Ion Acceleration in the Laser
Transparency Regime
vorgelegt von
Dipl.-Phys. Sven Steinke
Von der Fakultät II - Mathematik und Naturwissenschaften -
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzende: Prof. Dr. B. Kanngieÿer
Berichter: Prof. Dr. W. Sandner
Berichter: Prof. Dr. O. Willi
Tag der wissenschaftlichen Aussprache: 06.12.2010
Berlin 2010
D 83
For Kassian.
Abstract
In this work novel, approved approaches for the acceleration of ions by short,
intense laser pulse are investigated and characterized. The applied laser pulses with
relativistic intensities and ultra-high temporal contrast enabled the application of
foil targets with thicknesses below the collisionless skin depth of the laser ( nm)
and hence the rst experiments in the laser transparency regime.
In contrast to opaque, thick foils these targets allow the participation of all
electrons in the focal volume in the acceleration and a group velocity matching of
laser eld, electrons and ions. The presented results expand the known picture of
Target Normal Sheath Acceleration (TNSA) where the energy of the laser pulse
is transferred to kinetic energy of plasma electrons at front side of an opaque,
solid target. These "hot electrons" expand throughout the target and induce a
quasi-static electric eld at the rear side that in turn accelerates the ions. The
conversion eciency of this process is very low due to the high reectivity of the
laser and the lateral spreading of the electrons - typically 1%. Additionally, the
spectral shape of the accelerated ions exhibits an exponential slope.
With the help of a Double-Plasma-Mirror (DPM) the temporal contrast of the
laser pulse could be enhanced by approximately
4
orders of magnitude, without any
distortion of the wave front. In the rst place, the DPM allowed the deployment
of nm-thin foils. An energy throughput of the system of
60 −65
% was obtained.
Diamond-like-Carbon (DLC) foils with thicknesses down to 2nm were used as tar-
gets. They were illuminated with linear polarized laser pulses at normal incidence
and an optimum thickness for ion acceleration of
5.3
nm was demonstrated. At this
optimum thickness, the proton energy was enhanced by a factor of two (
13
MeV)
and in case of carbon ions by a factor of
20
(
71
MeV) compared to experiments
with similar laser parameters, accompanied by a signicant enhancement of the
conversion eciency up to values of about
10
%.
The existence of such an optimum is attributed to a pressure unbalance of the
ponderomotive force of the laser and the restoring electrostatic force raised by the
ions remaining at rest. If the ponderomotive exceeds the electrostatic force, the
electrons are expelled from the sphere of inuence of the ions and hence the ion
acceleration is less eective. In case of thicker targets, the ponderomotive force
iv
is not sucient to exert the maximum polarization between electrons and ions.
However the spectral shape of the ions was still exponential as in imprint of the
exponential distribution of the hot electrons.
Consequently, in the following circular polarization was used to suppress (
v×B
)
- heating as the dominant laser absorption process. According to that, the plasma
electrons are compressed and can be regarded as a mirror that gains more energy
in favor of less momentum if accelerated. Then the ion acceleration occurs in
a co-moving electrical eld and intrinsically leads to a mono-energetic spectrum.
This dominant acceleration by the laser radiation pressure could be experimentally
demonstrated for the rst time. The number and energy of accelerated electrons
could be reduced and a distinct peak in the carbon spectrum was obtained centered
around
30
MeV.
Furthermore, the harmonic radiation of the incident laser was measured giving
a detailed insight into the plasma dynamics during the acceleration. This allowed
the determination of the instantaneous plasma density by the spectral cut-o the
harmonics which could be ascribed to one dimensional plasma expansion.
List of Publications
Parts of this work have been published in the following references:
•
P. V. Nickles, M. Schnuerer,
S. Steinke
, T. Sokollik, S. Ter-Avetisyan,
W. Sandner, T. Nakamura, M. Mima, and A. Andreev. Prospects for ul-
trafast lasers in ion-radiography.
AIP Conference Proceedings
,
1024
, 183
(2008).
•
A. A. Andreev,
S. Steinke
, T. Sokollik, M. Schnuerer, S. Ter Avetsiyan,
K. Y. Platonov, and P. V. Nickles. Optimal ion acceleration from ultrathin
foils irradiated by a proled laser pulse of relativistic intensity.
Physics of
Plasmas
,
16
, 013103 (2009).
•
T. Sokollik, M. Schnuerer, S. Ter-Avetisyan,
S. Steinke
, P. V. Nickles,
W. Sandner, M. Amin, T. Toncian, O. Willi, and A. A. Andreev. Proton
imaging of laser irradiated foils and mass-limited targets.
AIP Conference
Proceedings
,
1153
, 364 (2009).
•
T. Sokollik, M. Schnuerer,
S. Steinke
, P. V. Nickles, W. Sandner, M. Amin,
T. Toncian, O. Willi, and A. A. Andreev. Directional laser-driven ion accel-
eration from microspheres.
Physical Review Letters
,
103
, 135003 (2009).
•
A. Henig,
S. Steinke
, M. Schnuerer, T. Sokollik, R. Hoerlein, D. Kiefer,
D. Jung, J. Schreiber, B. M. Hegelich, X. Q. Yan, J. Meyer-ter Vehn, T. Tajima,
P. V. Nickles, W. Sandner, and D. Habs. Radiation-pressure acceleration
of ion beams driven by circularly polarized laser pulses.
Physical Review
Letters
,
103
, 245003 (2009).
•
P. V. Nickles, M. Schnuerer,
S. Steinke
, T. Sokollik, S. Ter-Avetisyan,
A. Andreev, and W. Sandner. Generation and manipulation of proton beams
by ultra-short laser pulses.
AIP Conference Proceedings
,
1153
, 140 (2009).
•
S. Ter-Avetisyan, B. Ramakrishna, D. Doria, G. Sarri, M. Zepf, M. Borghesi,
L. Ehrentraut, H. Stiel,
S. Steinke
, G. Priebe, M. Schnuerer, P. V. Nickles,
vi
and W. Sandner. Complementary ion and extreme ultra-violet spectrometer
for laser-plasma diagnosis.
Review of Scientic Instruments
,
80
, 103302
(2009).
•
S. Steinke
, A. Henig, M. Schnuerer, T. Sokollik, P. V. Nickles, D. Jung,
D. Kiefer, R. Hoerlein, J. Schreiber, T. Tajima, X. Q. Yan, M. Hegelich,
J. Meyer-ter Vehn, W. Sandner, and D. Habs. Ecient ion acceleration by
collective laser-driven electron dynamics with ultra-thin foil targets.
Laser
and Particle Beams
,
28
, 215 (2010).
•
S. Pfotenhauer, O. Jaeckel, J. Polz,
S. Steinke
, H. P. Schlenvoigt, J. Hey-
mann, A. P. L. Robinson, and M. Kaluza. A cascaded laser acceleration
scheme for the generation of spectrally controlled proton beams.
New Jour-
nal of Physics
, 12(10):103009, 2010.
•
M. Schnuerer, T. Sokollik,
S. Steinke
, P. V. Nickles, W. Sandner, T. Ton-
cian, M. Amin, O. Willi, and A. A. Andreev. Inuence of ambient plasmas
to the eld dynamics of laser driven mass-limited targets.
AIP Conference
Proceedings
,
1209
, 111 (2010).
•
B. Ramakrishna, M. Murakami, M. Borghesi, L. Ehrentraut, P. V. Nickles,
M. Schnurer,
S. Steinke
, J. Psikal, V. Tikhonchuk, and S. Ter-Avetisyan.
Laser-driven quasimonoenergetic proton burst from water spray target.
Physics
of Plasmas
,
17
, 083113 (2010).
•
B. Ramakrishna, A. Andreev, M. Borghesi, D. Doria, G. Sarri, L. Ehrentraut,
P. V. Nickles, W. Sandner, M. Schnurer,
S. Steinke
, and S. Ter-Avetisyan.
Observation of quasi mono-energetic protons in laser spray-target interaction.
AIP Conference Proceedings
,
1209
, 99 (2010).
•
T. Sokollik, T. Paasch-Colberg, K. Gorling, U. Eichmann, M. Schnuerer,
S. Steinke
, P. V. Nickles, A. A. Andreev, and W. Sandner. Laser-driven
ion acceleration using isolated mass-limited spheres.
New Journal of Physics
,
12
, 113013, (2010).
•
A. A. Andreev,
S. Steinke
, M. Schnuerer, A. Henig, P. V. Nickles, K. Platonov,
T. Sokollik, and W. Sandner. Hybrid ion acceleration with ultra-thin
composite foils irradiated by high intensity circularly-polarized laser light.
Physics of Plasmas
,
17
, 123111 (2010).
•
R. Hoerlein,
S. Steinke
, A. Henig, S. Rykovanov, M. Schnuerer, T. Sokollik,
D. Kiefer, D. Jung, T. Tajima, J. Schreiber, B. M. Hegelich, P. V. Nickles,
vii
M. Zepf, G. D. Tsakiris, W. Sandner, and D. Habs. Dynamics of nanometer-
scale foil targets irradiated with relativistically intense laser pulses.
submit-
ted
.
•
M. Schnuerer, A. A. Andreev,
S. Steinke
, T. Sokollik,T. Paasch-Colberg,
P. V. Nickles, A. Henig, D. Kiefer, D. Jung, R. Hoerlein, J. Schreiber,
T. Tajima, D. Habs and W. Sandner. Optimization of electron density
distribution for laser driven ion acceleration.
submitted
.
•
S. Ter-Avetisyan, B. Ramakrishna, M. Borghesi, D. Doria, L. Ehrentraut,
A. A. Andreev, P. V. Nickles, G. Priebe,
S. Steinke
, M. Schnuerer, W. Sand-
ner, and V. Tikhonchuk. Negative ion laser accelerator.
submitted
.
•
S. Steinke
, M. Schnuerer, T. Sokollik, A. A. Andreev, P. V. Nickles, A. Henig,
R Hoerlein, D. Kiefer, D. Jung, J. Schreiber, T. Tajima, B. M. Hegelich,
D. Habs, and W. Sandner. Optimization of laser-generated ion beams.
sub-
mitted
.
viii
Contents
Introduction 1
I Basics 5
1 Interaction of Relativistic Laser Pulses with a Free Electron 7
2 Laser Plasma Interactions 11
2.1 Light Propagation in Plasma . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Ponderomotive Force . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Laser Absorption in a Plasma . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Resonance Absorption . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Brunel Absorption . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 HoleBoring........................... 16
2.3.4 Relativistic
j×B
Heating................... 17
3 Laser-Driven Ion Acceleration 19
3.1 Target Normal Sheath Acceleration (TNSA) . . . . . . . . . . . . . 20
3.2 Radiation Pressure Ion Acceleration . . . . . . . . . . . . . . . . . . 22
3.2.1 InitialStage........................... 23
3.2.2 Light Sail Stage . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Surface High Harmonic Generation by Coherent Wake Emission 31
II Hardware 33
5 Laser System 35
5.1 Ti:Sapphire High-Field Laser . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Laser Pulse Characterization . . . . . . . . . . . . . . . . . . . . . . 37
5.2.1 Spectral Phase Interferometry for Direct Electric-Field Re-
construction........................... 37
x CONTENTS
5.2.2 Scanning 3rd order Cross-Correlation . . . . . . . . . . . . . 40
5.3 Double Plasma Mirror . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Diagnostics 47
6.1 Thomson Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 XUV-Spectrometer........................... 50
III Experiments 53
7 Ion Acceleration with linearly polarized Laser Pulses 55
7.1 ExperimentalSetup........................... 55
7.2 Transition from TNSA to Enhanced TNSA . . . . . . . . . . . . . . 57
7.3 Nonlinear Laser Pulse Transmission of a Thin Plasma Layer . . . . 58
7.4 Maximum Ion Energies and Spectral Shape . . . . . . . . . . . . . . 59
7.5 Conversion Eciency . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.6 Simulation................................ 62
7.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 64
8 Ion Acceleration in the Radiation Pressure Dominated Regime 67
8.1 ExperimentalSetup........................... 67
8.2 Maximum Ion Energies and Spectral Shape . . . . . . . . . . . . . . 68
8.3 Simulations ............................... 69
8.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 71
9 High-Harmonic Generation in Ultra-Thin Foils 73
9.1 ExperimentalSetup........................... 74
9.2 Dynamics of Nanometer-Scale Foil Targets . . . . . . . . . . . . . . 74
9.3 Simulations ............................... 77
9.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 78
Summary and Outlook 81
List of Figures 86
Bibliography 99
Acknowledgments 101
Introduction
Beginning with the rst experimental realization of the laser in 1960 [1] an enor-
mous scientic activity with the intention of enhancing the power of the laser was
initialized. Particularly the approach of
Chirped Pulse Amplication
in 1985 [2]
has led to a continuous raise of the laser peak intensity. In a singular case val-
ues for the intensity of
1022
W/cm
2
have been published [3]. At the same time,
the laser pulse duration has been decreased to a few femtoseconds. The interac-
tion of such laser pulses with matter immediately creates hot-dense plasmas which
at intensities above
1018
W/cm
2
are characterized by relativistic velocities of the
plasma electrons accelerated in the laser eld.
In contrast to conventional accelerators where the accelerating electric elds are
limited to few tens of MeV/m which constrains the highest energy achievable to
∼1
PeV when the acceleration length is given by the equator circumference, plasma
acceleration does not suer from this connement. Already in 1956, the use of
collective elds
[4,5] provided by temporary electron-ion separation in longitudinal
plasma waves [6] has been suggested. These elds can principally exceed those of
conventional accelerators by four orders of magnitude. The possibility to excite
such plasma waves eciently with high intensity laser pulses has led to the proposal
of the
Laser Electron Accelerator
in 1976 [7]. Experimentally, quasi-monoenergetic
electron bunches with energies of
∼1
GeV have been demonstrated in 2006 [8] with
an eective acceleration length on the order of cm.
The current laser-induced acceleration of ions, on the other hand is a secondary
process since the required intensity for a direct acceleration of protons in the
laser eld (
5×1024
W/cm
2
) exceeds the ones currently available. Thus, electrons,
directly accelerated by the laser eld in a plasma are charge separated from the
ions and cause strong electrical elds which in turn accelerate the ions. These elds
reach eld strengths of
∼1012
V/m which is on the same order as the electric eld
of the laser itself. The resulting ion spectra exhibit a Maxwellian shape and a sharp
cut-o with energies up to
58
MeV [9] where a laser intensity of
∼1020
W/cm
2
and
an acceleration length of only a few
µ
m were used.
These beams have intrinsic qualities that distinguish them from beams pro-
duced by conventional accelerators, as there are very high laminarity (
∼0.003
mm
2 CONTENTS
mrad), an ultra-short duration (ps) when they are emitted and a high particle
number per bunch (
∼1011
).
The unique qualities of the laser induced particle beams oer a high potential
for future applications: time and space resolved radiography of dense matter, injec-
tion into conventional accelerators, medical applications (e.g. cancer therapy), ion
beam physics, spallation or transmutation. Moreover, they can be combined with
other laser generated radiation sources (e.g. electrons or high-order harmonics) to
realize pump-probe experiments.
In fact, the ion beam parameters are up to now deviant from those required for
the mentioned applications and thus, further investigations are needed to increase
the maximum energy and at the same time to narrow the energy spectrum to a
monoenergetic shape. In principal, two approaches are conceivable: To vary the
laser parameters and/or to change the target. These changes are then aiming at
changes of the principal acceleration process itself.
In this thesis novel, improved ion acceleration schemes are explored by com-
bining an advanced concept of laser contrast enhancement - the Plasma Mirror
with a target system consisting of free-standing
Diamond-Like-Carbon
(DLC) foils
with thicknesses below the skin depth of the laser, i.e. a few nanometers, devel-
oped by collaboration partners within the Transregio SFB TR18 at the Ludwig-
Maximilians-Universität München (LMU). This combination denotes the rst ex-
perimental demonstration of
Ion Acceleration in the Laser Transparency Regime
for macroscopic extended targets.
