Copyright
by
Stefan Kneip
2005
X-Ray And Hot Electron Enhancement With Advanced
Targets Irradiated by Ultra-High Intensity Laser
by
Stefan Kneip
Thesis
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Master of Arts
The University of Texas at Austin
December 2005
X-Ray And Hot Electron Enhancement With Advanced
Targets Irradiated by Ultra-High Intensity Laser
Approved by
Supervising Committee:
To my brother Peter
Acknowledgments
First, I would like to express my sincere appreciation to my advisor Todd Ditmire,
for giving me the unique opportunity to work in his group. I owe a great depth
of gratitude to him for his continuous encouragement and enthusiastic motivation
throughout and beyond this work. I am indebted to him for his excellent guidance
and advice, for providing excellent facilities and financial support. I am eternally
thankful to him for teaching me some very important soft-skills and supporting me
on my career goals in every imaginable way.
It is my very pleasure to thank Prof. Roger Bengtson for his fruitful input as a
reader of this thesis.
I owe a great depth of gratitude to Gilliss, Dan and Byoung-Ick. Neither would I
have been in the position to start nor to finish this work without them. It is my
innermost urge to laud our postdoc Dan, for being one of the strongest supporters
of me, for his patience with an unexperienced graduate student, for his assistance
during night-shifts and for being British. I wish to thank Andreas, Irina and Michael
for providing help during setup time and company during long runs. My gratitude
is extended to Ariel, especially for sharing his intellectual property on spheres. I
am lacking the words to honor Byoung-Ick’s 36 hour marathon shifts for target
fabrication. I deeply recognize many enjoyable and fruitful discussions with Dan,
Gilliss, Byoung-Ick, Ariel, Will, Greg, Jens, Matthias and both Aarons.
I promised to Allan Schroeder, that the machine shop crew would be the first to
v
mention, if I were to give a nobel laudation. Special thanks go to Donnie. He
shared a piece of his heart and soul with the construction of my spectrometer and
the outcome of the experiment.
I grateful acknowledge the individual help and support from our collaborators Prof.
E. Forster and Dr. O. Wehrhan from the X-Ray Optics Group at the University of
Jena in Germany. I owe them a great depth of gratitude for providing a powerful
spectroscopic tool and valuable assistance on its operation. The interpretation of the
pyramid data would not have been possible without the high quality PIC simulations
from our collaborator Prof. Y. Sentoku, University of Nevada at Reno. I am very
glad to thank Dr. S. Pikuz from Lomonov Moscow University as collaborator for
the investigations with the spherical crystal spectrometer.
Multiple cheers go to everybody in the group who was part of the by now legendary
atmosphere. This includes everybody’s respective role in facilitating this work, no
matter if in the lab or on 6th street. I could never leave unmentioned my homies
Andreas and Sebastian for their innumerable contributions to this work.
My most sincere gratitude goes to the Physics Department of the University of
Wurzburg and the German Academic Foundation Cusanuswerk. Without their help
and financial support, this stay would not have been possible for me.
Finally to my parents Ulla and Martin and to my brother Peter: Danke fur Eure
immerwahrende Ermutigung, Euer Verstandnis, Eure Unterstutzung, Aufopferung
und Euren Ruckhalt wahrend dieser Arbeit und in meinem ganzen Leben.
Stefan Kneip
The University of Texas at Austin
December 2005
vi
X-Ray And Hot Electron Enhancement With Advanced
Targets Irradiated by Ultra-High Intensity Laser
Stefan Kneip, M.A.
The University of Texas at Austin, 2005
Supervisor: Todd R. Ditmire
An experimental study of advanced target geometries for high intensity laser interac-
tion is presented with view to x-ray and hot electron yield enhancement. One target
family consists of guiding geometries such as pyramids and wedges that were etched
into silicon substrates. Another target family consists of monolayers of wavelength-
scale spheres that were laid down on silicon. A curved crystal spectrometer was
designed and employed to determine K-shell yields. Scintillator detectors were used
to determine the bremsspectrum and the hot electron temperature. In accordance
with recent literature, it is found that the open angle of pyramids is insufficient for
significant cone-guiding. A strong hard x-ray and Kα yield dependency was found
for wedges in s- and p-polarization. The results are explained with 2D PIC sim-
ulations. It was found that spheres-coated targets enhance the Kα yield by many
times. A sphere-size scan reveals a resonance-like behavior for 0.26 µm spheres.
vii
Contents
Acknowledgments v
Abstract vii
Contents viii
List of Figures xii
Chapter 1 Introduction 1
Chapter 2 Introduction to Plasma Physics and Cone-Guiding 3
2.1 Collisional and Collective Regime . . . . . . . . . . . . . . . . . . . . 3
2.2 Particle and Fluid Description of Plasmas . . . . . . . . . . . . . . . 4
2.2.1 Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Plasma Behavior and Laser-Plasma Interactions . . . . . . . . . . . . 7
2.3.1 Plasma Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Light Propagation in Plasmas . . . . . . . . . . . . . . . . . . 10
2.3.4 Resonance Absorption . . . . . . . . . . . . . . . . . . . . . . 11
2.3.5 Collisional Heating . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.6 Vacuum Heating . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.7 J x B Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.8 Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 X-Ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Line Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
viii
2.4.2 Continuum Emission . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Cone-Guiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 3 Experimental Setup 21
3.1 The THOR Laser Facility . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Bandwidth Limit and CPA . . . . . . . . . . . . . . . . . . . 22
3.1.2 Ultrashort Pulse Generation by Mode Locking . . . . . . . . 24
3.1.3 THOR Laser Layout . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Autocorrelation and Frequency Doubling . . . . . . . . . . . . . . . . 29
3.2.1 Nonlinear Wave Mixing and Phase Matching . . . . . . . . . 29
3.2.2 2nd Order Autocorrelation . . . . . . . . . . . . . . . . . . . 31
3.2.3 3rd Order Autocorrelation . . . . . . . . . . . . . . . . . . . . 32
3.2.4 Frequency Doubling . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Target Properties And Preparation . . . . . . . . . . . . . . . . . . . 35
3.3.1 Titanium Targets, Pyramids and Wedges . . . . . . . . . . . 35
3.3.2 Silicon and Sphere Targets . . . . . . . . . . . . . . . . . . . 38
3.4 High Resolution Bent Crystal X-Ray Spectrometer . . . . . . . . . . 39
3.4.1 Introduction to Bragg Reflection . . . . . . . . . . . . . . . . 39
3.4.2 PET as Crystal Material . . . . . . . . . . . . . . . . . . . . . 41
3.4.3 Properties of Cylindrical PET . . . . . . . . . . . . . . . . . . 41
3.4.4 Design and Alignment of Spectrometer . . . . . . . . . . . . . 44
3.5 The Solid Target Vacuum Chamber . . . . . . . . . . . . . . . . . . 46
3.5.1 Chamber Setup for Titanium Targets . . . . . . . . . . . . . 46
3.5.2 Chamber Setup for Silicon Targets . . . . . . . . . . . . . . . 51
3.5.3 Focal Spot Characterization . . . . . . . . . . . . . . . . . . . 52
3.6 Continuum Radiation Scintillation Detectors . . . . . . . . . . . . . 54
Chapter 4 Analysis of X-ray Film 57
4.1 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Overview of X-Ray Film Analysis . . . . . . . . . . . . . . . . . . . . 59
4.3 Digitalization of X-ray Film . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.3 Lineout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ix
4.4 Deconvolution of X-Ray Film . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Absolute Wavelength Calibration . . . . . . . . . . . . . . . . 61
4.4.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.3 Intermediate Result . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.4 Crystal Response and Filter Transmission . . . . . . . . . . . 63
Chapter 5 Experimental Characterization of Titanium Targets 66
5.1 Characterization of von-Hamos Spectremeter . . . . . . . . . . . . . 67
5.1.1 Accuracy of Data Reproduction . . . . . . . . . . . . . . . . . 67
5.1.2 Integrated Shot Number . . . . . . . . . . . . . . . . . . . . . 68
5.1.3 Accuracy of Wavelength Calibration . . . . . . . . . . . . . . 69
5.1.4 Plasma Emission . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.5 Spectral Resolution and Focusing Quality . . . . . . . . . . . 71
5.2 Angle Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Kα Yield Comparison of Flat and Micro-Shaped Targets . . . . . . . 72
5.3.1 Pyramid versus Flat Target . . . . . . . . . . . . . . . . . . . 72
5.3.2 P-Wedge versus S-Wedge . . . . . . . . . . . . . . . . . . . . 74
5.4 Hard X-Ray Yield Comparison of Flat and Micro-Shaped Targets . . 75
5.4.1 Dependency on Target Type . . . . . . . . . . . . . . . . . . 76
5.4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 PIC Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.1 Suprathermal Electrons . . . . . . . . . . . . . . . . . . . . . 79
5.5.2 Bremsspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.3 Electron Energy Density Plot . . . . . . . . . . . . . . . . . . 80
5.6 Spatial Kα Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Chapter 6 Experimental Characterization of Silicon Targets 84
6.1 Characterization of von-Hamos Spectrometer . . . . . . . . . . . . . 84
6.1.1 Spectral Characterization . . . . . . . . . . . . . . . . . . . . 84
6.1.2 Focusing Quality . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.3 Angle Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Kα Yield Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.1 Flat vs Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 88
x
6.2.2 Sphere Size Scan . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Hard X-Ray Yield Comparison . . . . . . . . . . . . . . . . . . . . . 90
6.3.1 Dependency on Sphere Size . . . . . . . . . . . . . . . . . . . 91
6.3.2 Hot Electron Temperature . . . . . . . . . . . . . . . . . . . . 92
6.4 Plasma Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.5.1 Local Field Enhancement . . . . . . . . . . . . . . . . . . . . 95
6.5.2 Multi Pass Vacuum Heating . . . . . . . . . . . . . . . . . . . 95
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 7 Future Prospects 97
Bibliography 99
Vita 105
xi
List of Figures
2.1 Theory Of Cone-Guiding . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Electrons can be accelerated along the surface for obliquely in-
cident laser fields. . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Self-consistent Surface Magnetic and Electric Fields allow for
Stable Surface Electron Current . . . . . . . . . . . . . . . . . 18
3.1 Chirped Pulse Amplification . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Oscillator output modes . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Mode-Locked . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Multi-Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.3 Continuous Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Kerr Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 THOR Laser Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 Non-Collinear Type-I Phase-Matching . . . . . . . . . . . . . . 30
3.5.2 Refractive Index Ellipsoid . . . . . . . . . . . . . . . . . . . . . 30
3.6 2nd Order AC Image And Trace . . . . . . . . . . . . . . . . . . . . 32
3.6.1 2nd Order AC Trace . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6.2 2nd Order AC Image . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 3rd Order AC Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.8 SEM Images of Silicon Targets . . . . . . . . . . . . . . . . . . . . . 36
3.9 Comparison of Possible Ti Target Geometries . . . . . . . . . . . . . 37
3.10 Illustration of Bragg Reflection . . . . . . . . . . . . . . . . . . . . . 40
3.11 3D Drawing of Spectrometer . . . . . . . . . . . . . . . . . . . . . . 45
3.12 Solid Target Interaction Chamber . . . . . . . . . . . . . . . . . . . . 47
xii
3.13 Spectrometer Setup for Titanium Targets . . . . . . . . . . . . . . . 50
3.14 Spectrometer Setup for Silicon Targets . . . . . . . . . . . . . . . . . 53
3.14.1 at 0⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.14.2 at 45⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.15 Focal Spot Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.15.1 Titanium Target . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.15.2 Silicon Target . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.16 Scintillation Detector Calibration . . . . . . . . . . . . . . . . . . . . 55
3.17 Absolute Energy Calibration . . . . . . . . . . . . . . . . . . . . . . 56
3.17.1 by Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.17.2 by Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1 Analysis of X-Ray film . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Density-exposure relation for X-Ray Film . . . . . . . . . . . . . . . 63
4.3 Efficiency of Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Kα Comparison of 30 and 90 Shot Runs . . . . . . . . . . . . . . . . 67
5.2 Full Spectral Comparison of 30 and 90 Shot Run . . . . . . . . . . . 68
5.3 Angle Scan with Copper . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Comparison of Kα Spectrum . . . . . . . . . . . . . . . . . . . . . . 73
5.4.1 11 µm Pyramid and Flat Target . . . . . . . . . . . . . . . . . 73
5.4.2 25 µm Pyramid and Flat Target . . . . . . . . . . . . . . . . . 73
5.5 Absorption of Ti K-Shell Radiation in Titanium . . . . . . . . . . . 73
5.6 Comparison of Kα Spectrum from Pyramid and Wedge Target . . . 73
5.7 Hard X-Ray Yields from Micro-Shaped and Flat Targets . . . . . . . 76
5.8 2D PIC Code: Electron and X-Ray Spectrum . . . . . . . . . . . . . 78
5.8.1 Electron Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 78
5.8.2 Bremsspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.9 2D PIC Code: Angular Dependency of Hard X-Ray Emission . . . . 79
5.9.1 Radial Plot of Hard X-Rays . . . . . . . . . . . . . . . . . . . 79
5.9.2 Interpretation of Radial Plot . . . . . . . . . . . . . . . . . . . 79
5.10 2D PIC Code: Electron Energy Density Plot . . . . . . . . . . . . . 81
5.10.1 P-Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.10.2 S-Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.11 Spatial Kα1 Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
xiii
6.1 Spectral Resolution for Silicon Spectroscopy . . . . . . . . . . . . . . 85
6.2 Focusing Quality of PET . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.1 Spatial Line Width . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.2 X-Ray Film Scans . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Kα Yield Flat vs Spheres Target . . . . . . . . . . . . . . . . . . . . 88
6.3.1 0⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.2 45⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Sphere Size Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.1 0⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.2 45⋄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5 Kα And Hard X-Ray Yield . . . . . . . . . . . . . . . . . . . . . . . 91
6.5.1 Kα Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5.2 Hard X-Ray Yield . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.6 Spectral Shape of Heα . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6.1 for Plane Target . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6.2 for Spheres Target . . . . . . . . . . . . . . . . . . . . . . . . . 93
xiv
Chapter 1
Introduction
Fusion powers the sun. The goal of a concerted, multidisciplinary research effort is
to utilize the same fusion reactions to feed the energy needs of mankind. State of
the art lasers can produce extreme states of mater, that are relevant to fusion energy
research [1]. Many obstacles have still to be taken until controlled thermonuclear
fusion can be realized. Nevertheless, several approaches have already passed the sta-
tus proof-of-principle and are ever more successfully implemented in laser-induced
fusion schemes [2] [3] [4]. One of the most crucial points toward laser-driven fusion
is the ability of transferring the provided pulse energy efficiently to the fuel. There-
fore, the interaction of short intense laser pulses with a variety of targets such as
atomic clusters [5], wavelength scale droplets [6] and solid guiding structures [7] has
been studied in the past. All three aforementioned states of matter have revealed
unique features and added to the knowledge of laser interaction regimes.
The work presented in this thesis is focused on two of these advanced target con-
cepts:
Guiding structures that were etched anisotropically into silicon wafers are equipped
with a high Z foil on the back side. When subject to ultra-short, ultra-intense laser
pulses, K-shell, continuum and electron radiation are investigated by means of sev-
eral detection systems. Pursuing this approach to guiding structures is motivated
as follows: The guiding geometry that has been used so far in laser driven fusion
experiments [3] are free standing gold cones, that are produced by General Atomics,
CA. Manufacturing free standing gold cones is complicated, expensive and intellec-
tual property of General Atomics. It is very desirable, to be able to produce guiding
1
structures that are based on silicon. The techniques of processing silicon wafers by
edging are well known and used intensively in the field of semiconductor industry.
Silicon cones can be produced fast, cheaply and with highest accuracy.
The second target concept employs uniform monolayers of wavelength-scale micro-
spheres that were deposited on silicon substrates. Again, x-ray continuum and line
radiation are measured while irradiated with high intensity laser pulses. The in-
teraction of ultra-short ultra-intense laser pulses with media of intermediate size
between gases and solids is of considerable interest. Clusters and microdroplets
have enabled new regimes of laser-matter interaction. In particular, hot electron
temperature enhancement was reported lately when microdroplets were irradiated
by ultra-short, ultra-intense laser pulses as compared to solids [9]. In order to turn
this electron enhancement into an ultra-bright x-ray source, one has to supply cold
target material for electrons to slow down and produce line and continuum x-rays.
This motivates the combination of electron-enhancing spheres on a solid substrate.
In Chapter 2, the basic theory of plasma physics will be reviewed with a view
to laser-plasma interaction regimes that are relevant to this work. Moreover, the
physics of line and continuum x-ray generation and cone-guiding will be introduced.
Chapter 3 presents the experimental apparatus. This includes the 20 TW laser fa-
cility THOR, the diagnostics, the target properties and preparation and finally the
experimental setup in the vacuum chamber.
The major diagnostic that has been developed for this work is a curved crystal x-ray
spectrometer coupled to scientific x-ray film. Chapter 4 reviews the details of x-ray
film analysis and data linearization.
Chapter 5 and 6 presents the experimental results and data interpretation that was
obtained with guiding targets and spheres targets respectively.
The work is concluded with some future prospects in Chapter 7.
2
Chapter 2
Introduction to Plasma Physics
and Cone-Guiding
All physical effects discussed in the presented work are based on the interaction of
ultra short laser pulses and plasmas created on the surface of solid targets. One
should therefore start introducing terms of the basic plasma theory.
2.1 Collisional and Collective Regime
A plasma is a system of N charges which are coupled to one another by their self-
consistent electric and magnetic fields. Even in an electrostatic simplification, one
would have to solve 6N coupled equations
miri = qiE(r)i (2.1)
E(ri) =∑
j
qj
|rirj |3(rirj) (2.2)
where mi, qi and ri are the mass, charge and position of the particle with index i.
This would be a very inconvenient approach.
Luckily, many physical situations do not require to track every single particle i for
itself. One can decompose the electric field E into two field E1 and E2, which have
distinct spatial scales. E1 has spatial variation on a scale length longer than the
Debye Length, which is the length over which the influence of a single charge is
shielded by the collective behavior of the surrounding charges (see section 2.3.2).
3
E2 represents the highly fluctuating microfield due to random encounters (collisions)
between particles.
The separation into a collective and collisional regime is very natural and simply
depends on the time scale of a typical collective or collisional process. In reference
[10], one can find a simple calculation comparing the rates (inverse time scales) for
the respective processes:
ν90
ωpe≃ const · Z
ND(2.3)
Here, ν90 is the rate of a typical random encounter (90 degree collision), ωpe is the
frequency of a typical collective phenomenon (longitudinal electron density fluctua-
tion, see section 2.3.1), Z is the atomic number and ND is the number of electrons in
a Debye sphere. From equation 2.3 it is obvious, that the fine scale behavior caused
by individual particles can be neglected over the behavior of the smoothed coarse
grain field if ND → ∞.
2.2 Particle and Fluid Description of Plasmas
The collective regime is the most relevant to this work. The next subsections will
present the Vlasov equation and the fluid equations which are the common descrip-
tion of a plasma in the collisionless regime.
2.2.1 Vlasov Equation
Complete information of all particles i, 1 ≤ i ≤ Nj of different families j (e.g. j = 1
for electrons and j = 2 for ions) is comprised in the distribution function f (j)(v, r, t)
[10]. The probability of finding a particular particle at time t between r and r + dr
with velocity between v and v + dv is given by f (j)(v, r, t)drdv/Nj .
Assuming neither ionization nor recombination, particles are neither created nor
destroyed. It follows, that the distribution functions f (j) are constant along their
trajectories in the 6Nj-dimensional phase spaces. The f (j) must obey the continuity
equation [10]:
∂f (j)
∂t+
∂
∂x·(
rf (j))
+∂
∂v·(
vf (j))
= 0 (2.4)
4
The laws of motion r = v and v =qj
mj
(
E + v×Bc
)
can be inserted in equation 2.4 to
obtain the equation of motion for the distribution function f (j), the so-called Vlasov
equation:
∂f (j)
∂t+ v · ∂f (j)
∂r+
qj
mj
(
E +v × B
c
)
· ∂f (j)
∂v= 0 (2.5)
To obtain a set of equations, which fully describes the dynamical evolution of the
plasma, we have to couple (2.5) self consistently to the Maxwell equations:
∇ · E = 4πρ (2.6a)
∇ · B = 0 (2.6b)
∇× E = −1
c
∂B
∂t(2.6c)
∇× B =4π
cJ +
1
c
∂E
∂t(2.6d)
Equations (2.5) and (2.6) comprise a complete description of collisionless plasma
behavior.
2.2.2 Fluid Equations
Derivation By taking different velocity moments of the Vlasov equation, one can
derive equations for density nj , velocity njuj and pressure Pj of each species in
space and time. The moments are defined as [10]:
nj =
∫
f (j)(r,v, t)dv
njuj =
∫
vf (j)(r,v, t)dv
Pj = mj
∫
(v − uj) (v − uj) f (j)(r,v, t)dv
and higher moments...
(2.7)
where j denotes the particle families and mj the mass of one particle of the jth
family.
As a next step, one calculates the same velocity moments of the Vlasov equation
5
(2.5):
∫
dv
[
∂f (j)
∂t+ v · ∂f (j)
∂r+
qj
mj
(
E +v × B
c
)
· ∂f (j)
∂v
]
= 0
∫
vdv
[
∂f (j)
∂t+ v · ∂f (j)
∂r+
qj
mj
(
E +v × B
c
)
· ∂f (j)
∂v
]
= 0
and higher velocity moments...
(2.8)
The integration of (2.8) can be performed using (2.7) and appropriate assumption
for the Lorentz force [10].
One ends up with a series of fluid equations, the first two of which are the continuity
and force equation for the density and mean velocity of particles with charge qj and
mass mj
∂nj
∂t+
∂
∂r· (njuj) = 0 (2.9)
nj
(
∂uj
∂t+ uj ·
∂uj
∂r
)
=njqj
mj
(
E +uj × B
c
)
− 1
mj
∂pj
∂r(2.10)
Here pj denotes the scalar pressure which emanates from the pressure tensor Pj in
the case of an isotropic fluid.
The continuity equation (2.9) is a differential equation for the density nj and brings
in the next higher velocity moment, the mean velocity nju. The equation (2.10) is
a differential equation for the density nj , the mean velocity nju and brings in the
next higher velocity moment, the pressure pj .
Truncation In order for this series to truncate, one can relate the pressure pj to
the density nj by a so-called equation of state (EOS). This is either an isothermal
or adiabatic equation of state, depending on reasonable assumptions for the heat
flow1:
• The isothermal equation of state is appropriate when ω/k << vj , where ω and
k are the frequency and wave number characteristic of the physical process
being considered and vj =√
kTj/mj =√
θj/mj is the thermal velocity of the
1The heat flow will appear in the third moment of the Vlasov equation [10].
6
particles:
pj = nj · Tj (2.11)
• In the opposite limit (ω/k >> vj), the heat flow can simply be neglected. This
assumption leads to an adiabatic equation of state
pj/nγ = const (2.12)
where γ = 2+NN with N being the number of degrees of freedom.
When ω/k ∼ vj , the details of the velocity distribution f (j) of the particles are
important and one has to return to the Vlasov equation (2.5) in the fully microscopic
picture.
Two-Fluid Description The density (2.9) and fluid equation (2.10) and appro-
priate equations of state (2.11) or (2.12) for the particle families j and an appropriate
form of the Maxwell equations (2.6) represent a complete self-consistent set of equa-
tions to describe the dynamic evolution of the plasma within the mentioned limits.
The two-fluid description of plasmas can be regarded as a special case of the fluid
descriptions, where the plasma consists of electrons (j = e) and ions (j = i). Due to
the origin of a laser-produced plasma, which is a neutral gas or solid, the two fluid
description and the assumption of quasi neutrality is valid.
2.3 Plasma Behavior and Laser-Plasma Interactions
The Two-Fluid Description will be used to investigate plasma behaviors such as
plasma waves (section 2.3.1), the propagation of electromagnetic waves in plasmas
(section 2.3.3) and laser absorption mechanisms (sections 2.3.4, 2.3.6 and 2.3.7)
relevant to this work.
