Characterization of ion acceleration with relativistic laser-plasma_+Diss... · 2017. 3. 28. ·...

Characterization of ion acceleration with

relativistic laser-plasma

Dissertation

zur Erlangung des akademischen Grades

Doctor rerum naturalium (Dr. rer. nat.)

vorgelegt dem Rat der

Physikalisch-Astronomischen Fakultät

Friedrich-Schiller-Universität Jena

von Dipl.-Phys. Oliver Jäckel

geboren am 10.02.1980 in Erfurt

To Katrin and Lasse

Zusammenfassung

Die vorliegende Arbeit untersucht die Prozesse, welche bei relativistischen Laser-Plasma-

Wechselwirkungen zur Erzeugung der Ionenstrahlen führen. Hierfür wurden verschiedene

Methoden zur Charakterisierung, Beobachtung und konsequenten Optimierung des Io-

nenbeschleunigungsprozesses entwickelt. Eine Serie von Skalierungsgesetzen wurde en-

twickelt, welche die Intensitätsabhängigkeit erstmals in ihren Komponenten Energie,

Laserimpulsdauer und Fokus�äche berücksichtigen. Ferner wurde eine vollständig op-

tische Beobachtungsmethode entwickelt, welche es erlaubt, die für die Ionenbeschleu-

nigung verantwortliche heiÿe Elektronenverteilung direkt und mit einer dem Laserim-

puls entsprechenden Zeitau�ösung zu analysieren, wodurch erstmalig eine vollständige

Charakterisierung und Optimierung der Beschleunigungsfelder, der Elektronentemper-

atur und der Konversionse�zienz von Laserenergie in kinetische Energie der Elektronen

möglich waren. Zudem konnte in einem einzigartigen kaskadierten Aufbau die Additiv-

ität des lasergestützen Ionenbeschleunigungsprozesses erstmalig nachgewiesen und für die

zur Verfügung stehende spektrale Kontrolle zur Erzeugung energetisch schmalbandiger

Protonenstrahlen ausgenutzt werden. Unterstützt durch numerische Simulation wird ein

detailiertes theoretisches Modell vorgestellt, welches in der Lage ist, die experimentellen

Beobachtungen dieser additiven Ionenbeschleunigung vollständig zu erklären. Der Ein-

�uss der vorgestellten Resultate reicht damit über die Grenzen der Grundlagenforschung

hinaus. Anwendungen für kompakte und kostengünstige lasergestützte Ionenbeschle-

uniger mit zum Teil einzigartigen Eigenschaften sind weit gefächert. Sie reichen von

der Kombination mit herkömmlicher Teilchenbeschleunigertechnik über die Erzeugung

sekundärer Strahlung und die Materialbearbeitung bis hin zur medizinischen Radiologie.

i

Abstract

The presented work investigates the processes which lead to the generation of ion beams

by means of relativistic laser-plasma interaction. For this purpose, speci�c methods have

been developed that serve to characterize, observe, and consequently optimize the ion

acceleration process. A series of scaling laws was derived which, for the �rst time, treat

the dependency of the ion acceleration on the laser intensity explicitly in its components

laser energy, pulse duration, and focus area. Furthermore, an all-optical observation

technique was developed which allows to directly observe and analyze the hot electron

population responsible for the ion acceleration with a temporal resolution matching that

of the laser pulse itself, enabling for the �rst time the complete characterization and opti-

mization of the accelerating �elds, the electron temperature, and the conversion e�ciency

from laser energy into kinetic energy of the electrons. With a unique staged acceleration

setup, the �rst proof of the additivity of the laser-based ion acceleration process was

achieved, granting reliable spectral control over the produced narrow-band beams. Sup-

ported by numerical simulations, a detailed theoretical model is introduced which allows

to explain the experimental observations of this additive ion acceleration satisfactorily.

The presented results have broad implications beyond mere fundamental research. Ap-

plications for compact and competitive laser-based ion accelerators with partially unique

properties are versatile, and include, for example, the combination with conventional

accelerator technology, the generation of secondary radiation, material processing, and

medical radiology.

iii

Contents

Zusammenfassung i

Abstract iii

Table of contents iv

Figures vii

Tables x

1. Introduction 1

1.1. Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Theory of laser ion acceleration 5

2.1. Laser matter interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1. Ionization processes and plasma generation . . . . . . . . . . . . . 5

2.1.2. Electrons in intense light �elds and ponderomotive force . . . . . . 7

2.1.3. Plasma properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4. Generation of hot electrons . . . . . . . . . . . . . . . . . . . . . . 12

2.1.5. Electron transport through the target foil . . . . . . . . . . . . . . 17

2.2. TNSA�Target normal sheath acceleration of ions . . . . . . . . . . . . . 18

2.2.1. 1D model of plasma expansion into vacuum . . . . . . . . . . . . . 19

2.2.2. Alternative approaches . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3. Generation of monoenergetic ion beams . . . . . . . . . . . . . . . 25

2.2.4. Electron's fountain e�ect . . . . . . . . . . . . . . . . . . . . . . . 26

3. Experimental scalings for laser ion acceleration 27

3.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2. Detailed intensity study of laser ion acceleration . . . . . . . . . . . . . . . 29

3.2.1. Dependence on laser energy . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2. Dependence on laser pulse duration . . . . . . . . . . . . . . . . . . 33

v

Contents

3.2.3. Dependence on focal spot size . . . . . . . . . . . . . . . . . . . . . 35

3.2.4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. Optical probing 41

4.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1. Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2. Ion accelerating electron sheath . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1. 3D-PIC simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2. Estimations regarding the proton acceleration duration . . . . . . . 47

4.3. Temporal evolution of the electron sheath . . . . . . . . . . . . . . . . . . 49

4.3.1. E�ciency of energy conversion . . . . . . . . . . . . . . . . . . . . 53

4.3.2. Multiple ion front expansion . . . . . . . . . . . . . . . . . . . . . . 55

4.4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5. Double stage laser ion acceleration 59

5.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2. Characteristic modulation of proton beam spectra . . . . . . . . . . . . . . 63

5.2.1. Spectral modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.2. Spectral control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.3. Energy gain of re-acceleration . . . . . . . . . . . . . . . . . . . . . 67

5.2.4. Energy loss via deceleration . . . . . . . . . . . . . . . . . . . . . . 68

5.3. Simulations and further explanations . . . . . . . . . . . . . . . . . . . . . 69

5.3.1. Phase space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.2. Coulomb mirror model . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4. Longitudinal proton probing of a TNSA �eld . . . . . . . . . . . . . . . . 76

6. Conclusion and future prospects 79

6.1. Summary of the results and recent prospects . . . . . . . . . . . . . . . . . 79

6.1.1. Intensity dependence of TNSA . . . . . . . . . . . . . . . . . . . . 79

6.1.2. Optical probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1.3. Double stage experiment . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1.4. Combination of experimental �ndings . . . . . . . . . . . . . . . . 83

6.2. Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 87

A. JETI�Jena Titanium:Sapphire TW laser system 101

vi

Contents

B. Ion spectrometer for online observation 105

B.1. Calibration of ion spectrometer at a conventional accelerator . . . . . . . . 105

B.2. Considerations on energy resolution . . . . . . . . . . . . . . . . . . . . . . 106

Danksagung 108

vii

Contents

viii

List of Figures

1.1. Photograph of a relativistic laser-matter interaction. . . . . . . . . . . . . 2

2.1. Ionization processes at high intensities. . . . . . . . . . . . . . . . . . . . . 6

2.2. Re�ection in exponential plasma pro�le. . . . . . . . . . . . . . . . . . . . 15

2.3. Plasma expansion into vacuum. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4. Proton energy spectrum for di�erent a0. . . . . . . . . . . . . . . . . . . . 21

3.1. Scheme of ion acceleration setup. . . . . . . . . . . . . . . . . . . . . . . . 28

3.2. Well-de�ned intensity variations. . . . . . . . . . . . . . . . . . . . . . . . 30

3.3. Energy and pulse duration dependence of ion acceleration. . . . . . . . . . 32

3.4. Intensity variation with pulse duration. . . . . . . . . . . . . . . . . . . . . 34

3.5. Intensity dependence of ion acceleration by varying the laser spot size. . . 36

3.6. TNSA �eld strength derived from the appearing ion species. . . . . . . . . 39

4.1. Sketch of the experimental setup for optical probing of the Debye sheath. 42

4.2. Target frame for optical probing. . . . . . . . . . . . . . . . . . . . . . . . 43

4.3. Electron density distribution in the TNSA sheath. . . . . . . . . . . . . . 45

4.4. Proton acceleration duration. . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5. Temporal evolution of the TNSA electron cloud. . . . . . . . . . . . . . . 50

4.6. Evaluation of plasma expansion into vacuum. . . . . . . . . . . . . . . . . 52

4.7. amount of energy of the hot electron cloud. . . . . . . . . . . . . . . . . . 54

4.8. Simulation of multiple expanding ion fronts [88]. . . . . . . . . . . . . . . 56

4.9. Ion front positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1. The schematical setup of the double stage experiment. . . . . . . . . . . . 60

5.2. Double stage setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3. Double stage target frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4. Characteristic modulation in the proton energy spectrum. . . . . . . . . . 64

5.5. Energy modulation tuning curve. . . . . . . . . . . . . . . . . . . . . . . . 66

5.6. Complete spectral control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ix

List of Figures

5.7. Energy gain from re-acceleration. . . . . . . . . . . . . . . . . . . . . . . . 68

5.8. Energy loss from action of Coulomb mirror. . . . . . . . . . . . . . . . . . 69

5.9. Phase space evolution of protons in a double stage con�guration. . . . . . 71

5.10. Double stage simulation results. . . . . . . . . . . . . . . . . . . . . . . . . 72

5.11. Simulation results by Velchev et al. [130]. . . . . . . . . . . . . . . . . . . 74

5.12. Energy gain as a function of the beam splitting ratio. . . . . . . . . . . . . 76

5.13. TNSA �eld probing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.1. Setup of the Jeti laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2. Autocorrelation trace of the Jeti laser. . . . . . . . . . . . . . . . . . . . . 103

B.1. Calibration of ion spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . 106

B.2. Energy resolution of the ion spectrometer. . . . . . . . . . . . . . . . . . . 107

x

List of Tables

3.1. Ionization energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1. Comparison of electron sheath parameters with theoretical predictions. . . 46

6.1. Scaling behavior of TNSA for intensity variations. . . . . . . . . . . . . . . 80

6.2. Properties of the hot electron population responsible for TNSA. . . . . . . 81

xi

List of Tables

xii

1. Introduction

At present, we experience the advent of laser-based accelerators for charged particles.