The thesis is structured as follows:
•
Chapter 1 introduces the most important properties of relativistic laser pulses
and gives a description of their interaction with a free electron.
•
Chapter 2 gives an overview of the physics being relevant for laser-plasma
interaction at relativistic intensities. Based on the description of the prop-
agation of the laser pulses in a plasma, several mechanisms of laser energy
transfer to plasma electrons are presented.
•
Chapter 3 provides a detailed description of several ion acceleration scenarios
being relevant for this thesis.
•
Chapter 4 presents the basics of high harmonic generation from solid density
targets.
•
Chapter 5 introduces the MBI High-Field Laser that was used for the experi-
ments. Here, the laser pulse is characterized in terms of pulse length and tem-
poral contrast. Additionally, the Double-Plasma Mirror System developed
within this thesis for contrast enhancement is introduced and characterized.
CONTENTS 3
•
Chapter 6 deals with the main diagnostics used to characterize the acceler-
ated ion beams and the high harmonic radiation.
•
Chapter 7 presents dierent interaction experiments of ultra-high contrast
laser pulses with DLC and Titanium foils of thicknesses ranging from
2.9
nm
to
5µ
m. The main focus lies on the ion acceleration, where the main property
of the transition from TNSA to enhanced TNSA as the dominant acceleration
mechanism, namely the symmetric acceleration to both sides of the foil was
revealed for the rst time and nally, rst signatures of a direct participation
of the laser radiation pressure were found. Furthermore, the transmittance
of the DLC foils was quantied and the spectral properties of the transmitted
laser pulse were analyzed. Parts of the results presented in this chapter have
been published in [10].
•
Chapter 8 presents experimental results on ion acceleration where the inu-
ence of the laser radiation pressure was emphasized by changing the laser
polarization from linear to circular. Thereby, the distribution function of
the electron is modied due to the absence of the longitudinally oscillating
component of the Lorentz force (for simplicity in the following referred to as
suppression of electron heating) what results in an ballistic acceleration of
the whole target foil in the focal volume as a quasi-neutral plasma bunch if
target thickness and density were chosen such that the radiation pressure on
the plasma electrons equals the restoring force given by the charge separa-
tion. Two-dimensional particle-in-cell simulations (by X. Q. Yan
1
,
2
) revealed
that the carbon ions are for the rst time dominantly accelerated by the laser
radiation pressure. The results have been published in [11].
•
Chapter 9 presents experimental results of high harmonic radiation gener-
ated at the DLC targets under normal incidence. The measured harmonic
spectra allow the extraction of the target density in the same scenario which
was used for the ion acceleration in Chapters 7+8. With the help of two-
dimensional particle-in-cell simulations the rst experimental observation of
predominantly odd-numbered harmonics could be addressed to a relativistic
generation by the longitudinally oscillating component of the Lorentz force.
Parts of these results have been submitted for publication [12].
•
Chapter 10 summarizes this work and gives perspectives for future experi-
ments on the basis of the ndings of this work.
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
State Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
4 CONTENTS
Part I
Basics
Chapter 1
Interaction of Relativistic Laser
Pulses with a Free Electron
The electric and magnetic elds
E
and
B
of electromagnetic waves can be derived
from Maxwells equations and written using a vector potential
A
[13] as:
E=−∂
∂tA,
(1.1)
B=∇×A.
(1.2)
An elliptically polarized plane wave in vacuum propagating along the x-direction
can then be expressed by:
A=0, δA0cos(φ),(1 −δ2)1/2A0sin(φ),
(1.3)
where
φ=ωt −kx
with the frequency
ω
, the wave number
k
and
δ
, a polarization
parameter such as
δ={±1,0}
for linear and
δ=±1/√2
for circular polarization.
Then, the electric eld can be calculated immediately for linear and respectively
circular polarization:
Elin =ωE0sin(ωt −kx)ey,
(1.4)
Ecirc =ωE0
√2(sin(ωt −kx)ey−cos (ωt −kx)ez).
(1.5)
The relation of the amplitudes
A0
and
E0
can be obtained from Eq. 1.1 and 1.2
and is
A0ω=E0
The vacuum intensity
I
of a monochromatic wave is given by the
cycle-averaged magnitude of the Poynting Vector
S
and is identical for linear and
circular polarization:
I=|S|=c
4π|E|2=1
20cE2
0,
(1.6)
8 Interaction of Relativistic Laser Pulses with a Free Electron
Figure 1.1:
Electron trajectory in case of (a) linear and (b) circular polarization
of the incident laser pulse, propagating in x-direction with a pulse duration of
τ= 15fs
and
a0= 5
. The net energy gain of the electron is zero in both cases, it
is only displaced in laser pulse propagation direction.
with the vacuum permittivity
0
and the speed of light in vacuum
c
.
In the following, the interaction of an electromagnetic plane wave with a single
electron is discussed. It is assumed that:
1. the laser eld is in vacuum with no walls or boundaries present,
2. no static electric or magnetic elds are present,
3. the transversal region of interaction is innite.
The motion of a single electron in an electromagnetic eld is described by the
Lorentz equation [14]:
dp
dt =d(γmev)
dt =−e(E+v×B),
(1.7)
with the relativistic factor
γ=1
1−v2/c2=r1 + p2
m2c2.
(1.8)
For non-relativistic electrons (
v/c 1
) the (
v×B
)-term can be neglected and the
maximum oscillation velocity of the electron
v0
is used to dene the normalized
amplitude
a0
:
a0=v0
c=eE0
meωc
(1.9)
9
which can be rewritten using Eq.1.6 as
Iλ2= 2π2a2
0
0m2
ec5
e2≈a2
0·1.37 ×1018 [
W/cm
2·µ
m
2].
(1.10)
The fully relativistic equation of motion of the electron can now be obtained
by substituting Eq. (1.1) and (1.2) in Eq. (1.7):
dp
dt =e∂A
∂t −v×(∇×A).
(1.11)
Following the derivation of [14] the trajectories in the laboratory frame are:
x=ca2
0
4ωφ+2δ2−1
2sin(2φ),
y=δca0
ωsin(φ),
(1.12)
z=−(1 −δ2)1/2ca0
ωcos(φ).
It has to be denoted that the longitudinal (x) component has (i) a quadratic
dependence on
a0
compared to the transversal (y, z) which only shows a linear
dependence and (ii) regardless of polarization, a component which will force the
electron to drift with an laser-cycle-averaged velocity:
vD=x
t=a2
0
4 + a2
0
,
(1.13)
where
x
is the cycle-average of
x
(Eq.1.12). In Fig.1.1 trajectories for an electron
being initially at rest are plotted in case of an interaction with a laser pulse of
nite duration and a Gaussian envelope. Since the electron is at rest again after
the interaction, its net energy gain by interaction with the electromagnetic eld is
zero, it has only been displaced in direction of laser pulse propagation.
However, most signicant for the experiments described in this thesis is the
fact that in case of circular polarization the electron does not exhibit an oscillating
motion in longitudinal direction as it does for linear polarization.
In order to directly accelerate an electron by an electromagnetic wave at least
one of the assumption (1-3) has to be violated. This violation is given in the present
experiments since the radius of the focal plane is in the order of the electrons
transversal deection (Ch. 2.2 and Fig. 1.1).
10 Interaction of Relativistic Laser Pulses with a Free Electron
Chapter 2
Laser Plasma Interactions
In this chapter the interaction of laser light with a plasma is described. In contrast
to Chapter 1.1 where the interaction with a single electron was discussed, the
collisionless plasma consisting of a cold electron gas (
Te= 0
) with a stationary
ion background is described based on a hydrodynamic theory without inherent
magnetic elds.
2.1 Light Propagation in Plasma
If a laser pulse interacts with such a plasma, electrons will be slightly displaced
while the ions rest. This charge separation creates an uniform electric eld and
causes a restoring force. The resonance frequency of the resulting oscillation is
known as the plasma frequency
ωp
:
ωp=snee2
0γme
,
(2.1)
where
γ
represents the cycle averaged mass increase of the electron [15]:
γ=r1 + a2
0
2.
(2.2)
With the help of the dispersion relation for electromagnetic waves in plasma [16]:
ω2=ω2
p+c2k2, k2=ω2
c2,
(2.3)
where the dielectric function of the plasma
is related to the complex refraction
index by
nR=√
, it becomes obvious that
ωp
divides the properties of the plasma
into transparent (underdense) if
ω < ωp
and opaque (overdense) if
ω > ωp
: The
wave vector
k
becomes imaginary if
ω/ωp<1
and hence electric- and magnetic
12 Laser Plasma Interactions
eld decay exponentially. An evanescent component will anyhow penetrate into
the overdense region up to a characteristic length, the collisionless skin depth
ls
:
ls=c
ωp
.
(2.4)
For the transition between dispersion and absorbtion where
ω=ωp
, the critical
density
nc
can be dened. Using Eq. (2.1) and Eq. (2.3) this critical density is
given by
nc=γomeω2
e2≈1.1×γ×1021 λL
µm−2
cm
−3.
(2.5)
By using this denition the intensity (Eq.1.10) can be further simplied to:
I=1
2mea2
0c3nc.
(2.6)
Finally, the refraction index can be calculated:
nR=r1−ω2
p
ω2=r1−ne
nc
.
(2.7)
The value for the critical density is
≈2.5
-orders of magnitude below the typical
solid density. Therefore, it is possible that the incident laser radiation will be
reected from such an overdense (overcritical) plasma surface. It is important
that an incident laser pulse with a temporal prole (e.g. Gaussian) will not be
reected totally. Since the target will be ionized and the plasma begins to expand
into vacuum, the main part of the laser pulse will interact with a partly underdense
plasma density gradient where dierent interaction/ absorbtion mechanisms take
place Ch.2.3.
But since the plasma frequency in the relativistic case (
γ > 1
) is becoming
dependent on the laser intensity and a focal plane of the laser with a Gaussian
intensity distribution is assumed, the reection index exhibits a spatial dependency
which acts as a focussing optic. The eect is called
Relativistic Self Focussing
[17].
In fact, the principal of the laser reection at an overdense surface is the key
principal of the Plasma Mirror [1821] as used for all the experiments presented
in this thesis (Ch.7-9) and of the generation of relativistic Harmonics [2224].
2.2 Ponderomotive Force
The pressure of electromagnetic radiation incident on a sharp boundary (surface)
is given by the radiation pressure
pL
:
2.2 Ponderomotive Force 13
pL=I
c(1 + nR),
(2.8)
where
nR
is the reection coecient and
c
the speed of light in vacuum. The
ponderomotive force per unit volume is related to the gradient of the radiation
pressure but Eq.2.8 becomes meaningless in case of a laser propagating in an
inhomogeneous plasma, since the electromagnetic eld cannot be calculated only
from the knowledge of
I
and therefore the light pressure is not known [16]. The
electromagnetic elds in the plasma have to be known in order to calculate the
ponderomotive force.
In case of an electron moving with non-relativistic velocity the
v×B
-term
in Eq.(1.7) is smaller compared to the
E
term, the electron oscillates parallel to
E(r)
. In rst order at
r=r0
this yields:
me
d
˙
r1
d
t+
e
E(r0) cos φ= 0,
(2.9)
then the solution is the displacement of the electron
r1=
e
E(r0) cos φ
meω2.
(2.10)
For the second order has to be considered that
E(r) = E(r0)+(r1·∇)E(r0),
(2.11)
and for the velocity d
r/
d
t=v=v1+v2
and the magnetic eld
B(r) = B(r0)
.
Substituting Eq.(2.9) and Eq.(2.11) into Eq.(1.7) results in the second order equa-
tion of motion:
me
d
v2
dt
=−
e
(r1·∇)E(r0) cos φ+v1×B(r0) sin φ
c.
(2.12)
Using Eq.(1.7), (2.10) into (2.12) and averaging the corresponding force over one
laser cycle leads to the ponderomotive force:
fP=−
e
2
2meω2[(E(r)×∇)E(r) + E(r)×(∇×E(r)]
1
=−e2
4meω2∇E2(r).
(2.13)
1
using:
E×(∇×E)=(∇E)·E−(E·∇)E=1
2∇·E2−(E·∇)E
14 Laser Plasma Interactions
Thus, the ponderomotive force in the non-relativistic case is proportional to the
gradient of
E2
and electrons will be pushed out of the region with higher intensities.
The fully relativistic description delivers an additional factor (
1/γ
) [25]:
fP,rel =−e2
4meγω2∇E2(r).
(2.14)
The ejection angle
θ
of the electron from the focus with respect to the laser propa-
gation axis is determined by the ratio of longitudinal and transversal momentum:
tan θ=py
px
=r2
γ−1
(2.15)
For free electrons in the laser focus the laser acts as an accelerator and a spec-
trometer, a fact which has been exploited experimentally at laser intensities of
I= 1018
W/cm
2
[26].
2.3 Laser Absorption in a Plasma
As it has been discussed in Section 2.1 an intense laser pulse incident on an over-
dense plasma will be partly absorbed. At moderate laser intensities the electron
absorbs a photon while colliding with an ion or another electron. The process is
called
Collisional
or
Inverse Bremsstahlung Absorbtion
.
Figure 2.1:
(a) Classical Rutherford scattering of an electron colliding with an
positive ion charge
Ze
.
b
is the impact parameter and
lca
is the closest approach
for
b= 0
. In (b) Eq. 2.16 is plotted
2.3 Laser Absorption in a Plasma 15
The electron-ion collision frequency
νei
is given by [27],
νei =4
3
(2π)1/2Z2
ie4niln(Λ)
(kBTe)3/2m1/2
e≃2.91 ×10−6Z2
iniln(Λ)
Te(eV )3/2[s−1],
(2.16)
with the ion density
ni
, the degree of ionization
Zi
, the electron temperature
Te
and the Coulomb logarithm
ln(Λ) = bmax/bmin
, where
bmax
and
bmin
are the
maximum, respectively minimum of the impact parameter
b
. In Fig. 2.1a the
scattering process for a single electron is sketched. In case of a plasma, the very
distant interactions are screened by the surrounding charged particles, so that
there is a nite value for the interaction range
bmax =λD
. The minimum distance
bmin
is given by the classical distance of closest approach
bmin =lca
(Fig.2.1b), so
that
Λ = λD
lca
=9ND
Zi
,
(2.17)
where
ND= (4πλ3
Dne)/3
is the number of particles in a Debye sphere.
Since the mean electron temperature is given by the Ponderomotive potential
(Eq.3.1) the collision frequency can be expressed as a function of the laser intensity
(Fig.2.1b). If the eective time between electron-ion collisions
τei
is longer than
the laser pulse duration
tp
τei = 1/νei > tp,
(2.18)
the interaction can be treated collisionless, because (i) collisions become increas-
ingly ineective for rising electron temperatures (Eq. 2.16) and (ii) the electron
quiver velocity exceeds their thermal velocity and hence the collective motions
are dominating the plasma kinematics [14]. This condition is fullled if a laser
pulse with a duration of
45
fs and an intensity
>1015
W/cm
2
is interacting with
solid density (
ni= 1023
cm
−1
) plasma (Fig.2.1b) and hence for all the experiments
presented in this thesis.
For this reason the main collisionless absorbtion mechanism shall be sketched
in the following sections.
2.3.1 Resonance Absorption
In case of a p-polarized laser pulse obliquely incident onto a plasma gradient, it is
reected at a density lower than the critical density:
ne=nccos2θ,
(2.19)
where
θ
is the angle of incidence. The electric eld perpendicular to the target
surface tunnels through the point of reection to the critical density surface and
16 Laser Plasma Interactions
resonantly drives an electron plasma wave. The energy distribution of the accel-
erated electrons corresponds to the Boltzmann-distribution [28].
