

60

65

70

75

Photon Energy (eV)Figure 6.10: Comparison of photon spectra measured using the magnetic bottle photoelectron
spectrometer and the Hettrick spectrometer.

Non-R esonant

A uger E lectron

R esonant
6p

5p,5s

A uger E lectron

5p,5s
E photon= <68 eV

E photon= >68 eV(68eV ) 4d
(68eV ) 4d

Figure 6.11: Diagram of resonant and non-resonant Auger processes in xenon.

101
using photon energies lower than 67.5 eV through a resonant Auger process where the xenon
atom is first excited to a Rydberg state and then undergoes Auger decay. The lifetime of theseprocesses is very short. Experimentally, Auger processes in atoms are measured by the energy
bandwidth, since lifetime broadening is the only broadening mechanism. Auger decay in Xe has
been measured to be ∼ 8 fs (extremely fast!).[38] Using a cross-correlation technique describedin the next section, it is possible to experimentally measure this process in the time domain,
and was done in the case of krypton in reference [24]. While this is not interesting in the caseof atoms, it may be a useful tool for measuring Auger lifetimes in molecules.
Figure 6.12 shows the experimental photoelectron spectrum from ionization of xenon
clearly showing the Auger electrons. When an Al filter is used to block all the harmonics withphoton energies above 65 eV, most of the Auger electrons disappear. Even with the Al filter,some Auger electrons are produced from resonant Auger processes. Since the photoelectron
spectrum of xenon was taken while ionizing with a broad spectrum of harmonics, the Auger
electron peak overlaps with electrons produced from direct ionization of the valence and innershells. However, from the absorption spectrum of xenon, Figure 6.13, it is apparent that certain
photon energies are preferentially absorbed so that the photoelectron spectrum does have recognizable features attributed to different ionization processes. The harmonic structure is not wellresolved since xenon has multiple energy levels in its valence and inner shells. In the case of thevalence shell, the energy levels are separated by 1.3 eV while the inner shell levels are separated
by 2 eV. Each harmonic peak creates two photoelectron peaks for both ionization processes,
adding additional structure to the spectrum.
To achieve a background free spectrum of the Auger electrons, a narrow-bandwidth soft
x-ray multilayer mirror was used. The multilayer was made by Yanwei Liu at Berkeley. The
reflectivity of the mirror is given in Figure 6.14. The mirror has a radius of curvature of 50 cm.
It was placed 25 cm after the detection region, and the reflected light was focused back onto thedetector. The detector was slightly offset so that the EUV light before the mirror passes to theside of the detection region and does not generate any signal. The time zero of the reflected beam

102

18

A uger E lectrons

16

Inner-shell(4d) Ionization

14

Counts12

V alence Shell(5p) Ionization

10

8

6

4

2

0

0

10

20

30

40

50

60

70

80

90

Photoelectron Energy (eV)Figure 6.12: Photoelectron spectrum of xenon using all harmonics (blue) and harmonics passing
through an Al filter blocking energies > 65 eV (red).

1
0.9

5p Valence Shell Absorption(Ip = 12.1 & 13.4eV)

0.8

Absorption

0.7
0.6

4d Inner Shell Absorption(Ip = 67.5 & 69.5eV)

0.5
0.4
0.3
0.2
0.1
0
10

20

30

40

50

60

70

80

90

100

Photon Energy (eV)Figure 6.13: Absorption of xenon gas as a function of photon energy. Data from the Center for
X-ray Optics, Berkeley Lab (www-cxro.lbl.gov).

103
is delayed by an additional 1.67 ns (50 cm/c). Figure 6.15 shows the harmonic spectrum before
and after the multilayer mirror using He gas to generate harmonics and Ne as the detection gas.
The peak photon energy is a bit lower than the measurement done in Figure 6.14 because theincident angle is closer to 89 degrees. The mirror does a good job of selecting out harmonics ina narrow bandwidth around 90 eV. Using this selection of harmonic energies, the photoelectron
spectrum of xenon was measured; see Fig. 6.16. In this spectrum, the photoelectrons generated
by ionization of the valence shell, inner shell, and from Auger processes do not overlap and canbe isolated. This makes it possible to do measurements of the lifetime of Auger decay in Xe
using cross-correlation techniques which will be discussed in the next section.
0.7

0.6

Reflectivity

0.5

0.4

0.3

0.2

0.1

0

88

89

90

91

92

93

94

95

96

Photon Energy (eV)Figure 6.14: Reflectivity as a function of photon energy for the Si/Mo multilayer mirror, measured at an 85 degree incident angle.

