Study of laser-acceleration of ion
beams using ultra-short, ultra-high
power lasers
Daniele Capelli
October 2010 - Academic Year 2009/2010
Sapienza Università di Roma
Thesis for obtaining the master degree for the course of Electronic
Engineering, Faculty of Engineering, Sapienza Università di Roma.
Tesi per il conferimento del titolo accademico di dottore in ingegneria del
corso di laurea specialistica in Ingegneria Elettronica, Facoltà di Ingegneria,
Sapienza Università di Roma.
This thesis was developed in the framework of a four months ERASMUS
exchange program within the École polytechnique in Paris, France, in the
LULI Laboratoire pour l'Utilisation des Lasers Intenses, and also at the
INRS-EMT Institut National de la Recherche Scientique Université du
Québec - centre Énergie, Matériaux et Télécommunications.
Thesis director at Sapienza Università di Roma: prof. Luigi Palumbo, Dip. di
Energetica.
Thesis director at École polytechnique Paris: prof. Julien Fuchs, LULI laboratory.
2
In such a case everybody should write:
this page intentionally left blank
IMHO: it's not blank at all.
3
4
Estratto
italiano
5
6
Abstract
inglese
7
8
Acknowledgements
I'm gratefull to Prof.
Julien Fuchs who invited me at LULI, and Prof.
Luigi Palumbo
who gave me the possibility to broaden my culture: thank you very much.
I would like to thank also Sebastien Buechoux, Sylvain Fourmaux, Bruno Albertazzi,
Anna Levy and all the ordinarily extraordinary people that I met daily during this 4
months experience: thank you for supporting and helping me understand.
I will never forget.
9
10
Role of the author
11
12
Contents
1
Introduction to plasma physics
15
1.1
17
1.2
1.3
1.4
1.5
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1
Kinetic temperature and other quantities related to the distribution
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.1.2
Plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.1.3
Debye length and Debye shielding . . . . . . . . . . . . . . . . . . .
21
1.1.4
Plasma coupling parameter
. . . . . . . . . . . . . . . . . . . . . .
23
Kinetic description
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Fluidic description
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.3.1
Fluidic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.3.2
Fluidic equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Charged particle motion
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.4.1
Particle motion in uniform elds . . . . . . . . . . . . . . . . . . . .
33
1.4.2
Method of averaging
34
1.4.3
Guiding centre motion
. . . . . . . . . . . . . . . . . . . . . . . . .
37
1.4.4
Particle motion in oscillating eld . . . . . . . . . . . . . . . . . . .
41
1.4.5
Relativistic regime
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Waves in plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
1.5.1
Electromagnetic oscillation in plasmas
. . . . . . . . . . . . . . . .
47
1.5.2
Dielectric response of a collisionless cold plasma . . . . . . . . . . .
49
1.5.3
Plane waves propagation in a cold homogeneous plasma . . . . . . .
50
1.5.4
Plasma wave in a collisionless warm plasma
1.5.5
Landau damping
1.5.6
Dielectric response of a collisional plasma . . . . . . . . . . . . . . .
64
1.5.7
Propagation in inhomogeneous plasma: WKB approximation . . . .
66
. . . . . . . . . . . . .
54
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Description of the laser facility
73
2.1
The 200 TW laser beam line at INRS . . . . . . . . . . . . . . . . . . . . .
73
2.2
Laser chain devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
2.2.1
. . . . . . . . . . . . . . . . .
76
Laser beam characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
2.3
3
Plasma parameters
Chirped-Pulse Amplifying technique
Ion beam generation by laser-matter interaction
3.1
3.2
84
Energy absorption processes . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.1.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
Ion acceleration mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.2.1
89
jxB heating
Backward acceleration
. . . . . . . . . . . . . . . . . . . . . . . . .
13
Contents
3.2.2
Forward acceleration and Target Normal Sheat Acceleration
. . . .
91
3.2.3
TNSA parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.2.3.1
Electron spectrum
. . . . . . . . . . . . . . . . . . . . . .
93
3.2.3.2
Electron sheat model . . . . . . . . . . . . . . . . . . . . .
95
3.2.3.3
Electron current and recirculation . . . . . . . . . . . . . .
98
Radiation Pressure Acceleration . . . . . . . . . . . . . . . . . . . .
99
3.2.4
3.3
Models used to interpret the acceleration mechanisms . . . . . . . . . . . . 100
3.3.1
Plasma expansion acceleration model . . . . . . . . . . . . . . . . . 100
3.3.1.1
Isothermal expansion . . . . . . . . . . . . . . . . . . . . . 100
3.3.1.2
Adiabatic expansion
3.3.1.3
Hot electrons and ions expansions and angular distributions
3.3.2
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Radiation Pressure Acceleration model
. . . . . . . . . . . . . . . . 108
3.3.2.1
Hole boring regime . . . . . . . . . . . . . . . . . . . . . . 109
3.3.2.2
Light sail regime
. . . . . . . . . . . . . . . . . . . . . . . 112
3.4
Characteristics of the ion source . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5
Applications for laser generated proton/ion beams . . . . . . . . . . . . . . 118
3.5.1
Isochoric heating
3.5.2
Inertial Connement Fusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.5.3
Hadrontherapy
3.5.4
Isotope production for positron emission tomography
3.5.5
Proton radiography . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5.6
Particle accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
. . . . . . . . . . . . . . . . . . . . . . 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
. . . . . . . . 119
Description and results of the experiments
4.1
4.2
5
. . . . . . . . . . . . . . . . . . . . . 105
Outline of the interaction chamber and diagnostics
. . . . . . . . . . . . . 122
4.1.1
Laser beam path and imaging line . . . . . . . . . . . . . . . . . . . 123
4.1.2
Shadowgraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.1.3
Time and space resolved interferometry (TASRI)
4.1.4
Time of ight measurements . . . . . . . . . . . . . . . . . . . . . . 130
4.1.5
Spectralon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.1.6
Other diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results and discussion
. . . . . . . . . . 129
141
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Appendix
5.1
121
145
Thomson's parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
14
1 Introduction to plasma physics
Plasma is an highly ionized gas in which the charge of the electrons is balanced by the charge of the positive ions, so that the system as a whole is
electrically neutral. [Strickland 1985]
Plasmas are considered by some to be the fourth phase of matter.
They
are closely related to gases. In a plasma, the particles are [...] electrons and
positive ions. Plasmas can be formed at very high temperatures, high enough
to ionize the atoms. The resulting electrons and positive ions can then move
freely, like the particles in a gas. [The Gale Encyclopedia of Science 2004]
[Plasma is] an almost completely ionized gas that contains equal numbers
of electrons and positive ions moving freely and independently of each other.
Although a plasma is electrically neutral [...] it is highly conductive. [Encyclopedia of Astronomy and Astrophysics 2001]
[Plasma is] a gas consisting of ions, electrons, and neutral particles; the
behavior of the gas is dominated by the electromagnetic interaction between
the charged particles. [Academic Press Dictionary of Science and Technology]
The rst use of the term
plasma was for the description made by the czech physiologist
Jan Evangelista Purkinje, of the uid which remains after removal of all the corpuscular
material in blood.
At the beginning of the 20th century the american scientist Irving
Langmuir proposed that the electrons and ions in an ionized gas could be similarly considered as particles embedded in some kind of uid medium that was termed in analogy,
plasma. But the fact is that in plasmas there's no uid medium where electrons and ions
freely move.
First investigation of plasma behaviours were made in 20's and 30's on radio wave
prapagation that was aected by the ionospheric plasma, and on gaseous electron tube
used in electronics before the semiconductor-era.
In the 40's Alfven developed the theory of the so called Alfven waves: it was crucial in
the astrophysical plasma understanding.
In the 50's researches on nuclear fusion started and were based on plasma pysics; the
develop of fusion science was slow until 1968 when the rst russian
tokamak was designed
and started to produce plasmas with better parameters unlike the previous decade.
At the and of 20th century the break-even point of producing energy with controlled
fusion in tokamak was ready to be reached at JET: the Joint European Torus tokamak in
Culham, U.K., in operation since 1983, in 1991 achieved the world's rst controlled release
of fusion power, and its best result is release of 70% of energy used to start the fusion
process. The most important project of 21st century is the european ITER (International
15
1 Introduction to plasma physics
Thermonuclear Experimental Reactor), expected to produce fusion energy with a gain of
10 times the input energy (http://www.iter.org/).
In the meanwhile some other schemes of fusion were investigated: unlike the magnetic
connement fusion of tokamaks, the inertial connement fusion (ICF) was proposed. Up
to now a small number of ICF facilities were, or are planned to be, built all around the
world: the most important and largest laser facility in the world is the one of Lawrence
Livermore National laboratory (LLNL), called NIF (National Ignition Facility) that during this summer begun the experimental phase of the campaign that will possibly lead
the physicists to reach a controlled ICF with energy gain (https://lasers.llnl.gov/).
Other important projects are: Laser Megajoule (LMJ) in Bordeaux (France) which system is expected to be completed in 2012 (http://www-lmj.cea.fr/); High Power laser
Energy Research facility (HiPER) that is a proposed European facility and is currently
being designed to go beyond the features of NIF (http://www.hiper-laser.org/).
Parallel to the fusion eorts, other research were developed for astrophysical plasmas,
plasma space-propulsion, dusty plasmas, non-neutral plasmas and so on...
The research topic of this thesis is in the framework of laser-plasma interaction experiments to produce high energy proton beams for several possible application such as
medical hadrontheraphy, isotope production for positron emission tomography, proton radiography, particle accelerator proton source.
Further details will be given in Chapter
3.
Another important topic in plasma physics research was also relevant for industry:
plasma processing.
It is intensively used by semiconductor industries in some critical
steps of the integrated circuits (IC) fabrication process, such as PECVD (plasma-enhanced
chemical vapour deposition).
Understanding of plasmas is developed by studying a lot of various and dierent points of
view, because plasmas are complex systems: in fact they exist in a wide variety of situations
and each of them diering by many orders of magnitude from the others. Therefore, plasma
physics is composed of a variety of descriptions each covering and modeling a small piece
of the puzzle: the entire complex behaviour is not modeled at once. For each phenomenon
under consideration appropriate approximations are made to simplify the description and
to lead to a solution of the problem ready understandable.
Chapter description
In this chapter we will introduce basic concepts of plasma physics talking about the fundamental parameter that can describe a plasma. We will also give a basic description of
kinetic and uidic approach. The kinetic approach gives a description of the system based
on a time evolution of the distribution function obtained solving the so called Vlasov equation. Another characteristic approach is the uidic one: it gives a portrait of the system in
terms of global uidic variables easier to understand than the kinetic approach. In fact,
using the kinetic or uidic approaches to describe the system, does not keep track of the
trajectories of individual particles, but rather characterizes dierent groups of particles
having the same position and velocity at a xed instant of time.
In this chapter our very basic approach tries also to describe the dynamics of the particles that constitute the plasma. Dynamics is determined by the so called self-consistent
interaction between electromagnetic elds and statistically large numbers of charged par-
16
1 Introduction to plasma physics
ticles.
A cross correlation exists between the set of Maxwell equations and Lorentz's
equation: the former furnishes the electric and magnetic eld expressions in function of
the charge distributions and currents; the latter rules the particle motion (and therefore
the distribution of the charges) for a given electromagnetic eld. This approach although
very clear and simple to understand, is very impractical because of the high number of
particles and of the complexity of electromagnetic eld: it can be studied, eciently modeled, and further implemented in some codes for purpose of simulation, only under some
approximations.
The last section introduces some basic concepts about electromagnetic waves in plasma:
we describe the dierences in waves propagation in unmagnetized collisional or collisionless,
cold and warm plasmas, calculating the dielectric response, using the set of Maxwell's
equation for the wave propagation.
Usefull references for this chapter were: [Strickland 1985] [Landau 1946] and [Mora course 2008].
1.1 Plasma parameters
1.1.1 Kinetic temperature and other quantities related to the
distribution functions
We refer to an idealized plasma that consists in a system in which there is an equal number
of ions and electrons: each ion has a mass mi and a charge +e, whilst each electron has a
mass me and a charge −e.
The kinetic temperature of each particle species is dened as follows:
1
Te = me ve2
3
(1.1.1)
1
Ti = mi vi2
3
(1.1.2)
where the paratheses hi denote an ensemble average. In a statistical mechanics framework, we refer to an ensemble average as the mean, over the whole group of possible states,
of a quantity that depends on the microscopic state of a system, according to the distribution function of the system over its micro-states in this ensemble. Here v denote the speed
magnitude of a single particle and if the plasma is supposed to be at thermal equilibrium,
the distribution for the speed follows the so called Maxwell-Boltzmann distribution:
f (v) = 4π
m
2πkB T
23
mv 2
v exp −
2kB T
2
(1.1.3)
or for velocities that are the scalar components of the speed of a particle, and in this
case we choose the simple set of cartesian coordinates:
f (vx , vy , vz ) =
m
2πkB T
32
"
m vx2 + vy2 + vz2
exp −
2kB T
17
#
(1.1.4)
1 Introduction to plasma physics
These results can be derived, recalling the Boltzmann's distribution for energies of the
state of canonical ensemble:
Ni
=
N
where
h
i
gi exp − kEBiT
(1.1.5)
Z
Z is the partition function given by its denition:
Z=
X
j
Here gi is the degeneracy of the
Ej
gj exp −
kB T
(1.1.6)
i -th possible state
P with energy Ei , that is occupied by a
number Ni of particle of the same species. N =
i Ni is the total number of the particles
2
of the system. Substituting the expression for the momentum Ei = pi /2m:
Ni
=
N
h
i
p2
gi exp − 2mkiB T
(1.1.7)
Z
We can consider that the distribution of the fractional number of particle in the
i -th
state Ni /N must be proportional to the probability for nding a particle with momentum
pi =
q
p2i,x + p2i,y + p2i,z :
2
pi,x + p2i,y + p2i,z 3
C
dp
fp (pi,x , pi,y , pi,z ) d p = gi exp −
Z
2mkB T
3
(1.1.8)
Integrating over the momentum space, we must nd 1 if we want to normalize the
distribution, hence the constant can be calculated in this way:
ˆ
C
fp (pi,x , pi,y , pi,z ) d p =
Z
ˆ
3
2
pi,x + p2i,y + p2i,z 3
exp −
d p=1
2mkB T
(1.1.9)
Seeing the integral as a product of three gaussians:
ˆ
ˆ +∞
2
3
pi,x + p2i,y + p2i,z 3
p2i,x
exp −
d p=
exp −
dpi,x
2mkB T
2mkB T
−∞
Substituting the dummy variable: x = √
ˆ +∞
p
2mkB T
p
, we nd the well known
2mkB T
√ p
exp −x2 dx = π 2mkB T
(1.1.10)
Gauss integral :
(1.1.11)
−∞
Hence:
c=
Dropping the
Z
(2πmkB T )3/2
(1.1.12)
i for ease of notation and substituting the constant c, we nally obtain
the distribution for the momentum:
18
1 Introduction to plasma physics
2
px + p2y + p2z
fp (px , py , pz ) =
exp −
2mkB T
(2πmkB T )3/2
1
(1.1.13)
The derivation of the distribution for the velocity fv is quite straightforward if we recall
that p = mv and if we state that:
3
fv (vx , vy , vz ) d v = fp (mvx , mvy , mvz )
dp
dv
3
d3 v
(1.1.14)
hence substituting in (1.1.13) we nd the result presented in (1.1.4). In fact fv (vx , vy , vz )
can be seen as product of three monodimensional normal distribution (gaussian distribution) each of them is given by:
f (vj ) =
m
2πkB T
12
mvj2
exp −
2kB T
(1.1.15)
Sometimes is more usefull, simple and understandable, a description in terms of the
speed:
v=
q
vx2 + vy2 + vz2
(1.1.16)
3
Using a set of spherical coordinates in the velocity space, and considering d v
dvx dvy dvz = v 2 sin θdvdθdφ, we can integrate over the two other velocity components
"ˆ
2π ˆ +π/2
#
fv (v, θ, φ) v 2 sin θdθdφ dv
f (v) dv =
=
(1.1.17)
−π/2
0
that gives the result in (1.1.3), that is the well known Maxwell Boltzmann distribution
for speed. At thermal equilibrium the ensemble average of particle speed coincides with
the second order moment of v
, hence we have:
v 2 = vx2 + vy2 + vz2 = vx2 + vy2 + vz2
(1.1.18)
The second order moment can be calculated with the denition of gamma-functions:
ˆ +∞
vj2
vj2 f (vj )dvj =
=
−∞
m
2πkB T
12 ˆ +∞
vj2 exp
−∞
mvj2
−
dvj
2kB T
(1.1.19)
Gamma functions are dened as follows:
ˆ +∞
1
x exp −x2 dx = Γ
2
−∞
n
n+1
2
Performing a change of variable as was done before, we set x
dvj =
q
2
(1.1.20)
mv 2
= 2kBjT , thus dierentiating
2kB T
dx, and substituting:
m
vj2
=
m
2πkB T
12 ˆ +∞
−∞
2kB T 2
x exp −x2
m
19
r
2kB T
dx =
m
1 Introduction to plasma physics
ˆ +∞
2+1
(1.1.21)
2
(m−2)!! √
m
π , hence
The gamma function calculated for an argument m/2 gives Γ
=
2
2(m−1)/2
2kB T 1
√
=
m
2π
2kB T 1 1
√ Γ
x exp −x dx =
m
π2
−∞
2
2
the remarkable results:
vj2 =
kB T
m
(1.1.22)
kB T
m
(1.1.23)
v2 = 3
Thus at thermal equilibrium Te = Ti = kB T .
An interesting quantity related to the kinetic temperature is the thermal speed. The
thermal velocity or thermal speed has a variety of denition; usually it is considered as
a measure of the speed of particles in a plasma, and is a sort of root mean square of the
speed:
r
vt =
kB T
m
(1.1.24)
Note that ions and electrons have dierent thermal speed, in fact we can write:
r
vt−e =
mi
vt−i
me
(1.1.25)
Another interesting quantity related to the particle distribution function f (r, v) in the
six dimension space (r, v) where r is the vector in the 3D geometrical space of the system
that is under consideration, and v is the speed of the particle in that point, is the particle
density.
The particle density is
n the number of a certain species of particle per cubic
meter. It is given by the integral over the velocity space of the distribution function:
ˆ
n (r) =
f (r, v) d3 v
(1.1.26)
At equilibrium we can assume that the distribution is:
eΦ (r)
n (r) = n0 exp −
kB T
(1.1.27)
where the function Φ (r)denotes the potential in the plasma.
1.1.2 Plasma frequency
The plasma frequency is a fundamental parameter and is dened as follows:
s
ωp =
20
e2 n
0 m
(1.1.28)
1 Introduction to plasma physics
and it has to be specied if this refers to electron or ion plasma frequency by using
appropriate density and mass.
It can be seen that the plasma frequency is the typical
electrostatic oscillation frequency of a particle in response to a small charge separation.
Suppose that a sheet of charge of only one species, e.g. ion, is displaced from its position
by a distance δx in the positive
x direction; the resulting surface charge density is:
σ+ = enδx
(1.1.29)
Due to the total charge neutrality, we can state that an opposite charge density appears
on the other surface:
σ− = −enδx