2.3.2 Brunel Absorption
If the amplitude of the oscillating electrons
xp
which oscillate along the density
gradient exceeds the plasma density scale length
L
the resonance breaks down.
The amplitude of the oscillations is approximated by
xp≃eE0
meω2=v0
ω
(2.20)
and abrogates if
v0/ω > L
. However, the electrons in the vicinity of a sharp
vacuum-plasma interface can be accelerated into the vacuum during one laser half
cycle and back when the eld reverses. They cross the critical density surface and
propagate into the overdense plasma, where they are out of the sphere of inuence
of the electric eld and thus, gain energy [29].
2.3.3 Hole Boring
The surface of the critical density will be modulated by the Ponderomotive Force
as discussed in Ch.2.2. In case of a sharp plasma boundary the pressure of the
incident electromagnetic radiation is given by the radiation pressure (Eq.2.8),
which causes
Hole Boring
into the overdense plasma surface. This results primarily
from a pressure imbalance of the momentum ux of the ion ow into the target with
the radiation pressure
pepL
[30], leading to the formation of an electrostatic
shock wave travelling into the target with a constant velocity. The velocity can be
estimated by the pressure balance:
nimiv2
i=I
c(1 + nR),
(2.21)
where
ni
,
mi
and
vi
are the ion number density, mass and velocity. Using momen-
tum and number conservation [30] the velocity can be obtained by substituting
the intensity according to Eq.2.6:
vi
c=sZmenc(1 + nR)
2mine
a0,
(2.22)
with the ion charge state
Z
. In case of the laser with
a0= 5
impinging on a target
with
ne/nc= 500
this results in a velocity of
0.02c
and thus moving
244
nm in
45
fs. This eect was observed experimentally by the Doppler-shift of the reected
laser pulse [31].
2.3 Laser Absorption in a Plasma 17
2.3.4 Relativistic
j×B
Heating
Another mechanism whereby electrons are directly accelerated by the a laser eld
incident on a step-like density prole is called
Relativistic
j×B
Heating
- origi-
nally pointed out by Kruer and Estabrook [32]. The main dierence to
Brunel
Absorption
is the driving term which is the
v×B
component of the Lorentz force
(Eq.1.7) in this case. It oscillates longitudinally at twice the laser frequency
ω
. A
linearly polarized wave of the form of Eq.1.4 gives rise to the longitudinal force:
Fx=−m
4
∂v2
0(x)
∂x (1 −cos 2ωt).
(2.23)
With the help of Eq.1.9 the rst term can be identied as the x-component of
the Ponderomotive Force (Ch.2.2) which pushes the electrons inwards the plasma
density gradient (Ch.2.3.3). The second component leads to heating of the elec-
trons. This works for any polarization except for circular and is most eective
for normal incidence together with relativistic quiver velocities of the electron, i.e.
a0>1
.
18 Laser Plasma Interactions
Chapter 3
Laser-Driven Ion Acceleration
In this chapter the mechanisms of laser-driven ion acceleration which are relevant
for this thesis will be discussed.
A direct acceleration of ions in a eld of currently available laser systems is
impossible due to their high mass. The average kinetic energy a charged particle
with mass
m
can gain, is dened by the Ponderomotive Potential
Φp
[14,30]:
Φp=mc2(γ−1) = mc2 r1 + a2
0
18362−1!.
(3.1)
This yields in case of protons
Φp≈3.5keV
if
a0= 5
what is neglectable in
context of multi-
MeV
ion acceleration. At laser intensities of
5×1024
W/cm
2
(
a0= 1836
) the ions nally reach relativistic velocities within one laser cycle and
the acceleration process enters the so-called Piston Regime [33].
But the laser energy can be eciently transferred to the plasma electrons by
various mechanisms leading to dierent ion acceleration scenarios, depending on
the experimental parameters e.g. intensity and temporal contrast of the laser pulse
as well as target properties. Several mechanisms will be sketched in the following
sections. An empirical scaling law has been obtained by Esirkepov
et al.
[34] in
case of laser pulses with an ultra-high temporal contrast (i.e. a Gaussian pulse).
They found by multiparametric PIC simulations that the maximum ion energy for
a given laser intensity is reduced to a dependence on the normalized areal density
σ
:
σ=πne
nc
D
λL
,
(3.2)
where
D
is the target thickness and
λL
the laser wave length. Then, the highest
ion energy occurs at the optimum areal density
σopt
:
20 Laser-Driven Ion Acceleration
σopt
!
≈3+0.4×a0.
(3.3)
Besides the Piston Regime a radiation pressure dominated regime can be achieved
at much lower intensities when the electron heating is suppressed (Ch.2.3), e. g. by
changing the polarization to circular and using normal incidence on the target [35].
Here, the maximum ion energy becomes proportional to the laser intensity. This
regime is up to now the optimum achievable with the current laser systems in
terms of the maximum ion energy, eciency and the most promising one with
regard to the scalability with the laser intensity.
Except those optimum conditions laser-driven ion acceleration is possible for
a vast set of parameters. Here, the route towards the outlined optimum will be
sketched, beginning with laser and target parameters deviant from those given in
3.3, the co-called
Target Normal Sheath Acceleration
(TNSA) up to a dominant
Radiation Pressure Acceleration
RPA
.
3.1 Target Normal Sheath Acceleration (TNSA)
As well as other laser ion acceleration scenarios discussed here, TNSA is an indirect
mechanism. The laser energy is transferred to kinetic energy of plasma electrons
which in turn accelerate the ions.
Based on the TNSA scheme, rstly suggested by Wilks
et al.
[36] scaling laws
were developed which describe the acceleration process as an isothermal expansion
of an collisionless plasma [37,38] and cover a broad range of parameters [39,40].
Basically, the TNSA regime is distinguished from
Enhanced TNSA
(Ch.3.1) by the
temporal laser pulse contrast ratio, i. e. the relation between the intensity of the
peak of the main pulse and the intensity of the pedestal 5.2.2. Since the inherent
mechanism in TNSA is the laser pulse absorption in an underdense (below the
critical density) plasma, the pedestal of the focussed laser pulse has to be intense
enough (
>1012
W/cm
2
) to ionize the atoms at the front surface of the target and
on the other hand weak enough to not destroy the target before the arrival of the
main pulse. This connes the possible values of the contrast ratio to
10−5−10−9
dependent on the peak intensity of the used laser system.
At the front side of the target, the pedestal of the laser pulse creates an under-
dense plasma and the main laser pulse then interacts with this pre-plasma which
has an exponential density prole. The characteristic scale length in which the
density drops from solid- to critical density is:
L=csτp
, where
cs
is the plasma
sound speed and
τp
the laser pulse duration.
L
is in the order of a few
µ
m. The
amount of laser energy transferred to the kinetic energy of the electrons
η
is esti-
mated by an empirical scaling law:
3.1 Target Normal Sheath Acceleration (TNSA) 21
η≈a(ILλ2)3/4≈1.68 ×10−15 ×I3/4
L
[W/cm
2
]
,
(3.4)
and attains 10-50 % [9,4143].
The electrons are accelerated in direction of the density gradient of the pre-
plasma while the ions are initially stationary. On their way through the target,
the hot electrons are scattered at the cold target electrons or nuclei resulting in an
opening angle of
θ≈1.3
sr [43]. When leaving the target at the rear side, most of
the electron are forced to turn around and re-enter the foil since their energy is not
sucient to escape their self-induced eld. This leads to an equilibrium situation
with a constant number of electrons outside of the foil.
λD=r0kBTe
e2ne
,
(3.5)
with the hot electron temperature
Te
which is usually estimated by the Pondero-
motive Potential (Eq.3.1). The electron density behind the target
ne
can be es-
timated by the number of hot electrons given by the eciency (Eq.3.4) and the
volume they occupy i.e. a cylinder spanned by Eq.3.5 and the opening angle of
the electron bunch. Then, the eld strength is
F0≈kBTe
eλD
,
(3.6)
which is in the order of TV/m for a laser with an intensity in the order of
1019
W/cm
2
and sucient to immediately eld ionize the atoms in the vicinity
of the target rear side.
The actual ion acceleration process is modeled as isothermal expansion of the
capacitor system
sketched above, resulting in a spectrum with a Maxwellian shape
and a sharp cut-o energy [37]
Emax ≈2ZkBTeln 2ωpit
√2eE2
,
(3.7)
with Euler's number
eE
and the ion plasma frequency
ωpi =pZne0e2/0mi
where
Zne
is the initial ion density and
mi
the ion mass. The acceleration time
t
is in
the order of the laser pulse duration
τL
. Fuchs
et al.
[39] found the best t to the
experimental data, with
t≈1.3τL
.
The rst experimental demonstrations [9,4446] were followed by detailed stud-
ies of the ion beam properties: The origin of the protons in the ion spectra is
attributed to a water/ hydrocarbon contamination layer on the foil surfaces [47],
the source size is in the order of
≈100µ
m and the emittance is
10 −100
times
lower compared to conventional accelerators [4851]. The conversion eciency
from laser- to the kinetic energy of the ions is
≤1%
[39].
22 Laser-Driven Ion Acceleration
Furthermore the dominant acceleration of target ions heavier than protons can
be achieved by removing the contamination layer of the targets [5254]. In order
to generate quasi-monoenergetic proton/ ion spectra with energy spreads
<20%
the target geometry has to be changed. This is realized by (i) using spherical,
electrically isolated targets with diameters smaller than the source size to reduce
the transversal hot electron spread at the rear side [5557] or by applying a small
hydrogen-rich dot on the back surface of a foil target to enhance the proton yield
in the central part [58,59].
Enhanced TNSA
As discussed above, the TNSA model is restricted to a certain range of the laser
contrast and is described by an isothermal expansion of a quasineutral plasma on
a time scale of the laser pulse duration [38,40]. In case of an ultra-high contrast
(
<10−11
) and target thicknesses on the order of a few
100
nm this description is
not well tted anymore, since the main stage of ion acceleration in the expanding
plasma begins after the end of the laser pulse and the plasma scale length is in the
order of the Debye length. A more suitable description of an adiabatic expansion
was found to account for this [21,60,61]. The electrons are (re-)circulating between
the target and two Debye sheaths, one at the laser irradiated target front and the
other at the target rear [62]. If a constant number of hot electrons permanently
outside of the target (as in TNSA) is assumed, this results in a dependence of the
maximum ion energy on the target thickness.
The ion acceleration in a thin target continues after the end of the laser pulse
almost symmetrically from both sides of the foil and drops down after the adiabatic
cooling of fast electrons. The aspects of symmetric acceleration and the appearance
of an optimum thickness as an indication for this regime as further discussed in
scope the of experiments in Ch. 7.2.
3.2 Radiation Pressure Ion Acceleration
During the last few years a novel mechanism of laser driven ion acceleration has
gained a lot of interest [35,6369], where the particles are directly accelerated by
the laser radiation pressure.
The basic principle of this idea was at rst proposed in 1966 by Marx [70]. He
raised the idea of a laser-driven vehicle by means of interstellar travel. The photons
reected from the moving vehicle/ sail will be red-shifted in the rest frame of the
laser and thereby transfer a smaller fraction of momentum but a larger fraction
of their energy to the sail. The eciency approaches 100% as the vehicle reaches
the speed of light. Hence, a 10GW laser powered over 10 years would accelerate a
3.2 Radiation Pressure Ion Acceleration 23
vehicle of 30kg to relativistic velocities, neglecting the inherent divergence of the
laser beam. Marx was indeed very optimistic since from his point of view, the
main diculty was the deceleration from relativistic speed at the nal destination.
This idea was transferred to a thin foil target being accelerated entirely as soon
as short pulsed high-power laser were available. It was shown that at intensities
of
1024
W/cm
2
the RPA is dominating other accelerations schemes and that ions
could reach relativistic velocities within one laser cycle [33].
Later Macchi
et al.
[35] found that RPA can be revealed at much lower intensi-
ties by using circular polarization and normal incidence. In this case the generation
of
hot
electrons (necessary in TNSA) is suppressed since the longitudinally oscil-
lating component of the Lorentz force vanishes (cf. Ch. 1.1). Resonance- and
Brunel Absorption are also disbanded due to the normal incidence of the laser (cf.
Ch.2.3).
In fact the so-called RPA-concept consists of two stages: (i) the initial stage,
where the electrons are piled up to an equilibrium since the target ions are still
immobile in that early stage and create a restoring electrical eld. Later, the ions
are set into motion layer-by-layer. And (ii) the light sail stage, where the target
ions begin to move ballistically due to the electrical eld created by the displaced
and compressed electron layer which acts as an accelerated plasma mirror. During
a few laser pulse cycles a quasi-stationary state is established and the plasma cloud
expands with its center of mass moving at an almost constant velocity.
3.2.1 Initial Stage
As discussed above, an intense, circularly polarized laser pulse normally incident
(along the x-axis) on targets with thicknesses below the collisionless skin depth
(cf. Eq.2.4) and with an ion density of
ni=n0
(Fig.3.1a) will ionize the the whole
target volume immediately.
The electrons are pushed inside the target by the ponderomotive force (cf.
Ch. 1 and 2.2), while the ions are still immobile due to their high mass. The
electrons are piled up to densities
ne=np
and leave behind a charge depletion
layer of thickness
d
. This gives rise to an restoring electrical eld
Ex
. They quickly
reach an equilibrium position where
Ex
balances the ponderomotive force (cf. Fig.
3.1a). As soon as the electron density exceeds the critical density, the condition of
a sharp boundary is given and the pressure imposed on the electron layer might
be described by the radiation pressure (cf. Ch. 2.2). Then, the balance condition
with the electrostatic pressure can be formulated [64]:
0E2
0
2=I
c(1 + nR).
(3.8)
24 Laser-Driven Ion Acceleration
Figure 3.1:
(a) Schema of the initial stage, where the electrons are in equilibrium
and the ions still stationary, with the ion density (green line), the electron density
(blue rectangle) and the electrical eld (red line) [35]. (b) Ion phase space exhibiting
characteristic loops when entering the unperturbed plasma region [64]
Assuming that all electrons are removed from the foil, the amplitude
E0
of the
electrical eld
Ex
is given by the Poisson equation:
E0=en0d
0
.
(3.9)
By combining Eq.3.8 and Eq.3.9, the depth of the depletion layer is estimated:
d=s2I0(1 + nR)
cn2
0e2.
(3.10)
An expression for the optimum condition is found by introducing Eq.2.6 into
Eq.3.10:
a0√1 + nR= 2πned
ncλL
(3.11)
Introducing the laser and target parameter given in Ch. 7.1 and assuming total
reectivity (
nR= 1
) results in
d≈2
nm. In case of a target with an initial thickness
below
d
the compressed electron layer will be expelled from the ions [71].
If the target thickness
D
meets the condition
D≥d
, the foil remains in its
initial shape while the ions respond to the strong electrostatic eld
Ex
. Then the
3.2 Radiation Pressure Ion Acceleration 25
Figure 3.2:
Evolution of the electron density (black) and the ion density (red) from
PIC simulation by A.A. Andreev
1
,
2
at (a)
2.8
fs (b)
5.6
fs (c)
8.4
fs and (d)
10.2
fs
after arrival of the trapezoidal pulse with a duration of
33
fs and an intensity of
2×1019
W/cm
2
. The electron temperature in (a) has been set to
3
keV to account
for expansion, since the simulation here starts at an laser intensity of
1017
W/cm
2
.
The main features of the acceleration process described in the text are reproduced
here: The compression of the electron layer in (b), the development of the left peak
in the ions density (c) attributed to Coulomb explosion and the ballistic acceleration
of the whole target in (d).
optimum condition can be written as:
1
Max Born Institute, 12489 Berlin, Germany
2
Vavilov State Optical Institute, Sankt Petersburg, Russia
26 Laser-Driven Ion Acceleration
σopt ≥a0/√2,
(3.12)
which is close to the empirical expression (Eq.3.3) found by PIC simulations.