2.3.1 Plasma Waves
A plasma without large imposed magnetic fields can support two types of collective
longitudinal modes, one with a high frequency density modulation (electron plasma
wave) and one with a low frequency density modulation (ion acoustic wave). Plasma
7
waves are of great importance because they accelerate captured electrons and con-
tribute to heating of the plasma through Collisional Damping or Landau Damping
[10] for example.
Electron Plasma Wave The massive ions are treated as immobile, uniform and
neutralizing background with density ni0. With an adiabatic EOS (ω/k ≫ ve), the
electron fluid is described by (2.9) (2.10):
∂ne
∂t+
∂
∂x(neue) = 0 (2.13)
∂
∂t(neue) +
∂
∂x
(
neu2e
)
= −neeE
me− 1
me
∂pe
∂x(2.14)
pe
n3e
= const. (2.15)
∂E
∂x= −4πe (ne − Zni0) (2.16)
Equation (2.16) is the Poisson equation. The density fluctuation is taken to be in
x-direction. With a time derivative of (2.13) and a spatial derivative of (2.14) one
can eliminate ∂2neue
∂t∂x so that:
∂2ne
∂t2− ∂2
∂x2
(
neu2e
)
− e
me
∂
∂x(neE) − 1
me
∂2pe
∂x2= 0 (2.17)
By considering small amplitude perturbations of density, velocity and electric field
(ne = n0 + n, ue = u, pe = n0θe + p and E = E) one can linearize the equations
(2.15), (2.16) and (2.17):
p = 3mev2e n (2.18)
∂E
∂x= −4πen (2.19)
∂2n
∂t2− n0e
me
∂E
∂x− ∂2p
∂x2= 0 (2.20)
Substituting (2.18) and (2.19) in (2.20) yields a wave equation for small longitudinal
density fluctuations of the electron density:
(
∂2
∂t− 3v2
e
∂2
∂x2+ ωpe
)
n = 0 (2.21)
8
Here, ωpe :=√
4πe2n0/me defines the electron plasma frequency for a plasma with
electron density n0 = Zni0. By trying a wave solution n = eikx−iωt one obtains the
dispersion relation for electron plasma waves
ω2 = ω2pe + 3k2v2
e . (2.22)
The frequency of these oscillations is almost the electron plasma frequency with a
small correction by the wavenumber k and the thermal velocity ve.
Resonance absorption of obliquely incident laser light of frequency ωl = ωpe (see
section 2.3.4) can stimulate such density fluctuations at the critical density. Another
way of exciting electron plasma waves is by Stimulated Raman Scattering (SRS)
where an incident photon decays into a scattered photon and a plasmon [10].
Ion Accoustic Wave Both the electron and the ion fluid has to be considered.
Yet simplifications arise because the response of the light weight electrons is fast
(with ωpe) compared to the inert ions. The dispersion relation for the longitudinal
ion acoustic wave is [10]
ω = ±kvs (2.23)
where vs :=√
Zθe + 3θi defines the ion sound speed with the ion mass M .
Ion acoustic waves can be excited by Stimulated Brillouin Scattering (SBS) where
an incident photon decays into a scattered photon and an ion acoustic phonon [10].
2.3.2 Debye Shielding
A plasma modifies and shields the electric potential of a discrete charge as indicated
in section 2.1. The Poisson equation for a particle of charge q at position r = 0 is
∇2φ = −4πqδ(r) + 4πe(ne − n0) (2.24)
where the ions are treated as a neutralizing background ni0 and the electron density
is initially uniform ne = n0. In a static limit (∂/∂t = 0 and ue = 0), the force
equation for the electron fluid (2.10) with an isothermal EOS reduces to
0 = neeE + θe∇ne. (2.25)
9
With E = −∇φ, the electron density becomes
ne = n0 exp
(
eφ
θe
)
. (2.26)
An equation for the electric potential φ can be obtained by expanding (2.26) for
small eφ/θe and substituting ne in (2.24):
∇2φ − φ
λD= −4πqδ(r) (2.27)
where λD :=√
θe/4πn0e2 defines the electron Debye length. From the solution to
(2.27)
φ =q
rexp
(−r
λD
)
(2.28)
the meaning of the Debye length becomes obvious. As indicated in section 2.1, the
usually Coulomb like potential of a single charge q is shielded out by the collective
effect of an electron density ne, where the Debye length is the characteristic length.
2.3.3 Light Propagation in Plasmas
A plasma modifies the propagation of electromagnetic waves. A wave equation for
the high frequency field
E = E (r) exp (−iωt) (2.29)
can be derived using the fluid equation for the force (2.10), Maxwell equations (2.6)
and appropriate assumptions for small quantities:
∇2E −∇ · (∇ · E) +ω2
c2ǫE = 0 (2.30)
Here, ǫ = 1 − ω2pe
ω defines the dielectric function of the plasma. For ∇ǫ = 0 and
∇ · E = 0, equation (2.30) gives the dispersion relation
ω2 = ω2pe + k2c2. (2.31)
10
The minimum frequency for propagation of light is ω = ωpe. For ω < ωpe, the
wave vector k becomes imaginary (reflection) and the dielectric function ǫ becomes
negative. This yields a complex index of refraction n =√
ǫ. The physical meaning
of a non-zero imaginary part of n can be light absorption, which will be discussed
in some more detail in section 2.3.4.
There is an intuitive interpretation for equation (2.31): Since the characteristic
response time for electrons is ω−1pe , the electrons shield out the light if ω < ωpe.
Therefore the condition ω = ωpe =: 4πe2n0/me defines the maximum plasma density
to which a light wave can penetrate. This density is also called critical density
nc = mω2pe/4πe2.
2.3.4 Resonance Absorption
A light wave that is incident under an angle θ with respect to a plasma slab with
density gradient ∇ne||z in z-direction can resonantly transfer energy into the plasma
in case of the right polarization via fast electrons [11] [12]. The plane of incidence
shall be the y-z-plane without loss of generality.
Obliquely Incident S-Polarized Light The electric vector points out of the
plane of incidence. With E = Exx, the wave equation (2.30) becomes
d2E(z)
dz2+
ω2
c2
(
ǫ(z) − sin2(θ))
E(z) = 0 (2.32)
which yields the dispersion relation
k2c2 = ω2(
ǫ(z) − sin2 θ)
. (2.33)
For ǫ(z) = sin2 θ, the wave vector k becomes imaginary. Since ǫ = 1−ω2pe(z)/ω2 re-
flection takes place for ωpe = ω cos θ which corresponds to a smaller critical density
ne = nc cos2 θ than for normal incidence as discussed in section 2.3.1.
No energy is transferred to the plasma because the field can not acquire an electro-
static component (E = Exx):
∇ · E = 0 (2.34)
11
Obliquely Incident P-Polarized Light The electric field vector lies in the plane
of incidence. In this case E · ∇ne 6= 0, so there is an electric field component that
can oscillate electrons along the density gradient ∇ne to generate plasma density
fluctuations δne. Because of these density fluctuations, the wave is no longer purely
electromagnetic but acquires an electrostatic component. This is again verified with
the Poisson equation for E = Eyy + Ez z which yields
∇ · E = −1
ǫ
∂ǫ
∂zEz. (2.35)
Resonant response occurs for ǫ = 0 (ωpe = ω). The z-component of the electric field
Ez can be related to the magnetic field [10]
Ez =sin θB(z)
ǫ(z)(2.36)
Evaluating B(z) at the critical density reveals under which circumstances the driving
is most efficient [10] [13]:
Ez =EL
√
2πωLn/cφ(τ) (2.37)
where EL is the laser field, φ(τ) ≈ 2.3τ exp(−2τ3/3), τ := (ω0Ln/c)1/3 sin θ, ω = ωpe
is the laser frequency of resonant driving and Ln := ne
(
dzdne
)
nc
is the density scale
length of ∇ne. By introducing a small amount of damping to the ǫ(z) with the rate
ν ≪ ω, the fraction of energy transferred to the electron plasma wave is [13]
fra ≈ φ2(τ)
2. (2.38)
The damping can be caused by collisional or collisionless effects. A more detailed
numerical calculation gives a slightly lower absorption with the same angular de-
pendency [13]. For a linear density profile, the absorption (2.38) peaks at
θmax ≈ sin−1(
0.8(c/ωL)1/3)
(2.39)
and is of comparable height for a range of ∆θ ≈ θmax.
Apparently, resonance absorption requires a non-vanishing plasma density gradient
12
∇ne (Ln 6= 0). Either the rising edge of a ns long pulse laser or prepulses of a fs
ultra-short pulse laser can cause such an expanding pre-plasma of scale length Ln.
By carefully characterizing the temporal pulse shape of a fs laser as done in section
3.2.3, pre-pulses can be determined. By doing that, one can estimate if resonance
absorption will contribute as a heating mechanism. For very clean pulses and ultra
high intensities another absorption mechanism which is called vacuum heating starts
to kick in and eventually dominates over resonance absorption (see section 2.3.6).
Even for normal incidence or s-polarization, resonance absorption can contribute
significantly to the laser absorption, if only the intensity exceeds 1019Wµm2/cm2.
At these intensities, rippling of the electron critical surface caused by plasma insta-
bilities can occur which effectively always creates non-normal incidence [13].
2.3.5 Collisional Heating
Inverse Bremsstrahlung (electron absorbs photon in the vicinity of a third particle)
is the most relevant collisional process. It dominates if there are only few particle
within one Debye sphere (see equation (2.3)). This corresponds to low density
plasmas.
Electrons that are oscillating in an electric field acquire an averaged kinetic energy
which is called the pondoromotive potential [10] [14]
Up =1
2me〈v2〉 =
e2E2L
4meω2=
1
2· 0.71 · 0.511MeV ·
(
IL
1018Wcm−2
λ2
µm2
)
(2.40)
where ω is the laser frequency, EL is the laser field and IL is the focused intensity.
This energy is then transferred to ions by collisions and the plasma heats up. The
fractional absorption is [13]
fib = 1 − exp
(
32
15
νei(nc)
cLn
)
(2.41)
where Ln is the density scale length of a linear density profile and νei(nc) is the
electron-ion collision frequency evaluated at the critical density nc. The collision
frequency (in the weak field limit) depends on θe, ne and Z as follows:
νei(nc) ∝ncZ
θ3/2e
(2.42)
13
Hence absorption by inverse Bremsstrahlung is large for long density gradients, low
temperatures and high Z plasmas. Inverse Bremsstrahlung and collisional processes
in general peak at 1014 − 1016Wµm2/cm2 [14] and will only be to minor relevance
to the experiments presented in this work.
The hot electron temperature that can be reached with collisional absorption scales
as [14]
Te ≃ 8(
I16λ2µm
)(1/3)(2.43)
where I16 is the focused intensity in 1016 W/cm2 and λµm is the wavelength in µm.
2.3.6 Vacuum Heating
Vacuum heating is also refered to as not-so-resonant resonance heating or Brunel
effect [13] [15]. For resonance absorption a gently increasing plasma with scale length
Ln > λ is necessary so that the field can drive a large plasma wave resonantly. For
vacuum heating however a the laser couples into a short scale length plasma Ln < λ
or overdense plasma interface and no large plasma wave can be driven. Instead,
single atoms dragged away from the plasma interface out into the vacuum from where
they are accelerated back into the overdense plasma with random phase. Because
the ponderomotive potential can reach & 1 MeV at intensities of 1019W/cm2 and
800 nm, equally fast electrons can be created with one kick by the laser. Vacuum
heating is an important absorption mechanism for ultra short ultra intense lasers.
The angular dependency of the fractional absorption is [13]
fvh∼= 8
√
〈v2〉c
sin3 θ (2.44)
where√
〈v2〉 is the square root of the averaged oscillation frequency leading to the
ponderomotive potential (2.40). Therefore, vacuum heating is favored by large an-
gles and should peak for grazing incidence. This of course is not practicable because
also the focused intensity would become zero for grazing incidence. Consequently,
an angle betwenn 0⋄ and 90⋄ should give the highest absorption.
14
2.3.7 J x B Heating
This effect is an interplay between the ponderomotive Force Fp = − e2
4meω2
l
∇E2(r)
(see equation (2.40)) that causes electrons to oscillate perpendicular to the laser di-
rection and the magnetic field B which comes into play for intensities > 1018Wµm2/cm2
[16]. Since the oscillatory motion and the magnetic field are perpendicular, the elec-
trons experience a force along the laser direction which is [13]
Fz = − ∂
∂z
(
m⟨
v2⟩
2
4ω2
ω2pe
e−2ωpez/c
[
1 + cos 2ωt
2
]
)
. (2.45)
This force is felt by an electron a depth z inside the plasma. Equation (2.45) shows,
that the electrons oscillate at the vacuum plasma boundary with twice the laser fre-
quency (in laser direction). If the magnitude of Fz is big enough, all electrons will
oscillate (non-resonantly) with some electrons having the right phase so that they
gain energy from this oscillation before they are kicked into the overdense plasma.
J x B heating increases if more electrons can be accelerated, i.e. if the laser pene-
trates more into the overdense plasma. The skin depth can be increased either by a
higher focused intensity or by an overdense plasma with lower density. The latter
is reflected by the dependency of Fz on nc/n (see equation (2.45)).
2.3.8 Other Effects
The various absorption mechanisms discussed above indicate, that there is no single
model or approach to laser plasma interaction. Dependencies on the pulse charac-
teristics and hence the plasma parameters lead to these different absorption regimes.
Even more exotic phenomenons have been theoretically predicted, simulated and /
or experimentally revealed, some of which are
• Hole boring due to the light pressure and ponderomotive force at intensities
Iλ2µm > 1018Wµm2/cm2 [17],
• Plasma density profile steepening by the light pressure, accompanied by inward
fast ion generation at intensities Iλ2µm > 1018Wµm2/cm2 [17],
• Collisionless skin effects where electrons from the plasma approach the skin
layer, gain energy from the laser and are reflected back into the plasma [18]
[19].
15
2.4 X-Ray Emission
Fast electrons that are created by either one of the above mentioned heating mech-
anisms (section 2.3.4, 2.3.5, 2.3.6 and 2.3.7) can lead directly (Bremsstrahlung,
K-Shell emission) or indirectly (Plasma Heating) to x-ray emission.
2.4.1 Line Emission
Line emission can be caused by hot electrons or thermal plasma emission:
Hot Electrons A fast electron knocks out an inner shell electron of an atom / ion.
This vacancy of the electronic configuration is filled by an electron of this atom /
ion coming from a higher energetic level. The difference in binding energy is emitted
as a photon of characteristic energy and can be estimated with Moseley’s law:
1
λ= R∞(Z − σ)2 ·
(
1
m2− 1
n2
)
(2.46)
Here, R∞ is the Rydberg constant, Z is the atomic number, σ is a shielding constant
and m and n are the initial and final level.
This process is often restricted to atoms in cold target layers behind the plasma but
can also occur in partially ionized atoms within the plasma.
Plasma Emission In a plasma of given temperature, distinct ionization stages
occur with certain probability. Transitions between different electronic configura-
tions of these ions cause characteristic line radiation.
These and the above mentioned plasma transitions can be broadened or shifted de-
pending on the density and temperature of the plasma. Moreover, the ratio of these
plasma lines allows for an estimate of the coronal / LTE plasma temperature [58].
2.4.2 Continuum Emission
There are two types of continuum radiation:
16
Blackbody Radiation This broadband feature is centered at low energies (< 1
keV) and is caused by the thermal radiation of the plasma. It is not important for
this work because the detection systems are not sensitive to this radiation.
Bremsstrahlung This broadband feature is caused by electrons that are emitting
a photon while decelerating in the vicinity of a third particle, that assures energy
and momentum conservation. Bremsspectra always have a cutoff at high frequen-
cies caused by the maximum energy of the electrons. Hence, by measuring the
Bremsspectrum, one can obtain information about the fast electron temperature.
2.5 Cone-Guiding
A new target geometry that is referred to as cones has recently caused excitement.
It has been studied in experiments [3] [7], simulations [20] and theory [8]. By using
this target, the neutron yield from inertial confinement fusion (ICF) experiments
could be enhanced greatly [3]. Gold cones were inserted into the fusion pellet to help
laser light and suprathermal electrons to be guided to the center of the compressed
pellet. In order to achieve a fusion yield, the center region of the pellet, which is also
known as the spark, has to be be compressed to 1000 times solid density and heated
to 108 Kelvin. The fast ignitor approach to ICF separates the steps of compression
and heating. Long pulse lasers are used for concentric compression and a short pulse
Petawatt laser is used for heating. The short pulse laser can not be used for heating
by itself, because it can not propagate through the long scale length plasma around
the pellet to the spark. Gold cones have the ability to deliver the laser energy close
to the spark. The most important features of cones shall be reviewed:
• Light Guiding: Cones provide a guiding structure for the laser light which
is free of any pre-plasma that could cause deflection and propagation losses of
the laser around the pellet [3].
• Light Focusing: The rising edge of the incoming short pulse ionizes the cone
walls and creates a plasma mirror that allows for focusing the laser down to
higher intensities than reachable with a flat target [7]. This gives rise to a
higher conversion efficiency of laser light into suprathermal electrons.
17
k=ky
k‘=-kykx’=0
“kx=ky=0”kx=0
k=kx+ky k‘=kx-ky
“ky=0” “kx>0”
normal incidence oblique incidence
e-
x
y
z
2.1.1: For an obliquely incident laser field, electrons can be acceleratedalong the target surface because of a non-zero component of kx.
Bz
Bz
e-
e-Ex/y
Ex/y
x
y
z
2.1.2: Electrons that are travel-ling along the cone wall generateself consistent surface-magneticand electrostatic fields, whichare balanced.
Figure 2.1: An obliquely incident laser accelerates e few electrons along the surface (upperfigure). Self-consistent magnetic fields decouple more electrons from the bulk plasma whichenhances the magnetic field even more. Charge separation causes an electrostatic field whichis pushing the electrons back towards the cone wall (bottom figure). Both fields are balancedand allow for a collective steady-state surface current toward the tip of the cone (bottomfigure).
18
• Electron Channelling: Cones allow for surface magnetic fields and electro-
static fields that are such that fast electrons are channelled along the cone
wall towards the tip of the cone [8]. Creating a surface current of electrons is
not a feature which is restricted to cones. It also occurs when a high intensity
laser is incident obliquely on a flat target as depicted in figure 2.1.1. At the
turning point of the light wave, a non-zero component of the wave vector kx
remains, which is pointing along the surface. In a simple picture, a few elec-
trons are accelerated along the surface, causing a surface magnetic field. The
magnetic field pushes the electrons into the vacuum and decouples them from
the bulk plasma. The surface magnetic field drags even more electrons away
from the plasma into the vacuum, which, in turn, increases the magnetic field
self-consistently. As more and more electrons are separated from the plasma,
the significant charge separation gives rise to an electrostatic field that pushes
electrons back toward the plasma as depicted in figure 2.1.2. A collective
steady state of channelled electrons is obtained, because electric and magnetic
fields are in balance. In a cone geometry, the surface current converges toward
the tip of the cone. This makes cones capable of producing a pointing source
of suprathermal electrons that can efficiently heat the fusion pellet [3].
• Cone Angle: Up to know, free standing gold cones from General Atomics are
the only structures that have shown experimental evidence of cone-guiding [3].
They are available with opening angles such as 30⋄, 52⋄ and 60⋄. The influence
of the opening angle of the cone was studied experimentally, were the smallest
angle (30⋄ cone) showed the most electron channelling [7]. Surface current
flow was studied in theory by Nakamura et al. [8]. The smaller the open angle
of the cone, the bigger is the angle of incidence for the laser. The bigger the
angle of incidence, the bigger is the component of the wave vector kx along
the target wall. A big kx facilitates the build-up of the surface current and
the self-consistent fields. In fact, Nakamura et al. found, that for angles of
incidence bigger than a critical angle θ > θc, all electrons can be forced into
the surface motion. The smaller the angle, the less electrons are part of the
surface current and the more electrons are transmitted perpendicular to the
cone wall. Clearly, these electrons do not contribute to guiding. For focused
intensities of 8 × 1018 W/cm2
19
Cones can produce more and hotter electrons and protons than flat targets [7]. This
has been explained by an interplay of the above mentioned features. Three dimen-
sional particle in cell (3D PIC) simulations of a cone geometries with 52⋄ open angle
have been published recently by Y. Sentoku et al. [20]. It was found that the light
is optically guided inside the cone and focused at the tip of the cone. The intensity
increases by more than an order of magnitude in a several micron focal spot. Surface
electron flow that is converging at the tip of the cone is observed as a result of self
consistent surface magnetic fields and electrostatic fields.
These hot electrons, that are converging toward the tip of the cone might be ex-
ploited to build an ultra-bright x-ray source. The idea of this work is to attach
a high Z metal foil2 on the back side of a self-made guiding pyramid (see section
3.3.1). The Kα yield will be measured with a x-ray crystal spectrometer (see section
3.4).
2i.e. titanium
20
Chapter 3
Experimental Setup
This Chapter will introduce the THOR laser facility, the diagnostics for its beam
characterization, the target properties and the experimental setup including the
main diagnostics.
3.1 The THOR Laser Facility
The Texas High intensity Optical Research facility comprises a 35 fs FWHM 801
nm center wavelength 10 Hz repetition rate 0.7 J pulse energy 20 TW laser system.
The beam can be directed to several interaction chambers where focused peak in-
tensity of up to 1019 W/cm2 can be reached.
Not until the technique of chirped pulse amplification (CPA) was invented by Strick-
land and Mourou [21] [22], it seemed to be impossible to reach such high laser powers.
In the mid 1980s it turned out to be a challenge to amplify short pulse laser beams
to focused intensities of more than 1015 W/cm2. This was due to the fact that
even unfocused lasers started to damage the gain material and beam optics while
being amplified. The only way to circumvent this problem was by increasing the
beam diameter to lower the beam intensity. Since optics for huge beam radii cost a
fortune, this has always been a big disadvantage.
Strickland and Mourou [21] [22] came up with a different idea to keep the intensity
of the laser pulse below the damage threshold of the optics while the pulse is still
in the amplification stages of the laser. This technique is CPA.
21
3.1.1 Bandwidth Limit and CPA
The envelope of the electric field E(t) and intensity I(t) of a gaussian shaped laser
pulse can be written as:
E(t) = E0e−2 ln(2)·t2/∆τ2
(3.1)
I(t) = |E(t)|2 = I0e−4 ln(2)·t2/∆τ2
(3.2)
E0 and I0 are the respective amplitudes and ∆τ is the FWHM pulse duration of
the intensity I(t). By means of a Fourier transformation of E(t), one obtains the
spectral pulse shape E(ν) and its square I(ν):
E(ν) = E0e−
π2∆τ2ν2
2 ln 2 (3.3)
I(ν) =∣
∣
∣E(ν)
∣
∣
∣
2= I0e
−π2
∆τ2ν2
ln 2 (3.4)
The bandwidth ∆ν of the laser is defined as the FWHM of the intensity in frequency
space I(ν). It turns out to be
∆ν =2 ln 2
π∆τ(3.5)
From equation 3.5 it follows that the so called bandwidth product ∆ν · ∆τ is a
constant. This constant depends on the shape of the time envelope. If the time
envelope is given by a squared hyperbolic secant (sech2), the bandwidth product
turns out to be even smaller than in the case of a gaussian profile:
(∆ν · ∆τ)gauss =2 ln 2
π≈ 0.441 (3.6)
(∆ν · ∆τ)sech2 =4 · arccosh2
√2
π2≈ 0.315 (3.7)
The meaning of this equation is twofold. On the one hand one can estimate a lower
limit of the pulse length ∆τ by measuring the laser’s spectral range ∆ν. On the
other hand, the bandwidth product requires that a certain pulse length ∆τ has a
minimum spectral bandwidth ∆ν. If the laser is working at its smallest possible
bandwidth product, it is called bandwidth limited.