6.3

Cross-correlation Measurements
Recently, there has be a great deal of interest in using the light from high harmonic

generation for very short time measurements. High harmonic generation is emitted in a seriesof attosecond bursts every half cycle of the driving field. Using HHG and an IR probe pulse,
it is possible to make cross correlation measurements of the HHG emission itself,[34, 57] or of

104

T o ta l S p e c tru m
A fte r M u ltila y e r M ir r o r
1 5 0

C o u n ts

1 0 0

1 0 0
5 0

5 0
0

0
-5 0
4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

P h o to n E n e rg y (e V )Figure 6.15: Spectra of high harmonics generated using He gas and detected with Ne gas before(black) and after (red) the narrow-band multilayer mirror.

2 5

C o u n ts

2 0

1 5

1 0

5

0
0

2 0

4 0

6 0

8 0

P h o to e le c tr o n E n e r g y ( e V )Figure 6.16: Photoelectron spectrum of Xe using a narrow bandwidth of harmonics selected by
the multilayer mirror.

105
fast electronic processes[24]. All of these cross correlation measurements are based on what issometimes termed laser-assisted ionization, whereby the momenta of photoelectrons or Auger
electrons emitted from an ionization process are modified in the presence of a weak non-resonant
field. Experimental measurements using this technique were first done by Schins et al. andGlover et al. in 1996.[64, 32] In laser assisted ionization, ionization of an atom by an EUV
photon is accompanied by the absorption or emission of IR photons, giving sideband peaks tothe photoelectron spectrum. If a weak probe field is used, the intensity of the first sideband is
proportional to the intensity of the EUV pump pulse and the intensity of the IR probe pulse
according to lowest order perturbation theory. The sideband intensity as a function of probe
delay gives the intensity cross-correlation of the two pulses.[64]Micro-channel Plate Detector

Side View

Top View

Low B-Field

Gas Nozzle

High Harmonic Generation Cell

IR Femtosecond Pulse

Gold Mirror
With Hole

Aluminum
FilterDelayed IR Pulse

Magnetic Bottle
TOF Detector

EUV

e-

Detection Region
Detection Region

N
S

Detection Region

High B-Field

Figure 6.17: Schematic of experimental setup for measuring the pulse duration of high harmonic
generation.

For the measurement, 10% of the 800 nm light is split off from that used to drive highharmonic generation. This probe light is delayed and then recombined co-linearly with the highharmonic beam in the interaction region. The experimental setup is shown in figure 6.17. Whenboth the EUV and IR probe pulses are present, the resulting photoelectron spectrum contains
sidebands as a result of laser-assisted ionization. These sidebands have energies equal to a

106

M e a s u r e d S id e b a n d s

2 5 0 0T im e D e la y , 5 fs s te p s

C o u n ts

2 0 0 0

1 5 0 01 0 0 0

5 0 0

0
6

8

1 0

1 2

1 4

1 6

1 8

2 0

2 2

P h o to e le c tr o n E n e r g y ( e V )Figure 6.18: Photoelectron spectra at different probe delays. The sidebands that are analyzed
are indicated.

2 4 0

2 2 e V
2 5 e V

8 5 0 0

2 2 0

In te g ra te d C o u n ts

8 0 0 0

2 0 0

7 5 0 0

1 8 0

7 0 0 0

1 6 0

6 5 0 0

1 4 0

6 0 0 0

1 2 0
1 0 0

5 5 0 08 0
5 0 0 0
6 0
-4 0

-3 0

-2 0

-1 0

0

1 0

2 0

3 0

4 0

5 0

D e la y ( f s )Figure 6.19: Integrated counts for the sidebands at 22 and 25 eV as a function of probe delay.