The ions initially in the depletion layer (
0< x < d
) will be accelerated by
Coulomb explosion. This feature is also observed in the the PIC simulation (cf.
Fig. 3.2).
The balance condition Eq.3.8 is fullled for ions being initially in the compres-
sion layer (
d<x<d+ls
) for all times since
Ex
is a linear function of
x
. All
these ions reach the point (
d+ls
) at the same time with the same velocity. Then
they cross the electron layer (indicated by blue-shaded area in Fig.3.1b) and travel
inside the target until they encounter the next layer of stationary ions. And since
the inertia of the electrons is much lower than that of the ions, the compressed
electron layer adjusts its positions immediately due to the light pressure. Now, the
next stationary layer of ions experiences the electrostatic eld and is accelerated
itself. Thus, the acceleration proceeds layer by layer resulting in the characteristic
loops in the phase-space as depicted in Fig.3.1b. This cyclic acceleration contin-
ues until the target rear side is reached. Now, all ions have the same velocity and
undergo the Light Sail Acceleration described in Ch.3.2.2.
3.2.2 Light Sail Stage
Marx vision raised a lot of controversy and six month later Nature published a
letter by Redding [72] that identies a mistake in one of Marx's equations but in-
troduces again statements concerning the eciency which turned out to be wrong.
Since this claim was still quoted in the literature, Simmons and McInnes published
a paper 26 years later [73], where this topic was discussed nally. They conclude:
But whatever his mistakes, at least Marx was more right than his critics!
Howsoever, applying this simple model of an accelerated mirror to the
optimum
parameters (Eq.3.11) it turns out to be very predictive to estimate the ion energies.
To obtain the equation of motion and an expression for the eciency, the retarded
time
w=t−x/c
is introduced to be aware of the delay between photon emission
and the moment they reach the target. With d
x/
d
t=w/c =β
, the target velocity
in the laser frame, it follows that
d
t
d
w=1
1−β.
(3.13)
The light pressure
p
on the target between two events
Q1(x, t)
and
Q2(t+
d
t, x+
d
x)
,
in the rest frame of the laser is
d
p=(1 + nR)I
c
d
w.
(3.14)
3.2 Radiation Pressure Ion Acceleration 27
The Lorentz transformation to the instantaneous rest frame of the target intro-
duces the Doppler shift
d
p=(1 + nR)I
c
d
ws1−β
1 + β=(1 + nR)I
c
d
wγ(1 −β),
(3.15)
with
γ= (1−β2)−1/2
. The
proper time interval
between
Q1
and
Q2
is d
τ=
d
t/γ =
d
w/γ(1 −β)
and hence,
d
p=(1 + nR)I
c1−β
1 + β
d
τ.
(3.16)
By transforming back to the laser frame and interpreting the light pressure as
hydrostatic pressure by applying Pascal's law one obtains the equation of motion:
d
d
τ(γβ) = (1 + nR)I
ρDc21−β
1 + β,
(3.17)
where
ρ
and
D
are the target density and thickness. The nal velocity of the foil
βf
can be found in case of (
nR= 1
) as a function of the pulse uence by using
Eq.2.6
F=Zw
0
I
d
w≃2πIt
ω=2πa2
0mencc3t
ω,
(3.18)
where
t
is the laser pulse duration in laser cycles and nally by integrating Eq.3.17:
βf=(1 + E)2−1
(1 + E)2+ 1,
(3.19)
where
E
is the total energy output of the laser
E=2F
ρDc2= 2πZmea2
0t
Amiσ,
(3.20)
with
mi
,
A
and
Z
being the ion mass, ion mass- and charge number.
The kinetic ion energy then calculates to
T= (γ−1)mic2.
(3.21)
Nonlinear reectivity
To include the eects of partial reectivity and relativistic Self-Induced-Transparency
into the Light Sail model, an analytical expression for the reectivity of an over-
dense plasma is introduced. According to Vshivkov
et al.
[74] the reectivity of a
28 Laser-Driven Ion Acceleration
Figure 3.3:
Maximum ion energy as a function of the incident laser intensity for
a reectivity of
nR= 1
(blue line) and
nR
given by Eq.3.22 (red line) at the
particular optimum areal density
σopt =a0/√2
and a pulse duration of
t= 15
laser cycles.
plasma density prole with a delta-functional shape is well approximated by the
normalized areal density
σ
(Eq.3.2) and is given in the restframe of the foil by,
nR=σ2
1 + σ2.
(3.22)
Using this expression of
nR
in Eq.3.17 gives rise to the following expression of the
nal velocity [69]:
βnR(w) =
(1 + E(w)−σ−2)21 + q1+4σ−2(1 + E(w)−σ−2)−2+ 2σ−2−2
(1 + E(w)−σ−2)21 + q1+4σ−2(1 + E(w)−σ−2)−2+ 2σ−2+ 2
(3.23)
In case of a
at-top
temporal laser prole, i.e. a constant intensity,
E(w)
can be
written as a function of time
E(tp) = 2πZmea2
0tp
AmiσT ,
(3.24)
3.2 Radiation Pressure Ion Acceleration 29
where
tp
and
T
are the durations of the laser pulse and the laser period.
In Fig.3.3 the kinetic ion energy is plotted as a function of the laser intensity
for a laser pulse duration of
tP= 45
fs (
t= 15
and
T= 2.8
fs). Here, the two cases
nR= 1
and
nR
given by Eq.3.22 are compared and the areal density was matched
to the particular optimum condition (Eq.3.12). In the limit
σ→ ∞
, i.e.
nR→1
,
Eq.3.20 is recovered.
The resulting ion spectra are intrinsically monoenergetic and an ion energy of
1
GeV might be obtained with a laser intensity of
∼2×1022
W/cm
2
.
Eciency of the Light Sail Acceleration
The eciency
η
, dened as the ratio between the kinetic energy of the acceler-
ated target over the electromagnetic energy of the laser pulse can be obtained by
the conservation of the photon number
N
reected by the moving target. The
total energy of the incident photons is
N~ω
and the reected photons is
N~ωr
respectively. The frequency of the reected photons
ωr
is given by the Lorentz-
Transformation (Doppler-shift) from the rest frame of the laser to the mirror and
back and thus:
ωr=1−β
1 + βω.
(3.25)
Hence the energy transferred to the target is
∆E=N~ω−N~ωr=2β
1 + β
|{z}
=η
N~ω.
(3.26)
It follows that
η→1
as
β→1
and that the velocity of the target depends only
on how much energy has been pumped and not how.
30 Laser-Driven Ion Acceleration
Chapter 4
Surface High Harmonic Generation
by Coherent Wake Emission
High-order harmonic generation (HHG) due to reection of an intense laser beam
from a steep, overdense plasma gradient has been observed since the 1970s [75
77]. In contrast to this early measurements, the generation process in case of
femtosecond laser pulses has been found recently [78,79].
The generation of high harmonic radiation from a solid density bulk [23,24,79
84] as well as foil targets [78,85,86] has been studied extensively in the last years
and is mainly motivated by the possibility to generate energetic short-wavelength
light pulses. Simultaneously, HHGs give a detailed insight into the laser plasma
dynamics as they allow the probing of the plasma density [12,79], magnetic elds
[87], surface dynamics [82,88] and electron heating [85]. Properties which are
highly interesting e.g. for the understanding of ion acceleration processes from
thin foils as done in Ch.9.
The mechanism of Coherent Wake Emission (CWE) is based on a phase match-
ing of the laser eld with plasma oscillations inside an overdense plasma gradient
which leads to the emission of harmonics up to the plasma frequency.
To achieve this regime a p-polarized, intense laser pulse has to obliquely inci-
dent onto a plasma with a nite scale length. If the assumptions of Brunel ab-
sorption (Ch.2.3.2) are fullled, electrons in the vicinity of the the sharp plasma
density prole are accelerated into the vacuum during one half laser cycle and back
into the plasma when the eld reverses. They pass the critical density surface into
the overdense region, where the laser eld is no longer able to act on them. On
their way, the electrons excite plasma waves in their wake eld. Under certain
conditions these plasma waves can undergo linear mode conversion and radiate
electromagnetic waves - in principal due to inverse resonance absorption [89,90].
The periodic repetition of this process, every laser cycle, leads to a spectrum of
harmonics of the incident laser frequency [79].
32 Surface High Harmonic Generation by Coherent Wake Emission
The unique properties of the high harmonics generated by CWE clearly distin-
guishes them from relativistic harmonics generated in the regime of the
Relativistic
Oscillating Mirror (ROM)
[24,80]:
(i) CWE harmonics can be generated at sub-relativistic intensities (
a01
) as
low as
1016
W/cm
2
[79].
(ii) CWE exhibit a cut-o in the harmonic spectrum which does not depend
on the laser intensity but on the density of the target material itself, since CWE
harmonics are generated by the mode conversion of the plasma waves as described
above. The wavelength of the emitted harmonic is given by the frequency of the
plasma wave which is a function of the local electron density. Therefore each
harmonic is created in a certain depth of the plasma gradient and the cut-o
harmonic number
qco
is determined by the maximum electron density:
qco =rne
nc
.
(4.1)
This dependence has been shown experimentally [91,92] with a typical value for
qco
of 20 for fused silica or 14 for Polymethylmetacrylate.
(iii) CWE harmonics show two kinds of chirps: the harmonic-chirp inside each
harmonic and the atto-chirp between the individual harmonics. The harmonic-
chirp is attributed to the fact that a fs laser pulse changes its intensity during
the pulse and therefore, the Brunel electrons being pulled in the vacuum spend a
dierent amount of time there before being re-injected into the plasma [79]. This
time spacing increases linearly during the laser pulse and is analogous to what
happens during the generation of gas harmonics [93].
The atto-chirp on the other hand is related to the fact that each harmonic
is created at a specic depth in the plasma gradient. Because, the resonance
condition of the created plasma wave is dependant on the local electron density,
which is higher for higher frequencies. Therefore, higher order harmonics have to
travel a longer distance to the plasma surface than lower order harmonics.
The atto-chirp has been quantied by means of XUV-autocorrelation [84].
Part II
Hardware
Chapter 5
Laser System
The fundamental study of laser-plasma interactions (e.g. laser-ion acceleration)
in the relativistic regime requires electromagnetic elds at ultra-high intensities
(
IL>1018
W/ cm
−2
). To reach this intensity, e.g. a laser pulse energy of a
few hundred millijoule with pulse duration of a few femtoseconds are needed.
Delivering such pulses is the primary purpose of the High-Field Laser System
at the Max-Born Institute.
In Ch.5.1 a short overview of the TW-Ti:sapph system is given. The key to
achieve such high intensities and avoid damage of elements in the amplied chain
at the same time, is a technique called
Chirped Pulse Amplication
(CPA) [2].
With this technique an amplication up to the petawatt level becomes possible.
Figure 5.1:
Principle of Chirped Pulse Amplication (CPA)
Before the invention of CPA the amplication of femtosecond laser pulse was
limited to a few millijoule because a laser pulse at intensities of
∼
GW/cm
2
causes
damage to the amplier crystals through non-linear eects such as self-focussing.
Additionally, it is not advising to allow its propagation in air since the laser pulse
36 Laser System
would immediately self-focus or cause lament propagation which would ruin the
original pulses qualities. However it is anyhow possible to allow propagation in air
if the beam diameter is expanded and hence, self-focussing and lamentation are
suppressed. But at some point the intersection to the vacuum has to be realized
in order to conduct laser matter interaction experiments. In order to minimize
any nonlinearities at the transmittance through a window into the vacuum, the
intensity should be as low as possible.
In order to keep the intensity of the laser pulse below the threshold of the
nonlinear eects, in CPA, the femtosecond pulse is rst stretched in time, then
amplied and nally re-compressed in vacuum (Fig.5.1).
In chapter 5.2 it is described how the laser pulse was characterized by means
of (i)
Spectral Phase Interferometry for Direct Electric-eld Reconstruction
(SPI-
DER) [94] and (ii)
scanning 3rd order cross-correlation
similar to [95]. These
techniques allow the (i) reconstruction of the temporal shape of the main pulse
by spectral shearing interferometry on a femtosecond time-scale and (ii) high pre-
cision measurements of the
Amplied Spontaneous Emission
(ASE) background
level on a nanosecond time-scale. The duration of the ASE pedestal is usually
several orders of magnitude longer than the main pulse but still carries a fraction
of the total pulse energy, resulting in intensities well above the ionization threshold
of any target in the laser focus.
Chapter 5.3 illustrates one possibility to enhance the temporal intensity con-
trast between laser pulse peak and ASE pedestal. To improve this contrast by 4-5
orders of magnitude combined with a steep rising edge, the use of a
Double-Plasma
Mirror
[19,21,96] is required. In fact, the plasma mirror is the key technology to
access the transparency regime of laser-ion acceleration.
5.1 Ti:Sapphire High-Field Laser
The High-Field Laser at MBI is a >30TW-Ti:sapph based on the CPA technique.
A schematic setup is shown in FIG. 5.2.
The primary femtosecond (17fs) pulses with an average power of
300
mW are
generated by a commercially available Kerr-lens mode-locked oscillator [97] with a
repetition rate of
81.5
MHz. In the stretcher a linear frequency chirp is introduced
onto the pulse by the use of an all-reective Oenertriplet telescope [98]. Due to
the combination of two spherical concentric mirrors (the rst concave, the second
convex) with a single grating, the stretcher is completely characterized by symme-
try and hence, symmetrical aberration and astigmatism are disbanded. Chromatic
aberrations are per se excluded, because all the optical elements are mirrors. The
result is a stretched laser pulse with a duration of
≈700
ps. Afterwards the pulse
is amplied in three consecutive multi-pass stages up to
4
J. Between those ampli-
5.2 Laser Pulse Characterization 37
Figure 5.2:
Schematic drawing of the MBI TW Ti:sapph laser.
er stages the beam diameter has to be extended -to avoid damage of the amplier
crystals- up to
∼100
mm for a reasonable ux on the compressor grating. Finally,
the laser pulse is directed in a vacuum chamber where the compression takes place.
The compressor basically consists of two parallel grating and an vertical roof mir-
ror retro-reector. Motorized stages serve for alignment of the grating in vacuum.
Online compensation of (high-order) phase dispersion becomes possible due to the
measurement of the spectral phase with a SPIDER as described in chapter (5.2).
The nal pulse lengths is in the range of
40 −50
fs.
5.2 Laser Pulse Characterization
5.2.1 Spectral Phase Interferometry for Direct Electric-Field
Reconstruction
Spectral Interferometry is a method to measure the phase dierence, of two laser
pulses travelling along dierent optical paths, in the frequency domain. The ex-
perimental setup only consists of an interferometer (e.g. Michelson) and a spec-
trometer to detect the spectral intensity of the two recombined laser pulses. The
resulting spectral interferogram exhibits a strong intensity modulation along the
frequency axis with and -in case of two identical pulses- a constant modulation
38 Laser System
period length
ω
which decodes the temporal separation of the pulses
τ= 2π/ω
.
In spectral shearing interferometry -such as SPIDER is- the frequencies of the
two identical replicas of the input pulse are shifted by a small amount
Ω
with
respect to each other. The measured spectral interferogram yields
S(ωc) =|˜
E(ωc)|2|+|˜
E(ωc+ Ω)|2+ 2|˜
E(ωc)˜
E(ωc+ Ω)|
(5.1)
×cos[φw(ωc+ Ω) −φw(ωc) + ωcτ],
where
˜
E(ωc)
is the electric eld in frequency representation of the variable center
frequency of the spectrometer
ωc
.
S(ωc)
consists of three summands. The rst
and the seconds represent the sum of the two spectra which do not contain any
phase information. The third is the result of interference and carries all of the
phase information in terms of the phase dierence.
A simulated interferogram in shown in Fig.5.3a.