The spectral width of THOR after amplification is ∆ν = 12.6 THz which can be
22
Stretcher Amplifier Compressor
Input PulseStretched
PulseAmplified
Stretched Pulse Output Pulse
Stretcher Amplifier Compressor
Figure 3.1: The basic idea of chirped pulse amplification is to separate the laser’s spectralcomponents spatially and stretch the pulse in time before amplifying it. This lowers theintensity and avoids damaging of gain media and optics. Finally the beam is re-compressed,e.g. in a grating compressor, which is a reflective optic with a high damage threshold.
compressed to 35 fs pulses. The minimum sech2 pulse duration would be ∆τ = 25
fs.
Strickland and Mourou [21] came up with an elegant way to safely amplify short
laser pulses without exceeding the damage threshold of the optics. Their approach
exploits the broad spectral bandwidth which is inherent to short laser pulses. In-
stead of amplifying every frequency component of the pulse at the same time or
in parallel, the frequencies are stretched out in space and hence amplified serially.
By introducing this so called spectral chirp to the pulse, the pulse intensity can be
reduced by many orders of magnitude. This amplifies the pulse energy by many
orders of magnitude without reaching pulse intensities that could destroy optics.
After the stretched pulse has been amplified, the frequency dependant delay of its
spectral components is reversed again by a pulse compressor.
Basically a stretcher disperses the beam, so that different wavelengths travel dif-
ferent path lengths and are spread out in space to allow for serial amplification.
The stretcher used by Strickland and Mourou consisted of a 1.4 km long fiber that
introduced a linear chirp to the pulse by group velocity dispersion and self phase
modulation [21]. Other CPA setups are based on grating [24] or prism [23] stretcher
/ compressor. The THOR laser uses the grating type in order to achieve higher max-
imum powers. Gratings are reflective optics, which can have much higher damage
thresholds than transmissive optics.
23
3.1.2 Ultrashort Pulse Generation by Mode Locking
A Laser (Light Amplification by Stimulated Emission of Radiation) can be con-
sidered as an optical Maser (Microwave Amplification by Stimulated Emission of
Radiation). In order to build either device, one needs two integral parts: A gain
medium that can provide a population inversion and a structure that can stably
confine the wave or beam. The main conceptual breakthrough that enabled the
invention of lasers is the idea of transversal open resonator made of mirrors can
stably confine gaussian beams as efficiently as a completely enclosed waveguide of a
Maser1.
In a laser cavity of given length l, only those modes can be amplified, that satisfy
the round trip phase condition2 φ(ω) = q × 2π = 2πνL/c [41]. This leads to the
axial mode spacing:
∆νax = νq−1 − νq =c
2nl=
1
TRT(3.8)
Here, q is an integer, n is the refractive index of the cavity, c the speed of light
and TRT is the round trip time of a pulse inside the cavity of length L. Tabletop
TeraWatt class high power laser systems are typically based on an oscillator with
a cavity length of order 1 m. This corresponds to a round trip time of TRT = 6.7
ns and a axial mode spacing ∆νax = 0.15 GHz. For creating ultrashort pulses,
Titanium doped Sapphire crystals (Ti:saph) are the preferred gain medium. Of the
many modes given by the axial mode spacing, only such modes can be amplified,
that fall within the gain bandwidth of the Ti:saph gain medium and are above the
threshold for lasing. An upper limit for the usable gain bandwidth of Ti:saph is
the FHWM of its inhomogeneously broadened atomic transition, ∆νa = 86 THz.
With a mode spacing of ∆νax = 0.15 GHz as given by a cavity of l = 1m, as
many as Nm = 5.7 × 105 modes can be amplified. This situation can be expressed
mathematically:
E(t) =∑
Nm
= En · exp [i (ω0 − 2πm∆νax) t + φm] (3.9)
1In 1957, Shawlow and Townes received the Noble Price for realizing this.2Atomic pulling effects of the gain medium are neglected in this simple model.
24
The cavity is populated by a sum of Nm standing waves of amplitude Em, frequency
ω0 − 2πm∆νax and phase φm. The laser output is given by a small fraction of E(t),
which is leaking through the output coupler of the cavity. Depending on the values
of Em and φm, the laser output intensity as function of time can look quite different.
0 1 2 3 4 5time [a.u.]
0
5
10
15
inte
nsi
ty [a
.u.]
3.2.1: mode locking: φ1 = φ2 = φ3 = φ4.
0 1 2 3 4 5time [a.u.]
1
2
3
4
5
6
7
8
inte
nsi
ty [a
.u.]
3.2.2: multi mode: φ1 6= φ2 6= φ3 6= φ4.
0 1 2 3 4 5time [a.u.]
0
0.2
0.4
0.6
0.8
1
inte
nsi
ty [a
.u.]
3.2.3: continuous wave: φ1(t) 6= φ2(t) 6=φ3(t) 6= φ4(t).
Figure 3.2: Four gaussian oscillator modes of same amplitude E0 are added up. Thegraphs show the oscillator output for different phase relations between the four oscillatormodes.
• In case all Em and φm are arbitrary and time dependant with respect to each
other, the laser operates in Continuous Wave (CW) Output as depicted in
3.2.3.
• In case all Em are equal and the φm are arbitrary but constant in time, the
laser operates in Multi-Mode Output as depicted in 3.2.2.
• In case all Em and φm are equal, the laser operates in Mode-Locked Output as
depicted in 3.2.1. Already a sum of four mode locked cavity modes produce a
25
Kerr mediumlens lens
input output
low intensity
high intensity
aperture
Figure 3.3: Ultrashort pulses, exploiting the complete bandwidth of the Ti:saph gainmedium can be achieved by passive mode-locking via optical Kerr lensing. Self focusinginside the Kerr medium favors modes to be build up in phase whereas low intensity CWmodes are suppressed.
pulse train of high contrast. The repetition rate of the pulse train is given by
the inverse round trip time, which is also the longitudinal mode spacing.
In order to increase the contrast and reduce the pulse duration, many modes have
to be locked inside the cavity. Using the bandwidth product as derived in section
3.1.1, we can estimate the shortest pulse length possible for a Ti:saph gain medium
with the upper limit of the gain bandwidth ∆νmax = ∆νa = 86 THz to be ∆τ = 5
fs3.
The above picture of standing waves bouncing back and forth inside the cavity
suggests a practical approach to forcing these modes to lock: insert a fast shutter
into the cavity. In case the shutter is open, the cavity loss is low and the laser is
above the threshold necessary for lasing. In case the shutter is closed, the cavity loss
is high and the laser is below threshold. One realization of active mode-locking via
loss modulation is by introducing an acousto-optic modulator [42]. This approach
can not exploit the whole bandwidth ∆νa of Ti:saph and yields only pulses of lengths
> 3 ps. The shortest pulses possible can be obtained by techniques of passive mode-
locking such as optical Kerr lensing. The nonlinear refractive index of the Kerr
medium inside the resonator causes intracavity beams to be focused. The higher
the intensity of the intracavity beam, the tighter is its focus after having passed
the Kerr medium. An intracavity aperture after the Kerr medium transfers this
intensity dependant beam size modulation into a subsequent modulation of the
3This is only a very rough estimation, since effects such as gain narrowing and the type ofmode-locking have been neglected.
26
cavity propagation loss. For a properly aligned cavity, a smaller loss is achieved for
the highly intense circulating pulse. The competing low intensity CW-modes are
attenuated and suppressed. Figure 3.3 shows a schematic drawing of optical Kerr
lensing.
3.1.3 THOR Laser Layout
THOR is a tabletop high intensity 20 TeraWatt CPA laser system capable of deliver-
ing 35 fs pulses with energy 0.7 J at 10 Hz (see fig. 3.4). It is based on a commercial
Femtolaser Femtosource Scientific s20 oscillator which is pumped by a solid state
Spectra-Physics Millennia Vs J laser at 532nm. The Ti:saph gain material inside
the oscillator allows to amplify a broad range of wavelength modes around 801 nm.
The used bandwidth is 12.6 THz. The cavity length of 2 m corresponds to an axial
mode spacing of 75 MHz and hence 1.7 × 105 modes can be passively mode-locked
via optical Kerr lensing. The mode locked oscillator emits 1 nJ 20 fs laser pulses
at a repetition rate of 75 Mhz. A Pockel’s cell slicer reduces the repetition rate
to 10 Hz before the pulses enter the stretcher. Inside the stretcher the pulse be-
comes spectrally chirped with the low frequency parts of the spectrum travelling
ahead, followed by gradually increasing frequency components. The stretched 600
ps FWHM pulse is now amplified in three consecutive amplification stages:
I. The first amplification stage consists of 20 passes inside a regenerative amplifier
stage (regen). Its gain medium consists of a Ti:saph crystal which is pumped
by a Quantel Big Sky q-switched Nd:YAG laser frequency doubled at 532 nm.
The amplified pulse is switched out of the cavity by a high speed Pockel’s cell
when the gain starts to saturate. The pulse energy has been increased to 3.5
mJ of energy, which corresponds to six orders of magnitude gain.
II. To allow for amplification in the following 4-pass amplifier, the beam diameter
is increased from 2 mm to about 4 mm. The 4-pass amplifier is also based on
a Ti:saph medium, which is pumped by the same Quantel Big Sky laser as the
regen. The seed pulses leave this part with about 20mJ of stored energy. A
spatial filter setup cleans the spatial beam mode and increases the beam size
to about 15mm in diameter.
III. In the final 5-pass Ti:saph amplification stage, two Spectra-Physics Pro Series
27
frequency doubled q-switched Nd:YAG lasers pump a 20 mm crystal from
both sides. By mistiming the pump lasers, energies of 10 mJ to 1.2 J can be
achieved.
Depending on what the experiment demands, there are several options to further
modify and guide the beam to the interaction chamber.
• In order to obtain maximum focused peak intensity, the fully amplified pulses
can be directed in a single grating vacuum compressor, which yields pulses of
700mJ and 35fs, which can be focused to intensities of > 1019 Wcm2 .
• In case the experiment requires the longest pulse possible the fully amplified
pulses of 1.2J and 600ps can be guided directly to the interaction chamber
without compression.
• In case the experiment requires a second pulse of certain pulse length and
energy to arrive delayed or coincident with the vacuum compressor pulse at
the interaction chamber, the air compressor setup can be used in parallel to
the main pulse.
• In case the experiment requires a frequency doubled pulse of 400nm and high
contrast ratio, a small amount of the fully compressed beam can be split after
the vacuum compressor to be frequency doubled in a KDP crystal on air.
Beam splitters and multiple shutter allow any combination of these four options.
The spatial beam profile is cleaned again in a spatial filter and the beam diameter is
telescoped to 7 cm to allow for save beam compression inside the 40 cm gold coated
single grating compressor. Because of gain narrowing (Ti:saph crystals) and self-
phase modulation and group velocity dispersion (transmissive optics), the pulses can
not be recompressed to its original 20 fs. This is true even though a custom made
fused silica fiber after the stretcher compensates for up to 5th order nonlinear effects
throughout the whole system. To minimize the nonlinear effects of the laser pulse
in air, the whole compressor setup has to stay in a 10−5 Torr vacuum chamber.
Consequently, 7 cm diameter, 700 mJ, 35 fs pulses are produced at 10Hz, which
makes THOR a 20 TW laser facility.
28
1.4J @ 10Hz YAG1.4J @ 10Hz YAG
1.4J @ 10Hz YAG1.4J @ 10Hz YAG
0.2J
@ 10Hz
YA G
0.2J
@ 10Hz
YA G
5W
Ar
Ion
5
W A
r Io
n
20
fs T
i:s
ap
ph
ire
os
cil
lato
r
20
fs T
i:s
ap
ph
ire
os
cil
lato
r
R egenerative amplifier (20 pas s es )R egenerative amplifier (20 pas s es )
4-pas s amplifier4-pas s amplifier
5-pas s amplifier5-pas s amplifier
P uls e s tretc herP uls e s tretc her
V acuumP uls e compres s or
V acuumP uls e compres s or
35fs , 700mJto
T arget
35fs , 700mJto
T arget
600ps ,1.2J to T arget
600ps ,1.2J to T arget
2‘‘ K DP2‘‘ K DP
120fs , 12mJto
T arget
120fs , 12mJto
T arget
Air P uls ec ompres s or
Air P uls ec ompres s or
10ps - 42fs , 0.5J – 0.1J to T arget
10ps - 42fs , 0.5J – 0.1J to T arget
1.4J @ 10Hz YAG1.4J @ 10Hz YAG
1.4J @ 10Hz YAG1.4J @ 10Hz YAG
0.2J
@ 10Hz
YA G
0.2J
@ 10Hz
YA G
5W
Ar
Ion
5
W A
r Io
n
20
fs T
i:s
ap
ph
ire
os
cil
lato
r
20
fs T
i:s
ap
ph
ire
os
cil
lato
r
R egenerative amplifier (20 pas s es )R egenerative amplifier (20 pas s es )
4-pas s amplifier4-pas s amplifier
5-pas s amplifier5-pas s amplifier
P uls e s tretc herP uls e s tretc her
V acuumP uls e compres s or
V acuumP uls e compres s or
35fs , 700mJto
T arget
35fs , 700mJto
T arget
600ps ,1.2J to T arget
600ps ,1.2J to T arget
2‘‘ K DP2‘‘ K DP
120fs , 12mJto
T arget
120fs , 12mJto
T arget
Air P uls ec ompres s or
Air P uls ec ompres s or
10ps - 42fs , 0.5J – 0.1J to T arget
10ps - 42fs , 0.5J – 0.1J to T arget
Figure 3.4: Schematic structure of the THOR laser: The oscillator pulse is stretched,amplified an then compressed again to guarantee ultra high output intensities (modifieddrawing by courtesy of Dr. Todd Ditmire).
3.2 Autocorrelation and Frequency Doubling
The physics of laser-target interaction strongly depends on the pulse properties of
the laser. 2nd and 3rd order correlation measurements can be used to determine the
pulse duration and the contrast ratio of main pulse to prepulse intensity. Ultrafast
photodiodes or optical streak cameras can not be used to characterize fs pulses since
their time resolution is too low. Instead, 2nd or 3rd order correlation measurements
are used to achieve fs time resolution and the desired dynamic range.
3.2.1 Nonlinear Wave Mixing and Phase Matching
These correlation techniques are based on nonlinear mixing of short optical pulses
in optical uniaxial birefringent crystals.
Sum Frequency Generation (SFG) is an effect caused by the nonlinear contri-
bution of the index of refraction. Every medium has a nonlinear index of refraction,
which has to be taken into account if the light intensity becomes high enough. SFG
29
c-plane
c-axis
propagation
axis
non-collinear
ordinary beams
ω1=ω2=ω
extraordinary
beam ω3 =2ω
θ
ω1ω3ω2
ω
ω
ω
3.5.1: Second harmonic generation by non-collinear type-Iphase-matching.
index
ellipsoid (2ω
c-axis
c-plane
n(ω )=n(2 ω )
index
ellipsoid ( ) ω
θ
)
3.5.2: Refractive index ellipsoid.
Figure 3.5: Sum frequency generation and phase-matching in nonlinear birefringent crys-tals is shown for the example of non-collinear type-I phase-matched second harmonic gen-eration. This scheme is used for background-free 2nd order autocorrelation measurements.
can be expressed mathematically by ω1 + ω2 = ω3.
• Second harmonic generation (SHG) is a special case of SFG, following the
equation ω + ω = 2ω.
• Third harmonic generation (THG) is a special case of SFG, following the
equation ω + 2ω = 3ω.
In order to maximize SHG in a medium, the frequencies ωi have to travel equally
fast through the crystal. If the frequencies don’t see the same index of refraction,
two or all of the frequencies get out of phase and back conversion occurs.
Phase matching in birefringent crystals is a common technique used to maximize
SFG. A birefringent medium is a medium that has a non isotropic refractive index. If
only two of the three linearly independent components of the refractive index tensor
are different, the medium is called optically uniaxial. In this case, the refractive
index becomes an ellipsoid with one symmetry axis, its c-axis.
• Light which is propagating through the crystal and is polarized along the
c-axis, is called ordinary light. It sees the ordinary index of refraction n0.
30
• Light which is propagating through the crystal and is polarized perpendicular
to the c-axis is called extraordinary light. It sees the extraordinary index of
refraction ne.
There is Type-I and Type-II as well as collinear and non-collinear phase matching
and any combination.
• In Type-I phase matching, ω1 and ω2 are propagating as ordinary beams and
ω3 is propagating as extraordinary beam.
• In Type-II phase matching, ω1 is propagating as ordinary beam and ω2 and
ω3 are propagating as extraordinary beams.
• Collinear and non-collinear phase matching refers to the propagation direction
of ω1 and ω2.
If the desired type of SFG and phase matching is given and the index ellipsoids
for the ωi are known, one can calculate the condition for which phase-matching is
obtained. This condition will be incident angles for ωi with respect to the c-axis of
the crystal.
3.2.2 2nd Order Autocorrelation
The 2nd order correlation function
G2ω(τ) ∼∞∫
−∞
Iω(τ)Iω(t − τ)dt (3.10)
is measured with a non-collinear second harmonic generation scheme using
a type-I phase-matched KDP crystal as depicted in figure 3.5. Two beams of 800
nm light can be delayed with respect to each other. They are overlapped in space
on a thin KDP crytal, having slightly different angles of incidence. With the non-
collinearity of the two beams, different points in time are projected onto different
positions on the crystal. The longer the pulses are or the smaller the angle between
the two pulses is, the bigger the pulse overlap in space turns out to be on the
crystal. The angles of incidence are a design parameter of the autocorrelator setup
and they are chosen for the desired range of pulse durations to be investigated. By
31
100 200 300 400 500 600time [pixel]
inte
nsi
ty [
a.u
]
FWHM =(38 2)fs+-
3.6.1: Lineout taken from the image below.
0 100 200 300 400 500 6000
50
100
150
200
space [pixel]
spa
ce [
pix
el]
3.6.2: Image taken with an 2nd order autocorrelator.
Figure 3.6: Image and lineout of a background-free 2nd order autocorrelation of a fullyvacuum-compressed 800 nm pulse. The x-axis can be calibrated to represent time space.Averaging over all lines yields a pulse length of 38 ± 2 fs
introducing a relative delay between the two pulses, spatial and temporal overlap
which are required for SHG, are met on a different position on the crystal. If one
images the back side of the crystal on a CCD camera, a line of certain width is
observed from where SHG occurs. This linewidth in space can be related to a line
width in time. The linewidth in time essentially is the FWHM pulse duration of the
800 nm beam. The non-collinear setup is also chosen because it allows for 800 nm
background free measurements of the 400 nm second harmonic. Figure 3.6 shows
400 nm light as imaged from the back side of the KDP when two pulses of 800
nm light are overlapped in space and time. The line width in pixel corresponds
to a pulse duration of 38 ± 2 fs. Obviously, the pulse is only slightly longer than
the optimum of 35 fs which can be realized for a perfect alignment of the vacuum
compressor.
3.2.3 3rd Order Autocorrelation
A 3rd order correlation measurement is used to identify prepulses and to determine
the contrast ratio of main pulse to prepulse intensity. 3rd order correlation schemes
32
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
0.00001
0.0001
0.001
0.01
0.1
1
time [ps]
no
rma
lize
d in
ten
sity
prepulsepostpulse
Figure 3.7: Result of a 3rd order AC. Black points mark pre/post pulses. Blue pointsmark the noise level. Red dots mark the main pulse. From the red line one can estimatethe time resolution of the measurement.
are superior to 2nd order schemes because of their higher dynamic range and the fact
that they can distinguish between pre- and postpulses. A thin KDP crystal is used
in a collinear type-I phase matching configuration to produce 400 nm light. The
remaining 800 nm and the generated 400 nm are separated by a dichroic mirror. The
beam lines for 400 nm and 800 nm light clean the pulse from parasitic frequencies by
using high reflectors. The red pulse can be delayed with respect to the blue pulse.
Both frequencies are superposed again in a second dichroic mirror and collinearly
overlapped on a thin nonlinear BBO crystal. This crystal is used in a collinear type-I
phase-matched third harmonic generation configuration. If the ω and 2ω light fulfill
temporal overlap in the BBO crystal, violet 3ω photons are created. The third
harmonic is cleaned from parasitic ω and 2ω by using an interference filter, high
reflectors and a prism. Finally, the third harmonic is detected in a photomultiplier
tube. The third harmonic signal is proportional to the 3rd order correlation function:
I3ω(τ) ∼ G3ω(τ) ∼∞∫
−∞
Iω(τ)I2ω(t − τ)dt (3.11)
With I2ω(t) ∼ [Iω(t)]2 [27], the signal to noise ratio of the blue pulse is the square
of the signal to noise ratio of the red pulse under investigation. Assuming the
bandwidth of the KDP is big enough to double every frequency component of the red
pulse, the blue pulse will be shortened by 1/√
2. One ends up with a rather clean blue
probe. The pulse length of the blue pulse determines the temporal resolution with
33
which features of the red pulse can be investigated. Furthermore, if one considers
only those features of the red pulse that are much longer than the FWHM of the
blue probe, the above equation 3.11 reduces to
G3ω(τ) ∼ Iω(τ)
∞∫
−∞
I2ω(t − τ)dt (3.12)
leading to
G3ω(τ) ∼ I3ω(τ) ∼ Iω(τ) (3.13)
Figure 3.7 shows a 3rd order correlation trace G3ω of an amplified fully compressed
beam, revealing several pre- and postpulses. Around 21ps before the main pulse, the
first measurable prepulse occurs. the detected Prepulses are only thousand times
smaller than the main pulse. If one is shooting at 1019 Wcm2 , a contrast ratio of 10−3
still causes a preplasma, that severely changes the absorption mechanism for the
main pulse from vacuum heating to resonance absorption. In fact, the threshold
intensity for creating a plasma is as low as 1012 Wcm2 for a pulse length of order 100 fs
[25]. Some experiments, such as the spheres runs presented in Chapter 6 require a
much better pulse contrast. For these experiments, frequency doubling of the 800nm
pulses is used to improve the contrast ratio.
3.2.4 Frequency Doubling
A two inch diameter, 1 mm thick KDP crystal is used in a collinear type-I phase
matched configuration, to frequency double 800 nm pulses with an energy efficiency
of 10-15 %. The nonlinear behavior of the frequency doubling process (I2ω ∝ [Iω]2
[27]) squares the contrast of the initial pulse. A contrast ratio of < 10−3 for the
800 nm pulse is transformed into a contrast ratio of < 10−6 for the 400 nm pulse.
The unconverted 800 nm light is filtered out by a series of three dielectric / dichroic
mirrors optimized for 400 nm reflection. This reduces the intensity of the 800 nm
main pulse by a factor of < 10−6 and the intensity of the 800 nm prepulse by a
factor of < 10−9. Shooting with fully amplified and compressed beams corresponds
to focused intensities of 1019 Wcm2 . Frequency doubling as described leaves 400 nm
prepulses of 1013 Wcm2 . This would still be above the ionization threshold which is
34
1012 Wcm2 [25]. By not fully amplifying the beam in the 5-pass, the focused intensity
can be limited to 1017 Wcm2 . This way, a significant preplasma formation can be
avoided. Nevertheless, the prepulse intensity is high enough to alter the interaction
of the main pulse with a solid target. Effects of nonionizing prepulses as low as
108 − 109 Wcm2 in high-intensity laser-solid interactions have been reported. Prepulse
heating and vaporization of the target at these intensities can lead to a preformed
plasma once the vapor is ionized by the rising edge of the main pulse [28].
The bandwidth of the KDP crystal is broad enough to double every color of the
∼ 38 fs 800nm pulse. In theory, the 400nm pulse should be shortened by a factor of
1/√
2. Since perfect phase matching can only be obtained for the center frequency
of the red pulse, the different colors get slowly out of phase along the 1 mm thick
KDP crystal. Essentially this causes the 400 nm pulse to be temporarily broadened
up to 100-150 fs.
The experiments on flat targets covered with spheres which are presented in section
6 required a clean pulse. They were performed with 400nm pulses from the doubling
scheme described here. The pulse contrast turned out to be good enough.
3.3 Target Properties And Preparation
Two completely different target concepts were developed and investigated. The
first type of targets consisted of guiding structures etched into silicon wafers with
titanium foils attached to the back side. The second type of targets consisted of flat
silicon wafers with monolayers of wavelength scale spheres deposited on the front
side.