Laser-based accelerators employ the ultra-strong electric �elds occurring during intense

laser-matter interactions to accelerate electrons and ions on a micrometer to millimeter

scale up to Multi-MeV energies. This development became possibly only by the inven-

tion of the chirped pulse ampli�cation (CPA) technique in 1988 by Strickland et al. [1,2]

which led to the development of table-top lasers providing �relativistic intensities�. In this

context, relativistic means that free electrons, created by the laser-matter interaction, os-

cillate in the electromagnetic �eld of the laser pulse with an average kinetic energy larger

than their rest mass equivalent of 0.511 MeV. This relativistic threshold is reached at anintensity of 1.37 × 1018 W/cm2 · µm2. The CPA technique has tremendously increasedthe availability of intense short-pulse laser systems, and has thus contributed signi�cantly

to the �ourishing of the �eld of laser-plasma interactions.

The sub-�eld of laser particle acceleration can be roughly divided into two domains

of interaction, depending on the target material used. Electron acceleration employing

gaseous targets (also known as �under-dense targets� for reasons to be explained below)

has �rstly been proposed by Tajima and Dawson in 1979 [3] and their con�rmation re-

sulted in the �rst milestone of the production of monoenergetic electron bunches [4�6].

Since then, electron beams have been produced with GeV energies [7] and maturedtowards their application to conventional accelerator techniques, for example the pro-

duction of synchrotron radiation with the help of a magnetic undulator [8, 9].

On the contrary, solid (or "over-dense") targets are mainly used for ion acceleration.

Due to their 1836 times higher inertia, ion acceleration relies presently on secondary

e�ect of the production of such a hot electron population since it is not possible directly

with current laser technology. One has to apply the overdense plasma target with a hot

electron population in order to equip at least one vacuum plasma interface with enormous

electric �elds (∼ TV/m). This phenomenon has �rst been observed by Gitomer et al. [10]as well as Wilks et al. [11], who introduced the name target normal sheath acceleration

(TNSA) for ion acceleration from the rear side of thin target foils. Very impressive results

have been obtained for TNSA ions, including for example a maximum proton energy of

1

1 Introduction

Figure 1.1.: Photograph of a relativistic laser-matter interaction. The picture shows theinteraction of a Jeti laser pulse with a thin Titanium foil. The picturehas been recorded with a digital re�ex camera (Canon 300 D). An o�-axisparabolic mirror visible in the upper right background focused the pulse ontothe foil target mounted in the frame in the center of the picture. Trajectoriestowards the right are due to glowing ballistic blow-o� from an additional foilstripe mounted in front of the frame and especially used for the double-stageexperiments to be presented in Chapter 5.

58 MeV [12,13], 430 MeV palladium ions [14], and excellent beam properties expressed bytheir longitudinal and transverse emmitances [15,16]. One milestone was again set by the

achievement of quasi-monoenergetic ion beams in 2006 with the help of micro-structured

or nano-layered target foil surfaces [17, 18], droplet targets [19] or a laser-driven micro

lens [20].

Since ion acceleration is not directly driven by the laser pulse itself, it relies mostly

on the parameters of the hot electron population. The basic theoretical models of the

ion acceleration processes [21�23] are typically 1D models and are in good agreement

with many experimental results. For more sophisticated and quantitative predictions, all

models have to refer to estimations about the major parameters electron temperature

and electron density as well as the according Debye length. In addition there is, despite

2

the numerical simulations, still a lack of knowledge about the temporal evolution of the

TNSA process and especially with regard to the above mentioned parameters.

Thus, experimentalists have been eager developing methods to observe the transient

processes in real-time�an inevitable presupposition if one wants to gain true insights into

TNSA. The currently most widely used methods are optical, interferometrical probing of

the electron population on the one hand [24�28], and transverse probing of the electric

�elds using proton de�ectometry on the other [29�35]. Those methods have only been

applied to long pulse laser systems and their resolutions are in the picosecond range,

whereas the acceleration time scales with the laser pulse duration to several 100 fs orthey do not provide an appropriate spatial resolution in the micrometer range.

This work presents several experimental techniques to fully determine the central pa-

rameters of the laser-plasma interactions leading to the generation of energetic ion beams,

as well as their evolution over time with a temporal resolution matching that of the laser

pulse duration. First, the produced ion beams are analyzed with respect to their depen-

dency on laser intensity. A detailed study of the three parameters laser energy, pulse

duration, and focal spot size contributing to the laser intensity has been performed and

provides sophisticated scaling laws. In addition, the electric �eld strengths, which are

generated at the rear surface and responsible for the ion acceleration process, are evalu-

ated.

Secondly, a direct optical probing of the electron sheath that is driving the TNSA

process is presented. The detection of the electron sheath with su�cient spatial and

temporal resolution allows the direct experimental determination of all properties of

the electron population driving the acceleration process. The conversion e�ciency of

laser light into hot electrons is derived directly as the temporal evolution of the electron

temperature and the Debye length of the electron population.

The third approach refers to the question whether the TNSA mechanism can be applied

in an additive manner. Or in other words: Is it possible to expose already accelerated

protons to a second TNSA interaction scheme in such a way that they gain energy in it?

It turns out that this is in fact possible, and that in addition a characteristic spectral

modulation can be imposed on the otherwise broad, quasi-thermal spectrum, leading

to the consistent production of quasi-monoenergetic features. This spectral modulation

re-accelerates some fraction of the proton population, while decelerating others, resulting

in a signi�cant shift of the respective proton energies. The reliability of this technique

is again proven by the derivation of energy scaling laws. Simulations carried out for the

exact experimental conditions support all the �ndings and con�rm the interpretation of

the characteristic spectral modulation resulting from staged laser ion acceleration.

3

1 Introduction

1.1. Thesis outline

The outline of the thesis will be as follows. After a brief introduction of the theoretical

basics of laser plasma interaction and ion acceleration at a plasma vacuum interface in

Chapter 2, the next three chapters concern the experiments.

Starting in Chapter 3, a detailed study of the intensity dependency of laser ion ac-

celeration is presented with special emphasis on the fact that the laser intensity is a

parameter de�ned by the ratio of laser energy and the product of focal spot size and

laser pulse duration. Well-established scaling laws have been con�rmed and expanded to

these three di�erent parameters.

In Chapter 4, the optical probing of the ion accelerating electron population is pre-

sented. For the �rst time, this hot electron population has been observed with 104 fstemporal resolution. A delay line allows to study the temporal evolution of the electron

distribution, and all central parameters�electron temperature, electron density, and De-

bye length�were measured with this direct all-optical setup.

Chapter 5 concerns the �double stage� experiment�a setup which allows for a longi-

tudinal proton probing of the electric �elds appearing during the whole ion acceleration

process. The observations yield a characteristic spectral modulation of re-accelerated

protons of the beam, which are well con�rmed by 1-dimensional simulations.

A conclusion will complete the thesis in Chapter 6 with special attention on future

experimental approaches and how to best implement the �ndings of the presented work

into the experimentalist's everyday life. Finally, the thesis will conclude by pointing out

ways how the �ndings relate to possible future applications.

4

2. Theory of laser ion acceleration

A profound understanding of the major aspects of laser plasma interaction as well as

laser particle acceleration is necessary in order to provide a basis for the discussions

in the following chapters, where the experimental work will be discussed. Thus, the

following sections will guide the reader through the processes of plasma generation using

an intense laser pulse, the heating towards a relativistic plasma, electron acceleration

within this plasma, and �nally ion acceleration as a secondary e�ect of the relativistic

electron population.

2.1. Laser matter interaction

A typical custom made intense laser pulse always contains a rising edge due to its pulsed

structure. Regarding this fact it becomes clear that the laser pulse will irradiate a target

with very high intensities already before the main pulse impinges on it. The following

sections will lead through the di�erent processes that happen to the target material being

imposed to the rapidly increasing light intensity, spanning several orders of magnitude

during the whole duration of the laser incidence.