Figure 5.3:
Simulated Interferogram (a) and its Fourier Transform including a 4th
order super Gaussian lter function (b).
The procedure of reconstructing the phase shall be sketched in the following:
At rst the Fourier transform of
S(ωc)
is calculated and one of the oscillating
terms is isolated by a lter function. This time series reveals three components
at
t=±τ
and at
t= 0
(cf. Fig.5.3b). The Filter function is typically a fourth
order super Gaussian of full width
τ
. The ltered time series is inverse fourier-
transformed afterwards. Secondly, the linear phase term
ωcτ
has to be removed.
For this purpose a spectral interferogram is recorded without adding a spectral
shear. Since the two pulses are identical now, the only phase contribution is
ωcτ
.
If this linear part is subtracted, the phase dierence is
5.2 Laser Pulse Characterization 39
Θ(ωc)≡φw(ωc)−φw(ωc−Ω).
(5.2)
If the shear is relatively small compared to the structure of the spectral phase,
Θ(ωc)
may be approximated with
Θ(ωc)≈Ω
d
φω(ωc)
d
ωc
.
(5.3)
With the help of this approximation, the spectral phase can be reconstructed by
integration
φω(ωc) = 1
ΩZ
d
ωcΘ(ωc).
(5.4)
Finally the spectral amplitude
|˜
E(ωc)|
has to be determined. This is realized by a
separate measurement of the laser pulse spectrum.
The approach of the SPIDER as introduced in 1998 by C. Iaconic and I. A.
Walmsley [94] uses two replicas of the original pulse which are delayed with respect
to each other with an interferometer. A third one is stretched in time and strongly
chirped. Afterwards they are recombined in a thin nonlinear crystal by Sum Fre-
quency Generation (SFG). By doing so each of the two replicas is up-converted
with a dierent monochromatic frequency component of the third pulse. Thus,
both pulses are frequency shifted to the second harmonic frequency range by a
slight dierent amount resulting in the necessary frequency shear (cf. Fig.5.4).
Figure 5.4:
Principle of the SPIDER setup
The resulting interferogram diers from Eq. (5.1) by only a frequency shift.
Thus, the pulse may be reconstructed as described above using only algebraic
operations on the interferogram.
40 Laser System
Fig.5.5a shows the interferogram recorded with the Spider setup as described
above of the Ti:Sapph Laser System at
500
mJ. In Fig.5.5b the reconstructed laser
pulse in the time-domain is plotted along with the reconstructed spectral phase. A
Gaussian Fit of the temporal prole yields to a laser pulse duration of (
45 ±1
)fs.
A constant phase would describe a fourier limited pulse and therefore an optimum
compressor alignment. The every day performance of the Ti:Sapph laser deviates
from this optimum, but the characteristics of the spectral phase reveal the potential
of optimizing the compressor alignment. Hence, laser pulse durations in the region
of
30 −35
fs are accessible with this laser system.
Figure 5.5:
(a) The interferogram as recorded with the SPIDER setup.(b) The re-
constructed laser pulse in the time domain (blue squares) and its Gaussian Fit (red
line), yielding to a laser pulse duration of (
45 ±1
)fs (FWHM). The reconstructed
phase is plotted as black line
5.2.2 Scanning 3rd order Cross-Correlation
In addition to the characterization of the laser main pulse on a fs time scale the
knowledge of the pulse pedestals intensity on a ns time-scale is crucial for the
experiments described in this thesis. The level of the pedestal is described by the
relation between the intensity of the peak of the main pulse and the intensity of
the pedestal itself i. e. the temporal intensity contrast. Since the pedestal carries
a signicant fraction of the total pulse energy, its intensity is sucient to ionize or
even destroy any material in the laser focus before the arrival of the main pulse.
The challenge of the temporal characterization of a femtosecond laser pulse is
to measure the actual prole of the pulse, e.g. to distinguish between pulse front
and tail on a ns-time scale with a fs-resolution and a dynamic range of
>10
orders
5.2 Laser Pulse Characterization 41
Figure 5.6:
(a) Principle setup of the 3rd order cross Correlator. (b) Measured
temporal prole of the MBI high-eld Ti:sapph. The artefacts occur due to reec-
tions in the correlator setup.
of magnitude. The use of an
3
rd order non-linear process is one possibility to fulll
these requirements.
In a
3
rd order cross-correlator (Fig.5.6(a)) a laser pulse with frequency
ω
is
split into two identical pulses by the use of a beamsplitter. One pulse is fre-
quency doubled by Second Harmonic Generation (SHG) in a non-linear crystal
and focussed into a second crystal together with the delayed second pulse. By
sum-frequency-generation the third harmonic (
2ω+ω
) of the original laser pulse
is generated. The linear superposition principal of electric elds is not valid in
non-linear crystals, the electric elds here superimpose multiplicative. The used
setup is a scanning alignment (multi-shot) with a translation stage to delay the
two pulses with respect to each other from
20
fs up to
1
ns. A photomultiplier is
used for the detection of the
3ω
pulse. The dynamic range (
10
orders of magni-
tude) is realized by adjusting the photomultiplier voltage, e.g. the amplication.
In addition a set of calibrated lters is used in the original beam path.
In Fig.5.6b a typical example for the obtained temporal pulse prole of the MBI
Ti:Sapph laser (Ch.5) is shown. Thus, the initial temporal contrast
30
ps before
the arrival of the main pulse is in the order of
≈5×10−7
, originating mainly from
amplied spontaneous emission (ASE) in the amplier crystals. It additionally
consists of two ramps, one beginning
≈5
ps before the main pulse where the
contrast linearly increases up to
≈10−7
at
1
ps before the main pulse and the
second beginning
<1
ps before the main pulse reaching up the main pulse where
42 Laser System
the contrast reaches values of
≈10−5
. This is owing to imperfect compression
of the laser pulse and are hence only erasable after the nal compressor, e.g. by
means of an Plasma Mirror which is described in the next chapter (Ch.5.3). The
pulse prole does not own any real pre-pulses, since they were originating from
a leakage in a regenerative ampliedr which is not applied in the current setup
(Ch.5.1)
5.3 Double Plasma Mirror
Essential for the generation of high energetic ion beams and high harmonics from
the interaction of a high intensity laser pulse with a solid target with nanometer
thickness is the improvement of the intrinsic contrast (Ch.5.2.2) of the Ti:sapph
laser system (Ch.5.1) by
4−5
orders of magnitude. Thereby it can be assured
that the main pulse not only interacts with an intact target but also with a steep
plasma gradient.
Hence, the research in the eld of contrast enhancement has been intensied
during the last years. Basically two approaches have been considered: Improve-
ment of the laser chain itself or the cleaning of the pulse after it has been com-
pressed. Electro-optical methods (e.g. Pockels Cells) are capable of suppressing
pre-pulses on a ns time scale [99] as well saturable absorbers [100]. More advanced
concepts use a double CPA [101] with additional elements based on cross-polarized
wave generation (XPW) [102] or Optical Parametrical CPA (OPCPA) with short
pulse (fs-ps) pump lasers [103].
Self-induced plasma switches (Plasma Mirrors (PM)) for contrast enhancement
were rst demonstrated with glass targets in 1991 [104] and with liquid targets in
1993 [105]. In 1994 the contrast enhancement was measured by means of
2
nd order
autocorrelation [18], a full characterization of a single PM was given in 2004 [19,20].
The demonstration of a Double Plasma Mirror (DPM) in 2006 [106]. In 2007 it
was shown that a PM is also applicable for sub-
10
-fs lasers with a kHz repetition
rate [107].
The PM technique is based on focussing a high intensity laser pulse on a trans-
parent medium in a way that most of the ASE-pedestal is transmitted while the
raising edge of the main pulse rapidly ionizes the surface of the medium. When
the electron density reaches the critical density
nc
(cf.Eq. 2.5), the reectivity
increases abruptly. To achieve the highest possible reectivity for the main pulse,
the incident intensity has to be adjusted in a way that the PM is triggered at the
raising edge of the main pulse. At the same time it has be assured that the reect-
ing surface is still planar, i. e. to prevent a long-scale pre-plasma. The contrast
enhancement
C
is simply given by the ratio of the reection coecients from the
overcritical plasma surface
np
and from the undisturbed
cold
medium
nm
:
5.3 Double Plasma Mirror 43
Figure 5.7:
(a) A laser beam is focused on an anti-reection coated medium, in a
way that most of the pedestal energy is transmitted. With increasing of the intensity
in time, electrons are exited in the medium . If the electron-density reaches the
critical density
nc
the reectivity increases abruptly, the laser pulse is reected.
(b)The rst o-axis parabolic mirror (OAP1) focuses the incoming beam between
the two Plasma Mirrors (PM1+2). The pulse is then re-collimated by the second
OAP (OAP2) and afterwards directed to the nal target chamber.
C=np
nm
(5.5)
Thus, the contrast enhancement is principally given by the quality of the anti-
reection coating of the used PM substrates. The
cold
reectivity is specied by
the vendor of the plasma mirror substrates to be
nm<0.5%
[108]. Then the
contrast enhancement by two successive PM is better than 4 orders of magnitude.
To be of any practical use, the reected beam quality should show no degra-
dation relative to the incident beam. Consequently the PM surface must remain
at compared to the laser wavelength. The plasma expansion time has to be re-
stricted, resulting in an upper limit
∆t
that can elapse between plasma formation
and the arrival of the main peak. A tolerable expansion of the critical surface of
λ/4
is assumed. Thus, the inequality
cs∆t < λ
4
(5.6)
has to be fullled. Here
cs
is the sound speed of the expanding plasma surface.
It has been measured from the blue-shift of the laser spectrum reected o the
PM surface and is quantied to
≈5×107
cm s
−1
for an incident intensity of
1016
W/cm
2
[18,20]. This results in
∆t≈400
fs.
44 Laser System
Figure 5.8:
(a) Beam prole incident on PM1, showing the near eld modulations
100
mm out of focus, with
I≈3×1016
W/cm
2
and (b) on PM2,
1
mm out of focus
with
I≈1018
W/cm
2
.
For the present pulse shape (cf.Fig. 5.6b) this suggests the use of a focussing
optic with a focal length of
10
m in order to lower the focus intensity to values
<1017
W/cm
2
. Since this is unpractical, o-axis parabolic mirrors (OAPs) with
a focal length of
1.5
m are used in this setup, what results in a focal intensity of
1018
W/cm
2
. Hence, the rst PM substrate has to be placed in near eld of the
incident beam to lower the intensity on the PM surface. This may result in a
distortion of wave-front since the near-eld exhibits modulation, but it has been
shown that the prole of the reected pulse is hardly disturbed and that the beam
is still very well focusable [20,109].
Figure 5.9:
(a) The imaged back-reections (CCD2 Fig.7.1) for 3 dierent con-
trast ratios were recorded with a 16-bit CCD camera at the same zoom level and
full dynamic range compared to a low-energy shot without plasma generation (a):
(b) initial contrast: without DPM (
D= 30
nm) C=
5×10−8
, (c) high contrast:
uncoated BK7 (
D= 5.3
nm) C=
10−10
and (d) ultra-high contrast: (
D= 5.3
nm)
C=
10−12
.
The second substrate is placed in the focus (or slightly after in the expanding
5.3 Double Plasma Mirror 45
beam) since the contrast here has been increased already and the 2nd PM thereby
acts as spatial lter [110]. The corresponding beam proles are shown in Fig.5.8.
The PMs are situated, both with an
45◦
angle of incidence, at distance vari-
able between
5
mm and
20
mm. Computer-controlled translation stages move the
Plasma Mirrors in 3 dimensions: (i) parallel to their surfaces (
x
) to provide a clean
surface for each shot, (ii) on the beam-axis (
z
), to nd the best focal position, (iii)
and for alignment purposes in (
y
) and (
∆y
) to vary the distance between the two
surfaces.
It has been shown [20] that the reectivity of a single PM suddenly increases
when the laser intensity reaches the collisionless regime (
I > 1015
W/cm
2
, Ch.2.3).
To exclude Resonance- and Brunel Absorbtion s-Polarization with respect to the
PM surfaces was used. For that reason the reectivity raises until relativistic
absorption eects (cf.Ch.2.3) become signicant and/or the reected beam quality
is degraded by deformations of the critical surface due to the radiation pressure
(Ch.2.3.3).
To control the high spectral reectivity near- and far-eld of the reected beam
is monitored by CCD1+2 (Fig.7.1). The near-eld camera is also capable to
measure the overall reectivity from the DPM-system, since it has been cross-
calibrated with a calorimeter in the main beam path (Fig.5.10). By monitoring
the far-eld by means of the backscattered light from the nal target it is also
assured that the main laser pulse interacts with an intact target (Fig.5.9).
The overall reectivity of the DPM system is shown in Fig. 5.10. For the
shots with negative shot numbers the intensity on the PM surfaces was adjusted
by varying their positions relative the the laser focus i. e. the intensity on the
surfaces. For the shots
1−23
the intensities were kept constant. The linear
decrease of the reectivity from
65%
to
50%
is attributed to the accumulated
debris originating from the plasma expansion. To minimize this eect a ceramic
aperture was placed between the two PM surfaces (Fig.7.1).
46 Laser System
Figure 5.10:
Overall reectivity of the DPM system: The negative shot numbers
mark the alignment procedure, where the intensities on the PM substrates was var-
ied. After optimizing (positive shot numbers), the intensities where kept constant.
The linear decrease of the reectivity due to accumulated debris is emphasized by
a linear t (red line).
Chapter 6
Diagnostics
48 Diagnostics
6.1 Thomson Spectrometer
The spectral properties as well as the number of accelerated proton and ions can
be measured with a Thomson Spectrometer. It consists of a pinhole and a magnet
combined with electric eld plates such as their eld lines are parallel to each other
and placed perpendicular to the direction of propagation of the ions and particle
detector [111]. Technical details of the used spectrometer can be found in [112].
Figure 6.1:
CCD image of the illuminated phosphor screen coupled to the MCP
from a
5µ
m Titanium foil. The zero point determines the projection of the pinhole
without any deection.
A small ion beamlet propagating in z-direction through the pinhole is deected
by the magnetic eld
By
x=QBylL
miv.
(6.1)
The dierent ion species, i.e. particles with dierent
Q/mi
are separated by the
deection of the electric eld
Ey
y=QEylL
miv2,
(6.2)
where
Q
is the charge of the ion and
mi
its mass,
l
the eective extension of
the electric and magnetic eld and
L
the distance between magnet and detector.
For particles with the same
Q/mi
, the parabolic equation can be obtained by
combining Eq.6.1 and 6.2
y=miEy
QB2
ylLx2,
(6.3)
6.1 Thomson Spectrometer 49
which has its vertex in the zero point. It has to be denoted that Eq.6.1-6.3 are
only valid for small angles of deections. The particles are detected with a Multi-
Channel Plate (MCP) with a diameter of
80
mm coupled to a phosphor screen and
an attached CCD camera. The resulting counts of the CCD image were calibrated
with the detection sensitivity of the MCP, by using
5.48
MeV particles emitted
from a
241
Am source [112].
In Fig.6.1 a typical picture of an ion spectrum from a Titanium foil with
a thickness of
5µ
m as target is shown. The energy resolution of the Thomson
Spectrometer is given by the width of the parabolic trace
δ
[113] which can be
extracted from the CCD image (Fig.6.1).
The energy uncertainty
∆E
is estimated by
∆E
E≈δ
x,
(6.4)
where
x
is the magnetic deection (Eq.6.1) and is dened by the spectrometer
setup:
∆E≈δ
a, a =QBylL
√2mi
.
(6.5)
In the used setup the detected solid angle is
114 ×10−9
sr and the energy is reso-
lution
∆E/E ≈3%
.