(1.1.30)
The electric eld produced by this displacement is simply given by:
Ex (t = 0) = −
enδx
σ+
=−
0
0
(1.1.31)
The Newton's law, i.e. the so called dierential equation of motion, applied to a particle
is:
d2
e2 n
m 2 x (t) = eEx = −
x (t)
dx
0
(1.1.32)
e2 n
d2
x
(t)
=
−
x (t) = −ωp2 x (t)
dx2
m0
(1.1.33)
and rewriting it:
The previous equation has a solution:
x(t) = δx cos (ωp t)
(1.1.34)
The simple oscillatory time evolution of the charge position stated by (1.1.34) leads us to
conclude that the plasma reacts to a charge separation with a typical periodic oscillation
p
2π
of τp =
= 2π e02m
around its equilibrium position. It is important to note that here we
ωp
n
supposed the absence of a mechanism of energy dissipation that can signicantly damps
down the oscillation. Another remark: the plasma oscillations can only be observed if the
system is studied over a period τ much larger than the plasma period τ τp .
1.1.3 Debye length and Debye shielding
Is usefull to dene the plasma Debye lenght λD as the distance traveled by a particle
with velocity equal to the thermal velocity of its species in the typical period of plasma
−1
oscillation that is proportional to ωp :
r
λD =
0 kB T
vt−e
=
2
en
ωp
(1.1.35)
Spatial observation of the system can only be done if the observation scale length L is
comparable to, or larger than the Debye length. Debye length is also reered to be as the
21
1 Introduction to plasma physics
scale over which mobile charge carriers screen out electric elds in plasmas and in other
words, the Debye length is the distance over which signicant charge separation can occur.
To conrm the previous statements, consider a plasma at the Boltzmann equilibrium
(the superscript
zero means that the corrrisponding quantity at the second member is the
equilibrium value):
n0i = ni0 exp
qφ0
−
Ti
(1.1.36)
qφ0
Te
(1.1.37)
is the ion density,
n0e = ne0 exp
0
0
is the electron density. Note that at equilibrium ni = ne = n0 and Ti = Te = kB T . We
have:
ρ0
∇2 φ0 = −
0
(1.1.38)
ρ0 = q n0i − n0e
(1.1.39)
with:
where we consider for simplicity that the ion species has Z = 1 (protons). Now suppose
that an elementary charge q in r = 0 perturbs the potential:
φ = φ0 + δφ
(1.1.40)
ne = n0e + δne
(1.1.41)
ni = n0i + δni
(1.1.42)
ρ = ρ0 + δρ
(1.1.43)
and the other quantities:
The incremented charge density is given by the following distribution:
δρ = qδ(r) + q (δni − δne )
(1.1.44)
where here δ (r) is the Dirac's delta function (i.e. the impulsive distribution function
for the charge in r = 0). The increments of electron and ion density are:
δne = ne − n0e = ne0 exp
0
q (φ0 + δφ)
qφ
qδφ
0
− ne0 exp
= ne exp
−1
Te
Te
Te
22
(1.1.45)
1 Introduction to plasma physics
δni = ni − n0i = ni0 exp
q (φ0 + δφ)
qφ0
qδφ
0
−
− ni0 exp −
= ni exp −
−1
Ti
Ti
Ti
(1.1.46)
for little perturbation we can write:
δne ≃ n0e
qδφ
Te
(1.1.47)
qδφ
Ti
(1.1.48)
n0 q 2
δφ
kB T
(1.1.49)
δni ≃ −n0i
and thus we can obtain:
δρ = qδ(r) − 2
The potential must verify the Poisson's dierential equation:
∇2 φ = −
ρ
0
(1.1.50)
δρ
0
(1.1.51)
and thus:
∇2 δφ = −
substituting:
n0 q 2
q
2
∇ −2
δφ = − δ(r)
0 kB T
0
(1.1.52)
and thus the perturbation of potential looks like a Green's function:
h √
i
r
q exp − 2 λD
δφ(r) =
4π0
r
So it is clearly seen that the potential of the perturbing point charge
(1.1.53)
q is shielded on
distances longer than the Debye length by a shielding cloud of approximate radius λD
because the perturbation δφ(r) has a normalized tendency:
1.1.4 Plasma coupling parameter
The plasma coupling parameter are dened as the ratio of the average interaction potential
energy between two particles of the same species and the kinetic temperature of that
species:
Γee = Ve (r = de )/Te
(1.1.54)
Γii = Vi (r = di )/Ti
(1.1.55)
23
1 Introduction to plasma physics
√
Figure 1.1.1: Normalized potential perturbation in function of
24
2 λrD .
1 Introduction to plasma physics
where:
1
de = (ne )− 3
(1.1.56)
1
di = (ni )− 3
(1.1.57)
are the average distances between two particles. The potential energies are simply the
Coulombian ones:
q2 1
4π0 r
(1.1.58)
(Zq)2 1
Vi (r) =
4π0 r
(1.1.59)
Ve (r) =
thus we have:
1/3
2 1/3
q ne ne
1
1
=
2
0 Te
ne
4π n 3 λ2
e D−e
!
1/3
1/3
(Zq)2 ni
1 (Zq)2 ni ni
1
1
=
=
Γii =
2
4π0 Ti
4π
0 Ti
ni
4π n 3 λ2
q 2 ne
1
Γee =
=
4π0 Te
4π
i
(1.1.60)
(1.1.61)
D−i
and thus:
23
1
1
Γee =
4π ne λ3D−e
23
1
1
= Γee Z 5/3
Γii =
4π ni λ3D−i
(1.1.62)
(1.1.63)
These parameters can also be rewritten in another, more understandable, fashion:
Γee =
Γii =
1
(4π)1/3
1
(4π)1/3
×
×
1
ne 4πλ3D−e
1
ni 4πλ3D−i
23
(1.1.64)
23
(1.1.65)
where the numbers in the denominators of the second factors are the total number of ion
or electrons contained in a sphere with radius equal to the corrispondent Debye length.
The strongly coupled plasma is a plasma with Γ 1, i.e.
a plasma with a very high
number of particle contained in a Debye sphere. Otherwise weakly coupled plasma occur
when Γ ≥ 1.
25
1 Introduction to plasma physics
1.2 Kinetic description
Plasmas can be considered as systems of
N particles that are completely described by the
ensemble :
equations of motion for each particle of the
dri
= vi
dt
(1.2.1)
qi
dv i
= ai =
[E (ri , t) + v i × B (ri , t)]
dt
mi
(1.2.2)
where i = 1, ..., N and the elds
E and B must satisfy the set of Maxwell's equations:
∇ × E (r, t) = −
∂
B (r, t)
∂t
(1.2.3)
∂
∇ × B (r, t) = µ0 0 E (r, t) + J (r, t)
∂t
(1.2.4)
ρ (r, t)
0
∇ · E (r, t) =
(1.2.5)
∇ · B (r, t) = 0
(1.2.6)
in which every particle contributes with its position and velocity in the charge density
and current density.
In fact this is a system of equations that is not so easy to solve,
N is a very high number.
especially for macroscopic systems where
The scheme in gure (1.2.1) depicts the cross correlation that exists between the set
of Maxwell's equations and the equation of motion (or Lorentz equation).
Usually, a
statistical description of the ensemble of particle, is even more usefull.
j
Every particle species
is described with a distribution function fj (r, v, t) that cor3
3
responds to the average number of particles per unity volume d rd v of the
phase space
(r, v) at the istant t.
The previous Maxwell equations are still valid for
E and B only if we suppose negligible
the dierence between the mean elds and the microscopic elds for a given ensemble's
conguration: this is called Vlasov approximation; in that case the average charge and
current densities are given by:
ˆ
ρ (r, t) =
X
qj
j
J (r, t) =
fj (r, v, t) d3 v
(1.2.7)
vfj (r, v, t) d3 v
(1.2.8)
ˆ
X
qj
j
The evolutions of the distribution functions can be found as follows.
particles in a given volume
d
dt
ˆ
V in the phase space, has an evolution:
The number of
ˆ
3
3
fj (r, v, t) d rd v =
V
26
∂
fj (r, v, t) d3 rd3 v
∂t
V
(1.2.9)
1 Introduction to plasma physics
Figure 1.2.1: Scheme of the cross correlation between Maxwell's equations set and equations of motion for the description of plasmas.
27
1 Introduction to plasma physics
We can relate this evolution with the ux of particles in the surface
boundary of that specic volume
d
dt
where V
ˆ
V:
S which is the
ˆ
3
3
fj (r, v, t) d rd v = −
V
fj (r, v, t) V · n̂dS
(1.2.10)
S
= v + a is the velocity vector in the phase space, and thus has six components.
S is a surface in the phase space. The previous equation simply states that
the positive variation over time of the total number of particles in V is equal to the ux
(number of particle per unit of time) that passes through the surface S (the minus sign is
Note that also
present because ux is taken positive when outgoing). Using the divergence theorem we
obtain:
ˆ
ˆ
∇ · [V fj (r, v, t)] d3 rd3 v
fj (r, v, t) V · n̂dS =
(1.2.11)
V
S
and thus:
∂
fj (r, v, t) + ∇ · [V fj (r, v, t)] = 0
∂t
(1.2.12)
that is a sort of continuity equation for the distribution function. Note also that the
divergence operator has six components because it operates in the phase space:
∂
∂
∂
∂
∂
∂
+ ŷ
+ ẑ
+ vˆx
+ vˆy
+ vˆz
∂x
∂y
∂z
∂vx
∂vy
∂vz
(1.2.13)
∂
fj (r, v, t) + ∇r · [vfj (r, v, t)] + ∇v · [afj (r, v, t)] = 0
∂t
(1.2.14)
∇ = ∇r + ∇v = x̂
Calculating:
and:
∂
(1.2.15)
fj + fj ∇r · v + v · ∇r fj + fj ∇v · a + a · ∇v fj = 0
∂t
but ∇r · v = 0 because v does not depend on r, and ∇v · a = 0 because ow in velocity
space under the Lorentz force is incompressible, thus:
∂
qj
fj + v · ∇ r fj +
[E + v × B] · ∇v fj = 0
∂t
mj
(1.2.16)
that is the Vlasov equation. The dierence between the mean elds and microscopic
elds is important in some situations where two particles are very close, with binary
interactions.
In cold plasmas where this kind of binary interactions is too weak, the
approximation is still valid. In cold plasmas we can take into account the uctuations of
the elds and other quantities between two dierent ensemble's conguration by writing
the
exact solution as a sum of the average solution found with Vlasov equation, and a
uctuation:
fjexact = fj + f˜j
28
(1.2.17)
1 Introduction to plasma physics
For instance the exact distribution functions are a sum extendend over the entire number
of particles of that species, of delta functions in the phase space:
fjexact (r, v, t) =
N
X
(j)
(j)
δ r − ri (t) δ v − v i (t)
(1.2.18)
i=1
Also the acceleration is perturbed because
E and B are not approximated with the
mean values and thus:
aexact = a + ã
(1.2.19)
∂ exact
f
+ v · ∇r fjexact + aexact · ∇v fjexact = 0
∂t j
(1.2.20)
So the Vlasov equation is:
and averaging it:
∂
fj + v · ∇ r fj + a · ∇ v fj =
∂t
∂
fj
∂t
(1.2.21)
coll
where the collision term is dened:
∂
fj
∂t
D
E
= − ã · ∇v f˜j
(1.2.22)
coll
which is very dicult to determine because it represents all the interactions between
the particles. More details are given in [Strickland 1985, Landau 1946].
1.3 Fluidic description
Although kinetic description of plasma is relatively rich, as it provides a direct access to
the ne structure of the system in the phase space, its approach is extremely unecient
because we have to manipulate for purpose of simulation, the distribution function in a
six dimensional space; the uidic description is more intuitive and simple than kinetic
because the signicance of uid quantities such as density, ux, pressure or temperature,
is immediately understandable, whereas the signicance of distribution functions is far less
obvious. Moreover, uid variables are relatively easy to measure in experiments, whereas,
in most cases, it is extraordinarily dicult to measure a distribution function accurately.
The uidic description involves the procedure of
taking the moments of the distribu-
tion function: these moments are used as variables for the uidic equations. The uidic
taking the moments of the Vlasov's equation.
One of the possible uidic descriptions is called the two-uid equations approach, be-
equations are equations obtained by
cause usually it is applied coupling the equation for electrons and ions using dierent variables for ions and electrons. Another approach is the
magneto-hydro-dynamics (MHD): it
consists in the usage of cumulative variables for ion and electrons that are linear combination of the variables used in the two-uid equations model. In the following two paragraphs
we will detail the derivation of the interesting variables and equations in a general uidic
29
1 Introduction to plasma physics
approach. The two uids equation can be obtained using the appropriate quantities for
ions and electrons in each general expression or equation. Usually the corrispondent MHD
equations can be derived summing each quantity or equation over the total possible species
contained in the plasma under consideration.
1.3.1 Fluidic variables
This description, in most of the cases of interest, involves a limited number of variables,
dened in the time and space domain, and whose evolutions are given by equations of
uids.
The variables of interest are the moments of the distribution function; the zero
order moment:
ˆ
fj (r, v, t) d3 v
nj (r, t) =
is the particle density of the species
(1.3.1)
j, and is related to the charge density of the j -th
species by:
ρj (r, t) = qj nj
(1.3.2)
The rst order moment:
1
v j (r, t) =
nj
ˆ
vfj (r, v, t) d3 v
(1.3.3)
j
is the average velocity for the species , that is related with other quantities such as:
φj (r, t) = nj v j
(1.3.4)
J j (r, t) = qj nj v j
(1.3.5)
termed ux, and:
the current density for each species. The second order moment in the rest frame of the
ux of particles is a tensor of rank 2 given by:
ˆ
P j (r, t) = mj
is called
v − vj
v − v j fj (r, v, t) d3 v
(1.3.6)
pressure tensor, and it can be related with a temperature tensor:
P j (r, t) = nj kB T j
(1.3.7)
If we do not take the second order moment in the rest frame of the species under
consideration, the tensor is called
stress tensor :
ˆ
p (r, t) = mj
j
vvfj (r, v, t) d3 v
and their relationship is quite simple to obtain:
30
(1.3.8)
1 Introduction to plasma physics
p (r, t) = P j (r, t) + mj nj v j v j
j
(1.3.9)
The third order moment in the rest frame of the ux of particles is a tensor of rank 3:
1
Q (r, t) = mj
2
j
and is called the
energy tensor :
ˆ
v − vj
3
fj (r, v, t) d3 v
(1.3.10)
heat ux. In another frame of reference it is normally called ux of
1
q (r, t) = mj
2
j
ˆ
vvvfj (r, v, t) d3 v
(1.3.11)
1.3.2 Fluidic equations
The dierential equations for these variables can be calculated using the moments of
Vlasov's equation. It is important to note that the
(
n-th equation will always contain the
n+1 )-th moment and the (n-1 )-th : this is a set of equation that forms a system with more
ad hoc closure equation,
unknown that equations. This problem can be avoided with an
relating the two higher moments of the distribution functions. Possible solutions are:
ˆ Cold plasma:
P j (r, t) = 0
(1.3.12)
T j = const
(1.3.13)
ˆ Isothermal plasma:
ˆ Adiabatic hypotheses:
γ
P j (r, t) ∝ nj j
(1.3.14)
ˆ The Fourier equation for the heat ux:
Q = −k∇T j
(1.3.15)
j
Let's now calculate the moments of the Vlasov equation; for each moment we obtain an
equation on the form:
∂
A = −∇r · B + C − D
∂t
(1.3.16)
where the several functions are dened:
ˆ
A=
ϕ (v) fj (r, v, t) d3 v
(1.3.17)
ϕ (v) vfj (r, v, t) d3 v
(1.3.18)
ˆ
B=
31
1 Introduction to plasma physics
ˆ
C=−
ˆ
3
∇v ϕ (v) · [afj (r, v, t)] d3 v
ϕ (v) ∇v · [afj (r, v, t)] d v =
ˆ
∂
fj
∂t
(1.3.19)
d3 v
(1.3.20)
fj (r, v, t) d3 v = nj (r, t)
(1.3.21)
vfj (r, v, t) d3 v = φj (r, t)
(1.3.22)
∇v [1] · [afj (r, v, t)] d3 v = 0
(1.3.23)
D=−
ϕ (v)
coll
First order: ϕ (v) = 1
ˆ
A=
ˆ
B=
ˆ
C=
ˆ D=−
Note that
∂
fj
∂t
d3 v = 0
(1.3.24)
coll
D = 0 because the collision term must undergo the principle of particle
conservation (if we make the hypotheses of absence of ionizing collision). The rst equation
reads:
∂nj
= −∇r · φj
∂t
multiplying for qj the species charge and summing over
(1.3.25)
j we obtain:
∂ρ
= −∇r · J
∂t
(1.3.26)
The other moments of the Vlasov equation can be calculated in the same way.
For
example the second order moment equation is:
∂
qj
1
1
v j + v j · ∇v j = −
E + vj × B −
∇ · Pj +
∂t
mj
mj nj
nj
ˆ
v
∂
fj
∂t
d3 v
(1.3.27)
coll
An usefull approximation for the collision term is the following:
1
nj
ˆ
v
∂
fj
∂t
d3 v = −υj v j − v j0
coll
where υj is the collision frequency.
32
(1.3.28)
1 Introduction to plasma physics
1.4 Charged particle motion
All descriptions of plasma behaviour are based, ultimately, on the motions of the single
constituent particles. We can distinguish three cases:
ˆ unmagnetized plasma: the motion of each particle is quite simple because it essentially moves in straight lines between collisions with other particles;
ˆ collisional magnetized plasma: in such plasmas the collision frequency greatly exceeds the gyrofrequency; in this case, the particles are scattered after executing only
a small fraction of their orbits thus still move essentially in straight lines between
collisions;
ˆ collisionless magnetized plasma: the typical gyroradius is much smaller than the
typical variation length-scale
L of electric and magnetic elds, and the gyroperiod is
much less than the typical time-scale of variation of these elds. Intuitively we can
infer that the typical path of such particles consist in the superposition of a rapid
B eld and a straigh line drift in the
E. Also the time/space variation of these elds of course inuences the
gyration on the plane perpendicular to the
direction of
particle motion.
1.4.1 Particle motion in uniform elds
Consider a particle motion in uniform time independent electric and magnetic elds
(electro-magnetostatic case):
m
d
v = e (E + v × B)
dt
(1.4.1)
Let's now analyse this problem along two principal directions: the direction b̂ parallel to
magnetic eld, and along the plane perpendicular to magnetic eld. We hence decompose
the velocity into v k = vk b̂ and v ⊥ in the plane perpendicular to magnetic eld:
h
i
d
m
v + v ⊥ = e E k + E ⊥ + v k + v ⊥ × B b̂
dt k
(1.4.2)
separating the two dierent components we have:
d
v = eE k
dt k
h
i
d
m v ⊥ = e E ⊥ + v ⊥ × B b̂
dt
m
(1.4.3)
(1.4.4)
Now integrating the rst equation we obtain the simple:
vk =
e
E t + v 0k
m k
Taking the second equation and rotating left of a vector B we have:
33
(1.4.5)
1 Introduction to plasma physics
m
d
B × v ⊥ = e [B × E ⊥ + B × v ⊥ × B]
dt
(1.4.6)
and therefore:
E⊥ × B
m d b̂
×
v
(1.4.7)
+
⊥
B2
eB dt
Dening a right-handed set of versors ê1 , ê2 , b̂ and the costant drift velocity due to
v⊥ =
the static elds v E =
E×B
, we can infer a solution like this:
B2
v⊥ =
E×B
+ ρΩ [ê1 sin (Ωt + γ0 ) + ê2 cos (Ωt + γ0 )]
B2
(1.4.8)
which in fact satises the dierential equation. Summing the two solution we obtain
the total velocity:
v(t) =
E×B
e
+ ρΩ [ê1 sin (Ωt + γ0 ) + ê2 cos (Ωt + γ0 )] + E k t + v 0k
2
B
m
(1.4.9)
Integrating to obtain the particle position we have:
r(t) =
E×B
e
t + ρ [−ê1 cos (Ωt + γ0 ) + ê2 sin (Ωt + γ0 )] +
E t2 + v 0k t + r0
2
B
2m k
(1.4.10)
The total motion of that particle consists in a superposition of gyration ρ(t) in the plane
orthogonal to
B eld and a motion R(t) with constant acceleration:
R(t) =
e
E×B
E k t2 + v 0k t +
t
2m
B2
ρ(t) = ρ [−ê1 cos (Ωt + γ0 ) + ê2 sin (Ωt + γ0 )]
(1.4.11)
(1.4.12)
R(t) is said to be the guiding centre position of the particle motion. To simplify the
problem when the elds are non uniform is sometimes important to know the motion
of the guiding centre; it can be found by simply
averaging the equation of motion over
gyrophase, so as to obtain a reduced equation of motion for the guiding centre.
1.4.2 Method of averaging
Consider the equation of motion of a particle rewritten in this fashion:
d
z (t) = f (z, t, τ )
(1.4.13)
dt
where f is a periodic function of the variable τ : f (z, t, τ ) = f (z, t, τ + 2π); τ in fact is
dened as to be a time variable but comparing with
τ = t/
34
t it has a dierent time scale:
(1.4.14)
1 Introduction to plasma physics
If we suppose that the motion consists in a superposition of a slow drift and a rapid
periodic oscillation, then the parameter can be considered quite small if compared to
t.
Let's seek a solution of the problem in the form z (t, τ ): note that τ is only an apparent
unknown. Dening the average motion as the average over the parameter τ :
z m (t) = hz (t, τ )iτ
(1.4.15)
we can seek a solution in the form:
z (t, τ ) = z m (t) + ζ (z m , t, τ )
(1.4.16)
whereζ is a periodic function of τ that has vanishing average:
ζ (z m , t, τ ) = 0
(1.4.17)
Substitution in the equation of motion (1.4.13) gives:
1 ∂
∂
z (t, τ ) +
z (t, τ ) = f (z, t, τ )
∂t
∂τ
(1.4.18)
and therefore:
1 ∂
∂ m
z (t) + ζ (z m , t, τ ) +
z m (t) + ζ (z m , t, τ ) = f (z, t, τ )
∂t
∂τ
(1.4.19)
or:
d
d m
z (t) + ζ (z m , t, τ ) = f (z, t, τ )
dt
dt
So considering the expansions in the small parameter , we set:
∞
X
d m
z (t) =
i F i (z m , t)
dt
i=0
m
ζ (z , t, τ ) =
∞
X
i ζ i (z m , t, τ )
(1.4.20)
(1.4.21)
(1.4.22)
i=0
Now substituting in (1.4.20):
∞
X
" ∞
#
X
d
i F i (z m , t) +
i ζ i (z m , t, τ ) = f (z, t, τ )
dt
i=0
i=0
(1.4.23)
Note that we can write:
d
ζ (z m , t, τ ) = dt
d m
1 ∂ ∂
z
ζ (z m , t, τ ) + ζ (z m , t, τ )
· ∇ζ +
dt
∂τ
∂t
Thus substituting:
35
(1.4.24)
1 Introduction to plasma physics
∞
X
i F i +
i=0
∞
X
!
i F i
i=0
"∞
X
"∞
"∞
#
#
X
X
∂
∂
i ζ i +
i ζ i = f (z, t, τ ) (1.4.25)
·∇
i ζ i +
∂τ
∂t
i=0
i=0
i=0
#
and rearranging:
∞
X
!(
i F i · 1 + ∇
i=0
"∞
X
#)
i+1 ζ i
i=0
"∞
#
"∞
#
∂ X i
∂ X i+1
+
ζ +
ζ i = f (z, t, τ ) (1.4.26)
∂τ i=0 i
∂t i=0
Expanding f too:
f=
∞
X
i di
i=0
i! dti
f
(1.4.27)
where the derivatives of f can be calculated with the well known
Faà di Bruno's formula
for the derivatives of composite functions:


!
kl
j
i  j
i

X
X
Y
D
h(t)
Dh 
(t)
i

D(t) {g [h(t)]} = i!
 j!

l!
j=1
k +...+k =j
l=1
1
(1.4.28)
j
(note that this formula is written for scalar functions for ease of notation: replace the
appropriate derivative with the directional derivative where it is necessary). We obtain
the nal equation:
∞
X
i
Fi
i=0
X
X
∞ ∞ ∞
∞ X
∞ X
X
∂ i
∂ i+1
i di
i j+1
F i · ∇ζ j +
ζi +
ζi =
f
+
∂τ
∂t
i! dti
i=0
i=0
i=0
i=0 j=0
(1.4.29)
∂
Note that averaging over τ the function
ζ we obtain zero because all the expansion
∂τ i
of ζ is periodic in τ ; therefore the condition of solubility is:
m
hF i (z , t)iτ =
1 di
f
i! dti
(1.4.30)
τ
hence:
m
F i (z , t) =
1 di
f
i! dti
(1.4.31)
τ
Equation (1.4.29) to the lowest order in gives:
F0 +
∂
ζ =f
∂τ i
(1.4.32)
whilst to the rst order it gives:
F 1 + F 0 · ∇ζ 0 +
∂
∂
ζ 1 + ζ 0 = ζ 0 · ∇f
∂τ
∂t
36
(1.4.33)
1 Introduction to plasma physics
The relative conditions of solubility from (1.4.31):
F0 = f
(1.4.34)
D
E
F 1 = ζ 0 · ∇f
(1.4.35)
∂
ζ =f
∂τ i
(1.4.36)
and therefore substituting:
f +
D
E
∂
∂
ζ 0 · ∇f + f · ∇ζ 0 +
ζ 1 + ζ 0 = ζ 0 · ∇f
∂τ
∂t
(1.4.37)
Integrating the rst equation:
ˆ τ
f − f dτ 0
m
ζ 0 (z , t, τ ) =
(1.4.38)
0
substituting in the second and further integrating it:
ˆ τ
D
E
∂
ζ 0 · ∇f − ζ 0 · ∇f − f · ∇ζ 0 − ζ 0 dτ 0
ζ 1 (z , t, τ ) =
∂t
0
m
(1.4.39)
And so on...
So we can write the coecient of the expansion of
d m
z and of ζ as functions of the f ;
dt
truncating the series at rst order we obtain:
D
E
d m
z (t) ≃ f + ζ 0 · ∇f
dt
(1.4.40)
ζ (z m , t, τ ) = ζ 0 (z m , t, τ ) + ζ 1 (z m , t, τ )
(1.4.41)
m
m
Recalling that z (t, τ ) = z (t) + ζ (z , t, τ ) we can evidently conclude that the function
z m (t) is a sort of guiding centre of the problem of motion and therefore its drift is
determined to lowest order by the average of the force f, and to next order by the
correlation between the oscillation in the position z and the oscillation in the spatial
gradient of the force . In fact what we notice is that in some problem of particle motion
in plasmas z corresponds to the velocity of the particle, f is the Lorentz force divided by
the particle mass. So it's easy to obtain a rst order solution of a motion problem of a
charged particle with this procedure: see next paragraph for a simple application.
1.4.3 Guiding centre motion
We are interested to obtain the solution of the dierential problem of motion:
d
r(t) = v(t)
dt
(1.4.42)
d
e
v(t) =
(E + v × B)
dt
m
(1.4.43)
37
1 Introduction to plasma physics
where we have introduced the small parameter . Seeking a change of variable in the
form:
r(t) = R(t) + ρ (R, U , t, γ)
(1.4.44)
v(t) = U (t) + u(R, U , t, γ)
(1.4.45)
where γ is the gyrophase and plays the role of τ , ρ e u the gyroradius and velocity of
gyration and they are periodic in γ with vanishing mean.
ρ=
∞
X
i ρi (R, U , t, γ)
(1.4.46)
i ui (R, U , t, γ)
(1.4.47)
i=0
u=
∞
X
i=0
Is also possible to expand in the same way the dynamical equation of γ
∞
X
d
γ=
i−1 ωi−1 (R, U , t)
dt
i=0
(1.4.48)
d
γ ≃ Ω hence γ ≃ γ0 + Ωt.
dt
Now substituting (1.4.44) and (1.4.45) in the equation of motion (1.4.42) and (1.4.43) :
but in fact we assume that
d R(t) + ρ (R, U , t, γ) = U (t) + u(R, U , t, γ)
dt
d
e
[U (t) + u(R, U , t, γ)] =
[E + (U (t) + u(R, U , t, γ)) × B]
dt
m
(1.4.49)
(1.4.50)
Therefore we can split the the rst calculating the relationships between R(t) and U (t)
and also between ρ (R, U , t, γ) and u (R, U , t, γ):
d
R(t) = U (t)
dt
(1.4.51)
d ρ (R, U , t, γ) = −1 u(R, U , t, γ)
dt
(1.4.52)
Using the consideration of the previous paragraph we can expand U (t):
∞
X
d
U (t) =
i F i
dt
i=0
(1.4.53)
and using other similar expansion we can rewrite (1.4.50):
∞
X
∞
∞
X
d X i
i−1 di
Fi +
ui (R, U , t, γ) =
f
i
dt
i!
dt
i=0
i=0
i=0
i
where the force term is given by:
38
(1.4.54)
1 Introduction to plasma physics
e
[E + (U (t) + u(R, U , t, γ)) × B]
m
f=
(1.4.55)
Calculating the second addend of the sum at rst member of (1.4.54):
∞
∞
X
d
d X i
d
∂
d ∂
i
ui (R, U , t, γ) =
R · ∇ui + U · ∇ui + ui + γ ui =
dt i=0
dt
dt
∂t
dt ∂γ
i=0
=
∞
X
i
U (t) · ∇ui +
i=0
∞ X
∞
X
i j
F j · ∇ui +
∞
X
i=0 j=0
i=0
i ∂
∞ X
∞
X
i j−1
u +
∂t i i=0 j=0
∂
u ωj−1
∂γ i
and back substituting in (1.4.54):
∞
X
i
Fi +
∞
X
+
i=0
U (t) · ∇ui +
i=0
i=0
∞
X
i
i j F j · ∇ui +
i=0 j=0
∞ X
∞
X
i ∂
∞ X
∞
X
i j−1
u +
∂t i i=0 j=0
∞
X
∂
i−1 di
ui ωj−1 =
f
i
∂γ
i!
dt
i=0
(1.4.56)
To the lowest order we obtain:
ω−1
∂
e
u0 =
[E + (U 0 + u0 ) × B]
∂γ
m
(1.4.57)
Averaging it over a period in γ :
E + U0 × B = 0
(1.4.58)
that is the rst solubility condition; rotating it left along the B eld direction, it gives:
U 0 = U0k b̂ +
E×B
= U0k b̂ + v E
B2
(1.4.59)
Note that the parallel velocity is not determined. Integrating (1.4.57), one obtains:
Ω
Ω
u0 = u⊥ ê1 sin
γ + ê2 cos
γ
+c
ω−1
ω−1
(1.4.60)
applying the periodicity constrain c = 0, and noting that ω−1 = Ω:
u0 = u⊥ (ê1 sin (γ) + ê2 cos (γ))
(1.4.61)
Note that also the amplitude of the velocity of gyration is not determined. Now we can
calculate the gyroradius:
"∞
#
∞
X
d X i
ρi (R, U , t, γ) =
i−1 ui (R, U , t, γ)
dt i=0
i=0
39
(1.4.62)
1 Introduction to plasma physics
to the lowest order:
Ω
∂
ρ = u0
∂γ 0
(1.4.63)
integrating, it gives:
ρ0 =
u⊥
(−ê1 cos (γ) + ê2 sin (γ))
Ω
(1.4.64)
that can be written:
u0 = Ωρ0 × b̂
(1.4.65)
Finally we have to calculate the mean value of the equation of motion for U, and at the
lowest order:
D
E d
e
U =
E + U 1 × B + u0 × ρ0 · ∇ B
dt 0 m
γ
(1.4.66)
The last term is:
D
E D
E
ρ0 × b̂ × ρ0 · ∇B
= b̂ρ0 · ρ0 · ∇ B − ρ0 · ∇ B ρ0 · b̂ =
D
E
= b̂ρ0 · ρ0 · ∇ B − ρ0 · ∇B ρ0 = b̂ · ρ0 ρ0 · b̂∇B − ρ0 ρ0 · ∇B =
E
D
= b̂b̂ − I · ρ0 ρ0 ∇B
(1.4.67)
calculating:
D
E
(u⊥ )2 ρ0 ρ0 =
I
−
b̂
b̂
2Ω2
γ
"
#
e
(u⊥ )2
d
U =
E + U1 × B −
∇B
dt 0 m
2Ω2
(1.4.68)
(1.4.69)
hence we obtain the nal:
d
e
e
(u⊥ )2
U = E + U1 × B −
∇B
dt 0 m
m
2Ω2
(1.4.70)
The component of this equation along the magnetic eld determines the evolution of
the parallel guiding centre velocity:
d
e
(u⊥ )2
U 0k = E k −
b̂ · ∇B
dt
m
2Ω2
(1.4.71)
Note that the rst-order correction to the parallel velocity, is undetermined to this order.
This is not generally a problem, since the rst-order parallel drift is a small correction to a
type of motion which already exists at zeroth-order, whereas the rst-order perpendicular
drift is a completely new type of motion. In particular, the rst-order perpendicular drift
diers fundamentally from the E × B drift, since it is not the same for dierent species,
and, therefore, cannot be eliminated by transforming to a new inertial frame of reference.
40
1 Introduction to plasma physics
1.4.4 Particle motion in oscillating eld
To lowest order, a particle moves along a simple harmonic-shape path in response to an
oscillating eld. However, to higher order, any weak inhomogeneity in the eld causes the
restoring force at one turning point to exceed that at the other. On average, this yields
a net force which acts on the centre of oscillation of the particle.
Consider a spatially
inhomogeneous electromagnetic wave-eld oscillating:
E (r, t) = E (r) cos (ωt)
(1.4.72)
suppose that Ω ω , the eld is thus rapidly oscillating while particle moves slowly
along the plane perpendicular to B. The equation of motion are (1.4.42) and (1.4.43).
Using the Faraday's law:
∂
1
∂
B (r, t) = B (r) sin (ωt)
∂t
ω
∂t
∇ × E (r, t) = [∇ × E (r)] cos (ωt) =
(1.4.73)
thus:
B (r) =
e
d
v(t) =
dt
m
1
∇ × E (r)
ω
1
E (r) cos (ωt) + v × B (r) sin (ωt)
ω
(1.4.74)
(1.4.75)
Using the method of averaging with variables:
r(t) = R(t) + ρ (R, U , t, τ )
(1.4.76)
v(t) = U (t) + u(R, U , t, τ )
(1.4.77)
we can calculate to the lowest order:
e
∂
u0 = E 0 (R) cos (ωτ )
∂τ
m
(1.4.78)
e
E (R) sin (ωτ )
mω 0
(1.4.79)
e
E (R) cos (ωτ )
mω 2 0
(1.4.80)
and thus:
u0 =
ρ0 = −
We can also dene an oscillation velocity:
v osc =
e
E
mω 0
Calculating the average of the equation of motion to the rst order it gives:
d
e D e
−
U (t) =
E (R) cos (ωτ ) · ∇ (E (R) cos (ωτ )) +
dt
m
mω 2 0
41
(1.4.81)
1 Introduction to plasma physics
1
+
E (R) sin (ωτ ) ×
∇ × E 0 (R) sin (ωτ )
mω 0
ω
e
(1.4.82)
The calculation results in:
m
d
U (t) = −e∇Φpond
dt
(1.4.83)
1 e
|E |2
4 mω 2 0
(1.4.84)
e
∇Φpond
m
(1.4.85)
where
Φpond =
is called ponderomotive potential.
U (t) = −t
The ponderomotive force is the product of the gradient of ponderomotive potential
and the elementary charge. This equation states that the oscillation centre experiences a
costant force that accelerate both ions or electron because the force is independent of the
charge sign:
Fpond = e∇Φpond
(1.4.86)
The total energy of the oscillation centre can be calculated:
εosc = kosc + Vpond
(1.4.87)
where Vosc is the potential ponderomotive potential energy:
Vpond = eΦpond
(1.4.88)
1.4.5 Relativistic regime
Consider a free electron initially at rest placed in an electromagnetic eld of a plane wave
in vacuum. From a relativistic point of view we can write the equation of conservation of
momentum and energy:
d
p=F
dt
(1.4.89)
d
E =v·F
dt
(1.4.90)
The equation of motion of a relativistic electron in an electromagnetic eld:
d
p = −e (E + v × B)
dt
(1.4.91)
We recall that in the relativistic regime the momentum is not the simple product of
mass and velocity but:
42
1 Introduction to plasma physics
p = γme v
(1.4.92)
Where γ is the Lorentz factor of that particle:
1
γ=q
1−
(1.4.93)
v 2
c
We can write:
v 2
c
=
p
cγme
2
(1.4.94)
and:
1−
v 2
c
=1−
p
cme
2
1
1
= 2
2
γ
γ
(1.4.95)
in which we use the denition of γ . So we can write also another relation:
s
1
γ=q
1−
v 2
c
1+
=
p
me c
2
(1.4.96)
The Maxwell's equation in the relativistic regime can be solved with the potential vectors
method, and in this simple case the expression of electric and magnetic eld are:
∂
A
∂t
(1.4.97)
B =∇×A
(1.4.98)
E=−
The equation of motion can therefore be rewritten:
d
∂
∂
∂
p = −e − A + v × (∇ × A) = e A − e (v × ∇ × A) = e A − e∇ (v · A) (1.4.99)
dt
∂t
∂t
∂t
d
∂
p − e A = −e∇ (v · A)
dt
∂t
(1.4.100)
The equation for the electric eld can be rewritten:
∂
d
E = −ev · (E + v × B) = ev · A
dt
∂t
(1.4.101)
x direction, thus the
electric and magnetic eld are vectors in the yz plane, therefore the potential vector A is
co-polarized with them:
If the electromagnetic wave is a plane wave propagating along
A = A⊥
43
(1.4.102)
1 Introduction to plasma physics
where the subscript ⊥ stands for plane perpendicular to propagation direction.
The transverse component of the equation (1.4.100) gives:
p⊥ = eA⊥
(1.4.103)
eA⊥
me γ
(1.4.104)
and:
v⊥ =
Substituting in the longitudinal component of (1.4.100) and in (1.4.101):
d
e2
∂
e2 ∂ 2
px = −
A A=−
A
dt
me γ ∂x
2me γ ∂x
(1.4.105)
e2
d
∂
e2 ∂ 2
γme c2 =
A A=
A
dt
me γ ∂t
2me γ ∂t
(1.4.106)
subtracting the rst equation, previously multiplied by
c:
d
e2
∂
∂
2
γme c − cpx =
+c
A2
dt
2me γ ∂t
∂x
For an electromagnetic wave traveling in vacuum the vector
(1.4.107)
A is only dependent on
τ = t − x/c, thus the right member of the previous equation is zero; for an electron
initially at rest we have:
γme c2 − cpx = me c2
(1.4.108)
and thus using equation (1.4.96):
px
=
γ =1+
me c
s
p
me c
2
2
1+
s
=
1+
p⊥
me c
2
+
px
me c
2
(1.4.109)
thus squaring:
px
1+2
+
me c
px
me c
=1+
p⊥
me c
2
2
+
px
me c
2
(1.4.110)
calculating γ :
1
γ =1+
2
44
eA
me c
(1.4.111)
1 Introduction to plasma physics
Linearly polarized wave
Supposing a linearly polarized wave we can write the potential
vector as to be:
eA
= a0 cos (ωτ ) ŷ = a0 cos (ωt − kx) ŷ
me c
ω
where k =
is the wave number and a0 is the normalized amplitude:
c
a0 =
p
eE
≈ 0.85 × λ[µm] × I18
ωme c
(1.4.112)
(1.4.113)
because:
∂ me c
ωme c
a0 cos (ωτ ) ŷ =
a0 sin (ωτ ) ŷ
(1.4.114)
∂t e
e
The normalized quantity a0 is an important parameter and is commonly used to check
E = E (τ ) ŷ = −
if the characteristics of a laser radiation lead to a relativistic laser-matter interaction: this
is the case where a0 & 1. In the non relativistic limit a0 1 the cross product term in
(1.4.91) can be neglected, and a0 coincides with the β factor v/c.
The oscillating momentum is given by:
p⊥ = me c [a0 cos (ωt − kx)] ŷ
the longitudinal momentum is (using also the independent variable τ = t −
1
1
px = me c [a0 cos (ωt − kx)]2 x̂ = me ca20 [1 + cos (2ωτ )] x̂
2
4
(1.4.115)
x
):
c
(1.4.116)
the γ factor is:
1
γ = 1 + a20 [1 + cos (2ωτ )]
4
(1.4.117)
1 2
E = me c 1 + a0 [1 + cos (2ωτ )] = me c2 + K
4
(1.4.118)
thus the total energy is:
2
The kinetic energy is:
1
K = (γ − 1) me c2 = a20 [1 + cos (2ωτ )] me c2
4
(1.4.119)
The particle trajectory can be found simply considering the proper time variable τ
=
t − xc (τ = t/γ ) and integrating the momentum in each dierent directions to obtain:
ky = a0 sin (ωτ )
(1.4.120)
a20
[2ωτ + sin (2ωτ )]
8
(1.4.121)
kx =
If we calculate the average of some quantities, we know the behavior of the centre of
oscillation; for example:
45
1 Introduction to plasma physics
1
hpx i = me ca20
4
(1.4.122)
hp⊥ i = 0
(1.4.123)
1
hγi = 1 + a20
4
(1.4.124)
hvx i = vdrif t =
hpx i
a20 /4
c
=
hγi me
1 + a20 /4
(1.4.125)
in the rest frame of the particle applying a Lorentz transformation, one can obtain
[reference needed]:
a0
k0y0 = p
sin (ω 0 τ 0 )
2
1 + a0 /2
(1.4.126)
a2
k 0 x0 = p 0 2 sin (2ω 0 τ 0 )
8 1 + a0 /2
(1.4.127)
the trajectory has an 8 shape qas shown in gure (1.4.1) in the rest frame of the
particle, caused by the doubled frequency in the x position formula.
Figure 1.4.1: Electron trajectory in the electromagnetic eld of a linearly polarized wave
in the frame of reference co-moving with the average velocity of the particle.
p
The quantity a1 is dened as a1 = a/
1 + a20 /2 where a20 is the normalized
0
0
amplitude of the wave. The quantity k = ω /c is the wave number in the
co-moving frame of reference. [Mora course 2008]
This motion induces a frequency doubled characteristic light emission, called
2
[Lau 2003] with hν me c .
Thomson scattering
46
nonlinear
1 Introduction to plasma physics
Circularly polarized wave
Supposing a circularly polarized wave:
eA
= a0 [cos (ωτ ) ŷ + sin (ωτ ) ẑ]
me c
(1.4.128)
we can obtain:
1
1
px = me c |a0 [cos (ωτ ) ŷ + sin (ωτ ) ẑ]|2 x̂ = me ca20 x̂
2
2
py = me c [a0 cos (ωτ )] ŷ
pz = me c [a0 sin (ωτ )] ẑ
(1.4.129)
(1.4.130)
(1.4.131)
And the particle trajectory is:
kx =
a20
ωτ
2
(1.4.132)
ky = a0 sin (ωτ )
(1.4.133)
kz = −a0 cos (ωτ )
(1.4.134)
1
γ = 1 + a20
2
(1.4.135)
1 2
E = me c 1 + a0 = me c2 + K
2
(1.4.136)
1
K = (γ − 1) me c2 = a20
2
(1.4.137)
the γ factor is:
hence the total energy is:
2
The kinetic energy is:
The drift velocity of the particle is:
1 2
a
px
= 2 10 2 c
vdrif t =
γme
1 + 2 a0
(1.4.138)
1.5 Waves in plasmas
1.5.1 Electromagnetic oscillation in plasmas
Now we consider the propagation of an electromagnetic wave, which elds are the solution
of the set of Maxwell's equations:
∇ × E = −jωB
47
(1.5.1)
1 Introduction to plasma physics
∇ × H = jωD + J
(1.5.2)
∇·D =ρ
(1.5.3)
∇·B =0
(1.5.4)
Using linear consitutive expressions we write:
D = 0 E
(1.5.5)
B = µ0 H
(1.5.6)
J = σE
(1.5.7)
Introducing them into the set of Maxwell's equations:
∇ × E = −jωB
∇ × B = jωµ0 0
σ
1−j
ω0
(1.5.8)
E
(1.5.9)
ρ
0
(1.5.10)
∇·B =0
(1.5.11)
∇·E =
We can dene the complex permettivity:
c = 0 − j
σ
ω
(1.5.12)
Taking the curl of the rst equation of the set (1.5.1) we obtain:
∇ × ∇ × E = −jω∇ × B
(1.5.13)
using the second equation of the set (1.5.2):
∇ (∇ · E) − ∇2 E = −jωµ0 c
(1.5.14)
hence the elctric eld must satisfy the so called wave equation:
∇2 E + ω 2 µ0 c E − ∇ (∇ · E) = 0
(1.5.15)
and the magnetic eld can be calculated using (1.5.2):
B=−
1
∇×E
jω
48
(1.5.16)
1 Introduction to plasma physics
1.5.2 Dielectric response of a collisionless cold plasma
We want to solve the electromagnetic problem of wave propagation in a plasma supposing
that its solution is like:
E (r, t) = < {E (r) exp (−jωt)}
(1.5.17)
B (r, t) = < {B (r) exp (−jωt)}
(1.5.18)
that is an oscillating electromagnetic eld at frequency ω . We suppose that the plasma
is cold and collisionless. Thus using a uidic approach, we can rewrite the moments of
Vlasov's equation as follows:
∂qj nj
= −∇ · Jj
∂t
(1.5.19)
qj
∂
v j + v j · ∇v j = −
E + vj × B
∂t
mj
(1.5.20)
where we have considered the hypotheses:
∂
fj
∂t
=0
(1.5.21)
coll
Pj = 0
(1.5.22)
If we consider a linear response the second term of the equation of motion is neglectable:
qj
∂
vj = − E
∂t
mj
(1.5.23)
In the frequency domain we obtain:
vj = −
qj
E
iωmj
iωqj nj = −∇ · J j
(1.5.24)
(1.5.25)
We have already seen that the density current for each species is given by:
J j = qj nj v j = −
qj2
qj2 nj
nj E = i
E
iωmj
ωmj
(1.5.26)
Using the denition of plasma frequency (1.1.28) we can write:
2
ωp−j
J j = −iω0 2 E
ω
(1.5.27)
Thus the conductance of the j-th species is:
σj = −iω0
49
2
ωp−j
ω2
(1.5.28)
1 Introduction to plasma physics
The main issue is to calculate the charge density, and the concentration of ions and
electrons. The equations that must be solved toghether are the continuity equation (1.5.25)
and the wave equation (1.5.15) recalling that ρ =
P
j qj nj and J =
P
j J j . If we consider
only protons and electrons then:
χe χp
(1.5.29)
because the plasma frequency of ions is much smaller that the plasma frequency of
electrons. Therefore one can assume:
J ≃ Je
(1.5.30)
And thus we can calculate the dielectric function of the plasma:
2 ωp−j
σ
σe
c = 0 − i ≃ 0 − i = 0 1 − 2
ω
ω
ω
(1.5.31)
is the permittivity function of the cold-plasma. The dielectric function r is the quantity
in parentheses and sometimes in the literature is noted as a simple variated epsilon ε ≡
r ≡ c /0 .
1.5.3 Plane waves propagation in a cold homogeneous plasma
Focusing on the case of an homogeneus, cold, collisionless, quasi-neutral plasma, where
∇ · E = 0, the propagation of electromagnetic waves is given by the Helmholtz equation:
∇2 E + ω 2 µ0 c E = 0
(1.5.32)
The solution of the dierential problem can be written in terms of a superposition of
plane waves oscillating with frequency ω with wave-vector k = k k̂ :
E (r, t) = < {E k exp [j (k · r − ωt)]}
(1.5.33)
B (r, t) = < {B k exp [j (k · r − ωt)]}
(1.5.34)
Substituting in (1.5.32) the wave vector must satisfy the equation:
k 2 = ω 2 µ0 c
(1.5.35)
In the frequency domain, it can be demonstrated that the ∇ operator, operates as the
complex vector
−jk (recall also that the time derivative operate like jω ).
In a more
complex problem, the plasma must be considered as an anisotropic medium.
We can
calculate the new set of Maxwell's equation that reduces to the simple following couple of
equation:
k × E = ωB
(1.5.36)
− jk × B = jωµ0 D + µ0 J
(1.5.37)
50
1 Introduction to plasma physics
The constitutive linear expressions for the plasma are:
D = 0 E
(1.5.38)
J =σ·E
(1.5.39)
where the conductivity tensor describe entirely the behavior of the plasma.
After a
simple substitution:
where the subscript
k × E = ωB
(1.5.40)
k × B = −ωµ0 0 T · E
(1.5.41)
k for the amplitude of the elds is suppressed for ease of notation.
T =1−j
σ
0 ω
(1.5.42)
is the complex dielectric permittivity tensor. Eliminating the magnetic eld from the
two equations we have:
k × (k × E) = −ω 2 µ0 0 T · E
(1.5.43)
Calculating:
k × (k × E) = −k
2
1 − k̂ k̂ · E = −ω 2 µ0 0 T · E
(1.5.44)
Hence:
h i
2
2
k k̂ k̂ − 1 + ω µ0 0 T · E = M · E = 0
(1.5.45)
Therefore we obtain the dispersion equation by noting that the previous equation has
a solution only if the determinant of the associated matrix is equal to zero:
det M = 0
(1.5.46)
Without losing generality, we can assume that the direction of propagation of the plane
wave is the z-axis.
Then in a simple Cartesian reference system we have wave vector
k = kẑ and the unity tensor 1 = x̂x̂ + ŷ ŷ + ẑ ẑ , and its associated matrix is the identity
matrix:


1 0 0
1 = 0 1 0 
0 0 1
(1.5.47)
If the conductivity tensor is diagonal (i.e. the conductivity has the simple value calcu2
ωp−e
):
lated in the previous paragraph σ ≃ −jω0
ω2
51
1 Introduction to plasma physics



ω2
−jω0 ωp−e
0
0
2
1
0
0
σ
j 

ω2
= 0 1 0 −
T = 1 −j


0
−jω0 ωp−e
0
2
0 ω
0 ω
2
ωp−e
0 0 1
0
0
−jω0 ω2

(1.5.48)
and:

  2

0
0
0
k
0
0
M = k 2 (ẑ ẑ) − k 2 1 + ω 2 µ0 0 T =  0 0 0  −  0 k 2 0  +
0 0 k2
0 0 k2
 ω2

p−e
1 0 0
2
ω2
ω 
ω

0 1 0 − 2  0
+ 2
c
c
0 0 1
0
2

0
0

2
ωp−e
ω2
0


0
2
ωp−e
ω2
(1.5.49)
nishing the calculation:

 2

2
2
0
0
ω
−
ω
k
0
0
p−e
1
2
=
0
ω 2 − ωp−e
0
M = −  0 k2 0  + 2 
c
2
2
0 0 0
0
0
ω − ωp−e

 2
2
− k2
0
0
ω − ωp−e
1
2

− k2
0
0
ω 2 − ωp−e
= 2
(1.5.50)
c
2
2
0
0
ω − ωp−e

then calculating the determinant:
2 2
2
det M c2 = ω 2 − ωp−e
− c2 k 2 ω 2 − ωp−e
=0
(1.5.51)
We see that the possible perturbations that can propagate are two; in the rst case we
must satisfy the equation:
2
ω 2 = ωp−e
+ c2 k 2
(1.5.52)
and the corrispondent wave are called properly electromagnetic plane wave; on the other
hand we can obtain:
2
ω 2 = ωp−e
(1.5.53)
that is the solution of the plasma wave oscillation, also termed electrostatic wave. The
two dierent complete solutions can be calculated from (1.5.45):
electromagnetic plane wave :
Ez = 0
B = Bx x̂ + By ŷ + Bz ẑ =
52
1
kẑ × (Ex x̂ + Ey ŷ)
ω
(1.5.54)
(1.5.55)
1 Introduction to plasma physics
therefore the components are:
k
k
Bx = − Ey By = Ex Bz = 0
ω
ω
Note that ∇ · E = 0 or k · E = 0.
(1.5.56)
plasma oscillation or plasma wave :
Ex = 0 Ey = 0
B=
(1.5.57)
1
kẑ × Ez ẑ = 0
ω
(1.5.58)
In other words, the wave is purely electrostatic. Note that:
Since ω is independent of
∇ × E = 0 or k × E = 0.
k, the group velocity is zero and thus it is clear that the plasma
wave is not a propagating wave, but instead has the property than an oscillation set up
in one region of the plasma remains localized in that region.
Now consider the dispersion relation (1.5.45).
The plasma frequency of the electrons
is somehow a cuto frequency, in fact if we have an electromagnetic wave oscillating at
frequency ω < ωp−e the wave doesn't propagate but is attenuated in space (i.e. the wave
in the plasma is evanescent), because:
2
ω 2 − ωp−e
k =
<0
c2
2
(1.5.59)
ω > ωp−e the plane wave is
propagated. This allow us to dene a critical density associated with ω using the denition
that results in a complex wave number.
Otherwise if
of plasma frequency:
nc (ω) =
−3 me 0 2 4π 2 me 1
−2
21
ω
=
=
1.1
×
10
×
λ
[µm] cm
e2
µ0 e2 λ2
(1.5.60)
The critical density is the density at which the wave vector is equal to zero, thus for
nc (ω) > ne the wave will propagate throughout the plasma otherwise it will be absorbed.
Plasma with electron densities ne < nc are called underdense, otherwise overdense plasma.
A diluted plasma is a plasma with ne nc . The permittivity function can also be
expressed in terms of densities:
c (ω) = 0 1 −
ne
nc (ω)
(1.5.61)
Adiabatic cloisure
In the case of electromagnetic plane wave we can also abandone the hypoteses of cold
plasma, introducing the adiabatic cloisure of the uidic equations by writing:
γ
P j = kB Tj nj j
(1.5.62)
The divergence of the pressure tensor is:
∇ · P j = γj kB Tj ∇nj
53
(1.5.63)
1 Introduction to plasma physics
The continuity equation is:
∂nj
= −∇r · nj v j
∂t
(1.5.64)
qj
1
∂
v j + v j · ∇v j = −
E + vj × B −
∇ · Pj
∂t
mj
mj nj
(1.5.65)
The equation of motion is:
we can study it with a linear perturbative approach:
nj (r, t) = nj0 + nj1 (r, t)
(1.5.66)
Substituting in the set of uidic equation the adiabatic cloisure and linearizing with the
previous expression of charge density we obtain:
∂nj1
= −nj0 ∇r · v j
∂t
(1.5.67)
∂
qj
1
vj = − E −
[γj kB Tj ∇nj1 ]
∂t
mj
mj nj
(1.5.68)
For an electromagnetic wave in an homogeneous plasma ∇ · E = 0 implies that ∇ · J = 0
and ∇ · v j
to zero.
= 0. The perturbation coecient nj1 must remain constant, and thus equal
The electromagnetic wave doesn't perturb the density of the species; there is
no current density and the particle motion is only governed by the E eld provoking
oscillations on parallels plane, perpendicular to the direction k̂ because the divergence of
the pressure tensor is equal to zero.
1.5.4 Plasma wave in a collisionless warm plasma
Now we want to investigate the plasma waves (longitudinal waves) in a collisionless warm
plasma. The main result is that, unlike the plasma waves in a cold plasma, these longitudinal waves acquire a non zero phase velocity, thus are waves that can propagate through
the plasma.
We have for the electrostatic wave ∇ × E
= 0 thus is an unmagnetized
plasma (B = 0). We refer to a warm plasma of electron with ions forming an immobile
background (ions are at rest). Using the equation (1.5.67), and the facts that ∇ −→ −ik
in the frequency domain we can write:
iωnj1 − inj0 k · v j = 0
nj1 =
nj0
k · vj
ω
(1.5.69)
(1.5.70)
Once we calculate the derivative of the equation (1.5.68) in the frequency domain we
obtain:
(iω)2 v j =
qj
γj kB Tj
iωE + ik
iωnj1
mj
mj nj
54
(1.5.71)
1 Introduction to plasma physics
Substituting (1.5.69) and calculating:
ω 2 v j = −iω
qj
γj kB Tj
E+k
nj0 k · v j
mj
mj nj
(1.5.72)
Without losing generality, let k = kẑ . In the case of plasma wave k × E = 0 the electric
eld and the velocity of the particle, that in this case are only the electrons, are parallel
to the wave vector, hence:
− ik · E = −ikEz =
−ene1
0
(1.5.73)
Calculating the eld:
Ez = −i
ene1
k0
(1.5.74)
Combining the previous equations we can obtain the dispersion relation:
ω2 =
e2 ne0
kB Te
+ γe k 2
me 0
me
(1.5.75)
k Te
e2 ne0
2
2
Note that ωp−e =
and that vt−e = B ; the exponent of the adiabatic expression
me 0
me
DOF +2
is a function of the degrees of freedom (
) of the particles γe =
. In this case
DOF
the electrons can move only by oscillation in the direction of the eld and thus DOF = 1.
DOF
The dispersion relation becomes:
2
2
2
ω 2 = ωp−e
+ 3k 2 vt−e
= ωp−e
1 + 3k 2 λ2D
(1.5.76)
This means that the phase velocity and the group velocity have non zero value:
ω
vφ = = ωp−e
k
r
ωp−e
1 + 3k 2 λ2D
≃
2
k
k
∂ω
3k 2 2
=
λ ω
≃ 3kλ2D ωp−e
∂k
ω D p−e
in the validity of the adiabatic approximation kλD 1.
vg =
(1.5.77)
(1.5.78)
1.5.5 Landau damping
The collisionless warm plasma and the study of electromagnetic oscillations can be also
carried out abandoning the uidic approach by using directly the Vlasov equation. One
great results that we can demonstrate is the presence of another eect that is not provided
by the uidic approach: with the term Landau damping we refer to the eect of damping (i.e. the decrease as a function of time) of longitudinal space charge waves (plasma
oscillation waves). The solution of the Vlasov equation given by Landau starts with the
assumption that the plasma consists in a uniform distribution of ions at rest and a distribution ef electrons; the plasma is supposed collisionless and unmagnetized; the initial
elds are supposed to be zero. The Vlasov equation for the electrons (1.2.16) is:
55
1 Introduction to plasma physics
∂
e
fe + v · ∇ r fe −
[E + v × B] · ∇v fe = 0
∂t
me
(1.5.79)
We consider the perturbation of the plasma wave acting as a deviation from the equilibrium distribution function fj0 :
fj (r, v, t) = fj0 (v) + fj1 (r, v, t)
(1.5.80)
Hence the equation becomes:
∂
e
fe1 + v · ∇r fe1 −
E · ∇v fe0 = 0
∂t
me
(1.5.81)
Let's now analyze the equation in the Fourier's transformed domain:
e
∂
fe1 + v · (ik) fe1 −
E · ∇v fe0 = 0
∂t
me
(1.5.82)
Note that we use the same simbols for the functions in both domains, but in fact the
following relations exist:
ˆ
E (r, t) eik·r dr
F {E (r, t)} = E (k, t) =
(1.5.83)
ˆ
fe1 (r, v, t) eik·r dr
F {fe1 (r, v, t)} = fe1 (k, v, t) =
(1.5.84)
Now let's dene:
v = v k̂ + vtransverse t̂
(1.5.85)
Focusing on the electrostatic waves ∇ × E = 0, or in the frequency domain k × E = 0,
we can write E = E k̂ ; substituting:
∂
e
∂
fe1 + ikvfe1 −
E fe0 = 0
∂t
me ∂v
(1.5.86)
Now for the purpose of studying the time evolution of the system caused by the initial
perturbation, we can usefully consider the equation in the domain of Laplace's transformation:
ˆ
L {E (k, t)} = E (k, s) =
E (k, t) e−st dt
(1.5.87)
ˆ
L {fe1 (k, v, t)} = fe1 (k, v, s) =
Recalling that L
d
dt
fe1 (k, v, t) e−st dt
(1.5.88)
f (t) = sL {f (t)} − f (0+ ), the equation (1.5.86) is transformed
to:
(s + ikv) fe1 −
e
∂
E fe0 = fe1 k, v, t = 0+
me ∂v
56
(1.5.89)
1 Introduction to plasma physics
Now we consider that the only force acting on electrons is the one derived from the
electrostatic eld induced by the perturbation of the distribution function.
If we note
with:
Φ (r − r0 ) =
e
4π0 |r − r0 |
(1.5.90)
the electrostatic potential of a single electron, we can write the total electric eld as follows:
ˆ
E (r, t) =
ˆ
dE = −
∇r Φ (r − r0 ) fe1 (r0 , v, t) dr0 dv =
ˆ
∇r Φ (r − r0 ) fe1 (r0 , v, t) dr0 dv
=−
(1.5.91)
0
where r is the position of a single electron charge. Calculating the divergence of E (r, t):
ˆ
∇·E =−
∇2 Φ (r − r0 ) fe1 (r0 , v, t) dr0 dv
(1.5.92)
we can note that:
∇2 Φ (r − r0 ) =
e
δ (r − r0 )
0
(1.5.93)
because of the Poisson's equation written for a single elementary charge distribution.
Substituting and using the denition of delta function:
ˆ
∇·E =−
e
e
δ (r − r0 ) fe1 (r0 , v, t) dr0 dv = −
0
0
ˆ
fe1 (r, v, t) dv
(1.5.94)
´ We can also obtain the same result recalling that, by denition of density ne1 (r, t) =
fe1 (r, v, t) dv , and that ρ = −ene .
Now we can move into the transformed Fourier domain by writing:
e
ik · E (k, t) = ikE (k, t) = −
0
ˆ
fe1 (k, v, t) dv
(1.5.95)
Laplace transforming:
e
ikE (k, s) = −
0
ˆ
fe1 (k, v, s) dv
(1.5.96)
From equation (1.5.89) we can calculate fe1 :
e
1
∂
+
fe1 =
E fe0 + fe1 k, v, t = 0
(s + ikv) me ∂v
(1.5.97)
Substituting in (1.5.96):
e
ikE (k, s) = −
0
ˆ
1
e
∂
e
E (k, s) fe0 dv −
(s + ikv) me
∂v
0
ˆ
1
fe1 k, v, t = 0+ dv
(s + ikv)
(1.5.98)
57
1 Introduction to plasma physics
After some calculations:
e2
ik 1 − i
me k0
ˆ
ˆ
1
∂fe0
e
fe1 (k, v, t = 0+ )
dv E (k, s) = −
dv
(s + ikv) ∂v
0
(s + ikv)
(1.5.99)
continuing the calculations:
E (k, s) =
h
2
0 1 − i meek0
1
´
e
ii
∂fe0
dv k
1
(s+ikv) ∂v
ˆ
fe1 (k, v, t = 0+ )
dv
(s + ikv)
(1.5.100)
We can interpret the function:
e2
d (k, s) = 1 − i
me k0
ˆ
1
∂fe0
dv = 1 + χe (k, s)
(s + ikv) ∂v
(1.5.101)
as the plasma dielectric function; this function is independent from the initial condition
+
of the perturbation in the istant t = 0 . The electric eld function is given by:
e
E (k, s) = i
k0 d (k, s)
ˆ
fe1 (k, v, t = 0+ )
dv
(s + ikv)
(1.5.102)
The inverse Laplace transform can be performed as follows:
1
E (k, t) =
lim
2πi ω→∞
ˆ σ+iω
E (k, s) est ds
(1.5.103)
σ−iω
The problems of this integrations arise in the values in C that are roots of the plasma
dielectric function (any innity of the numerator which are caused by the perturbation
can be in fact eliminated). This method of integration is termed Browich contour, and
in fact is dened as a path of integration in the complex plane running from σ − i∞ to
σ + i∞, where σ is a real, positive number chosen so that the path lies to the right of all
singularities of the analytic function under consideration. Hence the value of σ must be
greater than the real part of each complex pole. Supposing that E (k, s) is holomorpic in
C except that for its poles (i.e. the zeros of d (k, s)), we can apply the residue theorem on
an integration path γ that contains the poles:
˛
X
E (k, s) est ds = 2πi
I (γ, pj ) Res (E, pj ) epj t
(1.5.104)
γ
j
where
Res (E, pj ) = Rj =
is the residue of
lim [(s − pj )oj E (k, s)]
s→pj
(1.5.105)
E calculated in the j-th pole pj that has an order oj , and I is the
winding number of the integration path γ around the pole pj . The integral can be written
as the dierence of (1.5.104) and the integral made on the portion of the path γ (in gure
(1.5.1)) which does not include the straight line s = σ :
58
1 Introduction to plasma physics
Figure 1.5.1: Contour γ of integration for the Laplace inverse transform using Bromwich
contour.
The Bromwich integration path is the straight in the complex
planes = σ draw with solid line. The dashed line is the γ\s = σ integration
path.
ˆ
1
E (k, s) e ds −
lim
E (k, s) est ds =
ω→∞
2πi
γ
γ\s=σ
ˆ
X
1
p t
lim
E (k, s) est ds
=
Res (E, pj ) e j −
ω→∞ γ\s=σ
2πi
j
1
lim
E (k, t) =
2πi ω→∞
˛
st
(1.5.106)
The second term gives no contribution to the overall integral because the integrand
function must tend to zero in the limit of ω → ∞.
The poles are the set of values in the complex plane that satises {pj } ≡ {s ∈ C | d (k, s) = 0}
:
χe (k, s) = −1
(1.5.107)
or explicitly:
e2
i
me k0
ˆ
1
∂fe0
dv = 1
(s + ikv) ∂v
(1.5.108)
The integration can be carried out rst along the two transverse direction, u e w , with
v = v k̂ + uû + wŵ. The roots of (1.5.108) can be expressed in the form pj = −γj − iωj .
The electric eld evolution is:
E (k, t) =
X
Rj e−γj −iωj t
(1.5.109)
j
γj > 0 for each possible j
then the wave has dumped oscillation, otherwise if one of the possible γj < 0 the wave
The parameter
γj governs the behavior of the wave:
59
if
1 Introduction to plasma physics
increase its amplitudes and this case leads to instabilities.
The Landau solution shows
clearly that we can evaluate the behavior of the system knowing the initial perturbation
(its expression is carried by each of the residue Rj ) and the poles of the system:
the
uidic model cannot predict these kind of results. Equation (1.5.108) has simple solution
in two dierent cases; rst we can consider the low frequency limit k
´ +∞ 1 d
´ 1 ∂fe0
dv = −∞ (s+ikv)
F (v) dv and integrating by parts:
(s+ikv) ∂v
dv e0
e2
i
me k0
e2
−i
me k0
ˆ +∞
−∞
→ 0.
Dening
+∞
1
d
e2
1
Fe0 (v) dv = i
Fe0 (v)
+
(s + ikv) dv
me k0 (s + ikv)
−∞
ˆ +∞
−ik
e2
Fe0 (v)
dv
=
−
me 0
(s + ikv)2
−∞
ˆ +∞
Fe0 (v)
−∞
1
dv = 1
(s + ikv)2
(1.5.110)
Note that if we perform the limit k → 0 the integral of Fe0 (v) is the simple electron
density ne , so we obtain a well known result of a simple plasma cosine oscillation:
p1,2 = ±iωp−e
(1.5.111)
R1 = lim [(s − p1 ) E (k, s)]
(1.5.112)
R2 = lim [(s − p2 ) E (k, s)]
(1.5.113)
The residue are:
s→p1
s→p2
The evolution is:
E (k, t) =
2
X
Rj e−iωj t = R1 eiωp−e t + R2 e−iωp−e t
(1.5.114)
j=1
A dierent solution can be shown if we expand in power series the denominator of
(1.5.110):
+∞
1
1
1
1 X k n dn
=
=
s2 1 + ikv 2
s2 n=0 n! dk n
(s + ikv)2
s
"
#
1
1 + ikv
s
(1.5.115)
2
k=0
Calculating:
dn
dk n
"
#
1
1 + ikv
s
2
iv
= (n + 1)! −
s
n "
#
1
1 + ikv
s
n+2
k=0
iv
= (n + 1)! −
s
n
(1.5.116)
Substituting:
"
#
n
2
1
ikv
1
ikv
kv
(n + 1) −
= 2 1−2
−3
+ ···
2 = 2
s n=0
s
s
s
s
(s + ikv)
+∞
1 X
60
(1.5.117)
1 Introduction to plasma physics
The other reasonable assumption for carrying out this integral calculation is assuming
that the initial distribution for Fe0 is a maxwellian distribution.
even function of
powers of
The maxwellian is an
v : this leads to a non zero contribution to the integral only for the odd
v in the series expansion. Thus approximating to the second order we obtain
the equation:
ˆ +∞
2 #
kv
e 2 ne 1
e2 ne k 2 kB Te
Fe0 (v) 1 − 3
dv = −
=
+
3
s
me 0 s2
me 0 s4 me
−∞
2
ω2
ωp−e
2 2 p−e
(1.5.118)
= − 2 1 − 3k λD 2 = 1
s
s
e2 1
−
me 0 s2
Hence for small
"
k:
2
1 + 3k 2 λ2D
s2 ≃ −ωp−e
(1.5.119)
The nal result is:
p1,2 = ±iωp−e 1 + 3k 2 λ2D
1/2
(1.5.120)
with o1,2 = 2. This is the case of with the termal eect taken into account. The residues
are:
R1 = lim (s − p1 )2 E (k, s)
(1.5.121)
R2 = lim (s − p2 )2 E (k, s)
(1.5.122)
s→p1
s→p2
and the evolution of electric eld is:
1/2
E (k, t) = R1 eiωp−e (1+3k λD )
2 2
t
1/2
+ R2 e−iωp−e (1+3k λD )
2 2
t
(1.5.123)
Without any approximation is clearly seen from equation (1.5.108) or (1.5.110) that the
s
integrand function has a pole in v = i . This pole is on the path of integration and its
k
contribution can be taken into account with the extension of the integral in the complex
plane of
v as shown in gure (1.5.2).
Coming back to (1.5.108), now we can more easily understood his structure; separating
the electron density ne from the distribution function but mantaining the same simbol for
Fe0
ˆ
i e2 ne +∞
1
dFe0
χe (k, s) = −
dv =
k me 0 −∞ (s + ikv) dv
ˆ +∞
ˆ
i 2
1
dFe0
1
dFe0
= − ωp−e p.v.
dv + lim
dv =
ε→0 C (s + ikv) dv
k
−∞ (s + ikv) dv
ε
2
ωp−e
= 2 p.v.
s
ˆ +∞
1
i 2
Fe0 (v)
dv − ωp−e
lim
2
ε→0
k
−∞
1 + ikv
s
61
ˆ
1
dFe0
dv
Cε (s + ikv) dv
(1.5.124)
1 Introduction to plasma physics
Figure 1.5.2: Integration path for the velocity with extension in the complex plane to avoid
the singularity.
where the principal value of the integral can be calculated as in the previous approximation (where we didn't care about the pole) and the second term is given by the residue
of the integrand in the pole:
ˆ
s
dFe0
1
dv = 2πiI Cε , i
Res
lim
ε→0 C (s + ikv) dv
k
ε
1
s
dFe0
,v = i
(s + ikv) dv
k
(1.5.125)
the winding number of a half circle is of course 1/2. The residue is:
Res
1
s
dFe0
,v = i
(s + ikv) dv
k
= lims
v→i k
s
1
dFe0
1 dFe0
v−i
=
k (s + ikv) dv
−ik dv v=i s
k
(1.5.126)
Putting all toghether into (1.5.124):
2
2
ωp−e
ω2
ωp−e
dFe0
2 2 p−e
χe (k, s) = 2 1 − 3k λD 2 − iπ 2
s
s
k
dv v=i s
(1.5.127)
k
The poles have both real and imaginary part pj = −γj − iωj , and we can approximate:
χe (k, pj ) ≃ χe (k, −iωj ) + iγj
∂
χe (k, −iωj ) = −1
∂s
(1.5.128)
if |γj | ωj . Hence:
∂
1 + < {χe (k, −iωj )} + < iγj χe (k, −iωj ) = 0
∂s
∂
= {χe (k, −iωj )} + = iγj χe (k, −iωj ) = 0
∂s
(1.5.129)
(1.5.130)
Now examining equation (1.5.127) we see that rst term in the expression of χe evaluated
for s = iωj is a pure real function, while the second term is purely imaginary:
62
1 Introduction to plasma physics
χe (k, iωj ) =
"
2
ωp−e
(−iωj )2
1 − 3k 2 λ2D
#
2
ωp−e
(−iωj )2
2
ωp−e
dFe0
− iπ 2
k
dv v=i −iωj
(1.5.131)
k
We can reduce the previous two equation to the simple:
2
ω2
ωp−e
2 2 p−e
1 − 2 1 + 3k λD 2 = 0
ωj
ωj
2 2
ωp−e
ω2
dFe0
∂ ωp−e
2 2 p−e
−π 2
1 − 3k λD 2
+
+ γj = i
k
dv v= ωj
∂s s2
s
k
2
ωp−e
dFe0
−iπ 2
=0
ω
k
dv
v= j ,s=−iωj
(1.5.132)
(1.5.133)
k
We have already seen that the rst equation (1.5.132) has the approximate solution
(1.5.120):
ω1,2 = ±ωp−e 1 + 3k 2 λ2D
1/2
(1.5.134)
Further calculations for the second equation (1.5.133) give:
γ=−
3
dFe0
π ωp−e
2
2 ne k dv v= ω1,2
(1.5.135)
k
that is the Landau dumping coecient. This result, althought stems out from an approximated linearized analysis, is very remarkable because the eects of Landau damping
were also demonstrated experimentally and these results are in good agreement with theory [Landau 1946]; by the way, we can know the behavior of the system just knowing
the initial equilibrium distribution function. The pole has a positive real part only if the
derivative of the distribution function for the velocities is positive. In the special case of
a maxwellian equilibrium distribution function, we can write:
3
d
π ωp−e
γj = −
2 k 2 dv
r
=
r
=
3
π ωp−e
8 k3
m
kB T
32
3
π ωp−e
8 k3
(
m
2πkB T
m
kB T
32
21
mv 2
exp −
2kB T
)
2
ω1,2
m
ω1,2 exp − 2
k 2kB T
1/2
ωp−e 1 + 3k 2 λ2D
exp
=
v=
ω1,2
k
=
2
ωp−e
(1 + 3k 2 λ2D ) m
−
k2
2kB T
(1.5.136)
Thus calculating in the limit kλD 1 we obtain:
r
γj =
π ωp−e
1
3
exp − 2 2 −
8 λ3D k 3
2λD k
2
63
(1.5.137)
1 Introduction to plasma physics
The landau damping mechanism can be also understood intuitively introducing a typical
non-linear phenomena basic concept: the energy exchange between waves and particles can
happens when are met proper resonance conditions between phase velocity and particle
velocity. When a plasma oscillation occurs particles with velocity near the phase velocity
vf are forced to oscillate toghether with the same phase: the particle with v sligthly less
than vf are accelerated by the perturbation (substracting energy to it), and vice versa, the
other particles with
v more than vf will slow down losing energy. Therefore if the particles
with velocity v < vf are more numerous than the particles with v > vf (that is the case
dFe0
< 0) the wave globally will lose his energy resulting in a damped oscillation, and
of
dv
vice versa.
1.5.6 Dielectric response of a collisional plasma
In the previous paragraphs we have considered collisionless plasmas in which the only
kind of absorption present is the Landau damping of the electrostatic plasma wave. On
the contrary, in a dissipative plasma, the collisions can transfer eciently to the particles
the electromagnetic/electrostatic-waves energy.
Sometimes this collisional absorption is
bremsstrahlung (from german: braking radiation, deceleration radiation) but in
fact this is the exact inverse process of the typical bremsstrahlung of an electron moving
termed
and changing its path under the inuence of an ion. It can be seen also that in the limit of
a collisionless plasma (i.e. the limit where the collision frequency tends to zero) and under
certain conditions of incidence and inhomogeneity of the plasma, a resonant absorption
can occur.
If we consider a collisional plasma, and the case of electromagnetic wave, we can write
the Vlasov equation for electrons with the approximation of the collisional frequency, and
neglecting the thermal eect (i.e. P = 0) :
e
∂
v e + v e · ∇v e = −
(E + v e × B) − νei (v e − v e0 )
∂t
me
(1.5.138)
where the electron-ion collision frequency can be written in the case of coulombian
collision in several forms:
1/2
v2
1 2
1
Zne e4 ln Λ
=
νei (vt−e ) =
νei = νei (v) 2
3vt−e
3 π
3 (2π)3/2 20 me1/2 (kB Te )3/2
(1.5.139)
where νei (vt−e ) is only a denition of collision rate calculated for the thermal velocity.
[Antici PhD 2007]. Linearizing (1.5.138), in the rest frame of reference of the electron, we
obtain:
∂
e
v e = − E − νei v e
∂t
me
(1.5.140)
In the frequency domain, looking for an oscillating solution like the one in equations
(1.5.17) (1.5.18), we have:
ve = −
e
me (νei + jω)
64
E
(1.5.141)
1 Introduction to plasma physics
Recall that J = −ene v e :
J=
e 2 ne
E
me (νei + jω)
(1.5.142)
Thus using the well known denition of plasma frequency, the complex conductance is:
2
0 ωp−e
σe =
(νei + jω)
(1.5.143)
and the complex dielectric function is:
#
"
2
ωp−e
σ
c = 0 − j = 0 1 − 2
ω
ω 1 − j νωei
(1.5.144)
The dispersion relation (1.5.35) becomes:

ω
ν
1 − i ei 
k 2 = 2 1 −
2
ν
c
ω
ω 2 1 + ei
2

2
ωp−e
(1.5.145)
ω2
The wave vector in this case has a real and imaginary part k = β + iα with
α = = {k}
(1.5.146)
β = < {k}
(1.5.147)
both real vectors. Restricting the analysis to the electromagnetic waves we have ∇·E = 0
and thus k · E = 0, and therefore α̂ ≡ β̂ ≡ k̂ .

ν
ω
1 − i ei 
k 2 = k · k = β + iα · β + iα = β 2 − α2 + 2iαβ = 2 1 −
2
ν
c
ω
ω 2 1 + ei
2

2
ωp−e
ω2
(1.5.148)
Splitting it into its real and imaginary part:
"
#
2
2
ω
ω
p−e
β 2 − α2 = 2 1 − 2
ν
c
ω 1 + ω2 2ei
2αβ =
2
ωp−e
c2
νei
ν2 ei 1 + ω2 ω
(1.5.149)
(1.5.150)
The typical approximation is α β because in the plasma the collision frequency is
generally much smaller than the electron plasma frequency νei
ωp−e , (this is a valid
approximation only if the electron density doesn't approach the critical density):
ω
β≃
c
2 1/2
ωp−e
1− 2
ω
65
(1.5.151)
1 Introduction to plasma physics
2
1 νei ωp−e
α≃
2 c ω2
2 −1/2
ωp−e
1− 2
ω
(1.5.152)
In a simplied problem with k̂ = x̂, the electric eld assumes this behavior:
E (r, t) = < {E k exp [i (k · r − ωt)]} = E cos (βx − ωt) exp (−αx)
(1.5.153)
We can refer to the absorption length as the distance where the wave intensity (in fact
the wave intensity is proprotional to the squared electric eld) is attenuated of a factor
e
1/ ; it corresponds to the doubled inverse of the absorption coecient:
c ω2
Labs =
2
νei ωp−e
2 1/2
ωp−e
1− 2
ω
(1.5.154)
1.5.7 Propagation in inhomogeneous plasma: WKB
approximation
The wave equation in a inhomogeneous plasma is still valid, but we must take into account
of the inhomogeneity considering a dielectric constant that is variable with r :
k 2 = ω 2 µ0 c (r, ω)
(1.5.155)
We can distinguish two cases: plasma with or without collisions. In the case of collisionless plasmas the parameter of interest is only the complex dielectric function
2
(r)
ωp−e
σ (r)
c (ω) = 0 − j
= 0 1 −
ω
ω2
(1.5.156)
In the case of collisional plasmas we can take into account the collisions using the
electron-ion collision rate inside the expression of the complex dielectric function. It is the
same of (1.5.144) except for the spatial dependance:


2
ωp−e
(r)