3.3.1 Titanium Targets, Pyramids and Wedges
Idea Experiments [3] [7], simulations [20] and theory [8] have shown that guiding
symmetries such as the free standing gold cones produced by General Atomics have
the ability to increase and channel suprathermal electrons when irradiated with
ultra-short ultra-intense laser light as discussed in section 2.5. Manufacturing free
standing gold cones is complicated, expensive and intellectual property of General
Atomics. It is very desirable, to be able to produce guiding structures that are based
on silicon. Silicon can be processed fast, cheaply and with highest accuracy. The
techniques are intensively used in the field of semiconductor industry. This is why
35
<111>
<100>
54.7°70.5°
35.3°54.7°
Wedge
1000mm
100mm
1mm
Pyramid
500mm
1mm
500mm Silicon Waver
5mm
Figure 3.8: Micro-shaped silicon pyramid and wedge targets. Details are given in the text(Images courtesy of Byoung-Ick Cho, University of Texas at Austin).
36
Flat Titanium Foil Wedge P-polarizedWedge S-polarizedPyramid
Figure 3.9: Comparison of possible titanium target geometries. f.l.t.r.: stand alone ti-tanium foil (11 µm or 25 µm), pyramid with titanium foil (11 µm or 25 µm), wedge ins-polarization with 11 µm titanium foil and wedge in p-polarization with 11 µm titaniumfoil.
silicon pyramids have been developed and produced for this work by a collaborator,
Byoung-Ick Cho, University of Texas at Austin.
Production 500 µm thick silicon wafers serve as the substrate into which pyra-
mid and wedge-shaped dips are etched by means of anisotropic chemical etching.
Since this process is based on MEMS4 technologies that are intensively used in the
semiconductor industry, these targets can be produced cheap and with extremely
high accuracy.
Figure 3.8 shows a 4” wafer holding about 100 pyramid- / wedge-shaped dips and
several closeup pictures taken with an SEM microscope. Since the KOH etching rate
strongly depends on the lattice direction, etching alway occurs along 〈111〉 lattice
planes. The surface of the used silicon wafer is parallel to the 〈100〉 lattice planes.
Hence, depending on whether a square or rectangular mask is used, pyramid or
wedge-shaped tips can be obtained (see figure 3.8). Accurate control of the etching
parameters allows for pyramid tips of size less then 1µm2 with a silicon layer of only
several µm left underneath. Finally, a layer of either 11 µm or 25 µm thick titanium
foil is attached to the silicon substrate from the back side. Adhesion is accomplished
by a ∼ 1µm thick layer of super glue. Because of the diamond lattice structure of
silicon, the open angle of both pyramid and wedge targets is a fixed to 70.5⋄ (see
figure 3.8).
4Micro Electro Mechanical System
37
Goal Figure 3.9 shows the different target geometries that are possible. All of
them involve titanium foils, either stand alone or attached to the micro-shaped
silicon pyramid or wedge. The main interest is to proof experimentally whether or
not the micro-shaped targets support fast electron channelling and / or enhancement
as observed from free standing gold cones [3] [7].
• If electron enhancement occurs around the K-alpha cross section peak of ti-
tanium, a ultra-bright K-alpha source should be accomplishable. Therefore, a
von-Hamos crystal spectrometer will be employed from the back side (section
3.4).
• If electron channelling occurs, smaller K-alpha source sizes should be accom-
plishable with the guiding geometries than with flat foils. Therefore a 1D
spatial 1D spectral imaging spectrometer will be employed from the back side.
• It will be particularly interesting to find out if any result depends on the
polarization direction of the wedge as indicated in figure 3.9.
• Information about suprathermal energies can also be obtained via the detec-
tion of the bremsspectrum. Therefore an array of filtered scintillator / photo-
multiplier detectors (section 3.6) will be employed from a fixed direction.
All experimental results will be presented and discussed in Chapter 5.
3.3.2 Silicon and Sphere Targets
Idea Recent experiments suggest an enhancement of the laser absorption efficiency
and hot electron temperature when 1 µm droplets are irradiated with pulses of 35
fs, 820 nm, 120 mJ and 7 × 1017 W/cm2 peak intensity [9]. In another experi-
ment, ethanol microdroplets were irradiated with 100 fs, 1016 W/cm2 focused laser
pulses [6]. Protons with kinetic energies of up to 20 keV were produced in an
anisotropic microexplosion. Numerical modelling showed a strong spatial variation
of the electromagnetic field over the surface of the microdroplet causing local field
enhancement.
Searching for ways to exploit the properties of wavelength scale particles for hot
electron enhancement and ultra-bright x-ray sources, monolayers of spheres were
deposited on flat silicon wafer.
38
Preparation Mono-disperse spheres solutions from Duke Scientific with diameter
variations of 3% to 5% were diluted in Ethanol. The ratio of ethanol to spheres
solution was 1:0.02, 1:0.12, 1:0.18, 1:0.24 and 1:0.5 for 0.1 µm, 0.26 µm, 0.36 µm,
0.5 µm and 2.9 µm sphere diameter respectively. Large two dimensional arrays on
polished and unpolished silicon wafers were obtained by placing one drop of the
diluted solution on the wafer. The wafer was placed in a box with an incline of
∼ 10⋄ that was covered with a lid immediately after the deposition to prevent air
disturbance [30].
Characterization A careful characterization of the targets with an electron mi-
croscope is necessary to study the quality of the obtained spheres layer. It was
found, that the small sphere sizes 0.1 µm, 0.26 µm and 0.36 µm gave a several cm2
large, contiguous monolayer of spheres. This is why x- and y-alignment is not crucial
for these sphere sizes (see section 3.5.2). A more detailed analysis of the coverage
is underway to reveal its influence on the laser-induced x-ray yields.
Sphere sizes 0.5 µm and 2.9 µm gave patches of hexagonal close-packed monolayers.
The average patch size exceeds the focal spot size of the laser but uncovered target
regions can be found next to patches of monolayers. This is why x- and y- alignment
is crucial for these sphere sizes (see section 3.5.2).
3.4 High Resolution Bent Crystal X-Ray Spectrometer
A spectrometer using a curved crystal is the preferred choice for the wavelength
range λ < 1 nm = 10 A . A very common geometry employing a cylindrically curved
crystal is depicted in figure 3.10. Other types of spectrometers such as grazing
incidence spectrometers can not access λ < 10 A because of their low resolution and
reflectivity.
3.4.1 Introduction to Bragg Reflection
The reflectivity of a crystal increases with the number of lattice planes that are
involved in constructive interference of the scattered x-rays. A reflection builds up
if
2d sinα = nλ (3.14)
39
R=10cm
y
x
z
α
α
2Θ=33°
l=5cm w=6cm
x-ray source /plasma
size s of order 100µm
spatial linewidth
∆l=2R∆α/sin α2
line focus
Figure 3.10: Curved crystal spectrometer in Von Hamos geometry. X-Rays originatingfrom a point along the crystal axis (plasma) are spectrally dispersed and focused along thecrystal axis.
is fulfilled. The bragg angle α is measured between the x-rays and the reflecting
lattice planes as shown in figure 3.10. d is the lattice spacing and n is the order of
reflection. A crystal of given material has many different d’s. A particular d = dhkl
can be chosen by using the crystal in a distinct orientation (hkl). Changing the
order of reflection is accomplished by changing the orientation of the crystal: For
example (hkl) → (hk2l) causes d → d/2, which corresponds to n → 2n5. This is
why n is essentially redundant in equation 3.14.
Depending on the wavelength range of interest, one has to choose a suitable crystal
material, with a suitable lattice spacing dhkl. Since sinα ≤ 1, equation 3.14 requires
2d to fulfill 2d ≥ λ. However, not all angles α turn out to be useful for bragg
reflection:
• The reflectivity of the crystal increases with α. X-rays penetrate only a certain
distance into the crystal (6.2 µm for a Si Kα in PET [43]). For a large α, the
given penetration depth involves more lattice planes than for a small α. This
suggests a choice of a large bragg angle α.
• The linear dispersion of the crystal increases with decreasing α. From equation
5This simple relation is at least true for crystal systems with reciprocal lattice vectors that areorthogonal [31].
40
3.14 and x = 2R/ tan α, we obtain the linear dispersion
∆x = ∆λ4dR
λ2√
1 − λ2/4d2(3.15)
Here, R is the radius of a cylindrically curved crystal and x is the distance
between source and image if measured along the crystal axis (see figure 3.10).
Equation 3.15 shows, that the linear dispersion ∆x becomes big for small λ.
Since small λ fulfill the bragg condition for small α, this suggests a choice of
a small bragg angle α.
3.4.2 PET as Crystal Material
For the intended experiments, a crystal had to be found, That combines good per-
formance for two fairly different wavelength intervals:
• Silicon, wavelength range 6.6A to 7.2A (Heα, Li-like satellites, Kα).
• Titanium, wavelength range 2.2A to 2.8A (Kα, Kβ, Heα, Heβ and other plasma
lines).
A crystal made of PET (pentaerythritol, C [CH2OH4]) turns out to fulfill these
needs:
• The 2d spacing in (002) orientation is 2d = 8.742A. Therefore, the bragg
angles for silicon spectroscopy would range from α6.6A = 49⋄ ≤ α ≤ 55⋄ =
α7.2A
• The 2d spacing in (004) orientation is 2d = 128.742A = 4.371A. Therefore, the
bragg angles for titanium spectroscopy would range from α2.2A = 30⋄ ≤ α ≤40⋄ = α2.8A
Since PET consists of low Z materials, it produces a small amount of soft fluorescence
light when irradiated by x-rays [32]. Fluorescence is one of the major reasons for
background darkening of the x-ray film.
3.4.3 Properties of Cylindrical PET
Curved crystal vs flat crystal Curved crystals give much higher reflectivities
than flat crystals, because they are a focusing x-ray optic. The ratio is given by
41
[34]:
IHamos
Iflat= Θ sinα
√
RB
stan α (3.16a)
RB =
(
2d
λ
)2
R (3.16b)
Here, 2Θ is the open angle of the crystal, s is the source size and R is the crystal
radius as depicted in figure 3.10. With the numbers given in figure 3.10, the bent
PET has a 17 times higher reflectivity for Ti Kα and a 22 times higher reflectivity
for Si Kα. The ratio was calculated by using the measured source size of s = 100µm
(see section 5.6). This value was obtained for titanium targets at 1019 W/cm2 and
certainly overestimates the source size of silicon targets at 1017 W/cm2.
Spectral Resolution However, curving a crystal reduces its resolution λ∆λ . For
a perfect crystal, the resolution is limited by
I. Finite source size: This influence can be estimated with λ∆λ & R2/∆s2 = 108
[35]. It is negligible.
II. Offset of the reflected x-rays caused by a finite penetration depth into the
crystal: This influence can be estimated with equation 3.15 and turns out to
be of the order 106. This can be ignored.
III. Imaging faults because of the curvature of the crystal. This occurs even for a
perfectly cylindrical curved crystal. An estimation for this can also be found
in [35]. This effect is negligible.
IV. Width of the reflection curve ∆α: This turns out to be the limiting factor for
the resolution.
The width of the reflection curve has been calculated for a very similar PET crystal
in (002) orientation and Al Heα (7.75A), using the dynamic theory of x-ray diffrac-
tion. This calculation gives a FWHM of ∆αtheory = 0.39 mrad [25].
Using the derivation of equation 3.14, λ∆λ = tan α
∆α , one can calculate the resolution
for Si K-shell radiation. The resolution for Si Heα (6.648A) and Si Kα (7.127A)
is ∼ 3000 and ∼ 3600 respectively. For Al Heα (7.75A), the resolution would be
42
∼ 5000.
These calculated resolutions are not quite reachable in reality. The process of bend-
ing a flat PET crystal to obtain a cylindrical shape always involves lattice distur-
bance, such as creation of mosaic features [36]. In order to determine the actual
resolution λ∆λ of the crystal, an equivalent PET crystal has been characterized by
means of x-ray projection topography at the University of Jena in Germany [25][37].
The experimentally determined FWHM of the (002) reflection curve using Al Heα
(7.75A) turns out to be ∆αexp = 0.9mrad. This corresponds to a resolution of 2100
for Al Heα (7.75A) and 1600-1300 for Si K-shell radiation (6.6A to 7.2A).
Neither a calculation nor a measurement of the width of the reflection curve is avail-
able for titanium K-shell spectroscopy in (004) order of reflection (2.2A to 2.8A).
Chapter 5 and 6 will present the resolution that could be experimentally achieved
in the work presented here.
Focusing Quality The limited resolution of the curved crystal also limits its
ability to focus different colors along a tight line. The spatial width of the line focus
in the detection plane can be estimated by ∆l = 2R∆α/ sin2 α (see figure 3.10 [25]).
For ∆αexp = 0.9 mrad, ∆l turns out to be 270µm to 310µm. This effect is bigger
than the x-ray source size s ∼ 100 µm (see section 5.6). Therefore, this cylindrically
curved PET crystal can not be used for imaging purposes or to infer the x-ray source
size. The predicted spatial linewidth indeed could be verified experimentally (see
Chapter 6).
Integrated Reflectivity The determination of the integrated reflectivity com-
pletes the characterization of the PET. The integrated reflectivity is defined as:
Rint =1
I0
∫
∆α
I(α)dα (3.17)
where one has to integrate over the width of the reflection curve ∆α(λ) at every
wavelength.
The measurement has been carried out at the Synchrotron Facility BESSY in Berlin
by Foerster et al. [38] for an equivalent PET crystal with slightly smaller open angle
2Θ = 28⋄ in (002) reflection. Adding ∼ 20 % accounts for the PET crystal with a
slightly bigger open angle of 2Θ = 33⋄ which was used for this work. For Si K-shell
43
spectroscopy (6.6-7.2A) one obtains Rint = 0.3 − 0.35 × 10−3 rad.
No measurement is available for titanium spectroscopy in (004) reflection. However
the integrated reflectivity has been calculated for Ti Kα (2.75A) assuming a perfectly
bent PET crystal [44]. The obtained value Rint = 0.9 × 10−5 rad can be used as a
rough approximation.
Figure 4.3 in Chapter 4 provides a plot of these integrated reflectivities. They will
be used to obtain the absolute number of photons that are emitted from the source
per laser shot.
3.4.4 Design and Alignment of Spectrometer
The PET crystal was provided by Foerster et. al. from the X-Ray Optics Group
of the University of Jena in Germany. A crystal housing suitable for the intended
experiment on titanium spectroscopy had to be designed and manufactured.
The design of the spectrometer was influenced by the following points
I. The crystal has to be coupled to a scientific / industrial x-ray film. The x-ray
spectrum along the line focus of the spectrometer is rather long (2 − 7 cm x
1mm) because of the big linear dispersion of the crystal. X-ray CCD cameras
of this size are very expensive. Moreover, their resolution (∼ 25 µm) is worse
than the resolution of most x-ray films (order 5µm [39]). Since an x-ray film
is also sensitive to visible light, a visible light shielding for the x-ray film has
to be included in the spectrometer concept.
II. The alignment of the crystal with respect to the source and with respect to
the x-ray film is very critical. Only those x-rays are imaged perfectly, that
originate from a point along the axis of the cylindrical crystal [36]. Hence,
this axis has to intersect with the center of the x-ray source. Overlap has
to be accurate within < 0.5 mm to obtain satisfying imaging of the x-rays.
The result of poor and perfect alignment will be presented in Chapter 5 and
6. Moreover, a bad alignment also makes an accurate absolute wavelength
calibration of the spectrum impossible (see section 4.4.1). Both require a
spectrometer concept with accurate (built in) alignment aid.
III. K-shell line radiation is isotropic. But only those x-rays are reflected by the
crystal, that fulfill the Bragg-condition (3.14). Hence the crystal has to be
44
target shooter
spectrometer
alignment HeNe
target
PET crystal
x-rays
film holder
window covered
with Al coated mylar
Figure 3.11: Spectrometer housing. Details are given in the text.
placed in such a way with respect to the target, that the Bragg-condition can
be fulfilled for the desired wavelength range. This requires a spectrometer
concept with versatile mounting capabilities.
The best solution is a closed and light-tight crystal housing with a fixed position of
the crystal and the x-ray film. The housing guarantees, that the crystal and the film
are always aligned correctly with respect to each other. X-rays penetrate into the
box through a visible light-tight aluminum coated mylar window (see figure 3.11).
Two alignment holes are built into the spectrometer where the crystal axis intersects
with the box. If these holes clear an alignment HeNe, that is overlapped with the
focal spot on the target (according to II), the crystal axis is aligned. Figure 3.11
shows the spectrometer including the crystal and the alignment HeNe. The spec-
trometer housing is mounted on a magnetic base and a five axis aligner (NewFocus
9081). The spectrometer position is tweaked with the five-axis aligner to maximize
the thruput of the HeNe laser (see figure 3.11). The distance of the spectrometer to
the target has to fulfill the criterion given in III. The distance can be tweaked with
the five axis aligner. This concludes the spectrometer alignment. The alignment
has to be done only once and not for every run, because the magnetic base allows
to recover a previous spectrometer position very precisely. In section 3.5.1 and 3.5.2
this position will be given (according to II and III).
The entrance window was was closed with 2 foils of 12 µm mylar coated with 90-100
nm aluminum on either side. A plot of the transmission curve of that filter is given
45
in figure 4.3 as part of the x-ray film normalization. The spectrometer housing has
a lid on its back side so that precut pieces of x-ray film can be fed into the film
holder. For all experiments on titanium targets (see Chapter 5), the background
darkening of the film was a big issue. This is caused by fluorescence of the crystal
and all sorts of secondary radiation that interacts with the x-ray film. Therefore,
the inner wall of the spectrometer box had been equipped with a 3 mm led shielding.
In particular, K-shell radiation was prevented from hitting the film directly. The
flourescence was reduced by covering the film with an additional 30µm foil of mylar
inside the spectrometer.
Besides the newly designed spectrometer housing which is made for titanium spec-
troscopy (see figure 3.11), another box of similar design was used6. It is designed
for silicon spectroscopy and it was used in the second set of experiments presented
in Chapter 6.
3.5 The Solid Target Vacuum Chamber
This section will present the target chamber including the optical setup, the beam
paths and the arrangement and purpose of the diagnostics. The vacuum system
directly connects the solid target interaction chamber to the main pulse compressor.
Both chambers have their own roughing and turbo pumps. For experiments, the
pressure inside both chambers is almost equal at low 10−5 Torr. A manually driven
gate valve close to the target chamber can separate the target chamber from the
vacuum system. This allows for fast pump turn-arounds for target replacement.
3.5.1 Chamber Setup for Titanium Targets
Figure 3.12 (bottom chamber) shows the general setup as it was used for all experi-
ments with titanium target. This includes flat titanium targets, pyramid and wedge
targets. An average of 850 ± 50 mJ pulse energy was compressed to 38 fs as shown
in section 3.2.2. The throughput of the compressor is ∼ 65 %. Hence, 550 mJ are
focused down to a 10±2µm diameter spot by an F♯ = 3, 45⋄ off-axis gold parabola.
This leads to focused intensities of (1.8± 0.7)× 1019 Wcm2 on target. A detailed focal
spot analysis follows in section 3.5.3.
6provided by Foerster et. al., X-Ray Optics Group, University of Jena, Germany
46
Chamber
Setup
Spheres
Scatter
Diagnostic
Alignment
Screen
900 850
2000
1500
1000
500
0
Kα1,2
Heα
Li-like
350
300
2500200015001000500
Kα1,2Kβ
Spectrometer
Collinear
HeNe
Collinear
800nm
0.6J 40fs
800nm
0.1J 40fs
800nm
10mJ 150fs
400nm
DichroicKDP
Waveplate
Blue glass
High reflectors
F#=3
F#=2.8Spectrometer
Pyramid Target
Spheres Target
Thru Imaging
focus & wireChamber
Setup
Pyramid
Figure 3.12: This figure shows the chamber setup for titanium / pyramid / wedge ex-periments (bottom chamber) and silicon / spheres experiments (top chamber). Details aregiven in the text.
47
All shots on titanium targets were done with the laser being orthogonally incident
(0⋄). The target is fixed in an appropriate target mount which is attached to a x-y-
z-ω stage in the center of the chamber. The z-direction is along the laser axis. The
x-direction corresponds to left-right and the y-direction to up-down with respect to
the laser axis. The travel of the stage is 1 inch and 23⋄ respectively. The stage has
to be moved after each shot to place a fresh target in focus. This allows for about
70 shots on flat targets and 15 shots on shaped targets per pump turn-around.
Along with the main beam, there are two collinear alignment beams, a HeNe laser
and a 800 nm beam. They are used for the following target alignment diagnostics.
Scatter Diagnostic The HeNe is focused onto the target along with the main
beam. Both focal spots are overlapped in the focal plane of the main beam. The
HeNe is used as an alignment tool from shot to shot. The scattered light from the
HeNe is imaged onto a CCD by an f ∼ 7 cm lens under an angle of > 45⋄ with
respect to the target. The shape of the scatter is typically a spot or a line as shown
in figure 3.12.
It can not be avoided, that the target also changes its z-position, while it is moved
to a fresh spot. This is because the x-y-z stage is not perfectly orthogonal. If the
target drifts out of focus, while it is moved in x or y direction, the position of the
scattered HeNe changes on the camera. If the scatter camera is set up a few meters
away from the target, the change in position on the scatter camera is fairly big.
By moving the stage in z, the scatter can be brought back to its initial position on
the CCD. This technique allows to place the target in focus with an accuracy of
better than 10µm per shot. The scatter diagnostic is used for aligning flat targets
and shaped targets. In the case of flat targets, the scatter diagnostic is the only
alignment criterion. In the case of shaped targets, the scatter diagnostic is used in
combination with the alignment screen.
Alignment Screen The collinear 800 nm cw beam is focused along with the
main beam on target. Both focal spots are overlapped in the focal plane of the
main beam. The advantage of the 800 nm alignment beam is its stability. Within a
multi-day run, the 800 nm beam will stay on top of the main beam. This is because
the 800 nm alignment beam is fed into the compressor collinearly with the main
beam and experiences the same drifts as the main beam over time (see figure 3.12).
48
A stable 800 nm alignment beam is used to place pyramid and wedge targets in
focus. Alignment is extremely critical for these targets. The tip of a pyramid is
smaller than 1µm2. To study electron guiding effects on these targets, the laser has
to be centered accurately into the pyramid. Therefore the stable 800 nm cw beam
is used in combination with the alignment screen. The motorized screen is moved
into the beam to align a pyramid target. The screen leaves a hole for the HeNe
and the 800 nm cw beam. By eye, the target is moved to a fresh pyramid until the
alignment beams disappear inside the dip. The symmetric geometry of the silicon
pyramid with an open angle of 72⋄ reflects the alignment beams out of the pyramid
onto the screen. Only if the center of the pyramid overlaps with the center of the
alignment beam, the reflection on the screen forms a regular diamond shape pattern
as shown in figure 3.12. The HeNe is used for a rough alignment since it forms a
bright diamond pattern, which is easily visible on the screen. After that, the HeNe
is blocked. The room lights are turned of and the 800 nm cw beam is made visible
on the screen with an IR viewer. The target is carefully jogged in x and y direction
until the four reflections are balanced on the screen. This improves the accuracy up
to 2µm in x and y direction. This procedure takes 15 minutes per pyramid and 10
minutes per wedge target.