50 Diagnostics
6.2 XUV-Spectrometer
For the detection of the harmonic radiation presented in Ch.9 a modied ver-
sion of a commercially available
Acton Research VM-502
has been used. It is a
compact
20
cm focal length normal incidence vacuum spectrometer designed for
measurements ranging from UV down to the XUV spectral range. Due to the
normal incidence geometry the reectivity of the concave, iridium coated (1200
lines/mm) grating limits the detectable wavelength to
>50
nm, i.e. a harmonics
up to the order of 16.
Figure 6.2:
Schematic drawing of the modied Acton Research VM-502.
The principle setup is shown in Fig.6.2. The concave grating images the en-
trance slit onto the detector on the Rowland Circle [114] in a Seya-Namioka Ge-
ometry [115]. The detector consists of a CsI-coated Multi-Channel Plate (MCP)
coupled to a phosphor screen which is imaged using a ber taper and a 16-bit CCD
camera. The wavelength incident on the MCP can be adjusted by rotating the
grating which is realized by an external wavelength-calibrated controller. Thereby,
dierent wavelength are imaged to dierent positions on the MCP. By the use of a
large area MCP a spectral range of
50
nm can be acquired in a single shot, through
the expense of the spectral resolution, since the MCP is at and therefore does not
6.2 XUV-Spectrometer 51
account for imaging on the Rowland circle. Nevertheless, the spectral resolution
is still sucient to obtain high quality spectra of the individual harmonics.
52 Diagnostics
Part III
Experiments
Chapter 7
Ion Acceleration with linearly
polarized Laser Pulses
In this chapter experiments on ion acceleration from ultra-thin foils of thicknesses
ranging from
5µ
m down to
2.9
nm are presented. The targets were irradiated by
linear polarized pulses of 45fs FWHM duration focussed to a peak intensity of
up to
5×1019
W/cm
2
. Additionally, the spectral properties of the transmitted
laser pulse are analyzed. Parts of the results presented in this chapter have been
published in [10].
7.1 Experimental Setup
The experiments were performed at the MBI - TW Ti:sapph laser of central wave-
length 810nm delivering 1.2J in 45fs FWHM pulses with an ASE contrast ratio
smaller than
10−7
up to
∼10
ps prior to the arrival of the main peak. By means of
a re-collimating DPM 5.3, this contrast was increased by estimated four orders of
magnitude (Ch.5.3), which is essential for the suppression of pre-heating and ex-
pansion due to the pulse background. The energy throughput of this DPM system
was in the order of 60-65%, resulting in pulse energies of 0.7J. Finally, the laser
pulse was focused on Diamond-Like-Carbon (DLC) target with a f/2.5 parabolic
mirror down to 6
µ
m diameter and is diraction limited by 30% under normal
incidence. This corresponds to a peak intensity of
2.6×1019
W/cm
2
or a nor-
malized vector potential of
a0= 3.6
. For a second set of experiments the focussing
procedure was improved, leading to a focus diameter of 3.6
µ
m and therefore a
peak intensity of
5×1019
W/cm
2
or
a0= 5
. The resulting ions were detected with
a MCP coupled to a Thomson-Parabola as described in Ch.6.1. Additionally, the
transmitted laser pulse was registered with a 12-bit optical grating spectrometer.
DLC targets of thicknesses ranging from
2.9−50
nm were used, having a density
56 Ion Acceleration with linearly polarized Laser Pulses
Figure 7.1:
Setup of the experiment: The rst o-axis parabolic mirror (OAP1)
focuses the incoming beam (red) between the two plasma mirrors (PM1, 2). Then
it is re-collimated by OAP2. Afterwards it is directed to the nal target chamber,
where it is focussed by OPA3 on the target. The emerging ion beam (blue) is
analyzed and detected by a Thomson-Spectrometer and a MCP (Ch.6.1).
of
2.7
g/cm
3
. Owing to the high fraction of sp
3
-, i.e. diamond-like bonds of
∼
75%,
DLC oers unique properties for the production of mechanically stable, ultra-thin,
free standing targets, such as exceptionally high tensile strength, hardness and
heat resistance. The thickness of the DLC foils was characterized by means of an
atomic force microscope (AFM), including the hydrocarbon contamination layer
on the target surface which was present during the experiments [116]. In addi-
tion, in order to precisely determine the structure of the contamination layer, the
depth-dependend composition of the foil was measured via Elastic Recoil Detec-
tion Analysis (ERDA) [116]. From these measurements we obtain a thickness of
∼1
nm for the hydrocarbon contamination layer. In the following the combined
thickness of bulk and surface layer as it appears in the actual ion acceleration
experiment presented is referred.
7.2 Transition from TNSA to Enhanced TNSA 57
7.2 Transition from TNSA to Enhanced TNSA
As discussed in Ch.3.1 the transition from the
TNSA
to the
enhanced TNSA
regime
is characterized by a symmetric ion acceleration for both, front- and rear side of the
target and by a dependence of the maximum ion energy from the target thickness.
This dependence on the target thickness has been experimentally demonstrated
by several groups exhibiting an optimum thickness around the Debye length
λD
(e.g. a few hundred nm) [21,117119]. The aspect of a symmetric acceleration
lacks of an experimental proof so far and is the aim of the experiment presented
in this chapter.
To record ion spectra simultaneously from the front- and the rear side of the
target, the experimental setup (Ch.7.1) had to be varied. The target was rotated
to an angle of incidence of
45◦
and two ion spectrometers (Ch.6.1) were placed in
the target normal directions. Additionally, the incident intensity on the target was
reduced to
4×1019
W/cm
2
(
a0= 3.9
) for the shots without DPM and consequential
to
2.3×1019
W/cm
2
(
a0= 3.2
) with DPM. This enabled the use of Titanium targets
as thin as
1µ
m without the use of the DPM.
Figure 7.2:
Front side (black squares) and rear side (blue triangles) proton spectra
obtained from laser driven 1
µ
m Ti- foils when the intense driving laser pulse has
a medium contrast (a) and an ultra-high contrast (b).
In Fig.7.2 typical front- and rear side proton spectra are shown. Obviously,
the spectra obtained without DPM show a strong asymmetry between front- and
rear side with a cut-o energy of
<2
MeV for the front- and
6.5
MeV for the rear
side. The rear side spectrum also consists of a plateau region between
2−6
MeV
which approximately begins at an energy where the front-side spectrum has its
58 Ion Acceleration with linearly polarized Laser Pulses
energy cut-o. On the other hand, the spectra recorded with the use of the DPM
look almost identical concerning their maximum energy of
6
MeV and their overall
spectral shape. Their dierence in the number of ions is in the order of calibration
of the used spectrometer setup itself (Ch.6.1).
Together with measurement of the optimum thickness, obtained earlier [21,109]
this measurement undoubtedly represents the rst experimental demonstration of
ion acceleration in the
enhanced TNSA
regime.
7.3 Nonlinear Laser Pulse Transmission of a Thin
Plasma Layer
Since the partial transmission of the intense laser pulse through an expanding
target plays a decisive role to determine the dominant regime of ion acceleration
[120122], the opaqueness of the used foils is examined.
Figure 7.3:
(b) Measured, calibrated laser pulse spectra in transmittance of the DLC
foils. (a) shows the signal of a constant white light source used for calibration of
the optical path. (d) The resulting transmittance (
a0= 5
) of the DLC targets as
a function of the target thickness obtained by numerical integrating the spectra of
(b), compared to the analytical formula Eq.7.1. The errorbars are given by the
average shot-to-shot uctuation.
For the experiments presented in this chapter the target rear side was imaged
to a 12-bit optical grating spectrometer to (i) measure the transmittance of the
7.4 Maximum Ion Energies and Spectral Shape 59
laser pulse through the foil and (ii) obtain the spectral properties of this nonlinear
process. For calibration of the optical path and the spectrometer a commercial
white light source delivering a constant intensity over a signicant spectral range
was introduced in the setup. In Fig.7.3a the white light spectrum is shown which
was used for calibrating the spectra shown in Fig.7.3b.
Finally these calibrated spectra were numerically integrated to obtain the trans-
mittance of the foils. To have an absolute measure of the transmittance, the shots
were normalized to a shot without target. The experimental results for the trans-
mission
T
are plotted along with the values expected by the analytical formula,
here approximated for linear polarization [74,123]:
T∼
=1/[1 + (σ/γ)2].
(7.1)
The experimental results are in agreement with the analytical model.
To decrease the target thickness below
2.9
nm, they were heated by a cw-
laser with a powers in the range of
100 −500
mW for several seconds resulting
in foil temperatures of
1000 −3000
K. Thereby it is assured that the water and
hydrocarbon contamination layer is completely removed from the DLC foils. In
fact, the ion spectra measured from these targets do not show any protons. Since
the thickness of the contamination layer was determined to
∼1
nm, the obtainable
minimum thickness was
∼2
nm.
The situation changes drastically when these pre-heated targets were used in
the experiment. The targets are getting highly transparent and the spectrum
(Fig.7.3b) exhibits a strong modulation and is broadened signicantly. To have a
closer look at this feature, in Fig.7.4 several spectra, obtained for a thickness of
∼2
nm are plotted. They demonstrate the whole bandwidth of the modulation
and broadening at this thickness. The spectral broadening gives rise to the onset
of pulse-shortening due to the non-linear interaction: The transmittance Eq. 7.1
is a function of the laser amplitude
a0
and hence of the intensity. Thus, a small
fraction associated with the most intense part of the laser pulse is transmitted, the
rest reected or absorbed. The transmitted pulse can therefore be much shorter
than the incident pulse [124].
7.4 Maximum Ion Energies and Spectral Shape
The maximum detectable energy/nucleon (
E
max
) obtained for protons and fully
ionized carbon ions C
6+
is plotted as a function of the target thickness in Fig.7.5
for two dierent laser intensities. The cut-o energies for both ion species exhibit
a strong dependence on target thickness. In case of protons and
a0= 5
,
E
max
increases from around 7.5MeV for a 40nm foil up to 13MeV for a 5.3nm foil,
while for C
6+
the maximum energy rises from 26MeV for use of a 40nm foils
60 Ion Acceleration with linearly polarized Laser Pulses
Figure 7.4:
(b) Measured, calibrated laser pulse spectra in transmittance of the
DLC foils with
≈2
nm thickness revealing a huge bandwidth of strong modulation
and spectral broadening.
to 71MeV for 5.3nm. Taking into account the dierent
Z/mi
values for protons
and C
6+
ions, they are both accelerated to the same velocity. Further decrease
of the target thickness down to 2.9nm results in a steep drop of the observed
ion energies. The energy distributions of all species are continuous as expected.
7.5 Conversion Eciency 61
Figure 7.5:
(a) Maximum energy per atomic mass unit for both protons and carbon
ions plotted vs. foil thickness for a normalized laser vector potential of
a0= 5
. The
experimental results are in excellent agreement with numbers deduced from 2D-
PIC-simulations by X. Q. Yan
1
,
2
. (b) Typical ion spectrum obtained at a target
thickness of
5.3nm
.
However, especially the observed carbon C
6+
energy of 71MeV reaches for the
rst time a range of values that were previously only accessible by large single
shot Nd:glass laser systems with
30 −50
J pulse energy [85,125]. The shot-to-shot
variations of 10%, indicated by the error bars, arise mainly from uctuations of
the laser pulse itself and from macroscopic modulations of the target surface.
7.5 Conversion Eciency
The conversion eciency (Fig.7.6b) was calculated by numerically convoluting
the energy dependent divergence of the accelerated ion beam with the initially
measured (
a0= 5
) spectra. These values for the divergence (cf. inset in Fig.7.6a)
were extracted from PIC simulations and supported by experimentally obtained
proton beam proles using a stack of radiochromic lm (RCF) layers. Comparing
those to the divergencies of ion beams generated by other laser systems e.g. [126],
the divergence is smaller and the linear dependence on the energy is reasonable in
the considered high-energy part. The dependence of the CE on the target thickness
is showing analogous characteristics to
E
max
(Fig.7.5). A sudden rise is observed
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
State Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
62 Ion Acceleration with linearly polarized Laser Pulses
when entering the laser transparency regime (
D < ls
) reaching 10.5% for
C6+
and
1.6% for protons at the optimum thickness.
Figure 7.6:
(b) Calculated CE for protons and C
6+
ions as a function of the target
thickness, based on a energy dependent divergence angle of the ion beam, which
was extracted from PIC simulations by X. Q. Yan
1
,
2
and supported by measured
beam proles (a). The experimental data are in good agreement with the values of
the CE obtained from PIC simulations.
The CE drops down to below 1% when the targets thickness reaches the TNSA
dominated regime [39,118].
7.6 Simulation
The experiments were compared to 2D-PIC simulations, where the laser pulse
was modeled by a Gaussian shape in time with a FWHM of 16 laser cycles, a
Gaussian intensity distribution in the focus with a FWHM spot size of 4
µ
m and
an
a0= 5
(
a0= 3.6
). It interacts with a rectangular shaped plasma of initial
density
ne= 500 ×nc
consisting of 90% carbon ions and 10% protons (in number
density) to account for the presence of a contamination layer. The simulation box
is composed of 1200
×
10000 cells with 1000 particles per cell and a total size of
(10
×
20)
µ
m
2
. The total simulation time
τ
given in laser cycles is 120 relative to
τ= 0
when the laser pulse reaches the initial position of the target (
x= 3λ
).
Fig.7.5a shows that the simulated maximum proton energies as well as the
carbon energies are in excellent agreement with the experimental data. In par-
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
Sate Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
7.6 Simulation 63
Figure 7.7:
Electron (a) and carbon ion (b) density distribution vs.
x
at time
t= 40τ
. (Simulation courtesy of X. Q. Yan
1
,
2
)
ticular, the optimum target thickness of 5.3nm is reproduced. This thickness for
the peak ion energy is consistent with the thickness given by the empirical rela-
tion
σopt
!
≈3+0.4×a0
that was found in multiparametric PIC-simulation studies
by Esirkepov
et al.
[34]. For foil thicknesses below the optimum (
σ < σopt
), the
plasma becomes increasingly transparent (Fig.7.3b) and the pulse is more trans-
mitted than absorbed. Due to the low number of electrons in the focal volume
(
∼1011
) their electric current is no longer sucient to (i) reect the laser pulse and
(ii) to establish an eective longitudinal charge separation eld. This results in a
sudden drop in ion energies and
50 %
of the CE (the same amount as the reduction
of the target thickness), as it was observed in the experiment (Fig.7.5+7.6). In
case of
σ > σopt
, the laser intensity is not sucient to generate the maximum
possible displacement of all electrons within the focal volume which gives rise to
a decrease in the longitudinal charge separation eld. Note that the optimum
thickness for ion acceleration predicted theoretically and observed experimentally
is much smaller than previously used target thicknesses.
To illustrate the underlying ion acceleration mechanism in Fig.7.7a,b detailed
electron and carbon densities after 40 laser cycles as a function of the laser propa-
gation direction
x
and the transverse coordinate
y
in units of the laser wave length
λL
are depicted. It is immediately striking that ions are accelerated strongly
asymmetric, heavily favoring the direction of laser propagation. This is in strong
contrast to the model based on the self-consistent solution of the Poisson equation
presented in [61] which predicts a symmetric acceleration that happens primarily
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
State Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
64 Ion Acceleration with linearly polarized Laser Pulses
after the end of the laser pulse. From Fig.7.7a,b it can be seen that the carbon
ions are accompanied by co-moving electrons which accelerate the ions in forward
direction.
7.7 Summary and Discussion
In summary, the experiments using laser pulses of ultra-high contrast presented in
this chapter demonstrate increased values for the highest ion energies, two times
in case of protons (
13
MeV) and 20 times for carbon ions (
71
MeV) if comparing
to experiments performed with approximately the same laser parameters. It shall
be noted that for the generation of
70
MeV carbon ions so far huge single shot
Nd:glass laser facilities with pulse energies of
(20−50)
J were needed. And although
conducted in single-shot mode, the limitation of repetition rate is principally given
by the laser system which operates with
10
Hz.