c (r) = 0 1 −
νei (r)
2
ω 1−j ω
If we consider a simple plane wave propagating in the
(1.5.157)
z direction, the wave vector is
k = kẑ with wave number:
p
ωp
√
k = k (r) = ω µ0 0 ε (r) =
ε (r)
(1.5.158)
c
p
The dimensionless quantity
ε (r) is dened as to be the refractive index N of the
medium. Also a local wavelenght can be dened:
λ (r) =
We can rewrite the wave equation as follows:
66
2π
k (r)
(1.5.159)
1 Introduction to plasma physics
d2 E
+ k 2 (r) E = 0
dz 2
(1.5.160)
This problem can be studied separately for TEM, TE or TM electromagnetic waves.
WKB approximation for normal incident, s-polarized, electromagnetic wave on a
collisionless inhomogeneous plasma
An s-polarized electromagnetic wave is a TE (transverse electric eld) wave, in which
the oscillation direction of electric eld is perpendicular to the so called
incidence plane.
The incidence plane is the plane described by the wave propagation direction, and the
perpendicular direction of a target plane. Hence if we arbitrarily consider z axis as the
propagation direction, and if we consider normal incidence, we can assume the reference
in gure:
It means that the incidence plane is set to be parallel to the y-z plane, hence the spolarized TE electromagnetic wave has those components:
E = Ex x̂
(1.5.161)
B = By ŷ + Bz ẑ
(1.5.162)
For that wave moving toward a plasma with ne = ne (z), slowly varying, the equation
(1.5.15) projected along
x direction becomes:
d2
Ex + k 2 (z) Ex = 0
dz 2
(1.5.163)
and, of course, its solution is not so simple as the solution (1.5.33) because of the spatial
dependance of the wave number. Now we suppose that we can consider valid the following
inequality:
ε (z)
dε (z)
dz
λ (z)
(1.5.164)
that coincides with the assumption:
d
λ (z) 1
dz
(1.5.165)
If the medium was homogene, the product of the refractive index and the position
z
could give the information about the spatial dephasing between a reference position z0 ,
and the observation point
z and we could simply write this spatial phase as:
ˆ z
ˆ
ω z
ω
ϕ (z) =
k (z ) dz =
N (z 0 ) dz 0 = l
(1.5.166)
c z0
c
z0
´z
where the quantity l =
N (z 0 ) dz 0 = N (z − z0 ) is called optical path of the ray.
0
0
0
Similarly for the case of inhomogeneous medium we can write the solution in the form:
67
1 Introduction to plasma physics
ˆ z
0
Ex (z) = Ex0 exp [iωt] exp ±i
k (z ) dz
0
z0
ˆ
ω zp
0
= Ex0 exp [iωt] exp ±i
ε (z 0 )dz
c z0
(1.5.167)
The magnetic eld can be calculated throughout equation (1.5.16);
1
1
B =− ∇×E =−
iω
iω
d
d
d
1
d
d
x̂ + ŷ + ẑ
× Ex x̂ = −
−ẑ Ex + ŷ Ex =
dx
dy
dz
iω
dy
dz
ˆ
ω zp
0
0
exp ±i
ε (z )dz
=
c z0
ˆ
Ex0 p
ω zp
0
0
= ∓ŷ
ε (z) exp [iωt] exp ±i
ε (z )dz
c
c z0
d
Ex0
exp [iωt]
= −ŷ
iω
dz
(1.5.168)
The Poyinting's vector is:
ˆ
1
1
ω zp
0
0
P = E×H =
ε (z )dz x̂ ×
Ex0 exp [iωt] exp ±i
2
2
c z0
ˆ
ω zp
Ex0 p
0
ε (z) exp [iωt] exp ±i
ε (z 0 )dz
=
× ∓ŷ
c
c z0
ˆ
2 r
Ex0
0
ω zp
0
=∓
exp [iωt] exp ±2i
ε (z 0 )dz ẑ
2
µ0
c z0
The ux of the Poyinting vector through a surface z = z
∗
(1.5.169)
is clearly not constant, and
this means that the solution (1.5.167) is not valid if we suppose a non collisional plasma.
Otherwise if we look for a solution in the form:
i
h ω
Ex (z, t) = Ex0 exp [iωt] exp i f (z)
c
(1.5.170)
we can calculate the solution in a perturbative manner:
f (z) =
+∞ X
c j
j=0
ω
fj (z)
(1.5.171)
The dierential equation becomes:
h ω
io
h ω
i
d2 n
2
Ex0 exp i f (z) + k (z) Ex0 exp i f (z) = 0
dz 2
c
c
(1.5.172)
calculating the derivatives:
h ω
io
h ω
i df (z)
d n
ω
Ex0 exp i f (z) = i Ex0 exp i f (z)
dz
c
c
c
dz
h ω
io
h ω
i df (z) d2 n
ω
d
Ex0 exp i f (z) = i Ex0
exp i f (z)
=
dz 2
c
c
dz
c
dz
68
(1.5.173)
(1.5.174)
1 Introduction to plasma physics
h ω
i d2 f (z) ω 2
h ω
i df (z) 2
ω
= i Ex0 exp i f (z)
Ex0 exp i f (z)
+ i
c
c
dz 2
c
c
dz
(1.5.175)
Hence substituting this expression, (1.5.172) reduces to:
ω d2 f (z) ω 2
i
+ i
c dz 2
c
df (z)
dz
df (z)
dz
+ k 2 (z) = 0
(1.5.176)
ω
N (z):
c
Going on with substitution of (1.5.158) k (z) =
c d2 f (z)
i
−
ω dz 2
2
2
= −N 2 (z)
(1.5.177)
The perturbated dierential equation is:
" +∞
#!2
d X c j
fj (z)
=
dz j=0 ω
" +∞
#
c d2 X c j
i
fj (z) −
ω dz 2 j=0 ω
+∞
c X c j d2 fj (z)
−
=i
ω j=0 ω
dz 2
Using the property of series
i
+∞ X
c j+1 d2 fj (z)
j=0
ω
dz 2
P
−
+∞ X
c j dfj (z)
+∞
j=0 αj
ω
j=0
2
=
!2
= −N 2 (z)
dz
P+∞ P+∞
j=0
j=0 h=0
h=0 (αh αj ) :
+∞ X
+∞ X
c h+j dfh (z) dfj (z)
ω
(1.5.178)
dz
dz
= −N 2 (z)
(1.5.179)
To the lowest order, we obtain:
df0 (z)
dz
2
= N 2 (z)
(1.5.180)
that gives:
ˆ z
N (z 0 ) dz 0
f0 (z) = ±
(1.5.181)
z0
To the rst order (N (z) has no variation with the perturbation parameter
i
c
):
ω
c d2 f0 (z)
c df0 (z) df1 (z)
−2
=0
2
ω dz
ω dz
dz
(1.5.182)
df1 (z)
i 1 d
=
N (z)
dz
2 N (z) dz
(1.5.183)
Therefore:
Integrating:
69
1 Introduction to plasma physics
i
ln N (z)
2
f1 (z) =
(1.5.184)
Hence the approximation for the solution is:
ˆ z
ci
0
0
N (z ) dz +
±
ln N (z)
=
ω2
z0
ˆ
ω z
Ex0 exp [iωt]
0
0
(1.5.185)
exp ±i
N (z ) dz
=
N 1/2 (z)
c z0
ω
Ex (z, t) ≃ Ex0 exp [iωt] exp i
c
This wave will continue propagating if and only if:
ˆ z
N (z 0 ) dz 0 ∈ R
l=
(1.5.186)
z0
This means that the dielectric function ε (z) in (1.5.156) must be positive:
2
ωp−e
(z)
ε (z, ω) = 1 −
>0
ω2
(1.5.187)
recalling the denitions of plasma frequency (1.1.28) and critical density (1.5.60) we
have:
ne (z) < nc (ω)
(1.5.188)
A reection of the electromagnetic wave occurs on the critical surface, that is the surface where the electron plasma density equals the critical density. The so called WentzelKramer-Brillouin (WKB) approximation consist in neglecting a particular term that appears if we perform the verify of the solution substituting it in the wave equation. Writing
Ex (z, t) in the form:
Ex (z, t) = Ex0 exp [iωt] g (z) exp [h (z)]
(1.5.189)
with:
1
g (z) = (N (z))− 2
ω
h (z) = ±i
c
(1.5.190)
ˆ z
N (z 0 ) dz 0
(1.5.191)
z0
with the reciprocal relation:
h0 (z) = i
ω 1
c g 2 (z)
(1.5.192)
Verifying the solution:
d2
(g (z) exp [h (z)]) + k 2 (z) g (z) exp [h (z)] = 0
dz 2
First term:
70
(1.5.193)
1 Introduction to plasma physics
d2
d
[exp [h (z)] (h0 (z) g (z) + g 0 (z))] =
(g (z) exp [h (z)]) =
2
dz
dz
d 0
0
0
0
0
(h (z) g (z) + g (z)) + h (z) (h (z) g (z) + g (z)) =
= exp [h (z)]
dz
n
o
2
= exp [h (z)] g 00 (z) + 2g 0 (z) h0 (z) + g (z) h00 (z) + (h0 (z)) g (z)
(1.5.194)
2
g 00 (z) + 2g 0 (z) h0 (z) + g (z) h00 (z) + (h0 (z)) g (z) +
h00 (z) = −2i
ω 2
c
1
=0
(g (z))2
ω g 0 (z)
c g 3 (z)
(1.5.195)
(1.5.196)
The dierential equation becomes:
2
ω 2 1
ω 1
ω g 0 (z)
ω 1
g (z) = 0
g (z)+2g (z) i 2
+g (z) −2i 3
+ i 2
g (z)+
c g (z)
c g (z)
c g (z)
c g 2 (z)
00
0
(1.5.197)
Thus calculating:
g 00 (z) + 2i
ω 2 1
ω g 0 (z)
ω g 0 (z) ω 2 1
+
−2i
−
+
=0
c g 2 (z)
c g 2 (z)
c g 3 (z)
c g 3 (z)
This is true only if we neglect the term g
00
(1.5.198)
(z).
WKB approximation for oblique incident, s-polarized, electromagnetic wave on a
collisionless inhomogeneous plasma
The case of oblique incidence is a simple variation of the normal incidence: the propagation
direction is always given by the wave vector k , but now it has two components:
k = ky ŷ + kz ẑ
(1.5.199)
that can be calculated with ky = k · ŷ and kz = k · ẑ . Hence dening θ the angle between
k̂ and the z axis we can write:
ky = k sin θ
(1.5.200)
kz = k cos θ
(1.5.201)
The wave equation now reads:
∂2
∂2
+
∂y 2 ∂z 2
E + k 2 (r) E = 0
The electric eld can be written:
71
(1.5.202)
1 Introduction to plasma physics
E (y, z, t) = E x exp [i (ωt − k · r)] = E x exp [iωt] exp [−i (ky y + kz z)] =
= Ex (z) x̂ exp [iωt] exp [−ik0 sin θy]
(1.5.203)
(1.5.204)
Substitution into the wave equation gives:
∂2
∂2
2
+
+ k Ex (z) exp [−ik sin θy] = 0
∂y 2 ∂z 2
(1.5.205)
d2
Ex (z) + k 2 Ex (z) + (−ik0 sin θ)2 Ex (z) = 0
2
dz
(1.5.206)
ω2
d2
2
E
(z)
+
ε
(z,
ω)
−
sin
θ
Ex (z) = 0
x
dz 2
c2
(1.5.207)
thus calculating:
and:
that can be solved in the same way as we did in the previous paragraph.
The only
dierence is that now the propagation into the plasma can be possible only if:
ε (z, ω) − sin2 θ > 0
(1.5.208)
Recalling the equation (1.5.156) we have propagation only if:
ne (z) < nc (ω) 1 − sin2 θ
The electron density increase with
(1.5.209)
z because of the plasma gradient, then the previous
inequality states that the electromagnetic wave with oblique incidence is reected earlier
than in the case of normal incidence.
72
2 Description of the laser facility
2.1 The 200 TW laser beam line at INRS
The ALLS 200 TW laser system is a very compact laser system based on Ti:Sapphire
technology and the chirp pulse amplication (CPA) technique. It is located at INRS-EMT
(Institut National de la Recherche Scientique Université du Québec - centre Énergie,
Matériaux et Télécommunications).
The system consists in a 10-Hz CPA multi-stage
amplier that can deliver 6 J of energy in 30 fs pulses with a contrast ratio better than
1010 : 1. The intensity reached on target is thus around 1021 W/cm2 .
Three key technologies are include in this laser system:
ˆ a seed pulse contrast cleaning
ˆ a regenerative pulse shaping via an acousto-optical lter
ˆ a cryogenic cooling for large surface amplier.
The seed pulse delivered by the front-end is a large spectrum chirped pulse where the ASE
(amplied spontaneous emission) noise is highly ltered. During the amplication process
inside the regenerative cavity the gain narrowing is compensated by an acousto-optic
programmable gain lter (AOPGC). This enables the pulse to maintain a large spectrum
throughout the rest of the laser chain. The pulse energy obtained at the output of the
pre-amplication stage is about 700 mJ.
The last amplier stage is pumped by 16 Nd:YAG lasers delivering a total pump energy
of 16 J. The Ti:Sapphire crystal is maintained at 130° K in order to reduce the accumulation
of heat from the high average power pumping. The output gaussian beam has an energy
2
about 6 J distributed on a diameter of 120 mm (at 1/e ). Due to the large beam size
and large spectrum, the nal optic compressor uses 4 gratings that are placed in a mosaic
conguration. All components of the chamber are specially tted for vacuum and are kept
−8
under an ultra-high vacuum (10
Torr) to prevent any surface contamination that can
cause degradation of the devices.
The prepulse attenuation is a major concern in short-pulse, laser-solid experiments. The
temporal intensity contrast of the pulses is characterized by three components:
ˆ a 12 ns prepulse produced by leakage in the regenerative amplier,
ˆ an amplied spontaneous emission ASE,
ˆ a pedestal produced by the third and fourth-order aberrations in the stretcher.
The most signicant prepulse comes from the leakage pulse from the regenerative amplier.
9
Using two Pockels cells and polarizers, a contrast of Ileakage /Imain ≈ 10 is obtained.
[reference]
73
2 Description of the laser facility
2.2 Laser chain devices
The increase of the on-target laser intensity of Ti:Sapphire based systems is necessary to
reach the expected energy required for achieving the modern regimes of interaction such
TSNA or the expected RPA. We will describe these mechanisms of laser-acceleration of
ion beams in chapter (3). Attaining a very high intensity on the target surface it requires a
2
small focal spot. The peak focal spot intensity
is approximately proportional to E/w τL ,
I
where
E is the energy contained in the focal spot, w is the beam diameter at the focal
plane and τL is the laser pulse duration. Attaining high intensity by increasing laser energy
is costly and induces a large thermal heating eect (thermal lensing) in the amplication
crystals leading to deformation of the laser beam wavefront. It also implies the use of a
large beam due to the limited damage threshold of the grating optics in the compressor
system and large amplication Ti:Sapphire crystals which are costly and dicult to manufacture without defects (i.e. spatial variation in doping). Multiple passes through such
Ti:Sapphire crystals degrade the overall wavefront quality, limiting the maximum energy
that can be injected into the laser system to prevent from damaging optics and degrading
the focusing capacity of the beam. In the 200 TW beam line a deformable mirror is used
to perform a wavefront correction in order to reduce the wavefront degradation eects
caused by all the optics and devices of the laser chain.
Enhancement in the repetition rate and focusability is limited by thermal load in the
ampliers. Even on a single shot, thermal load induces wave front distortions and hence
reduces the beam focusability. If one tries to increase the repetition rate above the dissipation time of the thermal load, cumulative thermal load appears, the focusability worsens,
and the performance of the laser further degrades.
Summarizing, the ALLS 200 TW beam line consists in:
ˆ oscillator: Ti:Sapphire crystal used at 800 nm wavelength;
ˆ booster: it incorporates a solid state saturable absorber to increase the laser pulse
contrast;
ˆ stretcher: it's a device based on gratings where the laser pulse duration is increased
up to 350 ps;
ˆ regenerative amplier and two multipass ampliers: they provide a preamplication
increasing the energy of the stretched beam up to 700 mJ;
ˆ power amplication stage: the main beam it's pumped with 16 Nd:YAG Pro-pulse
pump lasers (aproximately 16 J total pumping energy) to amplify the laser pulse
increasing its energy up to 6 J;
ˆ vacuum compressor: is made with gratings and is used to compress the laser pulse
duration down to 30 fs;
ˆ deformable mirror: it provides a good wavefront correction.
The gure (2.2.1) explains the layout of the laser chain of 200 TW laser system.
74
2 Description of the laser facility
Figure 2.2.1: Laser chain schematic.
The laser oscillator are devices where the coherent laser light pulses are generated. In
the case of the 200TW system, a Ti:sapphire crystal is used as oscillator.
+
Titanium-doped sapphire (Ti3 :sapphire) is a widely used transition-metal-doped gain
medium for tunable lasers and femtosecond solid-state lasers. The Ti:sapphire crystals are
commonly chosen for several reasons:
ˆ Sapphire (monocrystalline Al2 O3 ) has an excellent thermal conductivity and thus
this feature can alleviate thermal eects even for high laser powers and intensities
ˆ The Ti3+ ion has a very large gain bandwidth, allowing the generation of very short
pulses and also wide wavelength tunability.
Other important oscillators are necessary in the beam line to provide the correct pumping
to the optical amplication stage: in the case of 200 TW ALLS system a set of 16 Nd:YAG
is used. Also Nd:YAG (neodymium-doped yttrium aluminium garnet; Nd : Y3 Al5 O12 ) is
a crystal that is used as a lasing medium for solid-state lasers.
The oscillator of 200 TW ALLS produces an initial 18 fs seed pulse with a 100 nm
spectrum bandwidth around 800 nm. Two acousto-optic programmable dispersive lters
75
2 Description of the laser facility
(AOPDF) (not shown in gure (2.2.1)) are used to control the gain narrowing that occurs in
the amplication stage and also control the spectral phase prole. The rst is placed before
the stretcher to control the spectral phase while the second is located in the regenerative
cavity (in the power amplier stage) to control the spectral amplitude of the pulse. A 55
nm bandwidth full width half maximum (FWHM) is achieved after compression with a
pulse duration similar to the seed pulse from the oscillator.
The seed pulse is therefore amplied using a CPA, and after the compression a bimorph
deformable mirror is used as a wavefront corrector to control optical aberration and optimize laser focusing on target. This mirror has a 120 mm clear aperture and 100 mm active
aperture. It is composed of 48 piezoelectric-ceramics electrodes controlled by high voltages
[Formaux 2008]. An example of deformable mirror is shown in gure (2.2.2) (taken from
[Antici PhD 2007]).
Figure 2.2.2: Example of deformable mirror.
Left:
partition of the surface of the de-
formable mirror into dierent areas, each controlled by a piezoelectric actuator.
Right: working principle of the deformable mirror; the wavefront
analyzer samples the wave and corrects via feedback loop the laser light
wavefront controlling the voltages on the electrodes of the piezo actuators.
2.2.1 Chirped-Pulse Amplifying technique
In ampliers for ultrashort optical pulses, the optical peak intensities that occur can become very high, so that detrimental nonlinear pulse distortion or even destruction of the
gain medium or of some other optical element may occur.
This can be eectively pre-
vented by employing the method of chirped-pulse amplication (CPA), which was originally developed in the context of radar technology, but later applied to optical ampliers
76
2 Description of the laser facility
[Strickland 1985].
Figure 2.2.3: CPA technique schematic (reprinted from Wikimedia commons).
Before passing through the amplier medium, the pulses are chirped and temporally
stretched to a much longer duration by means of a strongly dispersive element (the
stretcher, e.g.
a grating pair).
This reduces the peak power to a level where the non
linear eects in the gain medium are avoided. After the gain medium, a dispersive compressor is used, i.e., an element with opposite dispersion (typically a grating pair), which
removes the chirp and temporally compresses the pulses to a duration similar to the input
pulse duration. As the peak power becomes very high at the compressor, the beam diameter on the compressor grating has to be large. For the most powerful devices, a beam
diameter of the order of 1 m is required. In gure (2.2.3) is delineated the scheme of a
CPA based laser beam amplier.
The most common stretcher and compressor used in a CP amplier are devices that
spatially disperse the dierent wavelenght/frequency components of a laser light using the
so called
diraction gratings.
77
2 Description of the laser facility
A diraction grating is a optical dispersive component with a periodic structure, which
splits and diracts light into several beams travelling in dierent directions. The direction
of the diraction depends on the spacing between grooves, the shape of the grooves, and
the wavelength of the incident light; referring to gure (2.2.1) we can write the
equation :
d (sin ϑm − sin ϑi ) = mλ
gratings
(2.2.1)
where d is the spacing of the grooves, ϑi is the incident angle over the grating, λ the
incident light wavelength,
m is an relative integer that represents the diraction order, ϑm
m -th diraction order.
is the angle of the diracted ray corresponding to the
Figure 2.2.4: Diraction from a grating.
Rearranging (2.2.1) to calculate the several possible ϑm :
λ
ϑm = arcsin sin ϑi + m
d
(2.2.2)
The zero order that corresponds to m = 0 is the simple reection that of course occurs
for any given wavelength at an angle ϑ0 = ϑi . The rst order diraction is the most intense
λ
and occurs at ϑ1 = arcsin sin ϑi +
. The period of the pattern must be comparable with
d
the wavelength.
Stretchers and compressors are characterized by their dispersion. With negative dispersion, light with higher frequencies (shorter wavelengths) takes less time to travel through
the device than light with lower frequencies (longer wavelengths).
In a CPA, the dis-
persions of the stretcher and compressor should cancel out as depicted in gure (2.2.5).
Because of practical considerations, the stretcher is usually designed with positive dispersion and the compressor with negative dispersion.
78
2 Description of the laser facility
Figure 2.2.5: Diraction gratings based stretcher and compressor (reprinted from Wikimedia commons).
At the top of gure (2.2.5) there is a positive dispersion stretcher, because the distance
from the lenses to the gratings is choosen smaller than the focal length. Also the distance
between the lenses is equal to twice the focal length as in a telescope with magnication
M = 1. The case L > f corresponds to negative dispersion stretcher and the conguration
in which L = f is used only in the pulse shapers. At the bottom, a negative dispersion
compressor is designed.
2.3 Laser beam characteristics
The most used model for approximating the electromagnetic eld of a laser radiation is the
gaussian beam model. It comes out from the solution of the wave equation in the paraxial
approximation. The paraxial approximation consists in the following assumption: the wave
79
2 Description of the laser facility
vector k , that describes the propagation direction of the wave, is considered approximately
parallel to the optical axis of the system. In other words, the angle between the optical
axis
z and the wave vector must be negligible so that to the rst order (and with a good
approximation also for the second order):
sin θ ≈ θ
(2.3.1)
tan θ ≈ θ
(2.3.2)
cos θ ≈ 1
(2.3.3)
The electric eld is hence approximable as the eld of a plane wave modulated by a
complex envelope:
E (r, t) = M (r) exp [iωt] exp [−ikz]
(2.3.4)
Further substitution in the wave equation in cartesian coordinates gives:
2
∇ + k 2 (M (r) exp [−ikz]) = 0
(2.3.5)
Calculating the rst member:
2
∂2
∂2
∂
2
+
+ k (M (r) exp [−ikz]) = 0
(M (r) exp [−ikz]) +
∂x2 ∂y 2
∂z 2
(2.3.6)
∂2
∂2
2
dening the transverse operator of Laplace ∇⊥ =
+ ∂y
2 , we have:
∂x2
exp [−ikz] ∇2⊥ [M (r)] + k 2 M (r) exp [−ikz] +
∂2
[M (r) exp [−ikz]] = 0
∂z 2
(2.3.7)
For calculating the third addend of the rst member, let's take the rst derivative of
the eld:
∂
[M (r) exp [−ikz]] =
∂z
∂
−ikM + M exp [−ikz]
∂z
(2.3.8)
and taking its derivative:
∂2
∂
∂
∂2
[M (r) exp [−ikz]] = (−ik) −ikM + M + −ik M + 2 M exp [−ikz] =
∂z 2
∂z
∂z
∂z
∂
∂2
2
= −k M − 2ik M + 2 M exp [−ikz]
∂z
∂z
Thus substituting:
80
(2.3.9)
2 Description of the laser facility
∇2⊥ [M ] − 2ik
∂
∂2
M + 2M = 0
∂z
∂z
(2.3.10)
The paraxial approximation allows to drop the third term, and nally we obtain:
∇2⊥ M − 2ik
∂
M =0
∂z
(2.3.11)
It can be shown that the solution for this dierential equation guides us to write the
electric eld:
w0
r2
r2
E (r, z, t) = êE0
exp − 2
exp −ik
+ iζ (z) exp [i (ωt − kz)] (2.3.12)
w (z)
w (z)
2R (z)
p
that is the so called gaussian beam. Note that here r =
x2 + y 2 is the radial distance
z
from the center axis of the beam in a plane transverse to the propagation direction ( axis). The z = 0 position corresponds to the minimum of the beam transverse size, called
beam waist. The function w (z), called beam width or spot size, describes the variation of
the beam transverse size along z, and in fact for a xed z, it represents the radial distance
at which the electric eld falls to 1/e of its peak value (i.e. the radial distance at which
2
the intensity falls to 1/e of its peak value): thinking in terms of gaussian prole, the w (z)
can be seen as a measure of the lateral spread of the beam (i.e. it plays the same role that
standard deviation plays in the normal gaussian distribution) and it has its minimum at
z = 0, the waist w0 . The beam width is given by:
s
w (z) = w0
1+
z
λR
2
(2.3.13)
where λR is the Rayleigh length that depends on the wavelength of the laser radiation
πw2
λR = λ 0 . The denition of the Rayleigh length in (2.3.13) has this simple consequence:
it is the distance where the cross sectional area of the beam at 1/e doubles the value of
the cross sectional area of the beam at the waist A0 . The cross sectional areas for dierent
2
are circles: A0 = πw0 at the waist, while AλR = 2A0 at the Rayleigh distance, and its
√
between these two point is called
beam width is w (±λR ) = w0 2. The distance along
z
z
confocal parameter or depth of focus of the beam and corresponds to twice the Rayleigh
length. A picture of a possible beam waist is given in gure:
81
2 Description of the laser facility
Figure 2.3.1: Beam waist w0 , Rayleigh length λR , beam width w (z), and depth of focus
of the beam dof = 2λR . This graph is for w0 = 5 [µm], λ = 800 [nm], hence
λR ≈ 98.2 [µm].
Another parameter that appears in the expression (2.3.12) is the radial curvature of the
beam, and it gives the shape of the wavefront phase. Its expression is:
"
R (z) = z 1 +
λR
z
2 #
(2.3.14)
The other parameter is the Gouy phase, or phase longitudinal delay, ζ (z), and it is
given by:
ζ (z) = arctan
z
λR
(2.3.15)
Between the far eld before the focus, and the far eld after the focus, the Gouy phase
experiences a shift of
π radians.
It can be shown that this shift originates from the
transverse spatial connement of the beam, which, through the uncertainty principle,
introduces a spread in the transverse momenta and hence a shift in the expectation value
of the axial propagation constant [Feng 2001].
It is interesting to analyze the intensity prole of the gaussian beam; from the denition
of Poyinting vector we calculate the intensity as its length, hence using (2.3.12):
82
2 Description of the laser facility
|E (r, z, t)|2
|E0 |2
I (r, z) =
=
2η
2η
where η
=
q
w0
w (z)
2
2r2
exp − 2
= I0
w (z)
1+
1
2r2
2 exp − 2
w (z)
z
λR
µ0
is the vacuum characteristic impedance and I0
0
=
(2.3.16)
|E0 |2
is the peak
2η
intensity. At distance from the waist equal to one Rayleigh length, we have:
2
I0
r
I (r, ±λR ) = exp − 2
2
w0
(2.3.17)
Near the waist we can neglect the rst dumping factor, obtaining a perfectly gaussian
shape for the beam intensity spatial prole:
2r2
I (r, z → 0) = I0 exp − 2
w0
(2.3.18)
The full width half maximum FWHM distance in the orthogonal plane is, by denition,
∗
∗
where I (r ) falls to its half maximum I0 /2:
twice the distance r
√
F W HM = w0 2 ln 2
A quantity called often
(2.3.19)
eective beam waist is dened w = √w02 hence the intensity is:
2
r
I (r) = I0 exp − 2
w
83
(2.3.20)
3 Ion beam generation by
laser-matter interaction
Introduction
The rst highly collimated proton beams, obtained when ultra-intense laser pulse hits a
solid target, were published independently by three research groups in 2000.
They discover that protons can be accelerated in two direction referring to the laser
direction:
ˆ backward: these protons leave the front surface of the target and move toward the
laser
ˆ forward: these protons are generated in the rear or front surface of the target and
thus we distinguish:
FSA: front side (forward) acceleration
RSA: rear side (forward) acceleration
Figure (3.0.1) depicts those basic concepts.
Figure 3.0.1: Forward and backward proton acceleration
84
3 Ion beam generation by laser-matter interaction
It is important to note that direct ion acceleration is not possible due to the ion mass
unlike electron. In fact when the short and ultrahigh laser pulse irradiates the front of a
metal target, a part of the laser energy is absorbed by electrons, which acquire relativistic
velocities and then propagate through the target to the rear surface producing a hot
electron cloud at this point: the cloud creates an electrostatic eld strong enough in the
TV/m range to ionize atoms and accelerate ions from the rear surface into vacuum.
Now we want to distinguish the following key arguments that will be treated separately:
ˆ energy absorption process that heats up the electrons
ˆ electrons transport through the foil
ˆ ions acceleration
A remarkable note to do concerns the origin of the protons or ions that are accelerated
from the target. parlare dell'origine degli ioni contaminanti ecc h+ ecc ecc
3.1 Energy absorption processes
The initial stage that leads to electron heating is the creation of a so called preplasma in
front surface of the target due to the leading edge of the laser pulse. The coupling of the
laser pulse and the electrons, highly inuenced by this preplasma, results in the forward
acceleration of electrons inside the target.
As stated in [Lefebvre 1997], the possible mechanism that allows the electron heating
(i.e. the coupling between the laser beam and the cold electrons in the target) are:
ˆ resonance absorption
ˆ vacuum heating or Brunel absorption
ˆ j × B heating
ˆ dierent sorts of skin eect
The rst two mechanisms are absent in the case of normal incidence in one dimension,
since the laser eld has no longitudinal component to directly drive the electrons along
the density gradient.
The mechanisms of accelerating electrons depend strongly on the density gradient on
the front surface plasma. A key parameter that determine if the density gradient is long
or steep, is the plasma scale length
L [Santala 2000]. The steeper the leading edge, the
later the preplasma is formed and the steeper the density gradient. If L → 0 then the
plasma is said to be a sharp boundary plasma (i.e. a perfect step function density). It
can be observed that:
ˆ steep density gradient (L ≈ λD ) accelerates electrons mainly by:
vacuum heating [Brunel 1987, Gibbon 1992]
resonant absorption [Estabrook 1978]
85
3 Ion beam generation by laser-matter interaction
ˆ long density gradient (L λD ) created by the prepulse (i.e.
the ASE before the
main pulse) consist in a plasma that subsequently interacts with the main laser
pulse and results in a very complex variety of mechanisms all leading to higher
electrons temperature as stated in [Santala 2000].
Note that in the main part of literature's articles, the fast electrons produced in the
hot electrons because the main parameter that
hot ). The coupling of the
laser energy into fast electrons is therefore called electron heating or plasma heating.
laser-plasma interaction region are called
describes the electrons is the temperature Te also termed Th (
In the case of a laser eld incident on a very steep electron density gradient, suach as a
metallic surface, an absorption mechanism occurs that sometimes is also called "Bruneleect" or "Bruenel heating".
In that process, electrons are heated by the p-polarized
component of the laser eld at the sharp plasma vacuum transition. Within half a laser
cycle, electrons are stripped from the plasma into the vacuum. The strong electromagnatic eld at the border pushes the electrons again back into the plasma region with the
oscillation velocity of the electrons, i.e.
v ≈ vosc = eE/me ω .
Since the laser eld can
penetrate only into a limited part of the plasma (the skin depth), its electric eld (that
could retain otherwise the electrons) has no longer eect on the electrons that therefore
are accelerated inside the target.
Resonant absorption is a collisionless absorption process that occurs at intensities I >
1015 [W/cm2 ] and along steep density gradient. When an oblique incident laser beam with
a p-polarized component of the electric eld is hitting a steep gradient, the oscillations of
the electric eld excite the electron of the density gradient. Those oscillations generate
charge oscillations that are increased by the plasma by resonance. Part of the incident laser
light is therefore transferred to an electrostatical oscillation (electron plasma wave) and
the laser energy absorption is increased. Note that plasma waves generated by resonant
absorption decrease the plasma gradient.
It can be observed also a correlation between
L and electron angular distribution: study-
ing the angular distribution of γ -ray beams informations can be obtained about the fast
electrons. γ -rays can be produced in some experimental set-up like that one described in
[Santala 2000], by the conversion of kinetic energy of the fast electrons into high energy
bremsstrahlung radiation.
The correlation between
L and electron angular distribution
could be explained by Brunel-type resonance absorption being the dominant absorption
mechanism in plasmas with a steep density gradient. As the plasma scale length increases,
the JxB mechanism becomes the main hot electron production mechanism. When L λD
the converging laser beam has to traverse a large distance in coronal plasma which results
in an essentially random angular distribution.
This may be due to lamentation and
self-focusing of the laser beam in the underdense plasma [Santala 2000]. For very small
L < λD the γ -ray beam is normal to the target, for intermediate L the beam is along laser
direction, and for long
L the beam direction is uncorrelated.
For steep density proles the hot electron average energy is denitely below the ponderomotive potential and the absorption is low (<10%).
When the density gradient is
attened, a transition from surface to volume absorption is observed leading to a higher
coupling eciency (30%) [Lefebvre 1997].
86
3 Ion beam generation by laser-matter interaction
3.1.1 jxB heating
When a high intensity laser pulse hits the target, the ponderomotive force pushes the
electrons present at the front surface, due to the preplasma, inside the target.
This
process generates a strong electrostatic eld and a space charge distribution in the front
surface that:
ˆ retains some accelerated electrons; those electrons could:
escape from the front surface and being accelerated toward the laser
remain in the laser target interaction region, contributing to the electrostatic
eld
ˆ creates a sweeping acceleration mechanism that continuosly pushes the front surface;
this moving surface reects and accelerates the ions present in that region. The model
used to interpret this phenomenon that occurs at moderate intensities (reference
needed) is the piston acceleration and it will be discussed next.
The pondermotive force exerted by an oscillating eld can be seen in the same way of an
imposed constant electric eld [Kruer 1985]. This coupling mechanism that arise from the
oscillating component of the ponderomotive force is termed j × B heating. Recalling the
equation of motion:
e
d
v(t) =
(E (r, t) + v × B (r, t))
dt
m
(3.1.1)
E (r, t) = E (r) cos (ωt)
(3.1.2)
Suppose:
Using Faraday's law we can obtain:
B (r, t) =
1
∇ × E (r) sin (ωt)
ω
(3.1.3)
See chapter 1 for further details. To the lower order the equation of motion reduces:
d
e
v(t) =
dt
m
1 e
2 1 + cos 2ωt
−
∇ |E 0 |
2 mω 2
2
(3.1.4)
and thus rearranging:
m
d
e2
v(t) = −
(1 + cos 2ωt) ∇ |E 0 |2
dt
4mω 2
(3.1.5)
with:
f pond = −
1 e2
∇ |E 0 |2 (1 + cos 2ωt)
4 mω 2
the eective ponderomotive force is:
87
(3.1.6)
3 Ion beam generation by laser-matter interaction
F pond = hfpond i = −
1 e2
∇ |E 0 |2
2
4 mω
(3.1.7)
The intensity of the laser radiation is dened as the length of Poynting vector:
1
1
I0 = |E 0 × H 0 | = |E 0 |2
2
2
r
0
µ0
(3.1.8)
Calculating:
r
2
|E 0 | = 2I0
µ0
0
(3.1.9)
The critical density of the plasma is:
nc =
me 0 ω 2
e2
(3.1.10)
The eective ponderomotive force can be also expressed by:
F pond = −
1 e2 √
0 µ0 ∇I0
2 mω 2 0
(3.1.11)
1 ∇I0
nc 2c
(3.1.12)
and therefore:
F pond = −
From a relativistic point of view, one can write:
F pond = ∇ (γ − 1) me c2
(3.1.13)
where γ is the Lorentz factor of the oscillating particle:
1
γ =1+
2
F pond = −
eA
me c
2
(3.1.14)
e2
∇A2
2me c
(3.1.15)
The extimation of the heating of electrons in the plasma is made by the assumption
that the hot electron temperature is approximately equal to the ponderomotive energy
[Wilks 1992]:
Th ≈ φpond = (γ⊥ − 1) me c2
(3.1.16)
where γ⊥ :
γ
2
= γ⊥2 +
and therefore is equal to:
88
px
me c
2
(3.1.17)
3 Ion beam generation by laser-matter interaction
γ⊥ =
q
1 + a20
(3.1.18)
For a laser with a given intensity one can calculate the transverse gamma factor with
the formula:
a20 =
eA
me c
2
=
eE
λ
2πme c2
2
=
!2
p
e 4 µ0 /0
√
I0 λ2
2π (me c2 )
(3.1.19)
2
Considering the intensity expressed in unit of W/cm and the wavelength in unit of µm,
we have:
a20 ≈
I0[W/cm2 ] λ2[µm]
(3.1.20)
1, 37 × 1018
and thus:
s
Th ≈ φpond ≈ 0, 511  1 +
I0[W/cm2 ] λ2[µm]
1, 37 × 1018