Calibration of the Alignment Diagnostics Before the scatter diagnostic and
the alignment screen can be used, they have to be calibrated. This essentially is
done by overlapping them with the main beam. Therefore the focal spot of the
main beam is imaged with the thru imaging diagnostic which is shown in figure
3.12. The target is moved out of the way. A 20x microscope objective is attached
to another x-y-z-ω stage, which is suspended from the ceiling. It is moved into the
beam path. The objective’s z-position is adjusted until the focal plane of the main
beam is imaged onto the CCD camera which is placed after a window flange outside
of the chamber. An image of the focal spot is taken for later characterization as
discussed in section 3.5.3. Afterwards, a f = 3m lens is brought into the beam
path in front of the compressor to disperse the main beam. By doing that, the main
beam is transformed into a back-lighter. A 10µm wire attached to the target holder
is moved into the backlight. Its z-position is adjusted until the wire is in focus
and hence imaged onto the CCD camera (see figure 3.12). Its x- and y-position
is adjusted so that the wire overlaps with the main beam. The HeNe and 800nm
49
Bragg angle:
30°<α<40°Kα
Heβ
observation angles:
α=32°-37°
α=45° between crystal
axis & target surface
Figure 3.13: Detailed spectrometer setup for titanium experiments. Crystal is set up todisperse x-rays with Bragg angles from 30⋄ to 40⋄. Angle between crystal axis and targetsurface is 45⋄. Detectable x-rays exit the target at angles of 32 − 37⋄.
alignment beam are also overlapped with this spot. The position of the scattered
HeNe light is marked on the scatter camera. This concludes the calibration of the
alignment beams and diagnostics. The objective is pulled out. A fresh target is
brought into focus. One is ready to shoot.
Spectrometer The intention of these experiments was to investigate titanium K-
shell x-rays (0.22 nm < λ < 0.28 nm) escaping from the back surface of the various
titanium targets. Hence, the PET crystal ((004) order of reflection) in combination
with the newly designed spectrometer box was placed behind the target. The Crystal
is set up to disperse x-rays with Bragg angles from 30⋄ to 40⋄ which corresponds to
the desired wavelength range. An angle of 45⋄ has been chosen for the crystal axis
with respect to the target plane. Detectable x-rays exit the target with an angle
of 32⋄ − 37⋄ depending on their wavelength (see figure 3.13). The spectrometer is
mounted and aligned as described in section 3.4.4. The scientific x-ray film Kodak
RAR 2492 has been used to integrate 30 to 70 shots of one target type to visualize
its x-ray spectrum.
50
3.5.2 Chamber Setup for Silicon Targets
Figure 3.12 (top chamber) shows the general setup as it has been used for all ex-
periments with silicon targets. This includes flat silicon targets and silicon targets
covered with mono-layers of wavelength scale spheres. Sphere sizes of 0.1, 0.26,
0.36, 0.5 and 2.9 µm were used. Experiments on silicon and sphere targets were
done with the laser being incident both orthogonally (0⋄) and at 45⋄. It has been
found out, that a mono-layer of spheres is extremely delicate and can be destroyed
by a prepulse intensity of the order 1016 Wcm2 [45]. Therefore, frequency doubled 400
nm pulses with a high contrast ratio of ∼ 106 were used (see section 3.2.4). Fully
compressed 38 fs, 800 nm pulses of energy 100 to 150 mJ were doubled in air in
a 2 inch KDP and cleaned with a dichroic and two more 400 nm high reflectors.
Hence, 10-11 mJ, 400 nm pulses are focused down to a 8 ± 2 µm diameter spot
by an F♯ = 2.8, 45⋄ off-axis aluminum parabola. This leads to calculated focused
intensities of (1.3 ± 0.6) × 1017 Wcm2 on target, when a pulse length of 150 fs is used
as discussed in section 3.2.4. A detailed focal spot analysis follows in section 3.5.3.
The target is fixed in an appropriate target mount which is attached to the same
x-y-z-ω stage as the one used for the titanium experiments. Sphere targets allow a
much higher number of shots per area as compared to the micro-shaped pyramid
and wedge targets. Since spheres (0.1, 0.26 and 0.36 µm) form almost arbitrarily
big patches of mono-layers, these sphere targets can be shot in the same way flat
targets are shot. Essentially, only the target’s z-position is required to be precise,
whereas the x and y direction is arbitrary. For 0.5 and 2.9 µm spheres, also x- and
y-alignment is crucial. A collinear HeNe beam was incorporated in the following
three alignment diagnostics.
Scatter Diagnostic For the sphere experiments, two scatter diagnostics were set
up. Both follow the same principle as the one which was used for the titanium setup.
However, by observing the scatter from two independent directions, the accuracy
may be improved. From figure 3.12 one can infer the observation angles and imaging
lines. The scatter diagnostics were used for placing all sphere sizes and flat targets
in focus. X- and y-alignment of the 0.5 µm was achieved by moving the target in x
and y to maximize the intensity of the scattered HeNe. This assures that a patch
of hexagonal close-packed spheres is targeted.
51
Alignment Screen For the biggest sphere size (2.9 µm) x- and y-alignment is also
crucial and was assured with the help of an alignment screen. If a collinear HeNe
shines onto a uniform patch of 2.9 µm spheres a regular diffraction pattern appears
in reflection. This was made visible on a motorized screen, which was moved in
before every shot, comparable to the screen used for the titanium setup (see figure
3.12). By maximizing the number and brightness of regular spot, it was made sure
that the main beam is centered on one of the spheres patches.
Spectrometer The intention of these experiments was to investigate K-shell x-
rays (0.66 nm < λ < 0.72 nm) escaping the from various silicon and sphere targets.
Since the silicon substrate is 0.5 mm thick, the desired x-rays can only be observed
from the front side. Hence, the PET crystal ((002) order of reflection) in combination
with the provided spectrometer box was placed in front of the target. The box is
mounted and aligned in the same way as for the titanium experiments. The details
are as follows:
• For 0⋄ shots, the angle between the crystal axis and target surface was 29⋄.
Detectable x-rays are escaping under an angle of 15⋄− 20⋄ depending on their
wavelength (see figure 3.14.1).
• For the 45⋄ shots, the angle between the crystal axis and target surface was
16⋄. Detectable x-rays are escaping under an angle of 60⋄ − 65⋄ depending on
their wavelength (see figure 3.14.2).
Since K-shell radiation is isotropic, one can compare the yields of 0⋄ and 45⋄ shots
although the observation angle of the crystal is different. The industrial X-ray film
Agfa Structurix D7 has been used to integrate 70 shots of one target type to visualize
its x-ray spectrum.
3.5.3 Focal Spot Characterization
The spatial profile of the beam in the focal plane is measured before every run. This
is done to verify the quality of the parabola alignment. Only a consistent focal spot
profile guarantees comparability between different runs. The focal spot is imaged
with the 20x objective inside the chamber through a window flange onto a CCD
camera outside the chamber (see figure 3.12). The imaging system was calibrated
52
55°49°
29°15°20°
Heα
Kα
3.14.1: 0⋄ shots. Angle between crystal axisand target surface is 29⋄. Detectable x-raysexit the target at angles of 15 − 20⋄.
49°
65°
55°
60°16°
Heα
Kα
3.14.2: 45⋄ shots. Angle between crystal axisand target surface is 16⋄. Detectable x-raysexit the target at angles of 60 − 65⋄
Figure 3.14: Detailed spectrometer setup for silicon and spheres experiments. Crystal isset up to disperse x-rays with Bragg angles from 49⋄ to 55⋄.
with a standard Airforce resolution test target. For all experiments on Titanium
targets, a F♯ = 3, 45⋄ off-axis gold parabola was used. For all experiments on sphere
targets, a F♯ = 2.8, 45⋄ off-axis aluminium parabola was used.
From the physics of Gaussian beam propagation, one can predict the diffraction
limited focal spot size. The 1/e2 beam radius w(z) at the distance z away from the
beam waist w0 is given by [41]:
w(z) = w0
√
1 +z2
z2r
(3.18)
were zr =πw2
0
λ is the Rayleigh length. In the far field z ≫ rr, this equation reduces
to:
w0 · w(z) =λz
π(3.19)
Assuming the 99%-intensity criterion for the beam and parabola diameter one ob-
tains D = πw(f). At the same time, one usually adopts the 1/e criterion for the
focal spot diameter d0 = 2w0, since this is a diameter, which contains 86% of the
focused energy. Equation 3.19 simplifies to:
d0 = 2λf
D= 2λF♯ (3.20)
Hence, the diffraction limited focal spot diameter would be 4.8 µm for the 800 nm
titanium experiments and 2.2 µm for the 400 nm silicon experiments. Analyzing the
53
-15
-10
-50
510
15
01x10
192x10
193x10
2
-15-10
-50
510
15x [mu]
I [W/cm ]
y [mu]
19
3.15.1: For titanium targets, the averaged in-tensity in the two times diffraction limited fo-cal spot is (1.8 ± 0.7) × 1019 W
cm2 .
-12
-8
-40
48
12
02x10
174x10
176x10
2
-12-8
-40
48
12x [mu]
I [W/cm ]
y [mu]
17
3.15.2: For sphere targets, the averaged in-tensity in the 3-4 times diffraction limited fo-cal spot is (1.3 ± 0.6) × 1017 W
cm2 .
Figure 3.15: Typical image of the focal spot as it was measured before every run.
focal spot images one obtains an actual beam diameter of 10±2 µm for the titanium
experiments and 8±2 µm for the silicon experiments. This corresponds to averaged
focused intensities of (1.8 ± 0.7) × 1019 Wcm2 and (1.3 ± 0.6) × 1017 W
cm2 respectively.
Averaging is carried out over the 1/e area of the focal spot with 86 % of the total
pulse energy.
3.6 Continuum Radiation Scintillation Detectors
An array of three to six scintillator detectors have been set up, calibrated and
employed to determine the bremsspectrum of the various target types.
Each detector consists of a 1 inch diameter NaI scintillator crystal of thickness 1cm
to down-convert hard x-rays. The scintillation is measured by a photomultiplier
tube (PMT). The detector comes fully assembled in a 1mm thick aluminum housing
from the manufacturer Burle.
Matching The Detector Responses The calibration was performed with a
radioactive Sodium-22 and Cesium 137 source. Both deliver γ lines of characteristic
energies:
• Sodium-22: 90% of the decays are β+. The emitted positron recombines
with an electron which gives γ photons of energy 0.511MeV. Electron cap-
ture accounts for the remaining 10%. The characteristic γ photon energy is
54
Detector 1
Detector 2
Detector 3
Detector 4
Detector 5
Detector 6
Photon energy [MeV]
Co
un
ts
0.10 1.00 10.00
10
1000
100000
Figure 3.16: Using a radioactive Sodium-22 source, the response of all 6 hard x-raydetectors could be matched. Peaks (f.l.t.r.): 0.511 MeV, 1.274 MeV and sum peak 1.785MeV.
1.274MeV. A sum peak appears at 1.785MeV. The ratio of the three peaks is
roughly 100 ÷ 10 ÷ 1.
• Cesium-137: β− is the only significant decay mode. The characteristic γ
photon energy is 0.622MeV.
With the help of a multi-channel analyzer, spectra of both sources are taken with
every detector. This is done for an external detector voltage of 900 V. By tweaking
the gain level of each detector’s dynode, the response of the detectors are matched.
Figure 3.16 shows the result.
Absolute Energy Calibration After that, every detector is hooked up to an
oscilloscope. A sequence of 1000 radioactive counts is recorded for each detector
and radioactive source. From that sequence, a histogram of both pulse amplitude
(in Volts) and peak area (in Volts x seconds) is obtained. Since the response of
the detector had been matched before, all histograms peak at very similar values.
From the histogram one can obtain the amplitude / area, that corresponds to the
characteristic γ energies of the used radioactive sources. Three γ energies of the
three most likely transitions (0.511, 0.622 and 1.27 MeV) could be evaluated and
related to the amplitude / area on the oscilloscope. The bigger the γ energy, the
bigger the amplitude / area on the scope turns out to be. By fitting a straight line,
one has achieved an energy calibration for the combination of hard x-ray detector
55
0 0.2 0.4 0.6 0.8 10
1
2
3
4
en
erg
y [
Me
V]
amplitude [V]
3.17.1: Gamma photon energy versus sig-nal amplitude on scope.
0 2 4 6 8 100
1
2
3
45
6
area [Vs]
en
erg
y [
Me
V]
3.17.2: Gamma photon energy versus areaunder signal on scope.
Figure 3.17: Result of absolute energy calibration of hard x-ray detectors. The amplitudeand area of the signal on the scope can be related to the incoming photon energy.
and oscilloscope (at 50 MΩ coupling). Figure 3.17 shows the result. This result can
be used later to determine the conversion efficiency of the laser pulse into continuum
radiation.
Measuring Continuum Radiation Spectra A set of three to six calibrated
detectors was set up to measure the bremsspectrum of titanium targets. The detec-
tors are filtered by slabs of copper, iron, aluminum and lead. The energy dependant
x-ray attenuation is given by:
I
I0(E) = exp(−µ
ρρl) (3.21)
where l is the thickness, ρ the density and µρ the energy dependant mass attenuation
coefficient coefficient of the material as found in [46]. The cutoff energy of a filter
is defined as the energy for which the attenuation goes down to 1/e.
Throughout all experiments with titanium targets, a 3.3 cm lead filter, a 4.9 cm lead
filter and a 9.5 cm copper filter were used. The cutoff energies are 800, 1000 and
1200 keV respectively. Hard x-rays passing through these filters are an indication
of hot electron with at least the same energy. Therefore, the electron temperature
can be inferred from the measured hard x-ray yield.
56
Chapter 4
Analysis of X-ray Film
Scientific x-ray films (e.g. Kodak DEF, Kodak RAR series) have been used for
x-ray spectroscopy in the last few decades. With the advent of position sensitive
counters such as the x-ray CCD camera, many scientific x-ray films have been dis-
continued. This is also because of the fact, that film processing makes an x-ray film
an inconvenient detector. Moreover all films are nonlinear detectors and have to be
characterized carefully to infer quantitative information such absolute x-ray photon
numbers. The next section will briefly introduce the theoretical background about
(x-ray) films which is necessary to understand the analysis of the x-ray films in the
following sections.
4.1 Theoretical Considerations
The action of X-ray on photographic film is to turn silver bromide particles present
in the film into a state in which they can be reduced by the action of a suitable
developer into silver grain [40]. Silver grain causes the darkening of the film. With
the assumption that one photon is necessary to excite one AgBr particle, one can
derive a relation for the response of the film [40]:
dn = cn0 − n
n0dE (4.1)
Increasing the exposure E by dE photons/unit area causes the number of excited
AgBr n to also increase by dn. Here, n0 is the number of available AgBr particles
57
and c is some constant. The solution to equation 4.1 is
n = n0 (1 − exp (−cE/n0)) (4.2)
and expresses the nonlinear characteristic of any (x-ray) film. Since silver grain
causes darkening of the film, a measurement of the optical density of the film can
reveal the exposure. Optical density is defined as
D = − log10
i
i0(4.3)
were i/i0 is the fraction of light transmitted by the film. An infinitesimal change of
excited AgBr by dn causes i/i0 to change by d(i/i0)
d(i/i0) = fi
i0dn (4.4)
where f is the effective surface of one silver grain. Combining equation 4.2, 4.3 and
4.4 one obtains the density exposure characteristic
D
E= S0
(
1 − D
2Dmax
)
(4.5)
where S0 = cf/ log 10 is the speed of the film and Dmax = n0f/ log 10 is its satu-
ration density. In principle, every x-ray film is characterized by its S0 and Dmax.
However the simple model leading to equation 4.5 neither includes an energy de-
pendency for the incident x-ray photon nor any absorption edges of silver bromide.
This is why most of the frequently used x-ray films have been characterized experi-
mentally.
Last but not least, unexposed film shows a spontaneous optical density DF called
fog, which also has to be taken into account. The value DF can be related to a
virtual x-ray exposure EF through equation 4.5. In order to obtain the effective
x-ray exposure Eeff of a developed film, one has to subtract the fog value according
to Eeff = E − EF .
58
4.2 Overview of X-Ray Film Analysis
Figure 4.1 shows how the raw data (x-ray film) has to be post-processed in order
to obtain meaningful spectroscopic information. After the film has been developed
6.6 6.7 6.8 6.9 7 7.1 7.2
3x106
2x106
1x106
0
l [A]
ph
oto
ns/
sho
t/m
A/s
rad
6.6 6.7 6.8 6.9 7 7.1 7.2
0
0.001
0.002
0.003
ph
oto
ns/
sho
t/m
m2
l [A]
noise 0.00025 #/shot/mm2
>14 #/run/mm
6.6 6.7 6.8 6.9 7 7.1 7.2
background fit ax2+bx+c
ph
oto
ns/
sho
t/m
m2
l [A]
0.010
0.008
0.009
0.011
0.012
OD
l [A]6.6 6.7 6.8 6.9 7 7.1 7.2
0.35
0.4
0.45
0.5
0.55
Dl=4mA since DR=0.5mm
5 10 15 20 25 30 35x [mm]
0.3
0.35
0.4
0.45
0.5
0.55
OD
x0=3.667mm
0 500 1000 1500 200036000
38000
40000
42000
44000
i [1
6b
it]
x [mm]
12log
1610−
−=í
D
+=
0
2arctansin2)(
xx
Rdxλ
subtract background
smooth
−=
max0
21
D
DS
E
D
density-exposure
relation
Feff EEE −=
subtract fog
density-exposure
relation
crystal reflectivity
open angle
filter transmission
2
int
/)(/
)(/)(/
xT
R
∆
∆λλ
λθλ
Figure 4.1: Various steps of X-Ray film analysis. It starts with a line-out from the scannedfilm (top left) and concludes with the absolute number of photons as emitted from the sourceper mA and srad (bottom). Details are given in the text.
manually, it is digitalized with a 16 bit scanner. From that, a line-out in spec-
tral direction is taken and converted in optical density. The x-axis is re-calibrated
from spatial coordinates to wavelength. The film is linearized with the help of its
density-exposure relation. The background is subtracted. The influences of the
59
filter absorption (entrance window), crystal reflectivity and crystal solid angle are
reversed. One ends up with the absolute photon number per shot, solid angle and
wavelength interval.
4.3 Digitalization of X-ray Film
This section will discuss the digitalization of the x-ray films in some more detail.
4.3.1 Development
The Kodak RAR 2492 film was manually developed according to the processing
instructions given in [39]. The Agfa Structurix D7 film was manually processed
according to:
I. Development: 5 min in AGFA Industrial X-ray Developer for manual devel-
oping FC59P001, few agitation.
II. Rinse: 30 sec Kodak Indicator Stop Bath, constant agitation.
III. Fixing: 15 min in AGFA Industrial X-ray Fixer / Replenisher / Hardener
LXW7V000, few agitation.
IV. Wash: 30 min in running tab water, 30 sec in Kodak Photo-Flo 200 solution,
few agitation
V. Drying: At room temperature in still air.
Both room temperature and chemicals were at (20 ± 1)⋄C. A red light with Kodak
Safelight Filter No. 2 was used.
4.3.2 Scan
The developed films were digitalized with a Konica Minolta Dimage Scan Dual D IV.
Different scanner settings were compared such as resolution, exposure and sampling
depth. Kodak RAR 2492 was giving the best results for 1600 dpi, exposure -1 and
16 bit greyscale depth. Agfa Structurix D7 film was giving the best results for 1600
dpi, exposure 0 and 16 bit linear greyscale depth. A lower (higher) exposure control
results in a positive (negative) offset of the optical density. A resolution of 1600 dpi
60
resolves features of ∼ 16 µm which is close enough to the spatial resolution of the
used x-ray films (∼ 5 µm for Kodak RAR 2492 [39]).
4.3.3 Lineout
After scanning, a line-out was taken by integrating over the spatial coordinate. For
titanium spectroscopy (Kodak film), the integration was carried out over 75 lines
(∼ 1mm). For silicon spectroscopy (Agfa film), the integration was carried out over
50 lines (∼ 0.8mm). Both widths are bigger than the spatial linewidths ∆l (see
section 3.4.3). This way it was made sure to count all incident x-ray photons and
reduce the noise at the same time.
Since the scanner assigns a 16 bit value i to every pixel, one has to use equation 4.3
to obtain optical densities (see first and second subfigure of 4.1). Optical densities
were typically ranging from OD = 0.2 − 0.7 for both types of film.
4.4 Deconvolution of X-Ray Film
This section will discuss the deconvolution of the x-ray films in some more detail.
4.4.1 Absolute Wavelength Calibration
In order to obtain OD-wavelength traces, one has to recalibrate the x-axis from spa-
tial coordinates to wavelength. The bragg equation 3.14 and geometrical relations
give
λ(x) = 2d sin
(
arctan
(
2R
x + x0
))
(4.6)
where x is the spatial coordinate along the line focus of the film and x0 is the
distance from the source to the the front end of the film where x = 0. Since a direct
measurement of x0 is not accurate enough, one has to pick one line (x = xcalib) of
known wavelengths λcalib from the spectrum. By plugging xcalib and its λcalib into
equation 4.6, one can solve for x0 and lock the the wavelength axis. The preferred
choice for λcalib is a cold transition, for example Kα. A hot plasma transition like Heα
can not serve as a wavelength normal since its wavelength and width depends on the
temperature and density of the plasma [58]. The absolute wavelength calibration
61
is very accurate in the vicinity of the wavelength normal Kα. However due to
the limited alignment accuracy of the spectrometer (∆R = ±0.5 mm), the absolute
wavelength calibration becomes more and more inaccurate the further the transition
is away from Kα. For Si Heα which is ∼ 0.6 A away from the wavelength normal
Kα, the accumulated inaccuracy is already 4 mA (see third subfigure of 4.1).
4.4.2 Linearization
For the linearization of the scientific x-ray film Kodak RAR 2492 that was used for
titanium K-shell spectroscopy, the measured density-exposure characteristics were
taken from [39]. This paper provides density-exposure relations for a wide range of
wavelengths including 2.2 A to 2.8 A.
For the linearization of the industrial x-ray film Agfa Structurix D7 that was used
for silicon K-shell spectroscopy, the measured density-exposure characteristic was
taken from [40]. This paper only provides the density-exposure relation for copper
Kα (1.54 A). It is used because no data for silicon K-shell radiation (6.6 A to 7.2 A)
could be found. As a consequence, no intensity ratios within one silicon spectrum
will be meaningful. However, intensity ratios between different silicon spectra for a
given wavelength can be taken of course.
Figure 4.2 shows these density-exposure relations for both x-ray films and different
photon wavelengths. At copper Kα, Agfa Structurix D7 is about 8 times faster than
Kodak RAR 2492. The Agfa film is much faster because it has an emulsion layer
on either side. The two emulsion layers however make the film thick which reduces
its spatial resolution.
4.4.3 Intermediate Result
After the data has been linearized, one can subtract the background which is caused
by fluorescence and all sorts of secondary radiation. This is done by fitting the
polynomial function f(x) = ax2 + bx + c to the data (neglecting the peaks) and
subtracting f(x) (see 4th subfigure of 4.1).
Consequently one obtains the useful intermediate result shown in the 5th subfigure
of 4.1, namely the number of photons per µm2 that have interacted with the film per
laser shot. From this trace, one can see that the noise level for silicon spectroscopy
(Agfa film) is 2.5 × 10−4 photons/shot/µm2. Multiplying by 70 shots per run and
62
0.25 0.5 0.75 1 1.25 1.5 1.75 2optical density
5
10
15
20
25
30
ph
oto
ns
/mm
2
Cu Ka on Kodak
Ti Ka on Kodak
Si Ka on Kodak
Cu Ka on Agfa
8:4:2:1
Figure 4.2: Density-exposure relations for Kodak and Agfa x-ray film as taken from [39]and [40].
by the integration width of 50 lines, this yields a minimum of 14 interacting photons
per run and µm in the direction of dispersion on the film. If less photons interact
with the film, a peak would be below the noise level. The same calculation for
titanium spectroscopy (Kodak film) yields a minimum of 70 photons per run and
µm in the direction of dispersion to be above the noise level. The ratio 14/70 reflects
the sensitivity ratio 4/1 of the Agfa and Kodak film (see figure 4.2).
4.4.4 Crystal Response and Filter Transmission
In order to obtain the total number of photons that are emitted by the source per
shot, solid angle and wavelength interval, one has to reverse the influences of filter
absorption, the crystal reflectivity and open angle of the crystal. All three effects
are wavelength dependent and their contribution to the efficiency of the spectrom-
eter is shown in figure 4.3 for (002) reflection and (004) reflection. Obviously, the
efficiency for the (004) reflection (∼ 5× 10−5 srad) is about 10 times lower than the
efficiency of the (002) reflection (∼ 2.3×10−6 srad). Compared to other von-Hamos
spectrometers, an efficiency of ∼ 2.3×10−6 srad is an absolutely competitive result.