Additionally the conversion eciency from laser- to kinetic energy of the accel-
erated ions was improved more than
3−10
times for protons (1.6 %) and
≈100
times for carbon ions (10%).
A strong dependence of the resulting maximum energies on target thickness
was primarily observed experimentally, with a pronounced optimum for an initial
foil extension of
5.3
nm which is in good agreement with empirical scaling laws
(Eq.3.2) reduced from PIC simulation [34]. The experimental results are in ex-
cellent agreement with 2D-PIC simulations which also reveal that the ions are
accelerated strongly asymmetric, heavily favoring the direction of laser propaga-
tion in contradiction to the regime of
enhanced TNSA
.
In addition, previously published
TNSA
scaling laws based on analytical models
fail to interpret these results: The isothermal uid model [39] does not predict an
optimum target thickness, moreover it only predicts an maximum ion energy of
∼1
MeV for our parameters. More generally in the
TNSA
scheme, a dependence on
the target thickness is only given due to the transversal spreading of the electrons
at the rear side of the foil [40]. This is neglectable if comparing foil thicknesses
between
2−50
nm. The model of
enhanced TNSA
(Ch.3.1) [61] predicts also an
optimum thickness but fails to explain the results on the maximum ion energy and
its position with respect to the target thickness and the nature of the sharpness
of the maximum around several nm, when the target becomes transparent for the
incident laser.
Furthermore, the opaqueness of the used foils were examined. Here, the mea-
sured transmission is in good agreement with the values calculated analytically [74].
The spectral shape of the transmitted light includes strong modulations and ex-
hibits a signicant spectral broadening and hence the beginning of the generation
of ultra-short transmitted pulses [124] suggesting the use of double targets [127].
7.7 Summary and Discussion 65
In conclusion, the results of the experiments with linear polarization constitute
the onset of the laser radiation pressure participating in the acceleration process
which is indicated by:
(i) Protons and C
6+
-ions are accelerated to the same velocities.
(ii) The spectral shape of the C
6+
-ion at the optimum thickness shows signa-
tures of modulation.
(iii) The acceleration is strongly asymmetric favouring the laser propagation
direction.
(iv) The measured maximum ion energies and the conversion eciency as well
as the position of the peak energies with respect to the target thickness can not
be explained by any theoretical model in the
TNSA
scheme.
To reveal the inuence of the radiation pressure on ion acceleration, experi-
ments with circular polarization were carried out and are presented in Ch.8.
66 Ion Acceleration with linearly polarized Laser Pulses
Chapter 8
Ion Acceleration in the Radiation
Pressure Dominated Regime
In order to emphasize the inuence of the laser radiation pressure on the ion
acceleration process the laser polarization is changed to circular [35,6369,128,
129]. As discussed in detail in Ch.3.2 the absence of the longitudinally oscillating
component in the Lorentz force (Ch.1) leads to a suppression of electron heating.
Instead, the electrons are compressed to a highly dense electron layer piling up
in front of the laser pulse which in turn accelerates ions. By choosing the laser
intensity, target thickness and density such that the radiation pressure equals the
restoring force given by the charge separation, i.e. the conditions given by Eq.3.12,
the whole target foil in the focal volume propagates ballistically as a quasineutral
plasma bunch. This
ying mirror
(Ch.3.2.2) is gaining less momentum in favor
of more energy from the laser eld as it is accelerated. As long as the electron
temperature is kept low, the acceleration can be maintained, and the process is
expected to lead to very high conversion eciencies and ion energies scaling linearly
with laser intensity in contrast to TNSA where the ion energy is proportional to
the square root of the laser intensity. In this scenario, all particle species are
accelerated to the same velocity, which intrinsically results in a monochromatic
spectrum with the energy given by Eq.3.21. The results have been published
in [11].
8.1 Experimental Setup
The experiments described in this chapter were also carried out at the MBI - TW
Ti:sapph laser system as introduced in detail in Ch.5. DLC foils of thicknesses
D=
2.9−40
nm were placed in the focal plane at normal incidence. Furthermore, the
experimental setup shown in Fig.7.1 was extended by a mica crystal operating as
68 Ion Acceleration in the Radiation Pressure Dominated Regime
λ/4
plate placed into the beam path behind the plasma mirror to change the laser
polarization from linear to circular. In addition, a magnetic electron spectrometer
with a solid angle of
∼2×10−4
sr equipped with Fujilm BAS-TR image plates
was positioned behind the target at an angle of
22.5◦
with respect to the laser axis.
8.2 Maximum Ion Energies and Spectral Shape
The obtained maximum ion energies per atomic mass unit plotted over the target
thickness
D
are shown in Fig.8.1a. Compared to linearly polarized irradiation,
the maximum energies of proton and carbon are lower but a dependence on the
initial foil thickness is still observable, with an optimum at
D= 5.3
nm with
maximum energies for protons and carbon ions of
10
MeV respectively
45
MeV.
The theoretically obtained optimum foil thickness
D= 2
nm, given by Eq.3.12
was reproduced experimentally by the same order of magnitude.
Figure 8.1:
(a) Experimentally observed maximum proton (blue) and carbon
C6+
(red) energies per atomic mass unit over target thickness for circular polarization
and (b) the spectra corresponding to a thickness of
D= 5.3
nm.
The corresponding electron spectra for
D= 5.3
nm are shown in Fig.8.2. It
can be clearly seen that circularly polarized irradiation results in a pronounced
reduction in the number of highly energetic electrons as expected. To illustrate
the consequent impact on the acceleration of ions, experimentally observed proton
and carbon spectra are plotted in Fig.8.1b at the experimentally observed opti-
mum foil thickness. A monotonically decaying spectrum is obtained for protons
comparable to the case of linear polarization in Fig.7.5. In contrast, in case of
circular polarization the spectrum of fully ionized carbon reveals two components.
8.3 Simulations 69
Figure 8.2:
Measured electron spectra at the optimum target thickness
D= 5.3
nm
showing a strong reduction in electron heating for circularly polarized irradiation.
In addition to the continuously decreasing low energetic ion population reaching
up to
∼20
MeV, a distinct peak is observable around
∼30
MeV. To compare the
experimental ndings to the model approximations in Ch.3.2.2 the target thick-
ness of
(5.3±1)
nm has to be introduced into Eq.3.2, and Eq.3.23 and Eq.3.21
have to be evaluated, resulting in an energy of E=
(20 ±10)
MeV. This is in agree-
ment with the experimentally observed value. If only the thickness of the DLC
is considered (without contamination), i.e.
D= 4.3
nm the resulting energy is in
excellent agreement with the experimental data.
This spectral peak experimentally only occurs at a thickness of
D= 5.3
nm
and circular polarization, whereas the the shape of the proton spectrum was not
aected.
8.3 Simulations
To substantiate the experimental results further, two-dimensional PIC simulations
were carried out. The parameters were the same as described in Ch.7.6 but the
polarization was changed to circular.
To account for the small solid angle of the Thomson spectrometer (cf.6.1), only
particles propagating in forward direction with a cone of half angle
0.01
rad were
considered for simulated spectra of Fig.8.3. An isolated quasi-monoenergetic peak
emerges at the end of the laser target interaction at
t= 45
fs.
The corresponding carbon ion phase space shows a signicant amount of parti-
cles located in a discrete area, constituting a rotating structure. The series of loops
originate from the continuing front side acceleration and ballistic evolution of the
70 Ion Acceleration in the Radiation Pressure Dominated Regime
Figure 8.3:
(a) Temporal evolution of the C
6+
ion spectrum in case of circular po-
larized irradiation of a
5.3
nm thickness DLC foil obtained from 2D PIC simulation
(Simulation courtesy of X. Q. Yan
1
,
2
.)
target and thus giving a clear evidence of radiation pressure being the dominant
acceleration force [35,6467] as described in Ch.3.2.1.
Figure 8.4:
Electron (a) and carbon ion (b) density distribution vs.
x
at time
t= 40τ
(Simulation courtesy of X. Q. Yan
1
,
2
.)
The dierence becomes striking when comparing the acceleration dynamics by
examining the electron and carbon density distributions observed in the simula-
tion (Fig.8.4) and comparing them to the ones obtained with linear polarization
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
State Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
8.4 Summary and Discussion 71
(Fig.7.7). In the case of circular polarization, the electron population maintains
in its structure as a thin layer of high density being pushed by the laser in forward
direction. Consequently, the carbon ions co-propagate with the compressed elec-
trons and the whole focal volume is accelerated as a quasi-neutral dense plasma
bunch by the laser radiation pressure.
However, the isolated quasi-monoenergetic peak in the carbon spectrum at
the end of the laser-target interaction does not preserve its shape upon further
propagation (Fig.8.3). Even though the apex energy stays constant, the spectral
shape is broadened and partially merges with the low energy ion population which
still gains energy after the end of the laser pulse (Ch.3.1). This results in a spectral
shape comparable to the one observed experimentally (Fig.8.1). This is attributed
to the considerable deformation of the foil plasma by the focussed Gaussian laser
(Fig.8.4). Therefore, the laser is incident on a bent surface, particularly at the
end of the interaction. Thus, electric eld components perpendicular to this moved
surface are present. Those eciently heat the electrons in those regions (Ch.2.3)
and cause the peak in the carbon spectrum to broaden as well as the energy gain
of the ions at the low energy part of the spectrum. Additionally, this makes the
detection of the peak in the proton spectrum which is, since all ions move at the
same velocity expected around
2.5
MeV, impossible since it vanishes in the low
energy part of the spectrum.
8.4 Summary and Discussion
In this chapter experimental investigations on ion acceleration from nanometer
thin DLC foils irradiated by circularly polarized, highly intense laser pulses were
presented. A strong reduction in number and energy of hot electrons was observed
when compared to linear polarization. As a result a pronounced peak in the car-
bon
C6+
centered around
30
MeV at the optimum foil thickness of
5.3
nm was
observed. The measured optimum thickness is on the same order as the one pre-
dicted analytically by Eq.3.12. The central energy of the peak is in good agreement
with analytical result taking into account the partial transmittance of the target
(Eq.3.23). Two-dimensional PIC simulations give a clear evidence that those ions
are for the rst time dominantly accelerated by the laser radiation pressure. Being
recently widely studied in theory, the comparative measurements provide the rst
experimental proof of the feasibility of radiation-pressure acceleration to become
the dominant mechanism for ion acceleration when circular polarization is used.
Additionally, a strong denting of the target foil during the interaction was observed
which results in a spectral broadening of the peak in the carbon spectrum in time
and a late energy gain of the ions in the low energy part of the spectrum. In the
near future, this might be compensated by properly shaping the laser focal spot
72 Ion Acceleration in the Radiation Pressure Dominated Regime
distribution [130] or by shaping of the foil targets itself [131].
However, this is the approach for the next chapter, where the denting and the
expansion of the target is examined by analyzing the generated harmonic radiation
in the interaction process.
Chapter 9
High-Harmonic Generation in
Ultra-Thin Foils
In this chapter, the dynamics of foil targets with thicknesses on the nanometer
scale irradiated with laser pulses of relativistic intensities is examined by means of
the generation of surface high harmonics (SHHG). Their analysis allows the in-situ
extraction of the electron density at times when the peak of laser pulse interacts
with the plasma target, which is of utmost interest for laser ion acceleration in
the RPA regime as presented in Ch.8. The key to ecient RPA is to suppress the
heating of the target electrons and carefully control the targets areal density. It
will be shown that the SSHG on the front surface of a nm-scale foil detected in
transmission, allows detailed studies of the crucial interaction parameters such as
target deformation and plasma density when the peak of the driving laser pulse
interacts with the target, i.e. while the ion acceleration takes place. In addition,
the rst observation of odd-numbered relativistic harmonics is verifying a predicted
property of harmonics originating from solid targets [24] and thereby advance the
understanding of the harmonic generation process itself.
As discussed above two distinct mechanisms capable of generating high-order
harmonics in transmittance of nm-scale foils have been identied [86]. For sub-
and weakly relativistic intensities corresponding to a normalized vector potential
a0
below or on the order of unity harmonics are generated via CWE (Ch.4) in the
density gradient on the rear side of the foil. These plasma waves are excited indi-
rectly by attosecond-electron bursts propagating up the rear side plasma density
gradient. For higher intensities (
a01)
) harmonic radiation can also be emitted
by plasma electrons close to the critical density which oscillate coherently with ve-
locities close to the speed of light in the driving laser eld [23,24,80]. In the case
of very thin targets this mechanism can also generate harmonics emitted in the
forward direction that propagate in forward- (laser propagation-) direction [85,86].
To determine the predominant mechanism in a specic experiment one can ana-
74 High-Harmonic Generation in Ultra-Thin Foils
lyze the characteristics of the detected harmonic spectrum. The main dierences
are: (i) Relativistic harmonics exhibit an spectral cut-o that is dependant on
the laser intensity, while the CWE harmonic cut-o is determined by the maxi-
mum density in the target. (ii) CWE harmonics require oblique incidence, while
in contrast relativistic harmonics can be also generated under normal incidence so
that
2ω
electron oscillations are driven by the
v×B
component of the driving
force (Ch.1). Parts of the results presented in this chapter have submitted for
publication [12].
9.1 Experimental Setup
For the detection of the high harmonic radiation in laser propagation direction, a
spherical mirror with unprotected gold coating was positioned under an angle of
45
degrees behind the DLC targets ranging from
5.3
to
30
nm in thickness, which
were irradiated in the same setup as described in Ch.7.1 with an intensity of
2.6×1019
W/cm
2
or a normalized vector potential of
a0= 3.6
. The resulting line
focus was placed on the entrance slit of a normal incidence ACTON VM-502 VUV
spectrometer equipped with a MCP detector and a ber-coupled CCD-camera
(Ch.6.2). The spectrometer allowed the detection of radiation from
140
down to
50
nm corresponding to harmonics
7
to
16
of the fundamental laser wavelength.
9.2 Dynamics of Nanometer-Scale Foil Targets
In Fig.9.1 typical, normalized harmonic spectra obtained from targets with dier-
ent thicknesses are plotted. They show odd and even harmonics with a pronounced
enhancement of the odd harmonics (7th and 9th) independent of the initial tar-
get thickness. The spectra exhibit a strong dependance of the highest harmonic
generated on the initial target thickness. While the spectrum from the thinnest
(
5.3
nm) targets has a cut-o at the
9
th harmonic order, harmonics up to the
15
th
were generated from the
16
nm targets.
Actually, the cut-o in case of
30
nm targets was not resolvable with the used
spectrometer setup, i.e. is
>16
th order (not shown here).
These spectral properties suggest that the harmonics are generated by two dif-
ferent mechanisms, one generating predominantly odd harmonics originating from
the central part of the focal region and one producing all harmonics from regions
o central where the laser intensity is lower: The dependance of the harmonic cut-
o on the target thickness and moderate intensities on the sides of the Gaussian
focus give rise to a generation of this part of the spectrum by CWE as described
in Ch.4. This means that the target has to be dented signicantly to facilitate
9.2 Dynamics of Nanometer-Scale Foil Targets 75
Figure 9.1:
Measured high harmonic spectra with linear laser polarization from
targets with dierent thicknesses. From top to bottom:
5.3
nm,
8
nm,
12
nm and
16
nm.
motion of the plasma electrons in and out of the surface in the electric eld of the
laser, i.e. to provide a surface which is perpendicular to the electric eld of the
laser and hence full the requirements for the generation of CWE harmonics.
The fact that higher order CWE harmonics are generated from thicker targets
corresponds to an expansion of the target, since the harmonic cut-o is determined
by the the peak electron density (Eq.4.1). In Fig.9.2b the cut-o harmonic or-
der and the corresponding peak electron density are plotted versus the original
thickness of the target foil. The peak density in the interaction region as inferred
from the harmonic generation during the most intense phase of the interaction,
increases with the initial target thickness.