2.1.1. Ionization processes and plasma generation

Atoms exposed to an electromagnetic wave will be electronically excited if the absorbed

photon energy matches the energy di�erence between the ground state and the excitation

state. If the excitation energy exceeds the highest bound state, the electron is released

into the continuum and one speaks of the atom being ionized. During the presented

experiments, thin Titanium foils will be commonly used. The necessary photon energy

for the ionization of atomic Titanium is EIon = 6.8 eV [36]. However, a Ti:Sapphire laseronly delivers photons of 1.6 eV at a central laser wavelength of λL = 800 nm which is notsu�cient to ionize Titanium with a single photon only.

At intensities of the order of 1010 W/cm2 the photon density becomes high enoughthat already excited electrons can absorb additional photons with su�cient probability

if they are provided within the lifetime of their virtual states de�ned by the uncertainty

5

2 Theory of laser ion acceleration

U(x) U(x)

x xmulti-photonionization

above thresholdionization

Evirt

-eEx

n hwL

- eZ2

4pe0x

tunellingionization

(a) (b)- eZ

2

4pe0x

Figure 2.1.: Ionization processes at high intensities. (a) Multiphoton and above thresholdionization: γ > 1 (b) Barrier suppression ionization: γ < 1

relation. This process known as multiphoton absorption (MPI) is also illustrated in Fig.

2.1 (a) [37]. In case of Titanium, the absorption of 5 photons is necessary to ionizethe atoms just once. Note that electrons may actually absorb more than the number of

photons necessary for ionization and subsequently leave the atom with a certain amount

of energy, which is called above threshold ionization (ATI) [37].

At higher intensities, additional ionization scenarios come into play. A su�cient electric

�eld strength EL = |EL| of the external light �eld can cause a perturbation of the atomicpotential. Following the classical picture the atomic potential can be described as a

superposition of the atomic Coulomb potential and the external �eld

U(x, t) =Ze2

4πε0x− exEL(t) , (2.1)

as illustrated in Fig. 2.1 (b) with the underlying assumption that the external �eld stays

constant within the classical revolution time of the electron around the nucleus. Here,

e is the elementary charge and ε0 is the vacuum permittivity. The remaining atomicpotential may be suppressed by the external �eld and an electron may tunnel through

the barrier with a �nite probability: barrier suppression or tunneling ionization (BSI /

TI).

When EL is strong enough to suppress the barrier beneath the electron state, i.e. at a

�eld strength of the order of the atomic �eld strength Eat = e/(4πε0a2B) ≈ 5.1×1011 V/mat an intensity Iat = 1/2 · ε0cE2at ≈ 3.5× 1016 W/cm2, it leaves the atom spontaneouslyover the barrier (OTBI). Here, aB = 4π�0~2/(mee2) is the radius of the Bohr atom withthe electron mass me and c is the speed of light. To distinguish in which way an atom

6

2.1 Laser matter interaction

was ionized, one can make use of the Keldysh parameter [38]

γK = ωL

√2EIonIL

, (2.2)

whereas ωL is the laser frequency and IL = 1/2 · ε0cE2L is the laser intensity. This givesγK > 1 for multiphoton, and γK . 1 for barrier suppression or tunneling processes,respectively.

Using a more quantitative description of OTBI [37] one �nds

IOTBI ' 4× 109(E2ioneV

)4 1Z2

W/cm2 (2.3)

as the laser intensity which is necessary for OTB ionization. In terms of Titanium,

ionizing just once requires only IOTBI (Ti+) = 8.6× 1012 W/cm2, which is even less thanthe atomic intensity.

Now, the target material1 is ionized, a plasma is generated which comprises quasi-free

electrons and an inert ion background but the intensity remains increasing. Subsequently,

the interaction takes place between the plasma and the laser pulse. But before these

collective e�ects start to play a dominant role, the in�uence of the intense laser �eld on

a single electron and some central plasma properties shall be discussed.

2.1.2. Electrons in intense light �elds and ponderomotive force

The motion of an electron exposed to a plane electromagnetic wave with wave number k,

wavelength λL, and frequency ωL propagating in z direction is described by the Lorentz

equation

F L =dpdt

=ddt

(γmev) = e (EL + v ×BL) , (2.4)

γ =(1− v2/c2

)−1/2, v, and BL are the relativistic factor, the electron velocity, and the

magnetic component of the wave, respectively. To solve the equations of motion, Eq.

(2.4), one can make use of the energy equation

dEkindt

= mec2dγdt

= −e (v ·EL) , (2.5)

1For the ionization processes described here it does not matter whether the target is initially re�ective,transparent, or opaque until the light intensity su�ces for ionization.

7


in order to �nd the trajectories of the electron after some elaborate calculation [37, 39]

(considering at least only one transverse direction py = 0, y = 0; linear polarization)

px = a0mec sinφ (2.6)

x = −a0cωL

(1− cosφ) ∼ a0 (2.7)

pz =a204mec (1 + cos 2φ) (2.8)

z =a20c4ωL

(φ+

12

sin 2φ)∼ a20 , (2.9)

where φ = ωLt − kz is the phase of the linearly polarized wave. The amplitude of thenormalized vector potential or relativistic amplitude of the laser pulse

a0 =ELeωmec

=

√Iλ2L

1.37× 1018 Wcm2· µm2

, (2.10)

describes the ratio of the classical electron quiver velocity vosc,class = eEL/meωL and thespeed of light a0 = vosc,class/c and thus, indicates �how relativistic� the quiver motion ofthe electron is. Eq. (2.8) shows that the electron experiences an additional acceleration in

z-direction, which is due to the fact that the magnetic component of the electromagnetic

wave is not longer negligible against the electric component in Eq. (2.4). The kinetic

energy of an electron due to the acceleration in z-direction is determined by

Ekin,z = pzc (2.11)

Transforming Eqs. (2.6)−(2.9) into the average rest frame of the electron gives thewell known �gure-8 motion of the electron owed to the z-oscillation with twice the laser

frequency, while the x-component oscillates with the original laser frequency only.

Ponderomotive force After this idealized case of in�nitely extended plane waves one

must consider the more physical case of an impinging laser pulse with a certain intensity

shape and pulse duration. Here, the ponderomotive force comes into play

F pond = −e2

4meω2Lγ∇(E2L(x)

), (2.12)

which acts along the intensity gradient [37,40�43]. γ is the time averaged relativistic fac-

tor and γ ≈(1 + a20

)1/2[44]. Furthermore the oscillating electron experiences a reduced

restoring force due to the decreasing intensity outside the laser center and is consequently

8


shifted away from the region of higher intensity. The dependency on the �eld gradient

makes the ponderomotive force most e�ective in the laser focus, where the gradient is

steepest. A corresponding ponderomotive potential can also be found for Eq. (2.12)

Φpond = (γ − 1)mec2 =(√

1 + a20 − 1)× 0.511 MeV . (2.13)

Wilks et al. [11] predicted the e�ective temperature kBTe (kB is the Boltzmann constant)that an electron population present in this potential may gain

kBTe ≈ Φpond , (2.14)

which seems to be in good agreement with the experimental work of several groups

[45�52]. The con�rmation of this important and central relation is very hard to measure

and in most cases done via the detection of the x-ray emission. Thus, it will be subject

to the analysis of the direct optical measurement of laser produced hot electrons in Sec.

4.2 later in this work.

2.1.3. Plasma properties

A plasma can be understood as a compound of an electron and an ion �uid with the

according electron and ion densities ne = Zni. Z refers to the charge state of the ionizedtarget material. The basic �uid model assumes that the particle number is preserved,

and that the plasma is collisionless and isothermal [40]. A description may be obtained

from the Vlasov equation

∂fα∂t

+ v∂fα∂x

+qαmα

(E + v ×B) · ∂fα∂v

= 0 , (2.15)

for a given phase-space distribution fα (x, v, t) of the two �uid components, with chargeqα and mass mα, as well as E and B as the electric and magnetic �elds related to the

charge of the �uids. α can be either e or i, referring to the electron or ion population,

respectively.

Integration of Eq. (2.15) yields the continuity and force equation for the �uid particle

density nα =∫fα (x, v, t) dv and mean velocity vα =

∫v fα (x, v, t) dv/nα,

∂nα∂t

+∂

∂xα(nαvα) = 0 (2.16)

nα

(∂vα∂t

+ vα∂vα∂xα

)= nα

qαmα

(E + vα ×B)−1mα∇pα . (2.17)

9


Here, pα is the particle pressure de�ned by the equation of state which reads pα = nαkBTαassuming an isothermal expansion. In the next step, magnetic �eld contributions and

the electron inertia are neglected with respect to the ion mass and one obtains,

∂ni∂t

+∂

∂xi(nivi) = 0 (2.18)

∂vi∂t

+ vi∂vi∂xi

= − 1mi

(Z∇pene

− ∇pini

)= −c2s

1ni

∂ni∂x

, (2.19)

as the continuity and force equations of the ion population. The electron and ion density

are connected via the initial assumption of macroscopic charge neutrality ne = Zni. Therelations

vi = cs +xit, (2.20)

ni = ni,0 exp[− xicst

], (2.21)

for ion velocity and density are a self similar solution of Eqs. (2.18) and (2.19), where

cs ≈√Z·kBTemi

≈ 0.31× 106 ms·√

kBTekeV

· ZA, (2.22)

is the ion sound speed and A the ion atomic mass number. Thus, the one dimensional

description of the freely expanding plasma obeys an exponential decay, whose scale length

LP = cst increases over time with the ion sound speed.