Shevelko et al. [47] for example have presented a compact von-Hamos spectrometer
based on third order reflection off of Mylar, reaching an efficiency of ∼ 5.6 × 10−6
srad at 2.6 A.
The transmission of the filter (aluminum coated mylar foils) was calculated with
equation 3.21 using appropriate mass absorption coefficients for mylar and alu-
63
silicon (002) order titanium (004) order
6 6.5 7 7.5 8
0.2
0.4
0.6
0.8
1.0
wavelenght [A]
Rin
t [m
rad
], T,
q [
rad
]qRint
T
(a)
6 6.5 7 7.5 8wavelenght [A]
3.5
4.0
4.5
5.0
5.5
6.0
eff
icie
ncy
[1
0-5
sra
d]
(c)
0.2
0.4
0.6
0.8
1.0
Rin
t [1
0-5
ra
d],
T, q
[ra
d]
2.2 2.4 2.6 2.8 3.0wavelenght [A]
Rint
T
q
(b)
2.2 2.4 2.6 2.8 3.0wavelenght [A]
2.20
2.25
2.30
2.35
eff
icie
ncy
[1
0-6
sra
d]
(d)
Figure 4.3: (a) and (b) show Rint, filter transmission T and open angle θ as function ofλ for silicon and titanium spectroscopy respectively. Multiplying these three contributionsgives the efficiency of the spectrometer for (002) reflection (c) and (004) reflection (d).
minum as found in [46]. The angle of incidence as a function of the wavelength was
taken into account.
The open angle as a function of the wavelength is given by
θ(λ) = 2 arcsin
(
wλ
2 · 2dhklR
)
(4.7)
where w = 6 cm is the width of the crystal, R = 10 cm is the radius of the crystal
and 2d002 = 8.73 A and 2d002 = 4.365 A respectively.
The integrated reflectivity of the crystal was either measured ((002) order) or cal-
culated ((004) order) as discussed in section 3.4.3.
Last but not least, one has to switch from per ”µm2“ to per ”mA“ to allow for
comparison with other literature. This is done by
∆λ
∆x2=
12d
λ2
2R
√
1 − λ2/4d2
integration width · pixellength(4.8)
64
where the denominator is the linear dispersion of the crystal ∆λ/∆x (see equation
3.15) multiplied by the pixellength ∆x ≈ 16 µm (scanner resolution 1600 dpi). The
integration width is 50 and 75 lines multiplied by ∆x ≈ 16 µm for silicon and tita-
nium spectroscopy respectively.
The result of the complete deconvolution of an x-ray film is shown in the last sub-
figure of 4.1.
65
Chapter 5
Experimental Characterization
of Titanium Targets
This chapter presents the data that has been obtained with the von-Hamos spec-
trometer, the hard x-ray detectors and the spherical crystal imaging spectrometer.
A few preliminary experiments with flat foil targets were conducted to characterize
the performance of the spectrometer and find the best operating parameters. An
angle scan with flat copper targets was conducted to infer the angle of maximum
absorption and to reveal the scale length of the pre-plasma. Then, flat foil tita-
nium targets and micro-shaped pyramid and wedge targets of different thicknesses
were shot under similar conditions. The von-Hamos spectrometer reveals a strong
Kα yield dependency upon the target type. The bremsspectra that were obtained
with the hard x-ray detectors reveal the laser-absorption properties of the different
target types and suprathermal electron temperatures. In the case of flat targets,
the spherical mylar spectrometer reveals a concentric sidepeak. This sidepeak is
significantly away from the laser-target interaction and therefore gives insight into
suprathermal electron trajectories and ultra-strong self-consistent magnetic fields.
Pyramids and wedges failed to produce a brighter Kα source than flat targets. This
is explained by the pyramid open angle that is insufficient for cone-guiding. Kα
yields, hard x-ray yields and hot electron temperatures agree with PIC simulation,
that were conducted for the presented target geometry.
66
2.74 2.75 2.76 2.77
1.2x109
0.8x109
0.4x109
0
30 shots
30 shots
90 shots
wavelength [A]p
ho
ton
s/sh
ot/
mA
/sra
d
Figure 5.1: Kα signal of two 30 shot runs and one 90 shot run recorded with the von-Hamos spectrometer under identical target and laser conditions. The different yield has tobe attributed to the limits of the x-ray film linearization as discussed in the text.
5.1 Characterization of von-Hamos Spectremeter
Several runs with 11µm thick flat titanium targets were conducted to characterize
the performance of the spectrometer. The laser parameters were 800 nm, ∼ 38 fs,
> 1019 W/cm2 and zero degree incidence. The accuracy was determined with which
Kα yields can be reproduced. Furthermore, the reliability of the film linearization
was determined and the influence of the spectrometer alignment on its spectral
resolution was studied. Besides Kα also Kβ radiation and hints of plasma emission
are observed.
5.1.1 Accuracy of Data Reproduction
It is particularly important for this work to determine the accuracy with which
K-Shell yields can be measured (reproduced) with the von-Hamos spectrometer.
Therefore, two consecutive runs were conducted for exactly the same conditions.
11 µm titanium foil was incident at 0⋄. 30 shots were integrated. The result is
shown in figure 5.1. The upper two curves represent the Kα peaks of two identical
runs. The maximum deviation in peak height is about 5%. In conclusion, only yield
differences of > 5% between two targets will be considered as a real physical effect.
67
90 shot run
2.3 2.4 2.5 2.6 2.7
0
0.005
0.01
0.015
wavelength [A]
ph
oto
ns/
sho
t/µ
m2
noise: <0.001#/shot/µm2
2.3 2.4 2.5 2.6 2.7
0
0.005
0.01
0.015
wavelength [A]
30 shot run
ph
oto
ns/
sho
t/µ
m2
noise: 0.002#/shot/µm2
Figure 5.2: Comparison of two complete titanium spectra with 30 and 90 shots recordedwith the von-Hamos spectrometer under identical target and laser conditions.
5.1.2 Integrated Shot Number
Changing the number of shots, that are integrated on RAR 2492 film influences the
noise level and also reveals the limits of the film linearization:
Integrating 30 shots on RAR 2492 film results in an optical density of ∼ 0.2 − 0.4
which is far from saturating the film. By integrating more shots, the noise level of
the lineouts will decrease as 1/√
N where N is the shot number. Figure 5.2 compares
the lineout of a 30 shot run and a 90 shot run. The noise level of the 30 shot run
turns out to be about twice as high as for the 90 shot run. This results in a much
more distinct Kβ peak in the case of 90 integrated shots as it can be seen in figure
5.2. This is a strong argument for integrating rather 90 shots than 30 shots for all
further runs. However the targeting of pyramids is very time-consuming. This lead
to the compromise of 50 integrated shots for all target types.
Besides that, it was found, that linearizing two identical runs that only differed by
the number of shots that were integrated does not yield the same result. A run of
90 shots would result in a Kα photon number of 6.0 × 109/shot/srad as compared
to 4.9× 109/shot/srad for a 30 shot run. This can be seen both in figure 5.2 and as
a closeup in figure 5.1. The reason for that must be attributed to the linearization
of the film as discussed in Chapter 4. In particular, the advanced age of the already
expired film in combination with a not accurate enough density-exposure relation
as taken from [39] are likely responsible for that. The problem was solved by always
integrating the same number of shots (50) per target.
68
5.1.3 Accuracy of Wavelength Calibration
The accuracy of the absolute wavelength calibration was determined from the dis-
tance between Kα1 and Kα2 as well as from the position of Kβ . As it can also be
seen from figure 5.1 for example, the distance between the two Kα lines turns out to
be 3.7±0.3 mA on average. It is supposed to be 4 mA, with Kα1 at 2.750 A and Kα2
at 2.754 mA [32]. The major characteristic of the absolute wavelength calibration
is such that the further the line is away from the model point Kα, the less accurate
its position can be determined (see section 4.4.1). The position of Kβ turns out to
be 2.521 ± 0.003 A as opposed to its true position of 2.516 A [32]. Since the offset
is systematically towards smaller wavelengths, a misalignment of the spectromter
has to be taken into consideration. The spectrometer was realigned several times
within the ∼ 20 runs, without effecting the systematically wrong position of Kβ.
This suggests, that the position of the film with respect to the crystal inside the
spectrometer is off. By carefully measuring all dimensions of the spectrometer, it
was found, that the film was placed to close to the crystal and the alignment holes
of the spectrometer box are define an axis which is off of the real crystal axis in the
other direction. Equation 4.6 and simple imaging properties of the crystal support
the hypothesis of inaccurate spectrometer dimensions:
• To reproduce the effects of a too close film, one would have to replace R = 100
mm by a smaller value in equation 4.6. This indeed causes Kβ to shift back
towards its true value.
• The HeNe alignment holes being further away from crystal than the crystal
axis would give an object distance p1 that is bigger than supposed. With the
focusing condition of the crystal (1/p+1/q = 2/R), this would yield a smaller
imaging distance q than supposed. p < q indeed reduces the dispersion of the
spectrometer and hence causes Kβ to fall on a too big wavelength.
Last but not least, scanning the films can also contribute to a systematic error of
the wavelength calibration. Each piece of film was about 8 cm long but the scanner
does only handle 1” per scan. Consequently, several subscans had to be connected
together. In fact, a small gap can be seen in figure 5.2 at around 2.524 A, which
also effects the accuracy of the wavelength calibration.
1The object distance is the distance from the source to the crystal whereas the imaging distanceis the distance between the crystal and the film.
69
He/Li electronic Gabriel’s wavelengthlike configuration notation [A]
Li-like 1s22p(2P ) − 1s2p2(2S) mn 2.620 for m
Li-like 1s22p(2P ) − 1s2p2(2P ) jkl 2.6319 for k2.6355 for j
Li-like 1s22p(2P ) − 1s2p2(2D) abcd 2.6295 for a
Li-like 1s22s(2S) − 1s2p(1P )2s(2P ) qr
Li-like 1s22s(2S) − 1s2p(3P )2s(2P ) st
He-like 1s2(1S) − 1s2p(1P ) w 2.6106
He-like 1s2(1S) − 1s2p(3P ) y 2.6229
Table 5.1: Some plasma lines of highly ionized titanium. The electronic configuration wastaken from [51]. For Gabriel’s notation refer to [50]. The wavelengths were taken from Nist[49].
5.1.4 Plasma Emission
A closer look on figure 5.2 reveals a faint and broad peak around 2.60-2.66 A. This
peak is most likely caused by transitions of highly ionized titanium in the region of
the plasma on the front side of the target. The peak is low compared to Kα because
the radiation is produced in front of the target and consequently will be attenuated
by the target itself before being detected by the von-Hamos spectrometer behind the
target. The spectral window under discussion allows for He-, Li- and Be-like lines,
which corresponds to titanium ions with 2, 3 and 4 bound electrons left in the shell
respectively. Some of these lines and the corresponding electronic configuration are
summarized in table 5.1.4. The ionization energy of He- and Li-like titanium is 1.43
keV and 6.25 keV respectively [48]. This can be seen as a crude estimate of the
plasma temperature.
Similar measurements on a laser-produced plasma could identify some of these tran-
sitions if the von-Hamos spectrometer was employed at the front side of the target
(Shevelko et al. [47]). A 1 µm, 10 ns pulse with an energy of order 1 mJ was focused
to intensities of order 1014 W/cm2. In this experiment, even such low intensities
yielded 109 − 1010 photons/shot/mA/srad for some of the satellites. With equa-
tion 2.43, hot electron temperatures around 1.7 keV can be reached via collisional
absorption, allowing Shevelko et al. to observe these ionization stages.
70
0 302010 40
angle [degree]
ha
rd x
-ray
yie
ld [
a.u
.]Figure 5.3: The angular dependency of ultra-short laser pulse absorption has been mea-sured with 800 nm, 38 fs, 0.6 J pulses incident on copper foils. A hard x-ray detector servesas measure for the absorption. The peak at small angles is a sign of a large scale lengthplasma Ln ≈ 12µm (Figure courtesy of D.R Symes and A. Sumeruk).
5.1.5 Spectral Resolution and Focusing Quality
Kα1 and Kα2 are almost perfectly resolved in figure 5.1. Hence the resolution of
the spectrometer for (004) reflection at 2.75 A can be calculated with λ/∆λ ∼ 700.
The resolution is believed to be even higher in case the spectrometer was aligned
perfectly (see discussion in section 5.1.3).
5.2 Angle Scan
An angle scan has been conducted to infer the prevalent absorption mechanism.
800 nm laser pulses of duration ∼ 38 fs, energy ∼ 0.55 J were shot on planar slabs
of copper under different angles of incidence. Focused intensities of greater than
1019Wµm2/cm2 were reached. The contrast ratio of prepulse to main laser pulse
was ∼ 10−3 (see section 3.2.3) which allows for a rather significant pre-plasma. The
absorption was measured with a hard X-Ray detector with cutoff energy 600 keV
as discussed in section 3.6. The angle of maximal absorption occurred at ∼ 10⋄
(see figure 5.3) which corresponds to a plasma density scale length of ∼ 12 µm as
calculated with equation (2.39).
71
5.3 Kα Yield Comparison of Flat and Micro-Shaped
Targets
In this section the Kα yield dependency upon the target type is presented. This
data was taken with the von-Hamos spectrometer.
5.3.1 Pyramid versus Flat Target
The laser parameters for this series of experiments were 800 nm, ∼ 38 fs and 0.55
J on target. This gave focus intensities of > 1019 W/cm2 (see section 3.5.3). Both
pyramids and flat targets were shot with the laser normally incident with respect to
the target surface as discussed in section 5.2. A closeup of the spectra showing the
Kα transition is depicted in figure 5.4.1 and figure 5.4.2. These figures correspond
to targets based on 11 µm and 25 µm titanium foils respectively.
Result As it can be seen from these figures, the pyramid targets produce less Kα
photons than a flat foil target of same thickness. This is true for the 11 µm and
the 25 µm targets. Integrating over the 1/e width of the Kα signal yields 6.0 × 109
photons/shot/srad for the 11 µm flat target and 1.5 × 109 photons/shot/srad for
the corresponding pyramid target. This is a factor of 4 times less Kα for the 11
µm pyramid. The effect of the pyramid geometry is less distinct for the 25 µm
targets. Here, the 25 µm flat target produces 6.5 × 109 photons/shot/srad whereas
the pyramid produces 2.4×109 photons/shot/srad. This is a factor of 2.8 times less
Kα for the 25 µm pyramid.
Dependency On Target Thickness In the case of flat targets, hot electrons
are produced mostly via collisionless mechanisms such as resonance absorption (see
section 5.2). These electrons are injected into the cold target material behind the
plasma, where they produce K-shell radiation as discussed in section 2.4. The thicker
the target, the more Kα photons are produced. Therefore, thicker foils should
produce a brighter source. However, Kα has a self-absorption length in titanium.
Hence, the number of photons that can escape from the rear side decreases if the
thickness is increased. The self-absorption length for titanium Kα (2.752 A) in
titanium is 21 µm. Figure 5.5 shows the self-absorption length for the wavelength
window covered by the von-Hamos spectrometer. Apparently, an absorption edge at
72
2.74 2.75 2.76 2.77
1.5x109
0.8x109
0
wavelength [A]
ph
oto
ns/
sho
t/m
A/s
rad
11µm flat
11µm pyramid
5.4.1: Comparison of Kα Spectrum from 11µm Pyramid and Flat Target
25µm flat
25µm pyramid
2.74 2.75 2.76 2.77
1.5x109
0.8x109
0
wavelength [A]
ph
oto
ns/
sho
t/m
A/s
rad
5.4.2: Comparison of Kα Spectrum from 25µm Pyramid and Flat Target
Figure 5.4: Comparison of Kα Spectrum from Pyramid and Flat Target with titaniumfoils of different thickness.
2.2 2.3 2.4 2.5 2.6 2.7 2.8
5
10
15
20
25
wavelength [A]
thic
kn
ess
Ti [µ
m]
Heβ Kβ Kα
Figure 5.5: Self-absorption of Ti K-shell radiation in titanium. From this figure, the idealtarget thickness (21 µm for Kα) can be inferred (see text) [52].
2.74 2.75 2.76 2.77
1.5x109
0.8x109
0
wavelength [A]
ph
oto
ns/
sho
t/m
A/s
rad 25µm s-wedge
25µm pyramid
25µm p-wedge
Wedge
P-polarized
Wedge
S-polarized
Pyramid
Figure 5.6: Comparison of Kα spectrum from pyramid target (2.4 × 109 pho-tons/shot/srad), s-wedge target (3.3×109 photons/shot/srad) and p-wedge target (0.9×109
photons/shot/srad). Each target was based on 25 µm thick titanium foil.
73
∼ 2.5 A makes higher energetic photons such as Heβ invisible to the spectrometer.
Since the 25 µm target is closer to the ideal thickness of 21 µm than the 11 µm target,
the 25 µm target should yield a higher Kα photon number. This is in agreement
with the measurements for flat targets an for pyramid targets.
Dependency on Target Geometry An interesting feature of this data is the
Kα yield dependency upon the target geometry. Apparently, the pyramid targets
failed to produce a brighter source as indicated by the studies of electron energy
transport in cone guiding symmetries [7]. The first explanation that comes to mind
involves the dependency of the Kα production upon the hot electron temperature.
The Kα cross section2 peaks for electrons around 13-17 keV [53] [54]. Apparently,
fewer electrons of this energy are guided along the pyramid wall and injected into
the titanium foil. However, this does not disprove the guiding ability of pyramid
targets in general. Further experiments on micro-shaped targets and the results of
the other diagnostics are required to understand the details of hot electron transport
in pyramids.
5.3.2 P-Wedge versus S-Wedge
The next step towards a better understanding of pyramid guiding geometries is
accomplished by simplifying the geometry, e.g. by making it two dimensional. A
pyramid clearly is a three dimensional structure, whereas a wedge (see discussion in
section 3.3.1) can be seen as a two dimensional pyramid, with only one pair of op-
posing walls. These walls however extend to infinity (several mm) compared to the
spatial scale of the laser focus (several µm). Therefore a wedge is a target geometry
that allows to study the laser interaction with only one pair of walls. In contrary,
pyramid targets always integrate over two orthogonal pairs of walls. In conclusion,
a wedge target significantly cleans the experimental conditions.
Result A closeup of the spectrum which is showing Kα is depicted in figure
5.6 comparing wedge targets in s-polarization, wedge targets in p-polarization and
the pyramid targets. Apparently, a wedge target produces less Kα if shot in p-
polarization. Integrating over the 1/e width of the Kα line yields only 0.9 × 109
2i.e. the K-shell ionization cross section peak
74
photons/shot/srad for p-polarization whereas more than three times as many Kα
photons are produced in s-polarization (3.3 × 109 photons/shot/srad). For com-
parison, also the pyramid Kα line is included in figure 5.6. The Kα yield from the
pyramid falls in between the different wedge polarizations.
Dependency on Polarization The right half of figure 5.6 quickly reminds the
meaning of p-polarization and s-polarization. In the case of p-polarization, the
electric field of the laser has a component normal to the wedge wall, whereas in
the case of s-polarization the electric field is parallel to the wedge wall. The rather
low contrast ratio of the laser pulse causes a long scale pre-plasma (see section 5.2).
Therefore, resonance absorption is one of the dominant laser absorption mechanisms
for the experiments presented here. As discussed in the section about resonance
absorption (2.3.4), no such absorption should occur for s-polarization. This seems
to be in contradiction to the higher Kα yield obtained for s-polarization. Recently,
it was shown, that normal incidence or s-polarization can allow for absorption if
multidimensional effects such as surface rippling are taken into account [13]. But the
tremendous light pressures that are required for surface rippling can not be created
for a large angle of incidence as it is the case for s-wedges (54⋄). In consequence,
there must be a different explanation for the observed polarization dependency. The
data of the hard x-ray detectors will have to be consulted first.
5.4 Hard X-Ray Yield Comparison of Flat and Micro-
Shaped Targets
A set of three scintillator / photomultiplier detectors has been employed to infer in-
formation about the hard x-ray yield of the different micro-shaped and flat targets.
This was done by filtering each detector with slabs of lead and copper, with 1/e2
cutoff energies 800 keV, 1000 keV and 1200 keV respectively. The detectors were
placed 4 m away from the target with polar angles of 0⋄, 2⋄ and 5⋄ with respect to
the horizontal plane respectively3. The azimuthal angle was 0⋄ for all three detec-
tors, which means that their lines of sight almost fell together with the direction of
the incoming laser.
3The laser was always polarized horizontally for all the work presented here.
75
25µm p-wedges
25µm flat
25µm pyramid
25µm s-weges
700 800 900 1000 1100 1200 13000
100
200
300
400
500
600
700
1/e2 cutoff energy [keV]
ha
rd x
-ray
yie
ld [
a.u
.]
Flat Ti Foil
Wedge
P-polarized
Wedge
S-polarized
Pyramid
Figure 5.7: Hard x-ray yield from micro-shaped and flat targets. An array of threedetectors was employed with cutoff energies of 800 keV, 1000 keV and 1200 keV respectively.
5.4.1 Dependency on Target Type
Figure 5.7 shows the hard x-ray yield as obtained from the different target types.
Apparently, p-wedges show the highest yield, s-wedges show the lowest yield and
flat targets and pyramids fall in between. The graph includes the averaged result of
50 to several hundred shots depending on the target type. The error bars are quite
big for several reasons:
• Averaging was carried out over multiple runs that were conducted over several
weeks. This involves the influence of some variation in the laser parameters.
• Averaging was carried out over multiple targets. This involves the influence
of target-to-target variations. In particular, the process of laying down the
titanium foil was improved several times during this work4.
• The mechanisms of hard x-ray production, down-conversion in the scintillator
crystal and detection in the PMT are highly stochastic. Hence, this diagnostic
inherently has considerable error bars.
• Micro-shaped targets gave bigger error bars than flat targets. This is due to
the limits of targeting pyramids and wedges consistently.
Nevertheless, a clear hard x-ray yield dependency upon the target type persists.
4 For example, each pyramid was equipped with individual foils instead of one foil that isspanning across many dips. This was done to minimize the risk of having a gap between thesubstrate and the back foil.
76
5.4.2 Conclusions
One has to draw several conclusions from figure 5.7:
Why p-wedge gives more HXR than s-wedge P-wedges give a higher yield
than s-wedges. Therefore the laser absorption efficiency is higher for p-wedges. This
is in accordance with the theory of resonance absorption. The first two detectors
(800 keV and 1000 keV) both have lead filters. Hence, the slope of a strait line
which is connecting these two data points can be used as an easy way to estimate
the hot electron temperature qualitatively. A bigger slope corresponds to a higher
temperature. P-wedges have a significantly higher slope than s-wedges, and therefore
a higher temperature.
Why p-wedge gives less Kα than s-wedge On the other hand, this result still
seems to contradict with the Kα yields, that were showing the inverse dependency.
Although the hard x-ray detectors are suggesting a higher absorption for p-wedges,
less Kα is produced. This only allows for two conclusions.
I. Electron channelling along the walls of a (P-) wedge is not significant. No
significant amount of electrons is injected into the titanium tamper.
II. The more optimistic hypothesis would be: No electron channelling and injec-
tion occurs for such electrons that have an energy around the Kα cross section
peak. To put it in other words, electron channelling is energy sensitive and
only allows for suprathermal electron to be channelled. These electrons are
too hot to contribute to Kα enhancement.
Electron channelling requires self-consistent magnetic fields, that force electrons into
the surface layer [8]. Channelling is proportional to j × B and accordingly should
occur more likely for faster electrons. This is an argument for hypothesis II.
PIC simulations will have to be consulted to proof or disproof either hypothesis.