To model the expansion of the foil after ionization, a uniform energy deposition
in the foil is assumed [21]. The ion sound velocity rapidly increases over a time
76 High-Harmonic Generation in Ultra-Thin Foils
Figure 9.2:
(a) Model of an exponential foil expansion and the resulting densities
and cut-o harmonics as a function of the target thickness(b). The blue lines
corresponds to the density expected from a exponential expansion with a scale length
of
16
nm on each side.
of
70
fs to approximately
cs,peak = 5 ×107
cm/s corresponding to a hot electron
temperature of
Te≈5
keV [14] at which the plasma becomes non-collisional under
the given experimental conditions (Ch.2.3). The average sound velocity is then
cs,mean = 2.2×107
cm/s which results in an exponential expansion with a scale
length of
L= 15.4
nm. The corresponding electron density
n1
calculates to
n1(D) = n0D
D+ 2
L
Z0
e−x/L
d
x
−1
,
(9.1)
where
n0= 480 ×nc
is the initial density of the DLC targets. In Fig.9.2
n1(D)
is
plotted along with the experimental data demonstrating a good agreement. Thus,
in addition to the determination of the target density, the experiment gives infor-
mation about the target heating prior to the relativistic interaction and demon-
strates a fundamental limitation of the nm-foil density at the instance of the in-
teraction with the peak of the laser pulse which can be crucial when even thinner
targets are employed.
The information about the target heating can be used to further substantiate
the argumentation of a reduction of the electron heating when the laser polarization
is changed to circular (Ch.8), which has been proven by comparing the electron
spectra of both polarizations. In Fig.9.3 a typical harmonic spectrum from a
target with a thickness of
5.3
nm and circular polarization is shown. The density
9.3 Simulations 77
Figure 9.3:
Measured high harmonic spectrum with circular laser polarization from
targets with a thickness of
5.3
nm in laser propagation direction. The 7th harmonic
is already on the edge of the detector.
corresponding to the highest harmonic (12th) is approximately two times higher
compared to linear polarization. Additionally, the spectrum does not show an
enhancement of the odd harmonic order 7 and 9 which certainly is attributed to
the absence of
v×B
component.
9.3 Simulations
To verify the argumentation that the pronounced enhancement of the odd har-
monics (7th and 9th) independent of the initial target thickness is attributed to
the
v×B
component of the driving force, i.e. to a relativistic driver 2D PIC
simulation were conducted by S. Rykovanov
1
using the code PIGWIG [132].
As discussed above even for ultra-clean Gaussian laser pulses, nm-scale targets
expand during the rising edge of the
45
fs pulse with an
a0= 3.6
. Hence, the
simulations were initialized using a triangular density prole with a peak intensity
of
ne= 100 ×nc
and a linear ramp of
25
nm on each side. This corresponds to
solid target with a thickness of
5
nm and a density of
ne= 480 ×nc
. The laser
pulse was Gaussian both in space and time with a spot size of
3µ
m FWHM and a
duration of
15
laser cycles FWHM and is incident normally onto the target. The
size of the simulation box was
6λ
in laser propagation and
15 λ
in polarization
direction. The time step was
τ/400
in laser propagation direction, where
τ
is the
period of the driving laser.
The results of the simulation shown in Fig.9.4a where the electron density near
78 High-Harmonic Generation in Ultra-Thin Foils
Figure 9.4:
Results of 2D PIC simulations by S. G. Rykovanov
1
. The electron
density distribution (a) at the instance when the peak of the laser arrives at the
initial target position and the time integrated harmonic spectrum (b) emitted in
forward direction. The radially integrated spectra of (b) are shown in (c).
the instance when the peak of the pulse interacts with the target is shown together
with the resulting spectrum. The integrated spectrum contains all harmonic orders
but an enhancement of odd harmonics originating from the center of the generation
region is visible constituting a signature of relativistic harmonic generation on axis.
9.4 Summary and Discussion
In conclusion the experiments presented in this chapter constitute the rst experi-
mental demonstration of High Harmonic Generation in transmittance of nm-scale
foils targets irradiated at normal incidence. High harmonics were generated by
two dierent mechanisms in dierent regions of the laser focus. A denting of the
1
Fakultät für Physik, Ludwig-Maximilians-Universität München, 80333 München, Germany
9.4 Summary and Discussion 79
target leads to eectively oblique incidence of the driving laser on the sides of the
laser focus and the generation of all harmonics. Only odd-numbered harmonics
are generated exactly on the laser axis, which constitutes the rst demonstration
of relativistic harmonic generation at normal incidence, a feature which disappears
as expected when circular laser polarization is used. In the regions of oblique in-
cidence the harmonics are generated by Coherent Wake Emission. This allows the
determination of the instantaneous target density of the foil at times where the
relativistic interactions start to be signicant. The obtained densities are in good
agreement with a one dimensional expansion of the foil targets and are reduced by
a factor of two when the laser polarization is changed to circular.
Hence, the experiments demonstrate that this method is a powerful diagnostic
which can be employed in many experimental scenarios, e.g. laser particle accel-
eration as described in Ch.7 and 8, without any changes in the existing particle
acceleration experiments.
80 High-Harmonic Generation in Ultra-Thin Foils
Summary and Outlook
During the course of this thesis, novel, improved approaches for ion acceleration
with high intensity laser pulses were explored and characterized. Laser pulses with
relativistic intensities and an ultra-high contrast enabled the employment of targets
with thicknesses below the skin-depth of the laser and hence, transparent targets.
The rst experimental results together with their theoretical interpretations of
Ion
Acceleration in the Laser Transparency Regime
mark an important step towards
future applications of laser accelerated ion beams.
The presented results extend the common regime of
Target Normal Sheath
Acceleration
(TNSA) where the laser energy is transferred to kinetic energy of a
hot electron population at the front side of thick (
µ
m), opaque laser irradiated
solid targets. These electrons traverse the target and set up a quasi-static electric
eld, that accelerates the ions. Due to the lateral spreading of the hot electrons,
the conversion eciency (CE) of ion acceleration in the TNSA regime is very
low (typically
∼1%
) and their energy spectra exhibit an exponential shape as
an imprint of the hot electron population. These properties make the ion beams
accelerated within the TNSA regime inappropriate for most applications where a
high number of ions in a tailored energy spectrum are required.
One possibility to overcome the limitations of TNSA is the use of targets with a
limited mass. Inspired by PIC simulations [34] that predict an optimum foil thick-
ness of only a few nanometer for the laser parameters available at the Max-Born
Institut which is as well close to the optimum thickness for a dominant
Radiation
Pressure Acceleration
(RPA) [66] if the laser polarization is changed to circular,
free-standing
Diamond-Like-Carbon
(DLC) foils were employed as targets. They
were developed and characterized by collaboration partners within the Transre-
gio SFB TR18 at the Ludwig-Maximilians-Universität München (LMU) and were
available with thicknesses ranging from
2.9−50
nm. DLC oers unique properties
for the production of mechanical stable, ultra-thin, free-standing targets, such as
exceptionally high tensile strength, hardness and heat resistance.
To apply DLC foils as targets for the ion acceleration in the rst place, the
inherent laser pulse contrast, i. e. the intensity relation between the peak and
the background of the laser pulse had to be enhanced signicantly to suppress
82 High-Harmonic Generation in Ultra-Thin Foils
pre-heating and expansion of the target. This nanosecond background of the
femtosecond pulse originates mainly from
Amplied Spontaneous Emission
(ASE)
in the amplier crystals and was measured to be
<10−7
up to
∼10
ps prior to the
main pulse.
In the frame of this work a re-collimating Double-Plasma-Mirror system was
developed that is capable of increasing the contrast by estimated four orders of
magnitude, while at the same time preserving the quality of the wavefront so that
the laser pulse is focusable to a diameter of
3.6µ
m. The energy throughput was
quantied to
60−65
% resulting in pulse energies of
0.7
J and focus peak intensities
of
5×1019
W/cm
2
at a pulse duration of
45
fs.
In contrast to thick foils typically employed for ion acceleration experiments
within the TNSA regime, DLC foils of nanometer thickness become transparent
under the interaction with an intense laser pulse and hence, the whole target in
the focal volume is participating in the acceleration.
Besides the rst experimental characterization of the non-linear transmittance
of the relativistic laser pulses through the DLC targets that is in good agreement
with the values calculated analytically [74], the major ndings of this work are:
In case of linear laser polarization and normal incidence an optimum for ion
acceleration was observed at a target thickness of
5.3
nm, accompanied by increased
values for the highest ion energies: two times in case of protons (
13
MeV) and
20 times for carbon ions (
71
MeV) if comparing to experiments performed with
approximately the same laser parameters. Additionally, the conversion eciency
from laser- to kinetic energy of the accelerated ions was improved
∼100
times for
carbon ions (10%). Nevertheless, the spectral shape of the accelerated ion beam
exhibits a exponential slope as an imprint of the hot electron population generated
by the longitudinally oscillating component of the Lorentz force.
To suppress this electron heating, circular laser polarization was used in a
second set of experiments. Then, the plasma electrons are piled-up to an equi-
librium ahead of the laser pulse acting as a mirror that gains less momentum in
favor of more energy when being accelerated as long as the electron heating is
suppressed. The ions are therefore accelerated by a co-moving electric eld to an
intrinsically mono-energetic spectrum. This so-called
Radiation Pressure Acceler-
ations
(RPA) has been explored in reams of publications during the last years by
means of Particle-in-Cell simulations but lacked an experimental demonstration.
The rst experimental verication has been given by the experiments presented
in this thesis. A strong reduction in number and energy of hot electrons was ob-
served when compared to linear polarization. As a result a pronounced peak in
the carbon
C6+
centered around
30
MeV at the optimum foil thickness of
5.3
nm
9.4 Summary and Discussion 83
was observed. Two-dimensional PIC simulations (by X. Q. Yan
1
,
2
) giving a clear
evidence that those ions are for the rst time dominantly accelerated by the laser
radiation pressure. Moreover, a strong denting of the target foil was observed in
the simulations which in turn is called to account for the exponential, low energy
parts of the spectra obtained in the experiment.
To quantify this denting and the dynamics of the interaction experimentally,
the high harmonic content of the transmitted laser radiation was examined for
the rst time. The high harmonics were generated by two dierent mechanisms
in dierent regions of the laser focus. A denting of the target leads to eectively
oblique incidence of the driving laser on the sides of the laser focus and therefore
to the generation of all harmonics. Only odd-numbered harmonics are generated
exactly on the laser axis, which constitutes the rst demonstration of relativistic
harmonic generation at normal incidence, a feature which disappears as expected
when circular laser polarization is used. In the regions of oblique incidence the
harmonics are generated by
Coherent Wake Emission
. This allows the determina-
tion of the instantaneous target density of the foil at times where the relativistic
interactions start to be signicant. The obtained densities are in good agreement
with a one-dimensional expansion of the foil targets and are reduced by a factor
of two when the laser polarization is changed to circular.
To avoid this denting will be in the focus of future investigations of ion accelera-
tion in the Laser Transparency Regime. Since the denting has been quantied with
the help of the experiments presented in this thesis, target fabrication labs will
provide pre-shaped targets that are supposed to compensate the observed denting
and hence sustain the RPA for longer times, resulting in higher ion energies. On
the other hand
at-top
, in favor of
Gaussian
spatial intensity distribution would
be capable to avoid the denting as well. In addition, the mechanical target and
plasma mirror engineering have to be improved to exploit the repetition of the
laser system and hence, to pave the way for rst applications. Since, number and
energy of the accelerated ions presented in this thesis are sucient, the repetition
rate is the last obstacle for the exposure of cells (e. g. cancer cells).
Furthermore, experiments with the new dual beam line (
30
TW +
100
TW at
the Max-Born Institut will provide unique possibilities to conduct
Proton Imaging
experiments that will give a more detailed picture of the acceleration process in the
Laser Transparency Regime as well as a whole variety of pump-probe experiments.
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
State Key Lab of Nuclear Physics and Technology, Peking University, 100871 Beijing, China
84 High-Harmonic Generation in Ultra-Thin Foils
List of Figures
1.1 Laser interaction with a free electron . . . . . . . . . . . . . . . . . 8
2.1 Collisional Absorbtion . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 InitialstageofRPA........................... 24
3.2 RPA 1D PIC Simulation . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Maximum ion energies is the Light Sail Regime . . . . . . . . . . . 28
5.1 PrincipleofCPA ............................ 35
5.2 MBI TW Ti:sapph Laser . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Simulated Interferogram and its Fourier Transform . . . . . . . . . 38
5.4 Principle of the SPIDER . . . . . . . . . . . . . . . . . . . . . . . . 39
5.5 SPIDERMeasurement......................... 40
5.6 Contrast measurement with the 3rd order cross Correlator . . . . . 41
5.7 Double Plasma Mirror setup . . . . . . . . . . . . . . . . . . . . . . 43
5.8 Incident beam proles on the DPM surfaces . . . . . . . . . . . . . 44
5.9 Far-eld Diagnostic of the DPM . . . . . . . . . . . . . . . . . . . . 44
5.10 Reectivity of the DPM system . . . . . . . . . . . . . . . . . . . . 46
6.1 Typical ion spectrum from
5µ
mTitanium .............. 48
6.2 Acton Research VM-503 . . . . . . . . . . . . . . . . . . . . . . . . 50
7.1 ExperimentalSetup........................... 56
7.2 Comparison between front- and rear side ion acceleration . . . . . . 57
7.3 Transmittance of DLC targets . . . . . . . . . . . . . . . . . . . . . 58
7.4 Spectral properties of a laser transmitted through a DLC target . . 60
7.5 Maximum ion energies for linear polarization . . . . . . . . . . . . . 61
7.6 Conversion eciency . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.7 Evolution of electron and ion densities for linear polarization . . . . 63
8.1 Maximum ion energies for circular polarization . . . . . . . . . . . . 68
8.2 Comparison of electron spectra . . . . . . . . . . . . . . . . . . . . 69
86 LIST OF FIGURES
8.3 Simulated electron spectrum and ion phase space plot . . . . . . . . 70
8.4 Evolution of electron and ion densities for circular polarization . . . 70
9.1 HHG with linear laser polarization . . . . . . . . . . . . . . . . . . 75
9.2 Exponential foil expansion and the resulting densities . . . . . . . . 76
9.3 High harmonic spectrum with circular laser polarization . . . . . . 77
9.4 Simulation of HHG from thin foils . . . . . . . . . . . . . . . . . . . 78
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Acknowledgments
At this point I would like to thank all the people who have contributed to this
work. Without your support, enthusiasm and friendship this work would have
been impossible. Especially, the following people shall be recognized:
•
Prof. Dr. W. Sander for supervising this work and the opportunity to work
in this excellent research group.
•
Prof. Dr. O. Willi for being the second referee of this thesis and for his
wisdom as spokesman of TR18.
•
Prof. Dr. P. V. Nickles for his visionary thoughts that paved the way towards
the realization of this work from the very beginning and for the uncountable
discussions.
•
Dr. M. Schnürer for his unfailing support and mentoring with an outstanding
competence.
•
Dr. T. Sokollik for much more than his friendly support and many fruitful
discussions.
•
L. Ehrentraut and Dr. G. Priebe for peak performance.
•
Prof. Dr. A. A. Andreev for patiently supporting me theoretically.
•
Prof. Dr. D. Habs, Prof. Dr. T. Tajima, Dr. A. Henig, Dr. J. Schreiber,
Dr. R. Hörlein, D. Kiefer for the very fruitful cooperation.
•
All other administrative and technical employers: P. Friedrich, S. Szlapka,
B. Becker, G. Kommol, J. Meiÿner, J. Gläsel and D. Rohlo.
•
Last but not least, my family for their personal support during all the years
and especially Marzena for backing me up all the time, no matter how many
nights I spend in the lab.