Preplasma Now the geometric and energetic properties of the so-called preplasma pro-

duced by the Jeti laser right before the main pulse incidence shall be estimated. Fig.

A.2 shows this temporal intensity distribution of the laser pulse. Considering the peak

intensity in the range of 1...4 × 1019 W/cm2 and the temporal shape of the Jeti laserpulse as given in Fig. A.2, the �rst time an intensity su�cient to produce a preplasma

appears 30 ps before the main pulse, where a prepulse with I . 1015 W/cm2 occurs. Fol-lowing Eqs. (2.14), (2.21), and (2.22) one can calculate the properties of the preplasma

with which the main pulse will subsequently interact 30 ps after its creation. The appliedintensity may su�ce for an ionization up to the fourth charge state of Titanium Z = 4following Eq. (2.3), which describes the appearance intensity for over the barrier ion-

ization. The temperature of the electron population can be deduced to kBTe ≈ 0.1 keVwith the help of both the ponderomotive scaling Eq. (2.14) as well as the expression for

the surface temperature due to collisional skin e�ects Eq. (2.33) discussed in Sec. 2.1.4.

10


Hence, the plasma comprises a very steep gradient with LP = 1.3µm = 1.6λL scalelength that extents further with cs ≈ 4.2× 106 cm/s.

Optical properties The free electrons and ions that constitute the plasma will respond

to small-scale deviations of their distribution, e.g. displacement in an electromagnetic

wave of frequency ω, with collective oscillations at certain plasma frequencies ωp de-

pending on their mass m, charge Ze, and density n

ωp,e =

√nee2

ε0me, ωp,i =

√niZ2e2

ε0mi= ωp,e

√Zmemi

, (2.23)

driven by the restoring electrostatic force due to charge separation. Considering further

mi � me, only the electrons contribute to the dielectric function [53]

ε(ω) = 1− nee2

ε0meω2= 1−

ω2pω2

, (2.24)

(ωp = ωp,e is the plasma frequency). The refractive index reads

nr(ω) =√ε(ω) =

√1−

ω2pω2

=√

1− nenc, (2.25)

where the critical density

nc =ε0meω2L

e2=

1.1× 1021

(λL/µm)2cm−3 , (2.26)

has been introduced, which is the density where the plasma frequency is equal to the

laser frequency.

Assuming ω > ωp, Eq. (2.25) becomes real and the plasma dispersion relation ω2 =

ω2p +c2k2 with the wave number k describes the propagation of the electromagnetic wave

through the plasma. For ω < ωp, the refractive index is imaginary and re�ection as well

as absorption occurs. The remaining penetration is just evanescent and characterized by

the skin depth ls ≈ c/ωp = 0.16λL. A similar demarcation is possible using a criticalelectron density nc(λL = 800) = 1.7 × 1021 cm−3 instead of the plasma frequency asthe boundary between transmission and re�ection or absorption meaning that ne < nc

denotes an underdense plasma which supports propagation of the electromagnetic wave

while for ne > nc the electromagnetic wave is re�ected.

11


Shielding An other important property of the plasma shall be mentioned because of

its importance for the laser ion acceleration mechanism: The Coulomb �eld of a single

charge Ze in the plasma is modi�ed to

Φel =1

4πε0Ze

rexp

[−rλD

], (2.27)

by the free electrons that form an equilibrium between the electrostatic �eld and their

thermal motion around it [40]. Here,

λD =√ε0kBTe

e2ne≈ 0.24µm×

√kBTe/MeVne/1021cm−3

, (2.28)

is the Debye length describing the scale length of the remaining distance of the electric

�eld action, which is somewhat an e�ective length scale of the �eld action. Thus, any

unneutralized charges in the plasma are attenuated signi�cantly beyond the Debye length,

but may nonetheless cause strong electric �elds within this scale length, which depend on

the charge density and electron temperature only. The �eld strengths can reach 1012 V/mand are utilized for the laser ion acceleration (cf. Sec. 2.2 as well as Fig. 3.6).

2.1.4. Generation of hot electrons

In Sec. 2.1.3 the optical properties of a plasma have been introduced neglecting any

damping of the electromagnetic wave inside the plasma. Thus, the distinction was only

made between transmissive and re�ective behavior. In order to add absorption processes

to the laser-plasma interaction, one may introduce a damping term into the Lorentz

equation of the electron motion, that represents collisions between excited electrons and

ions. This so-called collision frequency obtains the form [40,54]

νei ≈ 2.91× 10−6Z ln Λ ·ne/cm−3

(kBTe/eV)3/2

s−1 , (2.29)

whereas ln Λ is the Coulomb logarithm, Λ = 9ND/Z, and ND = 4π/3λ3Dne is the numberof particles in the Debye sphere. For the preplasma conditions achieved in the former

Sec. 2.1.3, the laser pulse is incident on a four times ionized Titanium plasma with a

temperature of 0.1 keV and propagates up to the critical density. Hence, a collisionfrequency νei ≈ 0.04ωL follows from Eq. (2.29).It can be seen that νei ∼ (kBTe)−3/2 and accordingly it will become less e�ective for

rising electron temperatures starting in the range of 1015 W/cm2 [37]. Therefore, other

12


"collisionless" mechanisms have to be taken into consideration. For the conditions at

Jeti, the most relevant collisionless absorption processes are resonance absorption and

Brunel heating as well as the already introduced ponderomotive acceleration (cf. Sec.

2.1.2), which is closely related also to v ×B heating.

Collisional e�ects Before starting with the description of the relevant collisionless pro-

cesses when the laser intensity exceeds 1015 W/cm2, a short paragraph shall discusscollisional e�ects appearing for the intensity range 1012...15 W/cm2. Collisional e�ectsinside the skin layer like inverse bremsstrahlung which is also known as normal skin

e�ect�electrons oscillate in the incident laser �eld and dissipate energy during collisions

with ions or other electrons�are at �rst responsible for heating processes leading to the

generation of a preplasma but can also contribute substantially to the energy absorption

rate of the plasma.

The dielectric function of the plasma Eq. (2.24) changes to

ε(ω) = 1−ω2p

ω (ω + 2iνei)=

(1−

ω2pω2 + 4ν2ei

)+ i

(2νeiω2p/ωω2 + 4ν2ei

), (2.30)

when collision are considered. Here, i denotes the imaginary unit. The according disper-

sion relation obtains the form

ω2 = ω2p(

1− iγcω

)+ c2k2 . (2.31)

An expression for the energy absorption rate of laser light incident to an inhomogeneous

plasma can now be found by solving the Helmholtz equations for the dielectric function

Eq. (2.30) including collisions. In the long plasma scale length limit LP � λL, an analyticsolution can be derived using the WKB-approximation [37,40,55]. The absorption rate

ηcollabs = 1− exp{−8νeiLp

3ccos3 θL

}, (2.32)

is received assuming an exponential plasma density gradient with scale length LP for s-

polarized light incident to the plasma under the angle θL. In addition, Gibbon [37] notes

that this result holds true for plasma scale lengths down to LP ≈ 0.1λL. Consideringagain the preplasma conditions at Jeti discussed in Sec. 2.1.3 where a plasma scale

length of LP = 1.6λL was deduced and assuming a 45◦ incident angle, in order to makeuse of Eq. (2.32) this gives ηcollabs = 30 % for the collisional absorption.

From a sophisticated treatment of heat-�ow including a self-consistent collisional ab-

13


sorption rate, a formula for the temperature scaling of the electron population

kBTe = 119( ne

1023 cm−3)1/12

Z1/12(

IL1015 W/cm2

)1/3 ( τL100 fs

)1/6eV (2.33)

can be obtained [37, 56] and one �nds kBTe = 74 eV for the above mentioned preplasmaconditions. This value for the preplasma electron temperature at Jeti is also in good

agreement with kBTe ≈ 0.01 keV deduced from the ponderomotive scaling in Sec. (2.1.3).

The absorption rate of 30 % seems to be quite substantial but collisional absorptionbecomes less e�ective for intensities above 1015 W/cm2 as has already been mentionedabove. Thus it yield a good estimation for the preplasma conditions. However, for the

intensity range above 1015 W/cm2 collisionless scenarios need to contribute to the energytransfer from the laser pulse into the plasma respectively the temperature of the electron

population.

Resonance absorption and Brunel heating A p-polarized laser pulse with oblique inci-

dence on the target has an electric �eld component, which is directed into the preplasma

slab assumed with an exponential density pro�le of scale length LP ≈ 1.6λL, i.e. indirection of increasing density.

Under oblique incidence, due to the k-vector conservation [40], re�ection actually oc-

curs before the critical density is reached, so that the laser frequency does generally not

match the plasma frequency in the plasma pro�le, cf. Fig. 2.2. The electron density oc-

curring at the turning point of the incident wave is deduced by ne,refl = nc cos2 θL whichis less than the critical density for incidence angles larger than zero. For the preplasma

properties discussed for the Jeti setup one �nds ne,refl = nc/2, 0.9µm in front of thecritical density.