107
harmonic plus or minus one photon of the fundamental (800 nm). The cross-correlation signal
is obtained by integrating the counts in the sidebands as a function of probe delay. Figure 6.18
shows photoelectron spectra as a function of probe time delay. The harmonics are generated in
Ar gas and detected with Ne. Figure 6.19 shows the integrated counts for two of the sidebands
as a function of probe delay. The cross correlation signal is 28 fs (+/- 2 fs) FWHM when fit
to a Gaussian. This measurement is consistent with the duration of the probe pulse indicatingthat the HHG emission is significantly shorter than the fundamental pulse. One of the majordifficulties of this technique is that the exact intensity profile of the probe pulse must be known
to measure the high harmonic emission. Recently, it has been demonstrated that the carrier
oscillations of the probe pulse can be used for time measurement, giving a much improved
temporal resolution and making it possible to measure a single attosecond pulse.[34]Chapter 7

Conclusion

In conclusion, this work applies nonlinear optical techniques to significantly higher photon energies than before, enhancing the conversion efficiency and opening up the possibility ofgenerating multi-keV photon energies from high harmonic generation. The ideas in this work
have the potential to greatly increase the number of applications possible using HHG in many
areas of chemistry and biology by making it a practical, efficient, table-top light source. The
enhancements demonstrated in this work are still far from the theoretical limits. Future work on
quasi-phase matching over longer fiber lengths and using shorter modulations should improve thephoton flux from HHG even further. Quasi-phase matching techniques may also provide a meansof generating an enhanced attosecond duration pulse of high harmonic light. Currently, highharmonic generation is the only method for generating light pulses of durations in the attosecond regime (1 attosecond = 10−18 seconds). Already, initial experiments have been performed
using light from HHG to measure electronic processes on attosecond time scales. Therefore,
the advances demonstrated in this work have the potential to influence both technology andfundamental science.

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Appendix A

ADK ionization rate calculation

The numerical calculations of the ionization rate use the expression in Ref. [71] for theAmmosov-Delone-Krainov (ADK) theory including the Keldysh theory after Coulomb correction. The tunnel ionization rate is given by
ω(t) = ωp |Cn∗ |2 (4ωp 2n∗−1
4ωp
)
exp(−
)
ωt
3ωt

(A.1)

with
IP
h̄
eEl (t)ωt =(2mIp )1/2
ωp =n∗ = Z(Iph 1/2)
Ip|Cn∗ |2 = 22n∗ [n ∗ Γ(n ∗ +1)Γ(n∗)]−1
where Ip is the ionization potential of the gas species, Iph is the ionization potential of thehydrogen atom, El (t) is the electric field of the laser, and m is the mass of the electron.
From this rate, it is straightforward to calculate the ionization fraction as a function oftime during the laser pulse given by:
Z t
η(t) = 1 − exp[−−∞

Matlab Codedt0 ω(t0 )].

(A.2)

116

Eion = 0.58065; % Ionization Potential for Argon in a.u.
Eion2=27.6/27.21; % 2nd Ionization Potential for Argon in a.u.
%Eion = 21.5646/27.21; % Ionization Potential for Neon in a.u.
%Eion2=40.1/27.21; % 2nd Ionization Potential for Neon in a.u.
QN = 1./sqrt(2*Eion);
QN2 = 2./sqrt(2*Eion2);
Ain = 5.0E14; % Peak intensity in W/cm2
Ain = sqrt(Ain/3.51E16); % Electric field amplitude
Tpul = 30.; % Pulse FWHM in fs
Tpul = Tpul*1E-15*4.1322E16;
Tf = Tpul*0.84932;
wo = 0.05695;
TT = 2.*pi/wo;
Nx = 10000; % Number of time steps
HT = 4.*Tf/Nx;
for i=1:Nx
T = i*HT;E(i)=Ain*exp(-(T-2.0*Tf)^2/Tf^2)*cos(wo*(T-2*Tf));
Int(i)=(Ain*exp(-(T-2.0*Tf)^2/Tf^2))^2;
end
N(1)=1.;N1(i)=0.;N2(1)=0.;
T(1)=0.;
Rint=0.;for i=1:Nx-1
T(i+1) = i*HT;F = abs(E(i))+1.E-30;
C2=2^(2*QN)/(QN*gamma(QN+1)*gamma(QN));
C22=2^(2*QN2)/(QN2*gamma(QN2+1)*gamma(QN2));
Rate1(i)=Eion*C2*(4.0*Eion*sqrt(2.0*Eion)/F)^(2.0*QN-1.0)*...
exp(-4.0*Eion*sqrt(2.0*Eion)/(3.0*F));
Rate2(i)=Eion2*C22*(4.0*Eion2*sqrt(2.0*Eion2)/F)^(2.0*QN2-1.0)*...
exp(-4.0*Eion2*sqrt(2.0*Eion2)/(3.0*F));
Rint=Rint+Rate1(i)*HT;
N(i+1)=exp(-Rint); % Number of Neutrals
N1(i+1)=1.-N(i)-N2(i); % Number of Single Ions
N2(i+1)= N2(i)+Rate2(i)*N1(i)*HT;% Number of Double Ions
Rate1(i)=Rate1(i)*N(i);Rate2(i)=Rate2(i)*N1(i);endNeutral=1-N;
Ion1=N1;
Ion2=N2;
T=(T-2*Tf)/TT*2.67;% Convert to fs