Why cone-guiding has to have occurred Another conclusion can be drawn
from comparing flat targets and p-wedges. Apparently, a higher hard x-ray yield
is obtained from p-wedges than from flat targets. That is to say, one can couple
energy more efficiently into a p-wedge than into a flat target that is shot at 0⋄. The
77
p-wedge
s-wedge
electron energy [MeV]640 2
1
1x102
1x104
1x106e
lect
ron
co
un
ts [
1/5
0k
eV
]
5.8.1: More and hotter suprathermal elec-trons are created for p-wedges.
photon energy [MeV]4320 1
1
1x102
1x104
1x106
1x108
ha
rd x
-ray
co
un
ts [
a.u
.]
p-wedge
s-wedge
5.8.2: The PIC code includes a MonteCarlo Simulation for the bremsspectrum.
Figure 5.8: The 2D collisional particle-in-cell code PICLS has been run simulating wedgetarget geometries and THOR laser parameters (modified figures, courtesy of Y. Sentoku,University Nevada, Reno).
higher absorption efficiency has to be caused by the geometry of the wedge. It is
not caused by the the fact, that the laser is incident with 54⋄ on the wall of the
wedge. A flat target that is shot at 54⋄ still gives a lower hard x-ray yield than a
p-wedge (see angle scan in section 5.2). This means that cone guiding has to have
occurred for p-wedges. It may have also occurred for pyramids, but the error bars
of the hard x-ray yields are too big to be able to tell.
The PIC simulations in the next section will support these conclusions.
5.5 PIC Simulation
Two dimensional particle-in-cell (PIC) simulations have been carried out by Y.
Sentoku from the University of Nevada at Reno using the PICLS code which includes
a Monte Carlo calculation of the Bremsstrahlung. The laser parameters were chosen
to model the Thor laser (40 fs, 5 × 1018 W/cm2, 6 µm focal spot). The target was
simulated as a fully ionized deuteron with an initial (electron) density of 4 × 1022
cm−3. The target is 6 µm thick with a wedge-shaped dip with 71⋄ open angle. The
intention was to study the polarization dependency between s- and p-wedges.
78
2x1072x107 0
0
2x107
2x107
p-wedge
s-wedge
laser
43o
45o
5.9.1: P-wedges show a strong angular de-pendency of the hard x-ray emission (mod-ified figure, courtesy of Y. Sentoku, Univer-sity Nevada, Reno).
channelled
electrons
transmitted
electrons
laser
p-wedge(72o)
36o
36o
36o
5.9.2: This angular dependency can beexplained by the relativistically boosteddipole radiation of channelled and trans-mitted electrons.
Figure 5.9: The 2D collisional particle-in-cell code PICLS predicts some electron chan-nelling of suprathermal electrons which effects the radial dependency of the hard x-rayemission.
5.5.1 Suprathermal Electrons
As expected, p-wedges show a much higher laser absorption efficiency than s-wedges.
A snapshot of the electron energy distribution function for p-wedges and s-wedges
is shown in figure 5.8.1. P-wedges yield a hot electron temperature of 720 keV
whereas s-wedges yield 650 keV. This is in accordance with the slopes in figure 5.7.
The number of hot electrons is more than one order of magnitude less for s-wedges.
The simulations revealed, that cone guiding is insignificant. Only few electrons are
channelled along the wall. Their energy is of the order 100 keV. In particular, no
electrons of energy around the Kα cross section peak of titanium are captured in the
surface current. That explains why pyramids failed to produce a brighter source.
5.5.2 Bremsspectrum
The increase in suprathermal electrons for p-wedges translates into an enhancement
of the hard x-ray yield as depicted in figure 5.8.2. This is in accordance with the
hard x-ray yield dependency that has been measured with the scintillation detec-
tors. It is particularly interesting to focus on the radial plot of the hard x-ray
79
emission for p-wedges given in figure 5.9.1. The bremsstrahlung is predicted to be
highly anisotropic. This anisotropy is caused by the dipole radiation that is rela-
tivistically boosted in the direction of the electron propagation for multi 100 keV
electrons. Cone geometries have two directions of preferred electron acceleration.
These are along the cone wall for channelled electrons and perpendicular for trans-
mitted electrons [8]. As discussed in section 2.5, all electrons are forced into the
surface current if the angle of incidence is bigger than a critical angle θ > θc. For
intensities of 8× 1019 W/cm2, the angle is θc = 65⋄ [8]. Pyramid and wedge targets
have an open angle of 71⋄ which corresponds to an angle of incidence of 54⋄. This
is significantly below the critical angle and about 50% of the electrons should be
transmitted. The direction of transmitted and channelled electrons is depicted in
figure 5.9.2. If no channelling at all would have occurred, the relativistically boosted
dipole radiation of suprathermal electrons should peak at 54⋄ from the horizontal.
If all hot electrons would have been channelled, the peak should be at 36⋄ from the
horizontal. The radial plot of the bremsstrahlung gives an indirect indication that
some electrons are channelled. The dipole radiation of the relativistically boosted
suprathermal electrons peaks at ∼ 44⋄ which is right in between 36⋄ and 54⋄.
5.5.3 Electron Energy Density Plot
2D plots of the electron energy density (ne/n0) ǫ were also provided by Y. Sentoku’s
PIC simulation. They are displayed in figure 5.10 for s- and p-polarization. One
should emphasize, that the contour plot shows the electron energy density and not
the electron temperature. That is to say, white refers to few or cold electrons and
red refers to many or hot electrons per unit volume. The PICLS code allows for an
initially cold target (< 100 eV) so that the process of heating can be studied. At
the beginning of the simulation, all electrons are at 0 keV which corresponds to an
entirely white contour plot.
Hence, one can infer the directions of electron energy transport from figure 5.10. For
p-polarization, streams of hot electrons are generated inwards the target, perpen-
dicular to the walls. That is the result of resonance absorption and vacuum heating.
Although more than 10 times more hot electrons are generated for p-polarization,
less energy is transported towards the tip of the cone. Instead, energy is dispersed
over the whole target (see 5.10.1). In the case of s-polarization (see 5.10.2), a stream
80
0 2 4 6 8 100
2
4
6
8
10
30keV/n0
0
p-wedge
x [µm]
y [µ
m]
5.10.1: Most of the energy is dispersedinto the target via streams of hot elec-trons perpendicular to the walls.
0 2 4 6 8 100
2
4
6
8
10
s-wedge
x [µm]
y [µ
m]
30keV/n0
0
5.10.2: Less absorption occurs but thecompression of the walls causes a denseelectron layer along the walls thattransfers energy towards the tip.
Figure 5.10: Electron energy density plot ((ne/n0)ǫ [keV]) as obtained with the PICLScode 20 fs after the laser interaction. (modified figure, courtesy of Y. Sentoku, UniversityNevada, Reno).
of hot electrons is established along the wedge walls. This is caused by the laser that
is rather compressing the target walls than being absorbed. Significant amounts of
the s-wedge remain uncompressed / cold. Nevertheless, many heated electrons con-
verge towards the tip, were they eventually are injected into the titanium tamper.
This explains why a brighter Kα source could be observed for s-polarization.
5.6 Spatial Kα Imaging
Spatial Kα imaging of flat and pyramid targets has been accomplished with a 1D
spectral, 1D spatial imaging spectrometer. This scheme is also referred to as FSSR-
1D (Focusing spectrometer with spatial resolution in one dimension [55]). The
spectrometer is based on a versatile spherical mylar crystal and was operated by
S. Pikuz from Lomonosov Moscow State University. For flat and pyramid targets,
the Kα emission occurred from an area of diameter 100 µm, which is significantly
bigger than the focal spot size (12 µm). The most interesting feature however was
observed for flat targets. A sidepeak of lower Kα intensity was measured repeatedly
(see figure 5.11). This corresponds to a concentric area of diameter 500 µm around
81
25µm flat
25µm pyramid
0 0.5 1.0-0.5-1.0 1.5-1.5x [mm]
inte
nsi
ty [
a.u
.] sidepeak @+250µm−
Figure 5.11: Spatial Kα1 image obtained with the spherical mylar spectrometer. Only forflat targets, a sidepeak appears 250 µm away from the area of laser-target interaction. Themain peak is much wider than the focal spot (∼ 150µm > 10µm). The vertical offset wasintroduced on purpose for a better comparability.
the laser focus. The same phenomenon was observed by M. D. J. Burgess et al.
[56] at intensities of 1017 W/cm2. Penumbral imaging of 20 keV X-rays revealed a
concentric area of ∼ 200 µm diameter and ∼ 50 µm width. The phenomenon was
linked to self-generated magnetic fields of concentric structure. Hot electrons that
are created by the above mentioned mechanisms of laser-absorption escape from the
plasma (front side of the target). They are bent back towards the target by a strong
self-consistent magnetic field. The magnetic field has to be placed where the local
minimum in Kα intensity occurs (see figure 5.11). Magnetic fields on the order of
106 Gauss are necessary to bend electrons with velocities of order 10 keV on a radius
of order 100 µm.
By introducing a pyramid substrate in front of the titanium foil, the side-peak
disappeared (see figure 5.11). Radial energy transport via electrons is inhibited or
simply can not effect the emission pattern of Kα. In conclusion, undesired side-
peaks can be avoided with the pyramid scheme. This guarantees for a point-like Kα
source, as necessary for many application such as x-ray diffraction.
5.7 Summary
Measurements together with PIC simulations form a complete picture of the pre-
sented micro-shaped targets.
82
• Kα and hard x-ray spectra have been measured for pyramid, wedge and flat
targets. The results show a polarization dependency of the Kα and hard x-ray
emission for wedges. Pyramids and wedges failed to produce a brighter Kα
source. The hard x-ray yields reveal that p-wedges are superior to flat targets.
This indicates the advent of cone guiding.
• PIC simulations are in accordance with the measured hard x-ray yield. They
predict some electron guiding for the wedge geometry, although the pyramid
angle is insufficient for significant guiding. Especially no electrons of energy
around the Kα cross section peak are channelled. This explains why shaped
targets failed to produce a brighter Kα source.
• The presented pyramid allows for a point-like Kα source without undesired
sidepeaks.
83
Chapter 6
Experimental Characterization
of Silicon Targets
6.1 Characterization of von-Hamos Spectrometer
The performance of the spectrometer was characterized by carefully analyzing every
scanned film. It was found, that the spectral resolution and focusing quality is
in good agreement with the theoretical prediction. The limits of the wavelength
calibration are discussed and the density scale length of the plasma is determined
for the laser parameters that were used for all experiments on silicon and spheres
targets.
6.1.1 Spectral Characterization
Spectral Resolution The spectral resolution can be determined by the FWHM
spectral linewidth of the Kα1/2 doublet (7.126 A and 7.128 A). For Si, the Kα1/2
width is 2 mA respectively [57]. With a spectral resolution of λ/∆λ = 3600, the
doublet would be resolvable. The predicted resolution1, however is only 1600 for
Si Kα1/2, which is why, the lines could not be resolved in the presented experi-
ments. The spectral linewidth, of the doublet obtained by averaging over all scans
is (6.6±0.8) mA. Hence, the measured linewidth can be used to calculate the spectral
resolution λ/∆λ = 7127/6.6 = 1080. This agrees reasonably with the prediction.
1see section 3.4.3
84
6.6 6.65 6.7 6.75
0
0.001
0.002
0.003
6.8
wavelength [A]
ph
oto
ns/
sho
t/µ
m2
qr
abcdjkl
polished
rough
Si3N4
coated
Figure 6.1: The Heα line and Li-like satellite lines (qr, abcd, jkl) are shown for threedifferent flat silicon target configurations. A polished silicon wafer, a rough silicon waferand a 100 nm Si3N4 coated rough silicon wafer were shot at 45⋄. The spectral resolution(∼ 4 mA) can be estimated by the rising edge of the Heα line and by the distance betweenthe satellites.
Several runs revealed plasma lines from highly ionized silicon ions as it can be seen
in figure 6.1. In particular, Si Heα and Li-like satellite lines could be identified. As
shown in figure 6.1, several satellites are grouped together causing distinct peaks.
These peaks could be partially resolved in some runs. The spectral distance be-
tween the satellites is ∼ 10 mA, which corresponds to a lower resolution limit of
675. Last but not least, the rising edge of the Heα line can also be used to esti-
mate the achieved resolution. The short wavelength side of the Heα line is free from
satellites which only occur on the long wavelength side. Its rise is determined by
the line broadening. As it can be seen in figure 6.1, different target configurations
yielded different rises. The fastest rise (∼ 8 mA) was achieved by a plane, polished
Si wafer. This corresponds to a lower resolution limit of 830. This is also consistent
with the predicted resolution of ∼ 1300 for Heα (see section 3.4.3). Line broadening
will be discussed in more detail in Chapter 6.4.
Wavelength Calibration It is impossible to determine the accuracy of the wave-
length calibration. The spectral window under observation only contains one wave-
length normal. This is the model point Si Kα1/2 (7.127 A). The plasma lines Si
Heα and the Li-like satellites can not serve as wavelength normals because they
are effected by line broadening and by line shifts [58]. Experiments using a 130
fs 150 mJ, 400 nm, 1019 W/cm2 peak intensity laser of unknown contrast ratio at
85
Lawrence Livermore National Labs (LLNL) determined a wavelength of 6.648 A for
Si Heα [59]. The HULLAC atomic code calculates a wavelength of 6.648 A for the
unshifted Si Heα line [60]. All spectra presented in this work revealed a Si Heα
wavelength ≥ 6.650 A (e.g. figure 6.1), which could be seen as a real redshift or a
systematic error in the wavelength calibration. The plasma lines will be discussed
in more detail in Chapter 6.4.
6.1.2 Focusing Quality
The focusing quality was determined from the spatial linewidth of the line focus on
every film. As discussed in section 3.4.3, a linewidth of 310-270 µm can be predicted
for the spectral window from 6.6 A to 7.2 A.
The first set of runs revealed a fairly good focusing quality ranging from 400-450 µm
as depicted in figure 6.2.1. Realigning the spectrometer caused a worsening of the
focusing quality (500-600 µm). A final realignment improved the focusing quality up
to its prediction (300-250 µm). The outcome of every alignment is depicted in figure
6.2.1. Besides that, representative scans of poor and good alignment are displayed
in figure 6.2.2. The focusing quality is changing the spectral linewidth of the Si
Kα doublet to a little degree. The FWHM spectral linewidth is (6.7 ± 0.4) mA,
(7.0± 0.9) mA and (6.3± 0.9) mA for the 1st, 2nd and 3rd alignment respectively.
6.1.3 Angle Scan
Preliminary experiments on spheres coated targets were performed using ∼ 38 fs,
800 nm pulses with a contrast ratio of 10−3. Prepulses were found at 8 and 20 ps (see
section 3.2.3). An angle scan for such laser conditions revealed a rather significant
preplasma of scale length L = 12µm. This is much bigger than the diameter of
the biggest sphere size (2.9 µm) and makes it impossible to study an absorption
enhancement of spheres-coatings. A suppression of the prepulses was necessary to
reduce the density scale length. A clean (10−6) frequency doubled laser pulse (400
nm, ∼ 120 fs, 12 mJ) focused to ∼ 1017W/cm2 was incident on glass targets. An
angle scan2 revealed a peak absorption at 55⋄ which is a strong indication of a
short scale length plasma and vacuum heating contributing to the absorption. With
equation (2.39), a peak angle of 55⋄ corresponds to a plasma scale length Ln ≈ 60nm
2conducted by A. Sumeruk, High Intensity Laser Science Group, University of Texas at Austin
86
6.6 6.7 6.8 6.9 7 7.1 7.2
100
200
300
400
500
600
700
800
3rd
2nd
1st
theory
alignments2nd alignment
1st alignment
3rd alignment
and theory
wavelength [A]
spa
tia
l wid
th [µ
m]
KαHeα Li-like
6.2.1: Spatial line width as a function of thewavelength for three different spectrometer align-ments and theoretical limit.
900
850
22002100200019001800170016001500140013001200110010009008007006005004003002001000
Good Focusing Quality
line width w=270mm
850
800
22002100200019001800170016001500140013001200110010009008007006005004003002001000
Kα HeαLi-like Satellites
Bad Focusing Quality
line width w=600mm
6.2.2: Scanned x-ray films from blank polished silicon wafer (run 12, top) and 0.1 µmspheres target (run 22, bottom). The focusing quality was improved a lot by realigning thespectrometer.
Figure 6.2: A systematic analysis of the focusing quality (upper figure) and sample scansof bad and perfect focusing (bottom figure).
87
ph
oto
ns/
sho
t/sr
ad
/mA
wavelength [A]7.1 7.11 7.12 7.13 7.14 7.15
1.5 x 106
1.0 x 106
0.5 x 106
0
0.26µm
rough
9.6x106
<1x106
ph
oto
ns/
sho
t/sr
ad
9.3x106
6.3.1: Typical Kα yields from plane andspheres-coated targets at 0⋄.
1.4x107
6.1x1067.9x106
6.9x106
ph
oto
ns/
sho
t/sr
ad
ph
oto
ns/
sho
t/sr
ad
/mA
wavelength [A]7.1 7.11 7.12 7.13 7.14 7.15
3.0 x 106
2.0 x 106
1.0 x 106
0
0.26µm
rough
polished1.3x107
4.8x106
6.3.2: Typical Kα yields from plane andspheres-coated targets at 45⋄.
Figure 6.3: Spheres coated Si targets show a huge increase of the Kα yield compared toplane Si targets.
on the order of the smallest sphere size (100 nm). Consequently, frequency doubled
light allows to study the absorption properties of spheres coatings.
6.2 Kα Yield Comparison
All experimental results presented in this section were obtained with the following
laser parameters: 400 nm, 100-150 fs, 9.3-11.2 mJ pulse energy and focused inten-
sities of (1.3 ± 0.6) × 1017 W/cm2. For every target, 70 shots were integrated on
AGFA Structurix D7 film.
6.2.1 Flat vs Spheres
Plane Si targets and 0.26 µm spheres coated Si targets were shot at 0⋄ and 45⋄ angle
of incidence. Targets coated with spheres yield many times more Kα photons per
shot than plane targets.
Zero Degree Incidence As depicted in figure 6.3.1, 0.26 µm spheres enhance
the FWHM integrated Kα yield by 10 times compared to the uncoated target. This
result was obtained repeatedly. In case of the uncoated targets, the rough side of
the silicon wafer was shot. The surface roughness should allow for non-zero degree
angles of incidence although the target surface is normal to the laser. Therefore, a
rough uncoated target gives a higher absorption than a perfectly flat polished wafer.
88
Nevertheless, the spheres coated target, if shot from the rough or polished side, gave
a much higer Kα yield.
45 Degree Incidence, P-Polarization Since the angle scan revealed a maxi-
mum absorption for 55⋄ angle of incidence, spheres coated targets were also com-
pared to uncoated targets at an angle in favor of the plane target. For alignment
reasons, 45⋄ was chosen instead of 55⋄. This is close enough to the angle of maximum
absorption. As depicted in figure 6.3.2, 0.26 µm spheres enhance the integrated Kα
yield by a factor of up to 3 compared to rough or polished plane silicon wafer. Rough
Silicon targets gave a slightly higher integrated yield than polished wafers. As it
can be seen in figure 6.3.2, all results were reproduced repeatedly with very small
yield deviations.
One has to draw two conclusions from this data. Firstly, a silicon target coated with
a monolayer of 0.26 µm spheres gives a brighter Kα source than a plane target at
optimal angle of incidence. Secondly, target-to-target variations for the same target
type are not an issue since the data could be reproduced almost exactly.
[61] gives an idea about typical Kα yields at comparable laser conditions. Plane
silicon targets were shot at 45⋄ with a 100 fs, 620 nm, 2-4.5 mJ pulse energy, 10−7
contrast ratio and focused intensities of 1017 W/cm2. Kα yields of ∼ 5.0 × 106
photons/shot/srad/mA were obtained. This is very close to the results presented
in this work. The normalization of the AGFA film is based on a calibration with 8
keV3 instead of 1.7 keV4 photons, so the absolute photon yields presented in this
work systematically underestimate the real yields.
6.2.2 Sphere Size Scan
Having shown that a spheres coating greatly increases the Si Kα yield, a sphere size
scan was performed. A strong Si Kα yield dependency on the sphere size was found
both for 0⋄ and 45⋄ angle of incidence.
Zero Degree Incidence As depicted in figure 6.4.1, the Si Kα clearly depends
on the sphere size. The highest integrated Kα yield was obtained for a sphere size of
3Cu Kα4Si Kα
89
1.0x107
0.4x107
0.6x107
0.3x107
ph
oto
ns/
sho
t/sr
ad
p
ho
ton
s/sh
ot/
sra
d/m
A
wavelength [A]
0.26µm
0.36µm
0.50µm
2.90µm
7.1 7.11 7.12 7.13 7.14 7.15
1.5 x 106
1.0 x 106
0.5 x 106
0
6.4.1: Sphere size scan for 0⋄ angle of inci-dence.
7.1 7.11 7.12 7.13 7.14 7.15wavelength [A]
ph
oto
ns/
sho
t/sr
ad
/mA
4 x 106
3 x 106
2 x 106
1 x 106
0
0.26µm
0.50µm
0.36µm
0.10µm
2.90µm
2.2x107
1.6x107
1.7x107
1.5x107
0.3x107 ph
oto
ns/
sho
t/sr
ad
6.4.2: Sphere size scan for 45⋄ angle of inci-dence.
Figure 6.4: A sphere size scan reveals a great dependency of the Si Kα yield on the spherediameter. Both for 0⋄ and 45⋄ angle of incidence, the 0.26 µm sphere gives the highest andthe 2.9 µ sphere gives the lowest yield.
0.26 µm (1× 107 photons/shot/srad). The lowest yield was obtained with a sphere
size of 2.9 µm (0.3× 107 photons/shot/srad). Other sphere sizes (0.36 and 0.5 µm)
give Kα yields that fall in between. Even the sphere size with the lowest Si Kα yield
is brighter than a plane target.
45 Degree Incidence, P-Polarization Also for optimized incidence, the Si
Kα yield clearly depends on the sphere size (see figure 6.4.1). Again, the high-
est integrated Kα yield was obtained for a sphere size of 0.26 µm (2.2 × 107 pho-
tons/shot/srad). The lowest yield was again obtained with a sphere size of 2.9 µm
(0.3 × 107 photons/shot/srad). Other sphere sizes (0.1, 0.36 and 0.5 µm) give Kα
yields that fall in between. One should mention, that for 45⋄ angle of incidence in
p-polarization, the sphere size with the lowest Si Kα yield (2.9 µm) was no longer
brighter than a plane target. 2.9 µm spheres gave a slightly lower yield than an
average plane silicon wafer. For every sphere size, the Kα yield could be increased
when switching from normal incidence to oblique incidence in p-polarization. The
2.9 µm sphere size was the only exception to this observation.
6.3 Hard X-Ray Yield Comparison
The sphere size scan was done to find out whether the Kα yield enhancement is
caused by the added target roughness of a spheres coating or by the distinct geometry
90
0 0.5 1 1.5 2 2.5 3 3.5sphere size [µm]
Kα
yie
ld
[ph
oto
ns/
sho
t/sr
ad
]shot @ 45o
shot @ 0o
0.5x107
1x107
2x107
1.5x107
2.5x107
6.5.1: Kα yield as obtained from spheres-on-silicon targets (45⋄) with the von-Hamos spec-trometer.
0 0.5 1 1.5 2 2.5 3 3.5sphere size [µm]
ha
rd x
-ray
yie
ld [
a.u
.]
shot @ 0o
500
0
1000
2000
1500
6.5.2: Hard x-ray yield as obtained fromspheres-on-glass targets (0⋄) with HXR de-tector (22 keV cutoff energy, 54-55” away).
Figure 6.5: Kα and hard x-ray yield show exactly the same dependency on the spheresize. Flat targets are represented by a diameter of 0. The hard x-ray data is courtesy of A.Sumeruk.
of the sphere. Since the sphere size is on the order of the wavelength of the laser and
the plasma scale length, one might expect a sphere size dependant laser absorption
behavior. This could, but clearly doesn’t have to result in a Si Kα yield dependency
on the sphere size. Measuring the Si Kα yield is mostly sensitive to electrons around
the Si Kα cross section peak5 (5-7 keV [53] [54]). In order to obtain information
about the generation of hotter electrons, a set of six hard x-ray detectors has been
employed. They were filtered to obtain the 1/e2 cutoff energies 22, 32, 39, 52 65
and 75 keV respectively. The hard x-ray data presented in this chapter is courtesy
of A. Sumeruk.