At the turning point, the electric �eld component of the incident wave obtains a stand-

ing wave pattern oscillating in target-normal direction. However, since the distance be-

tween the re�ection point and the position of the critical density is small enough, the

evanescently decaying electric �eld su�ces to excite an electron oscillation in longitudinal

direction within the region of the critical density. This very localized and quasi-standing

plasma wave is resonantly excited by the laser �eld and can be driven so strong that it

breaks which happens when the electron's oscillation amplitude is larger than the wave-

length of the plasma wave and a su�cient number of hot electrons is pushed into the

target. This process is called resonance absorption.

An expression for the absorption rate via resonance absorption [40]

14


ncr ZnTine,refl

targetfoil

qL

incident laserpulse,

-polarizedp

plasma withexponentialdensity profile

Figure 2.2.: Re�ection in exponential plasma pro�le. A p-polarized laser pulse obliquelyincidences the preplasma pro�le at the front of a dense target foil. Due tothe interaction with plasma electrons, re�ection occurs already before thecritical density has been reached. If the distance between the standing wavepattern at the turning point and the position of the critical density is shortenough the evanescent decaying electric �eld can excite electron oscillationsat this position in longitudinal direction. Since this oscillation is excitedresonantly at the critical density the plasma wave can be driven so strongthat it breaks and a su�cient number of hot electrons is pushed inside thetarget.

ηraabs =12

(2.3τ · exp

{−2

3τ3})2

, (2.34)

is related to the laser's incidence angle and the preplasma scale length only and τ =3√kLP · sin θL. Considering the Jeti preplasma conditions, a reasonable rate of ηraabs =

5.1 % results, since the assumption of the small plasma scale length limit τ � 1 (LP �λL) is not strictly ful�lled. However, absorption rates up to 60 % are reported for opti-mized laser and plasma parameters [37,57].

In contrast, Brunel heating [58] takes place at much steeper plasma gradients and

bene�ts from the fact that the skin depth is smaller than the maximum displacement of

the oscillating electrons (ls = 0.2λL < xmax = 0.3...0.8λL for a0 = 2...4, cf. Eq. (2.7)).The acceleration process now takes place in the vacuum space in front of the preplasma

pro�le or the target foil, respectively. In the waves �rst half cycle, the electrons move

15


away from the target into the vacuum, turn around, and are accelerated back into the

target. However, once they enter the target, the electrons are immediately screened from

the restoring electric �elds, so that the electrons retain their kinetic energy and contribute

to the hot electron population. The absorption rate via Brunel absorption [37]

ηBrunelabs =4π sin θL tan θL

(π + sin θL tan θL)2 , (2.35)

just depends on the incidence angle of the laser (the plasma properties are more or less

�xed due to the assumption of a very steep plasma gradient). Hence, θL = 45◦ accordsto an absorption rate of ηBrunelabs = 60 %. This very large value has to be discussedrather critically: On the one hand the Brunel mechanism does not account for �nite

plasma scale lengths and the Jeti conditions do not reveal a very steep plasma gradient

and on the other hand numerical studies assuming very short but �nite preplasma scale

lengths [37,59] show that ηBrunelabs runs into saturation at 10...15 %.

Ponderomotive force and v×B heating Besides those processes of collective resonance�electron oscillation excited by the electric �eld of the laser pulse�the electrons of the

preplasma are also a�ected by two other processes.

The �rst is the ponderomotive force due to the ponderomotive potential of the focused

laser pulse: The electrons are pushed away by the intensity gradient, whereas the e�ective

temperature that can be attributed to their distribution function is related to the laser's

ponderomotive potential, kBTe = Φpond ∼ (ILλL)1/2 Eq. (2.14), leading in our speci�ccase to a temperature of kBTe(a0 = 2...4) ≈ 0.6...1.7 MeV [11, 41].The second is v ×B heating (also j ×B, with the current density j) [37]. Electrons

oscillating in the electric �eld of a laser pulse with relativistic intensity experience the

in�uence of the v×B term of the Lorentz force which is no longer negligible for vosc ∼ cand obtain a net drift in longitudinal direction with a dependence on twice the laser

frequency and a20, cf. Eq. (2.9). For steep plasma gradients, this e�ect can contribute to

the absorption rate analogously as the Brunel mechanism does: Within the �rst quarter

of the wave's cycle electrons are accelerated into the vacuum, turn around, and are

accelerated back into the target where they become screened from the electromagnetic

wave and retain with their kinetic energy. In contrast to the Brunel mechanism depending

on the electric �eld of the laser, v ×B heating acts longitudinally due to the magnetic�eld contribution. Hence, v ×B heating favors normal incidence angles. The electron'skinetic energy achievable via v × B heating is deduced by the ∼ a20 dependence ofthe electron momentum in longitudinal direction, cf. Eqs. (2.8) and (2.11), and the range

16


a0 = 2...4 denotes energies of 0.9...5.4 MeV. Due to the relation to the Brunel mechanism,similar absorption rates in the 10...15 % range are expected after reaching the regime ofrelativistic intensities [60].

An important fact to be mentioned is the favor for di�erent incidence angles of the

presented absorption processes. The ponderomotive force and the v×B heating both actin laser propagation direction and advise normal incidence on the target, whereas Brunel

heating and resonance absorption make use of the electric �eld component of the incident

laser pulse and require oblique incidence angles θL ≤ 90. Thus, the steep preplasmagradients as present in the Jeti experiments suggest a 45° incidence angle to be a good

compromise between the advantageous of all the described absorption mechanisms.

2.1.5. Electron transport through the target foil

The total number of electrons produced by the absorption and acceleration processes at

the target front side can be deduced with the help of a simple estimation. A 500 mJlaser pulse applying an intensity of 2 × 1019 W/cm2 to the target, will produce a hotelectron population with an e�ciency of about 20 %. From the laser's ponderomotivepotential Eq. (2.14) one can estimate the temperature of the electron population to be

1.1 MeV. Thus, the laser pulse's amount of energy results in a total of Ne = 5.7 × 1011

hot electrons.

If this total number of hot electrons attempts to travel through the target, that

corresponds to a current J = eNe/τL = 1.1 MA�a value exceeding the Alfvén limitJ = βγ · 17 kA of a free electron current propagating in vacuum substantially [61]. How-ever, in a plasma the propagation of the estimated hot electron current becomes possible

due to return currents that compensate the self induced azimuthal magnetic �elds around

the relativistic electron current. In addition, the background electron density is of the

order of the solid density ne ≈ 1023 cm−3 and consequently much higher than the densityof the relativistic electron beam ne < nc ≈ 1021 cm−3. Hence, the return currents areprovided by a cold electron population obtaining just a slow net drift.

With the use of conducting target material the in�uence of magnetic �eld �uctuations

or Weibel instabilities can be neglected, since quasi-free charges are already available

to provide the necessary return currents. Consequently, a smooth electron beam with

Gaussian temporal and spatial shape due to the laser's properties will propagate through

the target [44,62].

17


2.2. TNSA�Target normal sheath acceleration of ions

For currently available laser intensities, ion acceleration is still only a secondary e�ect

of relativistic laser plasma interactions, because the ions are, at least in the case of

protons, 1836 times more inert than the electrons and can not be accelerated directly

by the laser �eld to su�cient velocities so far. Hence, it is necessary to �nd a feasible

scheme to transfer the electron energy onto ions. For this purpose, it becomes necessary

to rectify the fast switching electron motion imprinted by the laser �eld and to enhance

the according time scale from the laser period to the laser pulse duration.

The physical picture of the underlying Target Normal Sheath Acceleration (TNSA) is

the following: The relativistic electron population that is produced at the front of a solid

target propagates through it. If the target is chosen thin enough (i.e. typically a foil

target of a few µm thickness) the electrons reach the back surface without considerableenergy losses. The fastest electrons escape the target which is then positively charged

up2 and the main part of the population is trapped by the electric �eld generated by the

charge separation forming a sheath of hot electrons. As already mentioned in Sec. 2.1.3,

the electron distribution can shield the positive charge of the target only to the order

of the Debye length λD. The available electric �eld strength due to this uncompensated

positive charge ranges in the order of 1012 V/m. Those �eld strengths su�ce to �eld ionizeatoms (cf. Sec. 2.1.1) present at the back surface of the target�the target material or

layers of contaminants comprising hydrocarbons and water vapor. As soon as ions are

generated, they are accelerated according to their charge to mass ratio q/m. Since the

electric �eld is oriented perpendicular to the conducting surface of the target, the ions

are accelerated in normal direction. Thus, this process became known as Target Normal

Sheath Acceleration [13,63].

In the following, a more sophisticated mathematical treatment of the TNSA process

shall be introduced. For that purpose, the model of plasma expansion into vacuum

established by Gurevich et al . [64] and Crow et al. [65] as well as recently advanced by

Mora [22] is followed. In addition to its illustrative character, this is also one of the most

successful models with respect to predictions of experimental results and it has recently

stimulated more sophisticated descriptions.

2This positive charge results also from the background currents discussed in Sec. 2.1.5. A returncurrent provided by a cold electron population is compensating the enormous magnetic �eld due tothe relativistic electron current in forward direction through the target foil. Since the number ofcold electrons contributing to the return current is much higher than the number of hot electronspropagating towards the back surface, a positive charge results at the rear surface of the target.