Appendix B

Magnetic Field Calculations

The calculation of the magnetic field of the magnetic-bottle spectrometer in Mathematica
is given on the following pages.

118

abbreviatedmagbottlecode.nb

1ü Magnetic Field for Permanent Magnet with Magnetization, M, Using the Magnetic
Scalar Potential
Phi1@r_, θ_, z_, r1_, θ1_, b_D =

M
4∗π

i
y
1
1
j
z
j
z
j
z
j
j è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
z;!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!! − è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! z
r1
^
2
+
r
^
2
−
2
r
r1
Cos@θ
−
θ1D
+
HzL
^
2
r1
^
2
+
r
^
2
−
2
r
r1
Cos@θ
−
θ1D
+
Hz
+
bL
^
2
k
{

Phi2@r_, θ_, z_, r1_, b_D = Simplify@Integrate@Phi1@r, θ, z, r1, θ1, bD, 8θ1, 0, 2 Pi<DD;
Phi@r_, θ_, z_, R_, b_D := NIntegrate@r1 ∗ Phi2@r, θ, z, r1, bD, 8r1, 0, R<D;

∆ = .0001 H∗cm∗L;

HR@r_, θ_, z_, R_, b_D := −1 ∗ HPhi@r + ∆ ê 2, θ, z, R, bD − Phi@r − ∆ ê 2, θ, z, R, bDL ê ∆HZ@r_, θ_, z_, R_, b_D := −1 ∗ HPhi@r, θ, z + ∆ ê 2, R, bD − Phi@r, θ, z − ∆ ê 2, R, bDL ê ∆BR@r_, θ_, z_, R_, b_D := mu ∗ HR@r, θ, z, R, bD;
BZ@r_, θ_, z_, R_, b_D := mu ∗ HZ@r, θ, z, R, bD;
mu = 4 Pi ∗ 10 ^ −1 H∗Gauss cmêAmp∗L;
b = 5.08 H∗cm− 2 in thickness∗L;
R = 1.27 H∗cm − 1 in diameter∗L;
M = 8000 H∗magnetization∗L;ü Magnetic field of a loop with current J and radius a at any pt. (r,z). (equations fromLandau and Lifshitz)
BLoopR@r_, z_, z1_, a_, J_D = J ∗ Hmu ê H2 PiLL ∗ HHz − z1L ê rL ê Sqrt@Ha + rL ^ 2 + Hz − z1L ^ 2D ∗H−EllipticK@ 4 a r ê HHa + rL ^ 2 + Hz − z1L ^ 2LD + Ha ^ 2 + r ^ 2 + Hz − z1L ^ 2L ∗EllipticE@ 4 a r ê HHa + rL ^ 2 + Hz − z1L ^ 2LD ê HHa − rL ^ 2 + Hz − z1L ^ 2LL ;BLoopZ@r_, z_, z1_, a_, J_D = J ∗ Hmu ê H2 ∗ PiLL ê Sqrt@Ha + rL ^ 2 + Hz − z1L ^ 2D
HEllipticK@ 4 a r ê HHa + rL ^ 2 + Hz − z1L ^ 2LD + Ha ^ 2 − r ^ 2 − Hz − z1L ^ 2L ∗EllipticE@ 4 a r ê HHa + rL ^ 2 + Hz − z1L ^ 2LD ê HHa − rL ^ 2 + Hz − z1L ^ 2LL;
a = 2.05 H∗radius of coil in cm∗L;
J = .9H∗current in coil in Amps∗L;x1 = 5 H∗distance of solenoid from origin in cm∗L;
L = 30 H∗length of solenoid in cm∗L;
x2 = L + x1;
mu = 4 Pi ∗ 10 ^ −1H∗Gauss cmêAmp∗L;