6.3.1 Dependency on Sphere Size
Figure 6.5.2 shows the hard x-ray yield that has been measured with a 22 keV filter
as a function of the sphere size. The data has been taken at 0⋄ angle of incidence but
shows exactly the same tendency at the optimized angle 55⋄. One can clearly see
a strong yield dependency on the sphere size. The > 22 keV x-ray yield is peaking
for a sphere size of 0.26-0.36 µm. This tendency was measured repeatedly.
The hard x-ray yield dependency is compared to the Kα dependency that was dis-
cussed in the last section. Figure 6.5.1 shows the Kα yield as a function of the sphere
5i.e. the K-shell ionization cross section peak
91
size for 0⋄ and 45⋄ angle of incidence. Every data run is represented by a filled or
open square. For most of the sphere sizes, multiple runs were conducted. In this
case, vertical lines are min-max error bars. The broken line is a linear interpolation
after averaging over multiple runs for one sphere size. Both for 0⋄ and 45⋄ angle of
incidence one can clearly see the Kα yield peaking at 0.26 to 0.36 µm. Much more
interestingly, the overall yield dependency follows exactly the same tendency as the
hard x-ray yield depicted in figure 6.5.2.
6.3.2 Hot Electron Temperature
Yield Every hard x-ray detector with the aforementioned cutoff energy gave the
same sphere size dependency as depicted in figure 6.5.2. For example, a signal height
of 200 mV was obtained from the hard x-ray detector, when the strongest of the
aforementioned filters (75 keV) was used for 0.26 µm spheres. With the absolute
energy calibration of the hard x-ray detectors (see section 3.6), this corresponds
to more than 10−5 percent of the laser pulse energy being converted to > 75 keV
x-rays although the ponderomotive potential is only on the order of 2-3 keV (400
nm, 1017 W/cm2). Apparently, a significant amount of electrons as hot as 75 keV
are produced for the optimal sphere size 0.26 µm.
Temperature Assuming a Maxwellian electron temperature distribution, and fit-
ting the corresponding x-ray spectrum to the measured bremsspectrum yields an
electron temperature of about 13 keV for 0.1 µm and 2.9 µm spheres. Again, the
electron temperature peaks for the optimal sphere of 0.26 µm, i.e. a temperature
of 20 keV is obtained. This is significantly above the average electron energy that
should be obtainable via vacuum heating, i.e. the ponderomotive potential.
6.4 Plasma Lines
Plasma transitions are subject to line broadening and line shifting. Both effects
depend on the plasma parameters, i.e. the temperature and the density (profile).
Besides, the yield of a distinct plasma line also depends on the aforementioned
plasma parameters. Quantitative plasma spectroscopy allows to determine the tem-
perature and the density from intensity ratios and line broadening of appropriate
lines [58].
92
7-9 mA
polished
rough
2x
uncoated
0.003
0.002
0.001
0
ph
oto
ns/
sho
t/µ
m2
6.646.60 6.666.62 6.68 6.70
wavelength [A]
6.6.1: A steep rise of the Heα line was mea-sured repeatedly for plane targets.
3x
0.26 µm
12-14 mA
ph
oto
ns/
sho
t/µ
m2
wavelength [A]
0.0015
0.0010
0.0005
0
6.646.60 6.666.62 6.68 6.70
6.6.2: A slow rise of the Heα line was mea-sured repeatedly for spheres targets, e.g. for0.26 µm.
Figure 6.6: Comparison of the spectral shape of the Heα line for plane and 0.26 µm spherestargets.
For this work, a qualitative analysis of one particular spectral feature will be suf-
ficient to explain, why spheres-coated targets show enhanced x-ray yields. The
spectral feature of interest is the rising edge of the Heα line.
Concept The denser the plasma, the broader a plasma line turns out to be. Un-
fortunately, one can not determine the width of the Heα line, because it has con-
tributions of satellites on the high-wavelength side (see figure 6.1). Nevertheless,
information about the plasma density profile can be inferred from the satellite-free
low-wavelength side. A plasma of low density would result in a fast rise. The slower
the observed rise, the less are the contributions of low density regions within the
plasma profile. Therefore, a slow rise has to be attributed to a steep plasma density
gradient. A steep rise, however, has to be attributed to an integration over different
plasma densities with significant contributions of low6 density regions. [25].
Observation The rising edge of the Heα from plane and spheres targets were
compared. It was found, that spheres targets showed slower rises that plane targets.
The spectral accuracy did not allow for a comparison of the different sphere sizes.
For plane targets and 0.26 µm sphere targets, the results are in figure 6.6. The upper
6One should keep in mind that this discussion qualitative. So is the use of low density and high
density.
93
subfigure shows the Heα line of three experiments with 0.26 µm sphere targets. By
averaging, one obtains a rise within 12-14 mA for the low-wavelength side. The
lower subfigure shows the Heα line of two different plane targets, one of which is
a polished wafer and one of which is a rough wafer. The average rise is 7-9 mA.
The difference between a slow and a sharp rise is very distinct and can even be
seen by eye on the scanned films, e.g. from the two scans shown above (see figure
6.2.2). The upper one corresponds to a flat target with sharp rise and the lower one
corresponds to a spheres target with soft rise. Apparently, a spheres coating alters
the plasma density profile in front of the substrate.
Conclusion A coating of spheres allows for a shorter plasma scale length than
an uncoated target. Somehow, a monolayer is capable of reducing the plasma ex-
pansion before the peak of the laser pulse arrives. According to the angle scan (see
section 6.1.3), vacuum heating contributes significantly to the laser absorption. The
fractional absorption as discussed in section 2.3.6 was derived empirically by Brunel.
Improvements by Kato et al. [13] revealed the influence of the the electron plasma
density n:
fvh ∝ 1
1 − ω2
ω2pe
(6.1)
With ω2pe ∝ n, the fractional absorption increases, if the laser interacts with a denser
plasma. A shorter density scale length as observed from the spheres spectra allows
the laser to interact with denser regions of the plasma. Layers of spheres optimize
the plasma profile for absorption via vacuum heating.
6.5 Interpretation
Although the mechanism responsible for the resonance like behavior at 0.26 µm has
not been found yet, the understanding of processes that may be involved will be
listed in this section.
94
6.5.1 Local Field Enhancement
Local field enhancement on the surface of microdroplets by a Mie resonance has
been used recently to explain strongly anisotropic ion and proton yields observed
from high intensity laser irradiated droplets [6].
To study the local field enhancement of a sphere, a Mie code was used to evaluate
the field strength in the vicinity of the sphere surface. The code was implemented
by A. Sumeruk and the results are courtesy of him. It was found, that a strongly
anisotropic field pattern built up in the near field of the spheres, with a local intensity
inhancement peaking for a sphere size of 0.1 µm. The maximum intensity was found
to be 15-20 times bigger than the incident intensity. For a sphere size of 0.26 µm,
the intensity enhancement dropped to a factor of 5-10. The model peaking at 0.1
µm is at odds with the experimentally observed x-ray and temperature resonance
at 0.26 µm.
6.5.2 Multi Pass Vacuum Heating
A plasma electron can obtain an average energy on the order of the ponderomotive
potential in a single laser cycle. With equation (2.40), the ponderomotive potential
is 2-3 keV for 400 nm and 1017 W/cm2. B. Breizman et al. [62] and H. M. Milch-
berg et al. [63] have proposed schemes, where electrons can acquire many times
the ponderomotive potential by receiving multiple kicks of the laser field. This
clearly requires that the electron’s phase with respect to the laser field is reset after
each kick. Otherwise, the electron would just be quivering. Clusters or microscale
particles such as the used spheres can allow for the required phase reset.
Resonant Heating One can imagine a situation, where the electron receives a
ponderomotive kick by the light field and then disappears into the overdense plasma
formed by the sphere. Appropriate dimensioning of the sphere size and spacing could
enhance such electrons re-exiting the sphere, that are in phase with the laser field to
receive repeated kicks. This hypothesis would follow resonant heating as proposed
by H. M. Milchberg [63].
Stochastic Heating A much weaker phase requirement is imposed on the electron
in what was termed stochastic heating by B. Breizman et al. [62]. An electron
95
doesn’t have to disappear from the influence of the light field for exactly have a
laser cycle to be heated again. It would be sufficient to reset the phase randomly
after every ponderomotive kick to be heated in a stochastic manner.
6.6 Summary
The results obtained can be summarized as follows:
• A target covered with spheres can give as much as ten times more Si Kα and
1000 times more hard x-rays (> 20 keV) than a plane target. The enhancement
is less distinct for oblique incidence than for normal incidence. An analysis
of the Heα line reveals, that spheres layers optimize the plasma profile for
absorption via vacuum heating.
• A sphere size scan always reveals the same yield dependency upon the sphere
size, no matter if normal or oblique incidence is chosen and no matter if the Kα
yield or the hard x-ray yield is regarded. Mie-enhancement failed to explain
the resonance-like behavior for 0.26 µm spheres. Multi pass vacuum heating
schemes are likely to have an effect.
• Electron temperatures inferred from the hard x-ray spectrum reveal Maxwellian
electron temperatures many times above the ponderomotive potential. Multi
pass vacuum heating schemes are made responsible.
96
Chapter 7
Future Prospects
Both target families presented in this work are promising approaches for plasma
heating and x-ray probing. More interestingly, they contribute to the understanding
of advanced target concepts that are currently under broad investigation.
Pyramid Targets To reveal the details of hot electron generation and electron
transport, further diagnostics have to be implemented. It is inconvenient to infer
such mechanisms indirectly via the measurement of x-ray spectra. A much more
convenient and illuminating approach would be the employment of an electron spec-
trometer.
At the same time, the two biggest deficiencies of the presented guiding geometry
have to be eliminated. That is to say, the pyramid angle has to be narrowed to
reduce the amount of transmitted electrons. A free-standing silicon pyramid with
thin walls would prevent heat from being deviated from the tip.
Spheres Targets The details of the resonance-like behavior is the most interest-
ing facette of this target concept. The study of the plasma density (profile) and
temperature has to be extended. A next-generation crystal spectrometer has to
be employed to resolve satellites. The spectral window of observation has to be
extended to pairs of plasma lines. Their ratio allows for the determination of the
plasma temperature. Accurate determination of the widths of the plasma lines is
necessary to infer the plasma density.
Substrates other than silicon may have to be considered for that. Besides, a higher
97
Z substrate should enhance the Kα yield even more if its Kα cross section peak
matches the observed electron temperature.
To study the influence of multi-pass vacuum heating, a laser pulse scan has to be
conducted.
It will be absolutely crucial for future experiments to assure a consistent quality
of the spheres monolayers. Pre-, insitu- and post-mortem diagnostics have to be
improved.
98
Bibliography
[1] J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, ”Laser compression of
matter to supreme-high densities”, Nature 239, 139-142 (1972)
[2] T. Ditmire, J. Zweiback, V. P. Yanovsky, T. E. Cowan, G. Hays, K. B. Whar-
ton, ”Nuclear fusion from explosions of femtosecond laser-heated deuterium
clusters”, Nature 398, 489-492 (1999)
[3] R. Kodoma, P. A. Norreys, K. Mima, A. E. Dangor, R. G. Evans, H. Fujita,
Y. Kitagawa, K. Krushelnick, T. Miyakoshi, N. Miyanaga, T. Norimatsu, S. J.
Rose, T. Shozaki, K. Shigemore, A. Sunahara, M. Tampo, K. A. Tanaka, Y.
Toyoma, T. Yamanaka, M. Zepf, ”Fast heating of ultrahigh-density plasma as
a step towards laser fusion ignition”, Nature 412, 798-802 (2001)
[4] R. Kodoma, ”Fast heating scalable to laser fusion ignition”, Nature 418, 933-
934 (2002)
[5] T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Falcone, M. D. Perry, ”In-
teraction of intense laser pulses with atomic clusters”, Phys. Rev. A 53, 3379
(1996)
[6] D. R. Symes, A. J. Comley, R. A. Smith ”Fast-Ion Production from Short-Pulse
Irradiation of Ethanol Microdroplets” Phys. Rev. Lett 93, 145004 (2004)
[7] Z. L. Chen, ”Enhancement of energetic electrons and protons by cone guiding
of laser light”, Phys. Rev. E 71, 036403 (2005)
[8] T. Nakamura, S. Kato, H. Nagatomo, K. Mima, ”Surface-Magnetic-Field and
Fast-Electron Current-Layer Formation by Ultraintense Laser Irradiation”,
Phys. Rev. Lett. 93, 265002 (2004)
99
[9] T. D. Donnelly, M. Rust, I. Weiner, M. Allen, R. A. Smith, C. A. Steinke, S.
Wilks, J. Zweiback, T. E. Cowan, T. Ditmire, ”Hard x-ray and hot electron
production from intense laser irradiation of wavelength-scale particle”, J. Phys.
B 34, L313-320 (2001)
[10] W. L. Kruer, ”The Physics of Laser Plasma Interactions”, Westview Press,
Boulder (2003)
[11] D. W. Forslund, J. M. Kindel, K. Lee, ”Theory of Hot-Electron Spectra at High
Laser Intensity”, Phys. Rev. Lett. 39, 5 (1977)
[12] K. Estabrook, W. L. Kruer, ”Properties of Resonantly Heated Electron Distri-
butions”, Phys. Rev. Lett. 40, 1 (1978)
[13] S. C. Wilks, W. L. Kruer, ”Absorption of Ultrashort, Ultra-Intense Laser Light
by Solids and Overdense Plasmas”, IEEE J. Quant. Electr. 33, 11 (1997)
[14] P. Gibbon, E. Foerster, ”Short-pulse laser-plasma interaction”, Plasma Phys.
Control. Fusion 38, 769-793 (1996)
[15] F. Brunel, ”Not-So-Resonant Resonant Absorption”, Phys. Rev. Lett. 59, 1
(1987)
[16] W. L. Kruer, Estabrook, ”J x B heating by very intense laser light”, Phys.
Fluids 28, 430 (1985)
[17] S. C. Wilks, W. L. Kruer, M. Tabak, A.B. Langdon, ”Absorption of Ultra-
Intense Laser Pulses”, Phys. Rev. Lett. 69, 9 (1992)
[18] P. J. Catto, R. M. More, ”Sheath inverse bremsstrahlung in laser produced
plasmas”, Phys. Fluids 20, 704 (1977)
[19] T.-Y. B. Yang, W. L. Kruer, R. M. More, A. B. Langdon, ”Absorption of laser
light in overdense plasmas by sheath inverse bremsstrahlung”, Phys. Plasmas
2, 8 (1995)
[20] Y. Sentoku, K. Mima, H. Ruhl, Y. Toyoma, R. Kodoma, T. E. Cowan, ”Laser
light and hot electron micro focusing using a conical target” Phys. Plasmas 11,
6 (2004)
100
[21] D. Strickland, G. Mourou, ”Compression of Amplified Chirped Optical Pulses”,
Opt. Comm. 56, 219 (1985)
[22] M. Pessot, P. Maine and G, Mourou, ”1000 Times Expansion / Compression of
Optical Pulses for Chirped Pulse Amplification” Optics Comm. 62, 419 (1987)
[23] R. L. Fork, O. E. Martinez, and J. P. Gordon, ”Negative Dispersion Using Pairs
of Prisms”, Opt. Lett. 9, 150 (1984)
[24] E. B. Treacy, ”Optical Pulse Compression with Diffraction Gratings”, IEEE J.
Quant. Electron., 5, 454 (1969)
[25] U. Andiel, ”Isochore Heizung von festem Aluminium mit Femtosekunden-
Laserpulsen: eine rontgenspektroskopische Untersuchung der K-
Schalenemission”, Max Planck Institute for Quantum Optics, 85740 Garching,
Germany, Report No. 270, (December 2001)
[26] G. Albrecht, A. Antonetti, G. Mourou, ”Temporal Shape Analysis Of Nd3+:
YAG Active Passive Mode-Locked Pulses”, Opt. Comm. 40, 1 (1981)
[27] R. W. Boyd, ”Nonlinear Optics”, 1st edition Academic Press, San Diego (1992)
[28] K. B. Wharton, C. D. Boley, A. M. Komashko, A. M. Rubenichik, J. Zweiback,
J. Crane, G. Hays, T. E. Cowan, T. Ditmire, ”Effects of nonionizing prepulses
in high-intensity laser-solid interactions”, Phys. Rev. E 64, 025401 (2001)
[29] Y. Sentoku, private communication
[30] R. Micheletto, H. Fukuda, M. Ohtsu, ”A Simple Method for the Production
of a Two-Dimensional, Ordered Array of Small Latex Spheres”, Langmuir 11,
3333-3336 (1995)
[31] K. Kopitzki, ”Introduction to solid-state physics”, 3rd edition Teubner Studi-
enbcher, Stuttgart (1993)
[32] D. Vaughan, ”X-Ray Data Booklet”, Center for X-Ray Optics, Lawrence Berke-
ley National Laboratory, Berkeley, California, (2001)
101
[33] L. v. Hamos, ”Rontgenspektroskopie und Abbildung mittels gekrummter
Kristallreflektoren. Kurze Orginalmitteilungen.”, Naturwissenschaften 20, 705
(1932)
[34] L. v. Hamos, ”Rontgenspektroskopie und Abbildung mittels gekrummter
Kristallreflektoren”, Ann. Phy. 17, 716 (1933)
[35] P. Beiersdorfer et al., ”High-resolution x-ray spectrometer for an electron beam
ion trap”, Rev. Sci. Instrum. 61, 9 (1990)
[36] U. Teubner et al., ”X-ray spectra from highly ionized dense plasmas produced
by ultrashort laser pulses”, Appl. Phys. B 62, 213 (1996)
[37] G. Hoelzer, O. Wehrhan, E. Foerster, ”Characterization of flat an bent crystals
for x-ray spectroscopy and imaging”, Cryst. Res. Technol., 33, 555-567 (1998)
[38] E. Foerster, J. Heinisch, P. Heist, G. Hoelzer, I. Uschmann, F. Scholze, F. Schae-
fers, ”Measurement of the Integrated Reflectivity of Elastically Bent Crystals”,
BESSY Annu. Rep. 481 (1992)
[39] B. L. Henke, F.G. Fujiwara, M. A. Tester, C. H. Dittmore M. A. Palmer, ”Low-
energy x-ray response of photographic films. II. Experimental characterization”,
J. OPt. Soc. Am. B 1, 6 (1984)
[40] C. G. Vonk, ”A Reavaluation of Film Methods In X-ray Scattering”, The
Rigaku Journal 5, 2 (1988)
[41] A. E. Siegman, ”Lasers”, University Science Books, Sausalito, California
[42] L. E. Hargrove, ”Locking of He-Ne Laser Modes By Synchronous Intracavity
Modulation”, Appl. Phys. Let. 5, 1 (1964)
[43] CXRO, Center For X-Ray Optics, X-Ray Interaction With Matter Calculator,
”http://www.cxro.lbl.gov/tmp/xray6023.dat”
[44] O. Wehrhan, ”private communication”, X-Ray Optics Group, University of
Jena, Germany
[45] A. Sumeruk, laboratory notebook, High Intensity Laser Science Group, Uni-
versity of Texas at Austin, USA
102
[46] NIST, National Institute of Standard and Technology, Tables of X-Ray
Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients,
”http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html”
[47] A. P. Shevelko, ”Compact focusing von-Hamos spectrometer for quantitative
x-ray spectroscopy”, Rev. Scien. Instr. 73, 10 (2002)
[48] B. E. O. Rourke, H. Kuramoto, Y. M. Li, S. Ohtani, X. M. Tong, H. Watanabe
and F. J. Currell, ”Dielectric recombination in He-like titanium ions”, J. Phys.
B, 37, 2343-2353 (2004)
[49] NIST, National Institute of Standard and Technology, Atomic Spectra
Database Lines Data, ”http://physics.nist.gov/cgi-bin/AtData/display.ksh”
[50] A. H. Gabriel, T. M. Paget, Mon. Not. R. Astron. Soc. 160, 99 (1972)
[51] A. Djaoui, R. Evans, T. A. Hall, A. Badger, F. B. Rosmej, ”Li- and He-like
satellite emission from aluminium plasma produced by a 12 ps KrF laser”, J.
Phys. B., 28, 1921-1929 (1995)
[52] CXRO, Center For X-Ray Optics, X-Ray Attenuation Length of Solids,
”http://www.cxro.lbl.gov/optical constants/atten2.html”
[53] E. Casnati, A. Tartari, C. Baraldi ”An empirical approach to K-shell ionisation
cross section by electrons”, J. Phys. B. 15, 155-167 (1982)
[54] E. Casnati, A. Tartari, C. Baraldi ”Corrigenda: An empirical approach to K-
shell ionisation cross section by electrons”, J. Phys. B. 16, 505-505 (1983)
[55] P. Monot, T. Auguste, S. Dobosz, P. D’Oliveira, S. Hulin, M. Bougeard, A.
Ya. Faenov, T. A. Pikuz, I. Yu. Skobelev, ”High-sensitivy, portable, tunable
imaging X-ray spectrometer based on a spherical crystal and MCP”, Nucl.
Instr. Meth. Phys. Res. A 484, 299-311 (2002)
[56] M. D. J. Burgess, B. Luther-Davies, K. A. Nugent, ”An experimental study of
magnetic fields in plasmas created by high intensity one micron laser radiation”,
Phys. Fluids 28, 7 (1985)
[57] M.O. Krause, J. H. Oliver, ”Natural Width of Atomic K and L Levels, Ka X-
Ray lines and several KLL auger lines”, J. Phys. Chem. Ref. Data 8, 2 (1979)
103
[58] H. R. Griem, ”Plasma Spectrocopy”, McGraw-Hill Book Company, New York
(1964)
[59] B. K. F. Young et al., ”High-resolution x-ray spectrometer based on sperically
bent crystals for investigation of femtosecond laser plasmas”, Rev. Sci. Intr. 69,
12 (1998)
[60] E. Behar, H. Netzer, ”Inner-Shell 1s-2p Soft X-Ray Absorption Lines”, Astro-
phys. J. 570, 165-170 (2002)
[61] T. Missalla, I. Uschmann, E. Foerster, G. Jenke, D. von der Linde, ”Monochro-
matic focusing of subpicosecond x-ray pulses in the keV range”, Rev. Sci. Instr.
70, 2 (1999)
[62] B. N. Breizman, A. V. Arefiev, M. V. Fomyts’kyi, ”Nonlinear physics of laser-
irradiated microclusters”, Phys. Plasm. 12, 056706 (2005)
[63] T. Taguchi, T.M. Antonsen Jr., H. M. Milchberg, ”Resonant Heating of a Clus-
ter Plasma by Intense Laser Light”, Phys. Rev. Lett. 92, 20 (2004)
104
Vita
Stefan Kneip was born in Lohr am Main, Germany, on June 15th in 1981 as the
son of Ulla and Martin Kneip. After passing the Abitur at the Gymnasium Lohr
am Main, Germany, in July 2001 he applied successfully for a scholarship of the
Bischofliche Studienforderung Cusanuswerk for the time of his studies. Stefan reg-
istered at the Bavarian Julius-Maximilians University in Wurzburg, Germany, in
October 2001 and finished his intermediate diploma examination in October 2003.
Following another year of studies in physics and numerical mathematics he entered
The Graduate School at The University of Texas at Austin, USA. Since August 2004
he is working in Dr. Todd R. Ditmire’s High Intensity Laser Science Group, which
is part of the physics department.
Permanent Address: Buchenstraße 27
97854 Steinfeld-Hausen
Germany
This thesis was typeset with LATEX2ε1 by the author.
1LATEX2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademarkof the American Mathematical Society. The macros used in formatting this thesis were written byDinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extendedby Bert Kay, James A. Bednar, and Ayman El-Khashab.
105