18

2.2 TNSA�Target normal sheath acceleration of ions

2.2.1. 1D model of plasma expansion into vacuum

The model starts out by assuming a hot electron population ne(z) with a constant averagekinetic energy described by the temperature Te and just one ion species�the one with the

highest charge to mass ratio�arranged in the half-space z < 0 with density ne0 = Zni0at the time t = 0 and ni = 0 for t > 0. Assuming the electrons to be in thermalequilibrium with the electro-static potential generated by the charge separation, they

can be described by a Boltzmann distribution,

ne(z, t) = ne0 exp[

eφ(z, t)kBTe

], (2.36)

with ne0 = ne(z → −∞) being the unperturbed hot electron density, and φ(z, t) theelectrostatic potential between the positive charges generated by the charge separation

and the negatively charged electrons. The corresponding potential obeys the Poisson

equation,∂2φ(z, t)∂z2

=1�0

e[ne(z, t)− Z ni(z, t)] . (2.37)

Integration of Eq. (2.37) gives a measure for the initial electric �eld at the target rear

surface,

ETNSA = −∂φ(z, t = 0)

∂z

∣∣∣∣z=0

=√

2eN

√kBTene�0

=√

2eNE0 , (2.38)

whereas eN is Euler 's number and E0 is de�ned as E0 = (kBTene/�0)1/2. For the given

Jeti laser intensities together with a typical electron density3 of 5×1020 cm−3, Eq. (2.38)leads to a �eld strength of ETNSA(a0 = 2...4) = 2.1...3.4 TV/m.

In the positive half space z > 0, Eq. (2.37) can be solved analytically [65],

eφ(z, t = 0)kBTe

= −2 ln[1 +

z√2eNλD

]− 1 , (2.39)

with the boundary conditions ∂φ(z → +∞, t = 0)/∂z = 0 and φ(z → +∞, t = 0) = −∞.Together with Eq. (2.36) this gives,

ne(z > 0) = ne01

eN

(1 +

z√2eNλD

)−2, (2.40)

as an expression for the drop of the initial electron distribution into the vacuum. Special

attention to this formula will be payed by Chapter 4, where the measurement of this

drop is discussed in detail.

3A more precise estimation will be given later in this section

19


n ni e, / cm

-3

z

ne0

= Zni0

0z

proton frontexponentialdensity profile

t > 0

0

net = 0

0

neutral neutral

n ni e, / cm

-3

Figure 2.3.: Plasma expansion into vacuum.

The evolution of the ion distribution driven by the electron distribution is described

by the equations of continuity and momentum,[∂

∂t+ vi(z, t)

∂

∂z

]ni(z, t) = 0 , (2.41)[

∂

∂t+ vi(z, t)

∂

∂z

]vi(z, t) = −

Zemi

∂φ(z, t)∂z

, (2.42)

with the ion velocity vi. A self-similar solution of Eqs. (2.41) and (2.42) valid for

z + cst > 0 is given by

ne(z, t) = Zni = ne0 exp[− zcst− 1], (2.43)

vi(z, t) = cs +z

t, (2.44)

Ess(z, t) =kBTeecst

=E0ωpit

, (2.45)

where Ess denotes an electric �eld that results from a negative and a positive surface

charge σ = ±�0Ess located at z = −cst and z = 0, respectively [22].Since the scale length LP = cst of the electron distribution, cf. Eq. (2.43), is smaller

than λD, which is the initial Debye length of the electron sheath, the self-similar solu-

tion does not have a physical meaning. Thus, the point where λloc = λD (ne0/ne)1/2 =

λD exp [(z/cst+ 1) /2] equals LP which is the position where the self-similar solution be-comes valid, is attributed to the ion front position [65]: 1 + z/cst = 2 ln (ωpit). Theelectric �eld at the ion front turns out to be Efront ≈ 2Ess = 2E0/ωpit after time inte-gration of Eq. (2.45). A physically reasonable interpolation gives the electric �eld valid

20


1 2 3 4energyMeV

log d /d / MeVN E-1

0

8

9

10

11

7

h = 10 %

a = 40a = 20

1 2 3 4 5 6energyMeV

0

8

9

10

11

7

h = 20 %

a = 20

a = 40

log d /d / MeVN E-1

Figure 2.4.: Proton energy spectra for di�erent a0. Eqs. (2.47) and (2.48) help to predictthe proton energy spectra observed for Jeti experiments (e.g. Fig. 3.3). Thespectra on the left hand side represent a conversion e�ciency η of 10 %whereas the rightward spectra stand for η = 20 %. Altogether, a very goodagreement is found for the considered conversion e�ciency range.

at any time t ≥ 0 [22],

Efront ≈2E0√

2eN + ω2pit2=√

2eN

E0√1 + τ2

, (2.46)

combining the initial electric �eld Eq. (2.38) and the one occurring at the expanded ion

front. Here, τ = ωpit/√

2eN is a normalized acceleration time for the ions.

From this isothermal plasma expansion, one can derive now the maximum ion energy

Emax(t) = 2ZkBTe[ln(τ(t) +

√τ2(t) + 1

)]2, (2.47)

via integration of the equation of motion combined with Eq. (2.46), as well as the ion

spectrum

dN

dE(E, t) =

Zne0t√mi√

2Eexp

[−√

2EZkBTe

], (2.48)

per surface unit.

In order to make use of the formulas (2.47) and (2.48), more detailed information about

two parameters are necessary, which are the acceleration time t = τacc on the one handand the initial or undisturbed electron density ne0 on the other.

21


Acceleration time The acceleration time τacc as one of those main free parameters is

just set to the laser pulse duration by Kaluza et al. [44, 66] and later by Mora [67] too.

In a comparative study, Fuchs et al . [68] tried to relate the acceleration time to the laser

pulse duration. For laser pulses shorter than 150 fs, an additional intensity dependenceof the acceleration time [69],

τacc = α(τL + 60 fs) , (2.49)

with α = 2.6 decreasing linearly from 1019 W/cm2 to the constant value of α = 1.3 forI = 3 × 1019 W/cm2, was found, that also contains a minimum contribution of 60 fs tothe acceleration time. Inserting all values into Eq. (2.49) one �nds the acceleration time

to range from 180 fs to 360 fs for the considered conditions.

Initial electron density To estimate the initial electron density, the ongoing processes

shall be considered step by step: A standard approximation assumes that the laser en-

ergy is converted into a total number of hot electrons Ne of temperature kBTe with acertain conversion e�ciency Ne = ηEL/ (kBTe), where the electron temperature equalsthe ponderomotive potential of the laser, Eq. (2.14). Taking into account the energy

absorption rates of the hot electron generation in Sec. 2.1.4, one can estimate the con-

version e�ciency to 10...20 %. In contrast, for slightly longer laser pulses a scaling lawfor the conversion e�ciency [47,70,71] has been established,

η = 1.2×(ILλ

2L

)3/4, (2.50)

which would predict an η ≈ 16...65 % for a0 = 2...4, but with a maximum limitedto 50 % [13]. Corresponding to the results of Chapters 3 and 4, this simple intensitydependence does not hold for the Jeti parameters in its present form and consequently

further development is necessary.

During their propagation through the target, the electrons spread over a certain angle.

As a reasonable assumption, an opening angle of 8◦ has been found [23,62,66]. Since theelectron source is related to the focal spot AFWHM = πr2FWHM one �nds the area at theback surface

ASheath = π (reff + deff tan[8◦])2 , (2.51)

which is occupied by the hot electrons. The e�ective thickness deff = dtarget/ cos[45◦] andreff = rFWHM/ cos[45◦] refer to the Jeti incidence angle in the experiments. Assuming

22


the electrons to form a bunch of length cτL, one derives the initial electron density,

ne0 =Ne

ASheathcτL, (2.52)

which attains values ranging from 0.8× 1020 cm−3 to 2.4× 1020 cm−3 after inserting allnumbers given above.

Replacing t in Eqs. (2.47) and (2.48) with the acceleration time τacc, and making use of

Eq. (2.52), one obtains the proton spectra displayed in Fig. 2.4 for a0 = 2...4. Beyond theexcellent agreement between the predicted and the measured maximum proton energies

(cf. Fig. 3.3), the limits of the TNSA description via plasma expansion into vacuum need

to be discussed. The 1-dimensional Mora model starts out from an extended hydrogen

plasma, whereas the at least 2-dimensional (radial symmetry is a reasonable assump-

tion) experiments employ just a hydrogen rich contamination layer of several nanometer

thickness. Hence, in the 1D model, the spectral proton energy distribution emerges from

the longitudinal position of the protons�deeper sited protons experience an electric �eld

which is partially screened by their predecessors and gain less energy from the accelerat-

ing �eld. For the formation of the calculated spectra screening e�ects on a micrometer

need to be considered. In the 2-dimensional consideration of the acceleration process, the

electric �eld is not homogeneously distributed over its radial extent. Due to the imprint

of the laser's pulse shape to the hot electron population driving the ion acceleration,

the electric �eld strength decreases starting from the center and protons located within

the center can achieve more energy than protons initially located at an outer position.

In addition, the thickness of the contamination layers in the nanometer range does not

su�ce to provide as much screening to deeper sited protons as would be necessary in

order to produce the thermal proton distributions observed [44].

However, the predictive power of the incipient Mora model [22] is great since the

achieved proton spectra compare very well with the spectra obtained in Jeti experiments,

regarding the limitations of the physical meaning discussed above. Particularly, in the

range of η = 10...20 % an excellent agreement is found for the predicted maximum protonenergies.