BSolenoidZ@r_, z_, a_, J_D = Sum@BLoopZ@r, z, i ê 9, a, JD, 8i, 18, 288<D;

BSolenoidR@r_, z_, a_, J_D = Sum@BLoopR@r, z, i ê 9, a, JD, 8i, 18, 288<D;
BtotR@r_, z_D := BSolenoidR@r, z, a, JD + BR@r, 0, z, R, bD;

119abbreviatedmagbottlecode.nb

BtotZ@r_, z_D := BSolenoidZ@r, z, a, JD + BZ@r, 0, z, R, bD;

ü Calculation of the adiabatic parameter∆ = .0001 H∗cm∗L;
m = .511 10 ^ 6 H∗ mass of e in eV ∗L;
En = 1 H∗ Energy of e in eV ∗L;c = 3 10 ^ 10 H∗cmês∗L;
v = Sqrt@2 En ê mD ∗ c;me = 9.11 10 ^ −31 H∗kg∗L;
e = 1.6 10 ^ −19 H∗coul∗L;
K = 2 10 ^ 4 ∗ Pi me v ê e;

χ1@r_, z_D := −K ∗ HBtotZ@r, z + ∆ ê 2D − BtotZ@r, z − ∆ ê 2DL ê H∆ BtotZ@r, zD ^ 2L

2

Appendix C

Micro-Channel Plate Detector

Electron detection for the magnetic bottle spectrometer was accomplished with a custom
designed micro-channel plate detector. The detector consists of two plates biased at ∼ 1 kV each.
The plates are made up of many micron-size diameter channels. Any electron that impacts thefront of the detector kicks off secondary electrons. These electrons are then accelerated towards
the anode, while building up signal with each additional impact to the channel walls, allowing ∼60 times signal gain. The micro-channel plate detector was designed to be compact enough to fit
on a NW40 flange. Figure C.1 shows the basic design. The two plates are separated by a stainlesssteel spacer. The front and outer part of the detector is grounded while the anode is held at ahigh voltage. Vespel is used as the insulator. The anode is connected to a feedthrough which
supplies the high voltage and transmits the fast current signal from the micro-channel plates.
Separation of the fast signal is accomplished with a Bias Tee (Picosecond Pulse Labs, Model
5531). The feedthrough was purchased from Ceramaseal, Inc. (model 1084-01-W), designed for
50 Ohm impedance. To prevent any signal reflection which could cause multiple counts for one
electron, the detector was designed to have 50 Ohm impedance from the feedthrough to theanode. The impedance of a coaxial waveguide is given by the expression,
276 b
Z0 = √ ln
a


(C.1)

where  is the permittivity of the insulator between the conductors, a is the inner conductor
radius, and b is the radius to the inner edge of the outer conductor. For Vespel, b/a=4.81 for 50

121
Ohm impedance.[52] Figure C.2 shows the detailed dimensions of all the parts of the detector.

0V
Micro-channel Plates

Anode2000 V
VespelInsulator

Figure C.1: MCP detector design. The micro-channel plates are in dark blue. The anode isbiased at 2 kV and is insulated from the grounded stainless steel housing.

122

.976
.731
.328
.020

.079
.318

.155

Anode

.100
.191

.731
.056

.030

.086

.194

.164

.972D

.250
40 TP

Stainless Cap

1.18501.291526.12

o1.2131
1.095

40 TP
.1750

.196

Stainless Housing
.919
1.095
.976
.328
.191
.167
.139

.550

.155

Vespel Insulator

.175
.191
.919

Spot Weld to Flange in 4 Places

Figure C.2: Detailed schematic of the MCP detector giving dimensions for each part. All
dimensions are in inches.