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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


December 2005



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


December 2005

vi



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


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


P uls e s tretc herP uls e s tretc her

V acuumP uls e compres s or


35fs , 700mJto

T arget

35fs , 700mJto

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


10ps - 42fs , 0.5J – 0.1J to T arget

10ps - 42fs , 0.5J – 0.1J to T arget



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 )



P uls e s tretc herP uls e s tretc her



35fs , 700mJto

T arget

35fs , 700mJto

T arget



2‘‘ K DP2‘‘ K DP

120fs , 12mJto

T arget

120fs , 12mJto

T arget



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

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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

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