Several contributions to the plasma expansion into vacuum by Mora [22] were made

with respect to the evolution of the electron distribution, which was initially assumed

to be isothermal. Thus, more realistic scenarios have been considered in order to take

into account the �nite amount of laser energy for the hot electron generation and to

�nally switch towards a model which intrinsically contains a limitation of the ion energy

compared to the more heuristic approach of the acceleration time [67,72�74].

23


2.2.2. Alternative approaches

Many theoretical works have contributed to an advanced understanding of the laser-ion

acceleration processes via TNSA. Thus, a multitude of alternative analytic models as

well as particle-in-cell (PIC) and TREE code simulations are available and shall only be

brie�y mentioned here.

One important and successful approach is to describe the electron distribution in more

detail and to take into account di�erent electron temperatures [21,75�79]. This includes,

for example, the addition of a cold electron population which carries background currents

in the target material that are necessary in order to assure current neutrality inside the

target.

Schreiber et al. [23, 80] provided a model to predict the maximum energy of the most

e�ectively accelerated ion species, e.g. the species with the highest charge to mass ratio

which are the protons, over a wide range of laser intensities. A laser produced hot electron

distribution Eq. (2.36) and its extension at the back surface Eq. (2.52) are considered in

the same coordinate system as in the plasma expansion into vacuum model. The electron

sheath induces a surface charge Qe, which in turn traps the electrons with mean energykBTe at z = λD. The potential,

φ(ξ) =Qe2

2�0√ASheath

(1 + ξ −

√1 + ξ2

), (2.53)

can be derived from solving the Poisson equation with ξ = z·(π/ASheath)1/2. Assuming ane�ective number of charges Q = 2NeλD/ (cτL) outside the foil in electrostatic equilibriumwith the surface potential, this leads to an electric �eld

ETNSA =kBTeeλD

(1− ξ√

1 + ξ2

). (2.54)

between the electron charge and Q induced positive charges. If an ion with mass mi is

accelerated in this potential, it could gain an energy Ei,∞,

Ei,∞ =ZkBTeλD

√ASheathπ

= 2Zmic2√

ηPL8.7× 109 W

, (2.55)

whereas PL = EL/τL denotes the laser power. The maximum ion energy is now deducedby integrating the equation of motion to insert the time dependence of the acceleration.

24


Consequently, one �nds an implicit expression for the maximum ion energy Emax,

τL =

√miASheath

2πEi,∞

(1 +

12

11−X2

)+

14

ln1 +X1−X

, (2.56)

where X = (Emax/Ei,∞)1/2. For the Jeti parameters this model gives a maximum

proton energy of Emax ≈ 4 MeV.

2.2.3. Generation of monoenergetic ion beams

An important parameter which has not been considered so far, is the number of ion

species, which are accelerated by the hot electron population. Here, the power of numer-

ical approaches comes into play. In a numerical study, one can easily add additional ion

species to the plasma�provided that the computing power is su�cient. Most of all PIC

and TREE code simulations are performed in order to investigate the TNSA process.

An illustrative example for the predictive and descriptive power of the numerical ap-

proach is the development of the production of narrow-band ion spectra with laser plasma

accelerators. It started out from the theoretical prediction of employing a special target

geometry in order to radially con�ne the ion source within the TNSA interaction [81].

This con�ned TNSA scheme has experimentally been con�rmed by Schwoerer et al. [18].

Further investigation of this specialized TNSA scheme with a con�ned ion source resulted

in the production of very reliable narrow band ion beams from a laser-based ion accel-

erator [19, 82, 83]. This development only became possible due to numerous theoretical

contributions based on multidimensional numerical simulations of the con�ned TNSA

process which provided with a profound understanding of the ongoing processes [84�88].

A qualitative explanation can be given regarding the two conditions that contribute

to the broad energy spectra of the ion beams, as has been discussed in Sec. 2.2.1: A

radial inhomogeneous electric �eld as well as the screening of deeper sited ions by their

faster predecessors from the initial electric �eld compose spectrally broad ion beams. To

overcome these limitations, on the one hand the radial extent of the ion source needs

to be con�ned to an area corresponding to the laser spot size and on the other hand

the screening e�ects can be utilized when at least two ion species are present. The

acceleration of two ion species present at the target back side leads to the formation

of two ion fronts, which consequently cause an additional charge discontinuity like the

initial one at the fast ion front described in the models above, cf. Fig. 4.8. Within these

discontinuities formed by the heavier ions, lighter ions experience a region of ballistic

�ow because the electric �eld is reduced due to the con�guration of the di�erent charge

25


distributions and tend to form quasi-monoenergetic peaks in their spectral distribution.

Since the narrow-band ion energy spectra are produced behind the light ion front, the

peaks in the spectra appear at ≤ 70 % of the maximum cut-o� energy for ordinaryTNSA [82�84,88].

2.2.4. Electron's fountain e�ect

The simulations do not only permit a detailed view into the ion acceleration processes

but do also reveal information about the electron population and the according electric

�elds. Thus, a 3D PIC simulation by Pukhov [89] predicted a maximum proton energy

of about 6 MeV for laser parameters quite similar to the Jeti, which �ts to the measuredand modeled data. Additionally, it revealed the e�ect of a fountain-like spreading of

magnetically collimated hot electrons out of the back surface of the foil. This feature,

which has also been observed by other groups, [21, 26, 62, 90, 91], will be discussed in

detail in Chapter 4 where the �rst experimental observation of it is presented.

26

3. Experimental scalings for laser ion

acceleration

In preparation of the upcoming chapters of direct observation of laser ion acceleration,

the process parameters shall be optimized. Into consideration come laser properties,

such as pulse duration, intensity shape in terms of prepulses, and maximum intensity

and target properties such as material and thickness. Their dependencies are su�ciently

known. One can �nd an optimum pulse duration for the TNSA process [23, 80] as well

as an optimum target thickness according to the pulse shape of the laser [44, 66]. For

Jeti parameters, also a favor to lower Z material has been found [92] in contrast to

predictions that prefer high Z material [81, 93]. Consequently, 2...6µm thick Titanium(22Ti) foil targets have been used during all experiments.

In order to estimate the potential of laser accelerators for future applications, it be-

comes necessary to test the scalability of the processes. A well-established scaling law is

expressed by the term

Emax ∼ (ILλ2L)0.5 , (3.1)

which has been veri�ed by many groups [14, 22, 23, 80, 84, 94�96]. However, as the laser

intensity depends on three di�erent experimental parameters (energy, pulse duration,

and focal spot size)

IL =ELτLAL

, (3.2)

it becomes obvious that it is important to note, which of these quantities has been varied

to change the intensity. It turns out that most groups have just made energy scans and

kept constant the pulse duration and the spot size of the laser. Furthermore some have

varied the pulse duration separately [23,80,94,96] and found not a strong impact on the

TNSA process despite a slight optimum in [23,80].

Despite this lack of a detailed study of the di�erent contributions to the parameter

laser intensity, Eq. (2.47) gives rise to the prediction Emax ∼ kBTe ∼ (ILλ2L)0.5, similar tothe established formula. Other examples for a di�cult comparableness are the conversion

27

3 Experimental scalings for laser ion acceleration

z

x

y

BE

JETI laser pulse45°, f/2.2

thin titanium foilwith PMMA layer

Thomsonparabola

detection on CR39 orbeam imagingsystem

Dz

hot electronsheath

Figure 3.1.: Scheme of the ion acceleration setup. The Jeti laser pulses are focusedf/2.2 by means of an o�-axis parabolic mirror onto a 5µm thick Titaniumfoil coated with 1µm PMMA, which is leading to a maximum intensity of2 × 1019 W/cm2. The ions are accelerated via the TNSA mechanism anddetected with an ion spectrometer comprising a Thomson parabola as dis-persive element, and MCP detector for online observation. Alternatively,CR39 can be used as an ion detector.

e�ciency of laser light into hot electrons which depends on the laser intensity Eq. (2.50),

too. In addition, the huge variety of lasers that operate in a similar intensity range but

with di�erent pulse lengths and energies as well as wavelengths, holds di�culties for the

comparableness of experimental results.

As the basis for the experiments to come, this complex, entangled intensity dependency

of the ion spectra by varying the pulse energy, the focal spot size and the pulse duration

is investigated separately. Regular TNSA models would predict that all three parameter

have an equal impact on the observed spectra, as the central TNSA processes are assumed

to scale with (ILλ2L)0.5. However, in the following we will see that this crude assumption

does not hold, and that a more advanced scaling model must be introduced.

3.1. Experimental setup

The experiments presented in the following chapters were carried out with the Jeti

laser system at the Institut für Optik und Quantenelektronik of the Friedrich-Schiller-

Universität in Jena. An extensive description of the Jeti laser is given in Appendix A.

After their generation these pulses are delivered to the experimental setup arranged inside

a vacuum chamber, where they are focused f/2.2 by means of an o�-axis parabolic gold

mirror (OAP) onto a 5µm thick Titanium foil coated with a 1µm thick layer of the proton

28

3.2 Detailed intensity study of laser ion acceleration

rich material polymethyl methacrylate (PMMA) in order to enhance the proton yield by

a factor of 10 [92,97�100]. The target foils are stretched in special frames, making use of

the di�erent thermal expansion of frame and foil: With the help of an ordinary freezer,

frame and foil are cooled down to about −18◦C. Now, the foil is clamped inside theframe by tightening its screws and during the warm-up the less expanding foil becomes

stretched by the stronger expanding frame.

These frames are mounted on a x, y, and z translation stages as well as a θ rotation

stage and are checked whether the foil plane and the x and y axes are exactly parallel

(cf. Fig. 3.1 for the orientation of the coordinate system). Once this parallelism has been

ensured the target foil can be moved such that the laser pulse hits a new and una�ected

region on the foil without leaving the formerly adjusted focal plane.

The ions are accelerated normally to the target and detected with an ion spectrometer

which consists of a Thomson Parabola as a dispersive element and a multichannel plate

(MCP) based beam imaging system or nuclear track detector plastics (CR39) for ion

detection. The spectrometer entrance is de�ned by a pinhole, which is aligned to the

target normal axis and available in di�erent diameters de�ning the energy resolution

determined by the geometric opening angle of the spectrometer, e.g. ∆E/E = 0.05 MeVat 1 MeV. Note, however, that since it is possible to take into account the point-spread-function due to the apertures, the energy resolution of the ion spectrometer is at least a

factor of 2...3 better for continuous signals. For calibration methods and a more technicaldescription please refer to Appendix B.

3.2. Detailed intensity study of laser ion acceleration

The experimental realization of a separate and independent intensity variation for all of

the three parameters was done via pump laser tuning for energy variation, compressor

grating de-adjustment for pulse duration, as well as a defocusing due to a certain distance

to the focal plane for the spot size. For the purpose of energy variation, the energy of

the pump lasers was decreased detuning their �ashing delay. Compressor grating de-

adjustment implies changing the distance between the two gratings of each pair.

Fig. 3.2 (b)−(d) displays the intervals of these variations, which always covered atleast one order of magnitude from 2 × 1018 W/cm2 to 2 × 1019 W/cm2. Note, that thesolid red line in Fig. 3.2 (c) represents the theoretical intensity dependence of Gaussian

beam1.

1The spot size AL of a Gaussian beam increases with the distance from the focal plane AL(z) =AFWHM

[1 + (z/zR)

2]. zR = 2AFWHM/ (ln 2λL) denotes the Rayleigh length of the focused laserpulse, which is 19.4µm for the f/2.2 focusing and the Jeti wavelength. Since AFWHM = AL(z = 0)

29


(a) (b)

(c) (d)

Figure 3.2.: Well-de�ned intensity variations. Fig. (a) shows a 10 bit image of the focalenergy distribution. Figs. (b)−(d) illustrate the intensity variations realizedby varying the laser energy EL, the focal spot size AL(z), and the pulseduration τL which were achieved by pump laser tuning, compressor adjust-ment, and the distance to the focal plane, respectively. The solid line in Fig.(c) denotes the theoretical behavior of the spot size taking into account themeasured minimum of the spot size at z = 0 and a Gaussian beam with therespective f/2.2 focusing and constant q. The q factor, which denotes theratio of the amount of energy within the FWHM area and the total amountof energy, increases during the measurement with increasing spot size due toclipping on the CCD camera.

Ion spectra of all three measurements are shown in Figs. 3.3, 3.4, and 3.5, where

ion numbers per energy interval and solid angle are plotted as a function of ion energy

is the measured optimum focal spot size, this method holds true to compare favorably with themeasured spot sizes (red squares). The slight overestimated intensities for the measures shall beattributed to the �xed CCD camera chip size, which starts clipping at the edges for increasing spotsizes.

30


(vertical axis) and intensity (horizontal axis). They spread from the bottom to the top

with increasing particle energy, and from the left to the right with the scanning range

of the observed parameter. A logarithmic, color coded scale represents the number of

ions, which is given per ∆E = 0.01 MeV and per ∆Ω = 1µSr. In terms of energy andpulse duration variation it is su�cient to discuss just protons and carbon C2+, whereasin terms of spot size variation the whole amount of ion species needs to be taken into

account.

3.2.1. Dependence on laser energy

Fig. 3.3 shows the variation of the maximum ion energy for di�erent laser pulse energies.

From the proton and carbon C2+ spectra presented in Fig. 3.3 (a) and (b) we denotea similar behavior: When the laser energy is reduced, corresponding to an intensity

decrease from 2× 1019 W/cm2 to 2× 1018 W/cm2, the maximum ion energy Ecutoff�thetransition between the colors blue and turquoise�decreases from 3.2 MeV to 1.0 MeVfor protons and from 1.7 MeV towards 0.5 MeV for the case of C2+.A closer look into the laser energy dependence of TNSA is available in Fig. 3.3 (c)

where the maximum ion energies are plotted against the laser intensity for all laser

energy increments. The performed power law �ts give Emax,H+ = (0.60± 0.11) MeV ·(IL/1018 W/cm2

)0.59±0.07and Emax,C2+ = (0.24± 0.05) MeV ·

(IL/1018 W/cm2

)0.60±0.20for the protons and carbon C2+ ions, respectively. Thus, the initially introduced scalingbehavior Eq. (3.1) of the maximum ion energy for varying laser intensity is con�rmed

in terms of a laser energy variation with constant pulse duration and spot size. This

is in good agreement with earlier measurements for proton energy scalings carried out

by [14,84,94,96].

The exponent found for the carbon C2+ ions compares very well to the one found forthe protons. The di�erence concerning the prefactors of both scalings shall be attributed

to their di�erent kinetic energy. For this purpose, the following, very simple estimation

is made. Integration of the equation of motion of a charged particle in the TNSA �eld

gives v ∼ q/m · ETNSAt which yields Ekin ∼ q2/m · (ETNSAtacc)2 for the kinetic energy.The electric �eld strength is assumed to be constant for all times, whereas its �time de-

pendency� is adopted by the acceleration time scaling with the laser pulse duration. The

ratio of the factors q2/m for protons and C2+ ions (1 and 1/3 in proton units) comparesgreatly to the ratio of the prefactors within the intensity scalings of the maximum ion

energy (0.6 and 0.24) implying the product ETNSAtacc to be constant. However, screen-ing e�ects due to faster ion species are expected to in�uence the acceleration of carbon

C2+ ions. This is most likely attributed to a longer acceleration duration tacc for the

31


(a)

H+

(b)

C2+

(c) (d)

Figure 3.3.: Energy dependence of ion acceleration. Figs. (a) and (b) show log scalecolor coded proton and carbon C2+ spectra for an intensity variation real-ized by changing the laser pulse energy. The number of ions is given per∆E = 0.01 MeV and ∆Ω = 1µSr. Fig. (c) shows the relation between laserenergy and maximum ion energy. The square root scaling Emax ∼ I2L ex-pected from Eq. (3.1) is con�rmed. Fig. (d) illustrates the dependency ofelectron temperature kBTe deduced from the spectral shape of the ion beamsaccording to Eq. (2.48). The �t in Fig. (d) directly con�rms the pondero-motive scaling of the electron temperature given by Eq. (2.14).

heavier carbon ions.

Considering the acceleration of the present mixture of ion species (protons and carbon

C1...4+), results in an ion distribution, where one �nds the proton front to be the leadingone before the carbon ion fronts with decreasing charge states. According to this picture,

the carbon C2+ ions are screened from the electric �eld by three faster ion populationsand its dependence on the hot electron population will be subsequently attenuated.

32


A second approach for an evaluation of the ion spectra is the comparison of their

spectral shape dN/dE with the one predicted by Mora [22]. Following Eq. (2.48), theelectron temperatures were deduced from the spectral shape of each ion spectrum and

displayed in Fig. 3.3 (d). With decreasing intensity, the electron temperatures are reduced

by a factor 3 within the considered intensity interval.Again, a power law �t was performed in order to describe the measured scaling

of the electron temperature with the laser intensity for an energy variation. Thus,

kBTe,H+ = (72.7± 8.4) keV ·(IL/1018 W/cm2

)0.83±0.04and kBTe,C2+ = (10.8± 1.5) keV ·(

IL/1018 W/cm2)0.79±0.08

were obtained for protons and C2+ ions, respectively. Theponderomotive scaling of Eq. (2.14) which obeys an exponent of 0.5 is not con�rmed byboth power law �ts, but a much stronger dependence on laser intensity variations has

been found experimentally for protons as well as C2+ ions.This analysis of the electron temperature employing the spectral shape of the measured

spectra is not absolutely free of doubt, since, as discussed already in Sec. 2.2.1, the

predictions of the Mora model [22] are very good for the maximum energies of the ion

beams but less powerful in terms of their spectral shape. The di�erence between the

origin of the spectral shape in the 1 dimensional model in contrast to the experiment

is the assumption of a continuous hydrogen plasma compared to screening e�ects and

the radial shape of the accelerating �elds. The Mora model is at least describing a

homogeneous plasma containing just one ion species respectively the leading one.

3.2.2. Dependence on laser pulse duration

An entirely di�erent picture is drawn by the pulse duration measurement, presented in

Fig. 3.4 (a) and (b). While the intensity is shifted over the same order of magnitude 2×1018...19 W/cm2 as for the energy variation, the ion spectra vary to a much smaller degree.For decreasing intensity, the maximum ion energy rises towards a weakly pronounced

optimum at 3.8 MeV proto

Characterization of ion acceleration with relativistic laser-plasma_+Diss... · 2017. 3. 28. ·...

Documents

Transcript of Characterization of ion acceleration with relativistic laser-plasma_+Diss... · 2017. 3. 28. ·...