− 1 [MeV]
(3.1.21)
We can also make a nal remark on this argument: not only the electrons are accelerated
by the ponderomotive force, but also the ions; the main dierence is the high ion to electron
mass ratio.
3.2 Ion acceleration mechanisms
3.2.1 Backward acceleration
The backward acceleration mechanism is based on the plasma free expansion model (because shock driven acceleration is not possible in the opposite direction of the impinging
laser beam). Before comparing backward accelerated protons and forward accelerated protons from rear surface, both modeled with plasma free expansion, it must be said that the
formers come from the front surface that is initially perturbed by the prepulse activities of
the laser beam (i.e. ASE pedestal of the laser pulse or other pre-pulses); otherwise, if the
prepulse has not a high intensity and the target foil is not thin enough, the shockwave of
the prepulse doesn't reach the rear surface and thus this side remains unperturbed. In fact
proton beams stemming out from the front surface are of a lower quality (lower emittance,
lower laminarity, lower energy) than those of the rear surface because the laser prepulse
changes and inuences the planarity of the target's front surface.
The backward acceleration can thus be improved by using a high contrast laser, in the
10
order of 10 : in that case both surfaces can be considered as unperturbed and then the
two beams produced by the main pulse look alike, and have about the same energy. As
shown in [McKenna 2004] the high laser contrast produces similar but not equal forward
and backward proton/ion beams: in fact the forward beam is the uppermost in terms of
maximum energy.
89
3 Ion beam generation by laser-matter interaction
Figure 3.2.1: Proton spectra from Fe 100 µm target foil irradiated with 45° incidence laser.
20
2
7
Laser characteristics: λL ≈ 1[µm], τL = 0.7 [ps], I = 2 × 10 [W/cm ], 10
13
2
contrast ratio (ASE pedestal ≈ 10 [W/cm ]), no prepulses within 19 ns,
The proton ux from the heated target (gray lines), measured at both the
front (broken lines) and rear (solid lines) of the Fe target foil, is reduced by
about 2 orders of magnitude compared to the unheated target (black lines).
90
3 Ion beam generation by laser-matter interaction
3.2.2 Forward acceleration and Target Normal Sheat Acceleration
As stated in a previous section and in [Fuchs 2005, Allen 2004], there are two main acceleration mechanisms that produce the
forward accelerated proton beam: FSA and RSA.
The forward front surface acceleration FSA mechanism relies on the acceleration of
electrons from the laser-target interaction region
The forward rear surface acceleration RSA mechanism of acceleration is still based
primarily on the acceleration of electrons from the laser-target interaction region due to
ponderomotive force ( i.e. what we called
heating of the electrons), but in this case:
ˆ a very large fraction of the electrons that travel through the target do not excape
from the rear surface
ˆ only a very small fraction of the electrons can leave the target before the resulting
Coulomb potential traps the rest; these electrons form a very dense
sheat at the rear
surface.
Figure 3.2.2: Schematic front surface and back surface forward accelerations
15
From the laser-target interaction zone a few 10 hot electrons will be generated [Hatchett 2000].
They y through the thin targets in a broad angular beam, be turned around near the
back, and bounce back and forth through the target drifting transversely. For thin targets
18
it can be exstimated that the electrons have a density of several 10
cm−3 . The hot
electrons tend to relax to Boltzmann equilibrium, and they will set up a sheath at the
target surface with a scale length of about a Debye length and in fact this sheat consists
in a small region with a very high electron density[Hatchett 2000].
91
It can be seen that
3 Ion beam generation by laser-matter interaction
the density of electron is proportional to the ratio of the laser energy and the hot electron
temperature; therefore in this acceleration regime the Debye length is roughly expected
to be proportional to:
s
λD ∝
Th2
φpond
∝√
≈ const
Elaser
Elaser
(3.2.1)
Thus the electrostatic eld at the rear surface has an extremely steep gradient; this eld
is in the order of TV/m:
E≈
Th /e [MV]
λD [µm]
(3.2.2)
and is very much larger than the eld created by the initially escaping electrons for which
the scale length in the denominator is the target size [Hatchett 2000]. Such eld strengths
is sucent to eld-ionize atoms present at the back surface of the target. Those atoms
can be the target foil material itself, but commonly the accelerated ions come from the
layer of contaminants (comprising hydrocarbons and water vapour) always present in the
target if not externally heated, or additionally deposited material.
Protons and in general ions are emitted perpendicular to the rear surface of the target: the so called Target Normal Sheat Acceleration takes place (TNSA). The ions TNSA
mechanism is thus the result of a population of hot electrons (generated in the plasma
created by the laser prepulse interacting with the front of the target) that go through
the target and ionize the proton monolayer on the back of the target [Wilks 2001]. For a
description of the electrons transport throughout the target see the next section. The subsequent ionization and ions acceleration is caused by the high electric eld (3.2.2) present
in the sheat and is essentially modeled as a plasma expansion into vacuum [Mora2003].
Extensive experimental campaings have been carried out in the most important laser
facilities to investigate the TNSA regime. A lot of experimental results can be found easily
in literature.
Some of the works are focused on quantitative determination of the respective contributions of RSA and FSA for high energy proton beams as in [Fuchs 2005]. With a laser of
20-30 J and for thin (20 µm) metal foils, the RSA produces a well collimated (≈ 20°) beam
of energetic protons (>16 MeV) while the FSA produces high-divergence, low-energy (6
MeV) beam. The FSA for E>3 MeV accounts for the 3% of total proton beam energy. So
it is clearly demonstrated that the FSA beam has better qualities than FSA beam. This
is due to the stochastic laser-plasma-target interaction that happens in the front surface
while the rear surface remains unperturbed. It was also demostrated in [Ceccotti 2007]
that proton energy increases with decreasing target thickness and that the TNSA concept
can also apply for both target surfaces.
As we will see in the next section, in the presence of plasma gradients the initial accelerating electric eld is reduced from the expression (3.2.2) to a new expression involving the
plasma scale lenght
l instead of the Debye lenght. Very long scale length plasmas on the
target rear surface can quench entirely proton acceleration. Similarly, proton acceleration
towards the vacuum from the target front side, where a small plasma gradient is present
due to the laser pedestal, yields lower maximum proton energies than acceleration taking
92
3 Ion beam generation by laser-matter interaction
place at the unperturbed target rear surface. Note that such reduction depends on the
target thickness since acceleration at the target rear surface reduces with target thickness
due to increasing electron spreading in the target volume [Fuchs 2007].
Summarizing, the main parameters which aect the ion acceleration at the rear side of
the target heated by the laser pulses are:
ˆ laser pulse absorption eciency ηlaser
ˆ opening angle of the hot-electrons θin
ˆ hot-electron recirculation
ˆ laser pulse duration τL
ˆ target thickness, d
For target thicknesses larger than a critical value, Lc , the energy conversion eciency is
saturated at a value of about 40%50% and the change of hot-electron density is so small
that the hot-electron recirculation can be neglected.
With the increase of the target thickness, the angular eect and the target thickness effect both lead to the decrease of hot electron density and a global dilution eect. Therefore
the maximum ion velocity has a small decrease with the increase of the target thickness
observed by several experiments
3.2.3 TNSA parameters
In this paragraph we will describe briey the models usually used to describe the parameters of interest and analyze results of the target normal sheat acceleration. We will
talk about the characteristics of the electron spectrum that arise from the absorption of
the laser energy. After the model of the electron sheat is investigated, and two possible
extimations of maximum ion energies are given therein.
3.2.3.1 Electron spectrum
The total number of electrons produced by the absorption and electron heating processes
at the target front side can be deduced with the help of simple extimations.
Supposing we can approximate the time and space dependance of the main lobe of the
focal spot (i.e. the rst disk of the Airy disks pattern given by a focused gaussian beam)
(see next chapter for a description of Airy disks) on target with a gaussian pulse-shape in
time and space, the intensity is:
2
r
t2
I (r, t) = Imax exp − 2 exp − 2
τ
w
(3.2.3)
where:
τ=
F W HM (time)
√
2 ln 2
93
(3.2.4)
3 Ion beam generation by laser-matter interaction
F W HM (time) can be extimated with a second order correlation of the laser pulse, and
it is said to be the pulse duration τL ;
w=
F W HM (space)
√
2 ln 2
(3.2.5)
F W HM (space) depends on the beam waist and can be approximated with the rst
zero of the Airy disk function (an example of it is plotted in gure (4.1.4)) only if the laser
near-eld has a top-hat shape:
F W HM (space) ≈ r0 ≈
where
1, 22λ
f
D
(3.2.6)
f is the focal lenght of the parabola that focuses the beam on target, D is the
beam diameter, and λ the laser beam wavelength.
Note that this approximation is not directly connected with the gaussian beam expression (2.3.12): here we are simply approximating the rst lobe of the intensity pattern of
the focal plane with a pulse that has gaussian shape both in time and space.
We can
obtain a connection between the two models by noting that a real laser beam, that has
not an innite wave front, is focused by a lens with circular aperture, and this operation
will cause the formation of the Airy disks pattern in the focal plane (see next chapter).
The energy encircled by the main lobe can be found by integrating the laser intensity
over time and space, but this is also a measurable fraction of the total laser pulse's energy.
´ +∞
exp [−x2 ] =
Performing the two integrals, using the property of the gaussian integral
−∞
√
π we obtain:
√
Eencircled = Imax π πw2 τ = flobe Elaser
and thus rearranging to obtain the maximum intensity in units of
Imax
(3.2.7)
W
:
cm2
W
0, 829 × 1023 × flobe Elaser
=
2
cm2
τL[fs] × r0[µm]
(3.2.8)
where:
ˆ flobe is the fraction of the incident laser energy Elaser contained in the rst lobe (if
the focal spot is a perfect Airy function the maximum possible value of the encircled
energy is 88%)
ˆ τL[fs] is the pulse duration full width at half maximum in femtoseconds
ˆ r0[µm] is approximately the diameter of the main lobe of laser pulse (focal spot diameter) in microns
ˆ Elaser is the laser energy (i.e. the total energy cointained in the pulse that is hitting
the target: if a plasma mirror is used to increase the contrast ratio, this energy must
take into account the possible reduction caused by a reection<100%) in Joule.
94
3 Ion beam generation by laser-matter interaction
Now if we can consider the approximation for the hot electron temperature we can write:

v
u
2
u
I
×
λ
[µm]
max[ W2 ]
t

cm
− 1
Th [MeV] ≈ Φponderomotive ≈ 0.511  1 +
18
1, 37 × 10
(3.2.9)
The electron spectrum [Schreiber 2006a] is:
dN
E
= N0 exp −
dE
Th
(3.2.10)
The total energy of electrons can be calculated as follows:
ˆ N0
electron
=
Etot
0
ˆ 0
dN
EdN =
EdE = −N0
+∞ dE
ˆ +∞
0
E
dE = N0 Th2
E exp −
Th
(3.2.11)
A very interesting parameter of the coupling between laser pulse and electrons is the
laser to electron conversion eciency dened as to be:
ηe =
tot
Eelectron
Elaser
(3.2.12)
The total number of electrons can be calculated with the integral:
ˆ
Ne =
dN
dE = N0
dE
ˆ
+∞
E
E
E electron
exp −
dE = N0 −Th exp −
= N0 Th = tot
Th
Th 0
Th
(3.2.13)
The total energy of the electron is also a percentage of the energy laser encircled in the
focal spot:
electron
Etot
= felec Eencircled
(3.2.14)
where this fraction felec has been found to depend on the laser incident intensity as
follows [Fuchs 2006]:
felec = 1.2 × 10−15 × I[0.74
W/cm2 ]
(3.2.15)
3.2.3.2 Electron sheat model
A simple extimation of the electron density leads to write:
ne0 =
Ne
cτL Ssheat
(3.2.16)
where Ssheat is the area of the electron sheat at the rear surface of the target given by:
Ssheat = π (r0 + d tan θin )2
95
(3.2.17)
3 Ion beam generation by laser-matter interaction
where r0 is the radius of the laser focal spot, d the thickness of the target, θin the half angle of divergence of the electron beam [Schreiber 2006] [Fuchs 2006]. The formula (3.2.16)
takes into account that during the laser-target interaction, the relativistic electrons, once
heated up, start to travel inside the target at the speed of light, and when the interaction
is over the electron bunch has a length of cτL . Expression (3.2.17) simply models the sheat
surface with a circle, with radius B = r0 + d tan θ that depends on the lateral spread of
the electron beam: it can be seen that for very thin target the electron transport through
the foil doesn't suer of a large divergence angle, leading to smaller sheat surface and,
therefore, higher electron sheat density.
Now let's consider that a portion Q of the total hot electrons can cross the plasmavacuum boundary: this behavior induces in the target a positive charge surface layer with
density:
σ+ =
Qe
Ssheat
(3.2.18)
The potential Φ is a potential well that traps the electrons that wants to escape from
the target. The potential seen on the z axis can be calculated with:
1
Φ (z) = −
4π0
where
ˆ ˆ ˆ
ρdV
R
(3.2.19)
R is the distance on the z-axis from an innitesimal point charge in the surface
distribution σ+ at z = 0. Going on with calculation:
Qe
Φ (z) = −
20 πB 2
ˆ B
√
0
i
Qe h√ 2
r
2−z −Φ
dr − Φ∞ = −
z
+
B
∞
20 πB 2
z 2 + r2
setting Φ (z = 0) = 0, we have Φ∞ = −
"
Φ (z) = Φ∞
(3.2.20)
Qe
, hence:
20 πB
z
1+ −
B
r z 2
B
#
+1
(3.2.21)
The potential energy of the well is:
U = −eΦ (z) = U∞ s (ξ) =
(3.2.22)
where we set three denitions [Schreiber 2006][Zeil 2010]:
ξ=
z
B
s (ξ) = 1 + ξ −
(3.2.23)
p
ξ2 + 1
(3.2.24)
Qe2
20 πB
(3.2.25)
and:
U∞ = −eΦ∞ =
96
3 Ion beam generation by laser-matter interaction
Qe2
can escape the boundary, oth2π0 B
erwise they are retained in the sheat. As a reference we can take the electrons with the
Only the electrons with energy exceeding U∞ =
mean energy kB Te : if this energy is less than U∞ , they cannot escape to the potential and
the can reach the position ξˆ where their energy balances the potential energy of the well
at that point:
− eΦ ξˆ = kB Te = U∞ s ξˆ ≃ U∞ ξˆ
(3.2.26)
the approximation is valid only if ξˆ =
ẑ
1 hence only if ẑ B . For these electrons
B
Q
is valid this denition of electron density: n0 =
. Hence we have:
πB 2 ẑ
r
ẑ =
20 kB Te
= λD
n0 e2
with the implication λD B and U∞ =
(3.2.27)
kB Te B
. The electric eld is:
λD
"
#
dΦ
1 dΦ
kB Te
ξ
E (z) = −
=−
=
1− p
dz
B dξ
eλD
ξ2 + 1
in the surface z = 0 and for distances less than the Debye length ξ <
kB Te
eλD
Esheat ≃
(3.2.28)
λD
1
B
(3.2.29)
result yet presented in (3.2.2). The model of [Schreiber 2006] calculates also the max
energy that an ion could gain if it is accelerated by this electric eld, completely escaping
the well (z → ∞):
2
Ei,∞ = 2Zmc
PL
PR
1/2
(3.2.30)
3
where PL is the laser power, and PR = 8.71[GW] is the relativistic power given by mc /re
2
1
e
). The ion energy results solely
(re is the so called classical electron radius re =
4π0 me c2
from the repulsion due to surface charges
because the inuence of the hot electrons
Qe
density is neglected becuse the positive charge on the surface is more localized than the
electron sheat total charge. The electron centroid of charge in the sheat can be assumed
on the axis at a distance of about λD , and its longitudinal width is of the same order.
Thus, the forces of the electrons on an ion at some distance from the surface somehow
compensate each other [Schreiber 2006] [Zeil 2010]. For completeness we report another
model for the maximum proton energy scaling, that we will further detail in next section.
A uid isothermal expansion model [Mora2003] of the plasma can predict both maximum
energy and proton spectrum only if we set a limit time of acceleration:
h i2
√
max
Eproton
= 2kB Te ln τ + τ 2 + 1
≃ 2kB Te [ln (2τ )]2
ωp−i t
with normalized time τ = √
, and
2eN
97
(3.2.31)
3 Ion beam generation by laser-matter interaction
" r
#
dNp
ne0 cs tacc Ssheat
2E
exp −
=
dE
kB Te
(2EkB Te )1/2
(3.2.32)
tacc = 1.3 × τL
(3.2.33)
where:
This value is the best t value found by [Fuchs 2006] comparing both the results of
calculation of the laser to proton conversion eciency, from experimental data and the
isothermal model of [Mora2003]. The total laser to proton conversion eciency is dened
as:
ηp =
tot
Eproton
Elaser
(3.2.34)
and can be calculated by integrating the proton spectrum between a minimum xed
value of energy and the cut o given by (3.2.31).
3.2.3.3 Electron current and recirculation
With the extimation of the number of electrons in the bunch heated by the laser pulse we
can also approximately calculate the electron current that ows in the target:
J=
eNe
τL
(3.2.35)
This current exceeds the Alfven limit of the maximum possible electron current in
vacuum [Alfvén 1939]. However, the propagation of such a hot electron current becomes
possible in a conductor due to return currents that compensate the self-induced magnetic
elds around the relativistic electron forward current. Furthermore, by using a conducting
target material, the inuence of magnetic eld uctuations can be neglected since a big
amount of free charges is available to provide the necessary return current. Consequently,
a smooth electron beam with Gaussian temporal and spatial shape due to the laser pulse
properties will propagate through the target [Honrubia 2005]. However, ohmic losses via
collisions with the target foil atoms are still present and lead to a reduction of the electron
temperature at the rear side and a spread-out of the electron bunch, leading to a decrease
of the maximum achievable proton energy.
Decreasing the target thickness will improve a particular mechanism called electron
recirculation [Mackinnon 2001], leading to a better eciency in the ion acceleration. The
recirculation of the hot electrons is important in thin target because during the laser-target
interaction electrons can travel inside the target many times before the end of the pulse
being reected by the Debye sheaths at the target front and rear surfaces: this behaviour
aects the electron tranport inducing a transient electron density enhancement.
From simulations in [Buechoux 2010] is clearly visible that the enhancement in ion
acceleration can take place only if the electrons can come back from the point of reection
in the sheat at the rear side, to the target center, if it takes only an interval of time less
than the pulse duration. An interesting model called step model on electron recirculation
is given in [Huang 2007].
98
3 Ion beam generation by laser-matter interaction
3.2.4 Radiation Pressure Acceleration
In the recent years, with the further increase of laser pulse power and focused intensities,
another acceleration mechanism appears as promising candidate for the laserion acceleration:
the radiation pressure acceleration RPA. The idea to accelerate an object (in
the scenario of the article mentioned after, the object to accelerate is a spacecraft for
interstellar travelling!) using light was described rst in [Forward 1962] and successively
futuristic problem
lightsail concept is quite simpler on
analitically treated in [Marx 1968]. Other further observation on this
were made in [Forward 1982]. The application of the
small object like micrometric targets, and the regime of radiation pressure acceleration of
thin foils has been investigated more deeply.
21
The laser ponderomotive pressure at intensities of 10
W/cm2 is of the order of 300
Gbar, a value that is impossible to achieve on Earth by any other means. This pressure
can be used for ion acceleration: the momentum of the laser is imparted directly to the
object to be accelerated. The specicity of this process of radiation pressure acceleration
(RPA) is that it requires the electrons to remain cold, which is seems to be possible only for
circularly polarized laser pulses. The laser primarily interacts with the plasma electrons,
and usually the heating of the electrons leads to TNSA dominating.
This prevents the
entire foil from being uniformly accelerated. For any RPA scheme it is essential that the
electric eld is dominated by a single positive spike to ensure that the front and back
sheaths are relatively small.
Two circular polarization RPA regime were investigated by some researcher (Macchi
2008)
ˆ hole boring also termed thick regime
ˆ light sail also termed thin regime
23
2
Although the laser intensity requirements (exceeding 10 [W/cm ]) are not met yet, a
dominance of RPA over TNSA may be already obtained at much lower intensity if a laser
pulse with circular polarization (CP) instead of linear polarization (LP) is used: that's
because in such condition the acceleration of fast electrons at the laser-plasma interface is
almost suppressed, clearly excluding TNSA which is driven from the space charge produced
by energetic electrons escaping in vacuum [Macchi 2008].
Radiation Pressure Acceleration using Circular Polarization (CP-RPA) was rst investigated in [Macchi 2005] by considering thick targets, such that during the laser pulse
only a nite layer of the target at its front surface is accelerated, forming a dense bunch
of ions (neutralized by cold electrons) entering the target. Later on other groups studied
the acceleration of thin foils, to obtain the whole target acceleration. In this acceleration
regime, the use of CP is particularly important to prevent the foil expansion due to the
pressure of electrons, allowing the acceleration of the target as a single rigid object and
preserving the inherent monoenergeticity after the acceleration stage. The light sail regime
of CP-RPA is very attractive mostly for the possibility to accelerate a large number of
ions to GeV energies. The acceleration scheme described in [Macchi 2008] requires the use
of ultra high contrast (UHC) laser pulses. The thick target or hole boring regime allows
to reach much lower energies with present-day intensities and solid target densities but
99
3 Ion beam generation by laser-matter interaction
this regime requires less critical conditions than the light sail. Further details and analytic
descriptions of the RPA is given in the next section.
3.3 Models used to interpret the acceleration
mechanisms
3.3.1 Plasma expansion acceleration model
The plasma expansion into vacuum is a model that well predicts the behavior of ions and
electrons in a plasma where electrons are heated at a temperature Th by an ultra intense
laser pulse. The plasma expansion has been studied theoretically for several decades since
the rst paper by Gurevich et al.
in [reference needed].
Most of the studies describ-
ing the expansion concern the semi-innite plasma rarefaction, also called the isothermal
model.
It implies an innite reservoir of energy that maintains constant the electrons
temperature during the expansion i.e. the electrons are always
hot electrons because are
continuosly replaced by other electrons coming out from the unperturbed body of the
plasma [Crow 1975]. A more realistic model is described in [Mora 2005] for a plasma slab
with nite thickness: this is the adiabatic expansion model because conservation of energy
is provided in it. In this model the hot electrons cool down during the expansion while
they progressively give their thermal energy to the ions during acceleration. For further
details see the next paragraphs and [Grismayer 2006].
The electrostatic acceleration of protons, in the frame of TNSA (reference needed) is
induced by the expansion of electrons into vacuum. These models assume that electrons,
escaping from a sharp boundary plasma, have quasi-maxwellian distribution during all
the expansion process. This is a reasonable assumption for the rear surface of the target
because it remains unperturbed if the target is not too thin, but is also valid for the front
target only if the incident laser has a large contrast ratio that avoids the expansion of
preplasma.
3.3.1.1 Isothermal expansion
This model, explained in details in [Mora2003, Crow 1975], treats the expansion into
vacuum of a semi innite plasma that occupies the left hal-space x < 0. The ions in it are
initially cold and at rest:
ni (x, t = 0) =
ni0
0
this is called sharp boundary approximation.
x<0
x>0
(3.3.1)
Electrons are in equilibrium with the
electrostatic potential, i.e. are distributed with a Boltzmann-like functions, and supposing
that this equalibrium holds also for t > 0 (i.e. the expansion takes place witha time scale
longer than the time of an electron plasma period of oscillation):
eφ (x, t = 0)
ne (x, t = 0) = ne0 exp
kB Te
100
(3.3.2)
3 Ion beam generation by laser-matter interaction
The gure summarizes these statements.
The time evolution of the system is given by this complete set of dierential equation
for this problem:
∂
e
∂
v+v v =
E
∂t
∂x
mi
equation of motion
(3.3.3)
∂
∂
ni +
(ni v) = 0 continuity equation
∂t
∂x
φ
Boltzmann equilibrium solution
ne = ne0 exp
kB Te
0
d2
φ = e (ne − ni )
dx2
Poisson's equation
(3.3.4)
(3.3.5)
(3.3.6)
Initial conditions
The electrostatic potential must satisfy the Poisson's equation, so the initial condition
in t = 0 for the potential is:
0
d2
φ = e (ne − ni )
dx2
(3.3.7)
Note that in the limit x → −∞ the eld must be zero and the potential is chosen to be
zero; on the other hand in the limit of x → ∞ the eld must be zero and the potential tends
to−∞ ; for the equilibrium of the plasma ne0 = ni0 (supposing
Z = 1). The dierential
equation reduces to:

h
i
eφ

en
exp
−
1
0
d
kB Te
i
h
0 2 φ =
eφ

dx
en0 exp kB Te
2
x<0
x>0
(3.3.8)
integrating in the respective domains [Crow 1975]:

i
h
eφ
eφ

−
1
−
n
k
T
exp
0
B
e
1
kB Te
h
i kB Te
0 E 2 =
eφ

2
n0 kB Te exp kB Te
x<0
x>0
(3.3.9)
the boundary condition in x = 0 for the continuity of electric eld and potential gives:
φ (0, t = 0) = φ0 = −
kB Te
e
(3.3.10)
Integration of the dierential equation for x > 0 gives:
φ (x, t = 0)
x
√
= 2 ln 1 +
+1
φ0
λD 2eN
(3.3.11)
and thus the electric eld is:
d
2φ0
E=− φ=− √
dx
λD 2eN
101
x
√
1+
λD 2eN
−1
(3.3.12)
3 Ion beam generation by laser-matter interaction
The so called self-similar ion front electric eld [Mora2003]o [Antici PhD 2007] is the
value of the electric eld at time zero in the position of the ion front at t = 0 (i.e. x = 0):
2φ0
kB Te
Ef ront,0 = − √
=
eλD
λD 2eN
where is given another denition, the eld E0
r
2
= E0
eN
r
2
eN
(3.3.13)
B Te
= keλ
= − λφD0 : this is the eld that
D
initially drives the proton acceleration at the target vacuum interface. Formal integration
of the d.e. for x < 0 gives [Crow 1975]:
1
x
=√
−
λD
2
ˆ α
dα
−1 exp α − 1 − α
(3.3.14)
The expression of potential combined with the Boltzmann approximation of electron
density gives:
n0
ne (x, t = 0) =
eN
1+
x
√
λD 2eN
−2
x>0
(3.3.15)
Very roughly, the electron density has a sort of square hyperbolic decay. FIGURE
The quasi-neutral self-similar solution
Approximate solutions of various plasma expansion problems can be found by setting
simply ni = ne , thus abandoning Poisson's equation. The solution for the free expansion,
in this simple plane geometry is self similar:
x + cs t
n = n0 exp −
cs t
v=
where cs
=
q
x + cs t
t
(3.3.16)
(3.3.17)
kB Te
is the acoustic ion velocity, and ωp−i is the ion plasma frequency
mi
[Mora2003, Crow 1975]. A simple substitution can be made in the equation (3.3.4) as a
check of the propriety of solution. Subsituting the solution in (3.3.3) retrieves the denition
k Te
of E0 = B . Using (3.3.5):
eλD
φ = φ0
x + cs t
cs t
(3.3.18)
The electric eld can be calculated as the spatial derivative of potential:
E=−
d
E0
φ=
dx
ωp−i t
(3.3.19)
The self-similar eld corresponds to a positive charge surface distribution at position
x = cs t and a negative charge surface distribution at the plasma edge.
An important
parameter is the plasma scale length dened as to be:
∂
l = − ln
∂x
n
n0
102
−1
= cs t
(3.3.20)
3 Ion beam generation by laser-matter interaction
Another parameter is the
r
λD =
0 kB T
=
e2 n
r
local Debye lenght given by:
0 kB T
x + cs t 1
1
x
exp
= λD0 exp
1+
e 2 n0
cs t 2
2
cs t
(3.3.21)
One considerable remark is that the self similar solution has a valid meanings only when:
x + cs t > 0
(3.3.22)
l > λD
(3.3.23)
ωp−i t < 1
(3.3.24)
and:
or, in other terms:
The self-similar model predicts a velocity increasing without limit for
physically it must be limited by a certain value.
x → ∞ but
Thus using this expansion model we
cannot know the nal value of the ion velocity, and thus their energy. Approximation of
this velocity can be obtained if the
acceleration time (i.e. the instant of time when the
acceleration process stops) is a known parameter.
A correct estimation of the ion front x-position can be found by interpreting the previous
results.
The model breaks when the plasma scale length equals the local Debye length
l = λD ; at this instant of time we have:
xf ront
1
1+
cs t = λD0 exp
2
cs t
(3.3.25)
hence:
xf ront
1+
= 2 ln
cs t
cs t
λD0
(3.3.26)
and:
xf ront = cs t 2 ln
cs t
λD0
−1
(3.3.27)
and the velocity is:
vf ront =
xf ront + cs t
= 2cs ln (ωp−i t)
t
(3.3.28)
The electric eld at the front can be calculated using the equation of motion (3.3.3)
written in the form:
d
e
v=
E
dt
mi
Thus rearranging:
103
(3.3.29)
3 Ion beam generation by laser-matter interaction
Ef ront =
mi d
E0
vi−f ront = 2
e dt
ωp−i t
(3.3.30)
A more accurate result can be obtained numerically integrating the equation (3.3.3)
(3.3.4) (3.3.5) (3.3.6) with the initial solution for Ef ront given by the integral (3.3.13) of
Poisson equation:
kB Te
Ef ront,0 =
eλD
r
2
eN
(3.3.31)
The numerical integration furnishes an electric eld in function of the time that is very
well tted by the formula:
2E0
Ef ront ≃ q
2
2eN + ωp−i
t2
(3.3.32)
FIGURE
Integrating equation (3.3.29) and further integrating the result we obtain the ion front
ωp−i t
:
velocity and position at any position of the normalized time τ = √
2eN
√
vf ront ≃ 2cs ln τ + τ 2 + 1
(3.3.33)
h
√
i
√
√
2
2
xf ront ≃ 2 2eN λD0 τ ln τ + τ + 1 − τ + 1 + 1
(3.3.34)
More details on the electron and ion concentration and on the asymptotic ωp−i t 1
behaviour can be found in [Mora2003]. One of the most important parameter that this
model retrieves is the maximum energy for protons:
Emax ≃ 2kB Te [ln (2τ )]2
(3.3.35)
We already introduced it while talking about the parameters of TNSA in the previous
section.
The maximum velocity in the isothermal model diverges logarithmically with
time, while the total energy in the fast ions diverges linearly, so that to be able to apply
the model to the interpretation of experiments, one has to determine a characteristic time
at which the acceleration ends. A natural choice for that is the laser pulse duration but,
is not ensured that the acceleration stops eectively when the pulse ends.
As already
introduced in the previous section, in [Fuchs 2006] the eective acceleration time (or limit
time) is:
tacc = 1.3 × τL
(3.3.36)
tacc = α (τL + 60 [fs])
(3.3.37)
19
2
with α ranging between 2.6 for intensities ≈ 10
[W/cm ] and 1.3 for I > 3 ×
19
2
10 [W/cm ]. Using that approximation we can calculate also the proton spectrum:
" r
ne0 cs tacc Ssheat
dNp
=
exp −
dE
(2EkB Te )1/2
104
2E
kB Te
#
(3.3.38)
3 Ion beam generation by laser-matter interaction
which symbolism is described in a previous paragraph.
Two electrons populations
The expansion process of plasma into vacuum is dominated by the electrons heated
by the laser beam. In the previous paragraph we didn't take into account that there is
another electron population which, regarding its density, is not negligible comparing to hot
electrons: the cold electron density is usually greater of about 3 order of magnitude than
the hot electron density. It can be shown by simulation that this cold electron population,
doesn't aect the ion nal energy, thus the contribution of cold electrons can be neglected.
More detailed and very interesting discussions are given in [Antici 2008, Mora 2005].
3.3.1.2 Adiabatic expansion
The isothermal model presented in the previous paragraph assumes a constant electron
temperature, which can be a reasonable assumption only during the heating by the laser
pulse, but is certainly violated for late times, because the electrons progressively give their
energy to the ions and cool down in the expansion. These consideration were developed
in [Mora 2005] in a more realistic model which takes into account that the plasma in
expansion is not innite (i.e.
the plasma has not a innite reservoir of energy):
the
electrons cool down due to the energy transfer to the ions and the charge separation
eects. Other important kinetic approaches of this problem were succesfully attempted
by Kovalev et al.
in [Kovalev 2001, Kovalev 2003] generalizing the particular solution
given by [Dorozhkina 1998] using the
renormalization group method proposed in [Kovalev
1998 bis MANCANTE]. All these previous mentioned approach give an exact solution in
the quasineutral limit for the adiabatic plasma expansion of plasma bunches into vacuum.
[Dorozhkina 1998]obtained an exact self-similar solution to the three-dimensional Vlasov
equation for electrons and ions in the quasineutral approximation but this kinetic solution
describes the expansion of a plasma bunch into vacuum with arbitrary initial distribution
function, only in the particular case in which the electrostatic potential is quadratic in
the spatial coordinate: although the distribution function have an arbitrary shape, this
quadratic dependence implies the same dependence for the electron and ion distribution
functions on the coordinate and velocity. [Kovalev 2001]
The renormalization group approach allows the derivation of an exact solution to the
Vlasov equations for the electrons and ions in the quasineutral collisionless approximation.
The solution in [Kovalev 2001] describes the one-dimensional nonisothermal expansion of
a plasma bunch with arbitrary initial velocity distributions of the electrons and ions, and
with adiabatic cooling of the particles. However, as stated in [Mora 2005], these solutions
require specic initial conditions correlating the spatial and the velocity space; they do
not describe the charge separation eect and the structure of the ion front, which is crucial
in the determination of the maximum velocity of accelerated ions.
Let's consider a thin foil collisionless adiabatic expansion into vacuum. The thickness
L
of the foil is
and the initial position of the edges are in x = ± : this allows to study
2
only the half space x ≥ 0 for evident reasons of simmetry. At time zero the ions are
L
cold and at rest, and the plasma is supposed neutral with sharp boundary edges, as in
the isothermal expansion. The equations that rule the expansion are always the same of
(3.3.3) (3.3.4) (3.3.5) (3.3.6). Of course the dierences between the semi-innite plasma
105
3 Ion beam generation by laser-matter interaction
expansion model and the thin foil arise from the boundary conditions: the electric eld
must be zero in x = 0 and in the limit x → ∞. Also vi must be zero in x = 0 for every
instant of time. Furthermore the electron temperature is a function of time Te = Te (t)
and can be calculated with the energy conservation equation:
dUe dUi dUf ield
+
+
=0
(3.3.39)
dt
dt
dt
where Ui is the kinetic energy of the ions, Uf ield is the electrostatic energy of the eld,
and Ue is the thermal energy of the electrons, that is:
Ue = ne0 L × g (θ) × kB Te
(3.3.40)
The function g (θ) takes into account the physical ultrarelativistic or classical limit with:
θ=
kB Te
me c2
(3.3.41)
and
g (θ) = 21 θ = 0
g (θ) = 1 θ → ∞
(3.3.42)
As an alternative, the thermal energy of the electrons can be calculated throughout the
word done by the electric eld on the electron uid:
dUe
=e
dt
ˆ +∞
φ
−∞
∂ne
dx
∂t
(3.3.43)
If the electron temperature was maintained constant by an external source of energy, it
would take a time tL = L/2cs0 for the rarefaction wave to reach the center of the foil. It
means that tL can be an appropriate parameter to characterazie the expansion time of the
foil. For t tL the model gives results very close to the isothermal model, but for later
time, t tL , the cooling of the electrons is eective. In that limit the velocity saturates
x
in fact as can be seen in the graph:
to the value v ≃
t
In gure is also shown the solution of [Kovalev 2001]: in [Mora 2005] the self-similar
character is only acquired in the limit t tL in contrast to the selfsimilar character of
other solution proposed.
The asymptotic behavior of the electron temperature can be obtained from (3.3.43) as
to be:
dTe
Te
=−
dt
g (θ) t
(3.3.44)
thus separating the variables integrating:
1
Te (t) ∝ t− g(θ)
(3.3.45)
−2
In the classical limit the electrons exhibit a fast cooling proportional to ∝ t ; otherwise in the ultrarelativistic limit cooling is slower with a hot electron temperature
−1
decreasing proportional to∝ t . [Mora 2005]. Always in the limit of late time t tL ,
106
3 Ion beam generation by laser-matter interaction
also the density prole becomes self-similar, with n =
f (x/t)
, and the characteristic length
t
dened in (3.3.20)
∂
le = le (x, t) = − ln
∂x
n
n0
−1
(3.3.46)
becomes a linear function of time, when considered as a function of
x, except for the ion
front for which the local Debye length is the relevant characteristic length.
The electric eld is given by
E=
kB Te
ele (x, t)
(3.3.47)
and as can be seen in gure NEEDED, initially has linear increasing, after a large
plateau, and nally at the ion front it reaches the value:
Ef ront =
kB Te
eλD
(3.3.48)
where λD is the local Debye length.
A rened model presented by [Grismayer 2006] takes into account a plasma with a
smooth boundary edge: it means that the plasma scale lenght is nite and can inuence
the maximum electric eld.
(
ni (x, t = 0) =
ni0
ss
ni0 exp − x+l
lss
−L/2 < x < −lss
x > −lss
(3.3.49)
For the adiabatic case, the eect of the initial scale length is to delay the decrease of the
temperature. This is due to the reduction of the transfer of energy between the electrons
and the ions for increasing values of
3.3.1.3 Hot electrons and ions expansions and angular distributions
[Roth 2005, Cowan 2004, Wilks 2001]
107
3 Ion beam generation by laser-matter interaction
3.3.2 Radiation Pressure Acceleration model
As already introduced in the previous section, in addition to the target normal sheat
acceleration TNSA scheme, there are other promising technique that can allow us to obtain
laser-driven highly collimated high and mono-energetic proton/ion beams: the radiation
pressure acceleration technique (RPA). In fact RPA is presented by [Macchi 2008] as the
best substitute to the classical, widely and extensively studied TNSA.
A conspicuous number of papers on this topic shows by the use of simulations that
23
2
the RP scheme can dominate at both very high intensities (I ≥ 10 [W/cm ]) and lower
intensities, but in the latter case only using a circularly polarized laser beam. The early
studies of RPA suggested that the target foil can be fruitfully accelerated as a whole rigid
object, leading to quasi mono-energetic ion/proton spectra, and also that the acceleration
process is very ecient so that the characteristic parameter
energy per nucleon scales
in a good way with the laser pulse energy leading naturally to the relativistic domain
(GeV/nucleon).
Two of the most important requirements, that must be met, are: manufacturing very
thin target (few nanometers thin), achievement of ultra-high-contrast ratio (UHC) laser
pulses using plasma mirror technique in order to not disrupt destructively the target itself.
We will describe and detail the operation of plasma mirrors later on in the next chapter.
We already said that two RPA regimes exist: hole boring regime and light sail regime. We
will describe them in detail in the next two paragraph.
Now we want to demonstrate the role of the pulse polarization in laser-plasma interactions.
Recalling that the electromagnetic elds wave propagating in vacuum can be
expressed by a potential vector A, the general solution of wave equation of frequency ω
for the potential vector is:
A (r, t) = A0 exp [i (k · r − ωt)]
If we consider an e.m.
wave with elliptical polarization propagating along
(3.3.50)
x toward
an homogeneous overdense plasma, we are forced to write the expression of the potential
vector as follows:
A0
A (x, t) = √
exp [ikx] (ŷ cos ωt + εẑ sin ωt)
1 + ε2
(3.3.51)
with ε ellipticity factor of the wave. The range of variation of the ellipticity is, of course,
0 ≤ ε ≤ 1: with ellipticity approaching zero we obtain a linearly polarized wave, otherwise
a perfect circularly polarized wave. In an overdense plasma, where ω < ωp−e , or using
the densities ne > nc , the wave vector is purely imaginary because:
2
c2 k 2 = ω 2 − ωp−e
(3.3.52)
1q 2
iq 2
2
k=
ω − ωp−e =
ωp−e − ω 2
c
c
(3.3.53)
and:
Thus:
108
3 Ion beam generation by laser-matter interaction
A (x, t) = √
i
h xq
A0
2
2
exp −
ωp−e − ω (ŷ cos ωt + εẑ sin ωt)
c
1 + ε2
(3.3.54)
It can be demonstrated throughout the formula of the ponderomotive force (3.1.7) that
the longitudinal force along
x direction is equal to:
2x q 2
ωp−e − ω 2
Fx = F0 exp −
c
1 − ε2
cos 2ωt
1+
1 + ε2
(3.3.55)
With this force, the electric eld perturbation and the electron density are:


q
2
2x
1 − ε cos 2ωt 
F0
2
exp −
− ω 2 1 +
Ex =
ωp−e
e
c
1 + ε2 1 − 4 ωω2 2
(3.3.56)
p−e
δne = −
1 ∂
Ex
4πe ∂x
(3.3.57)
Integrating over the space (supposing innite extension of the plasma region) we obtain
the periodical variation of number of electron given by:
ˆ +∞

δne dx ∝ 1 +
∆Ne =
0

2
1 − ε cos 2ωt 
1 + ε2 1 − 4 ωω2 2
(3.3.58)
p−e
If this quantity is ∆Ne > 0, it means that electrons are accumulating inside the plasma,
and a surface positive charge density appears to balance this increment. If ∆Ne < 0 it
means that electrons are escaping in the vacuum at x = 0; this can occur when ε = 0,
i.e. in the presence of a linearly polarized electromagnetic wave, because the oscillating
factor becomes, in periodical intervals of time, greater than unity: this is a clear signature
of the onset of self-intersection of electron trajectories in vacuum leading to the electron
heating and to the appearance of fast electron jets twice per laser cycle [Macchi 2008].
The model predicts an
ellipticity threshold εT that is the limit beyond which (ε > εT ) an
elliptical polarization permits to the electrons to pile up in the plasma giving ∆Ne > 0
at every instant of time, and theoretically inhibiting the heating, during the laser pulse
duration: the circular polarization with ε = 1 is the limit case. So is clearly demonstrated
that in presence of circular polarization laser beam the oscillating term of the v × B force
is eliminated.
3.3.2.1 Hole boring regime
The term
hole boring is referred to a process of ion acceleration that is simply related to
the formation of an electrostatic eld due to the electrons displacement by ponderomotive
force that arises from CP laser light; that force pushes electrons away and consequently
causes the ions to move, reacting to this charge displacement. The gure extracted from
[Macchi 2005] gives evidence, with a 1D simulation, of ion acceleratio in this regime.
FIGURE 1 macchi 2005
109
3 Ion beam generation by laser-matter interaction
The ion density prole shows a clear separation of two ion peaks after a certain period
of time. The preceding bunch has an higher velocity, as is shown in the phase space, comparing to the following bunch. Also 2D simulations are in agreement with this behaviour
and the name hole boring clearly arises from the ion bunch shape that is shown in gure
below:
All these simulations are in fair agreement both qualitatively and quantitatively with the
model that we are going to explain. In a simple monodimensional electrostatic scenario,
the supposed, simplied, electron and ion densities are:
The distance

0<x<d
 0
n
d
<
x < d + ls
ne =
p

n0
x > d + ls
(3.3.59)
ni = n0
(3.3.60)
d represents the point, called evanescence point, in which the laser ceases
to inuence the plasma, in fact is the point where starts to be located the electron density.
Using the equation of the divergence of the electric eld we have:
∇·E =
ρ
0
(3.3.61)
that in the three dierent intervals, with ρ = e (ni − ne ) simplies into:
e
n
0<x<d
0 0
dE
e
(n0 − np ) d < x < d + ls
=
 0
dx
0
x > d + ls


(3.3.62)
Integrating we have:


en0
x
0
E (x = d) + e0 (n0 − np ) (x − d)
E (x) =

0
0<x<d
d < x < d + ls
x > d + ls
(3.3.63)
The continuity of electric eld requires:
en0
d = E (x = d) = Ed
0
(3.3.64)
e
(n0 − np ) ls = 0
0
(3.3.65)
and:
Ed +
thus leading to the simple charge conservation:
(np − n0 ) ls = n0 d
(3.3.66)
that coincides with the equivalence of the two rectangles in gure: (gure densities
and electric eld needed (macchi 2005))
110
3 Ion beam generation by laser-matter interaction



Ed
0<x<d
dx x−d
E (x) =
d < x < d + ls
Ed 1 − ls


0
x > d + ls
(3.3.67)
This electrostatic eld is due to the balance between the electrostatic pressure in the
so called compression layer (d < x < d + ls ) on the electrons and the radiation pressure
roughly given by 2I/c. The electrostatic pressure is:
ˆ d+ls
ˆ d+ls
−eEd
−eE (x) ne (x) dx =
P =
d
d
x−d
1−
ls
np dx = −eEd np ls =
e2 n0 np
ls d
0
(3.3.68)
The electrostatic eld on the ions starting at initial positions
x0 < d is a constant
over the trajectories of the ions, and increases with dierent initial but increasing initial
position. These ions will never reach the ions with initial position x1 > x0 because:
a (x) =
qi
E (x)
mi
(3.3.69)
Considering two dierent particle with dierent initial position x01 < x02 , both in[0, d],
the relative accelerations follow the linear law (3.3.67), and thus a1 < a2 , thus the particle
1 will never reach particle 2.
Since the electrostatic eld is a linear function of the initial position as stated by (3.3.67),
all the ions in the compression layer will reach point d + ls at the same time (that is
the reason for naming this zone
compression layer ). Supposing, as before, two dierent
particles, now in in[d, d + ls ], their dierent position in time are described by the simple:
1
x1 (t) = a (x01 ) t2 + x01
2
1
x2 (t) = a (x02 ) t2 + x02
2
The instants of time at which they arrive at d + ls :
s
v
u 2m (d + l − d − ξ )
2mi ls
u i
s
01 =
t∗ = t
qi Ed
q E 1 − ξ01
i
d
(3.3.70)
(3.3.71)
(3.3.72)
ls
where we have set x01 = d + ξ01 for simplicity. This time is independent on the initial
position hence all the ions reach the nal position at the same time. The ion density will
assume an innite value there: this model breaks, but after that point the ion bunch is
injected into the plasma, and travels ballistically in this unperturbed region, because of
the charge unbalance is neutralized by electrons accompanying the bunch.
The nal velocity is dierent for each particle, thus leading to a spread in the spectrum
of the ions that form bunch:
r
v (t∗ ) = (ls − ξ0 )
111
2Ed qi
mi ls
(3.3.73)
3 Ion beam generation by laser-matter interaction
and its maximum value is:
r
vmax =
2ls Ed qi
2ls
= ∗ = 2va
mi
t
(3.3.74)
where va is the average ion velocity. During the compression of the ion uid the electric
eld will tend to penetrate deeper into the plasma, keeping the eld at the surface and the
total electrostatic pressure constant. Thus, beyond
d position, ions will be accelerated by
a eld decreasing in time and will get to the breaking point later. This eect causes the
ion bunch to be more localized both in coordinate and velocity space.
If the pulse is not yet over when the ion bunch creation occur, then electrons that have
penetrated into the plasma rearrange to nd another equilibrium, and suddenly can occur
another ion bunch creation [Macchi 2005]. Another interesting way to see the expressions
(3.3.72) and (3.3.74) is by using the laser parameters:
s
Z me nc
aL
A mp ne
r
1
A mp
t∗ ≃
ωL aL Z me
va
=
c
(3.3.75)
(3.3.76)
eEL
is the normalized eld amplitude of the laser light.
me ωL c
Discussion on those topics can be found in [][]. The 1D and 2D PIC simulations proposed
with aL =
therein show a good agreement with the models.
3.3.2.2 Light sail regime
The equation of motion for a thin foil of thickness l , mass density ρi , reectivity
R under
the radiation pressure of a coherent laser light is:
p
p 2 + σ 2 c2 − p
2I
dp
= Rp
dt
c
p 2 + σ 2 c2 + p
where σ = ρi l is the areal mass,
(3.3.77)
p is the areal momentum, I the laser intensity, c the
velocity of light. Another general expression is:
d
(βγ) =
dt
(3.3.78)
It can be shown that this equation has an analytical solution [Esirkepov 2004, Robinson 2008]
if we suppose constant intensity and unity reectivity:
p = σc sinh (u) −
where the variable
1
4 sinh (u)
(3.3.79)
u is dened:
6I
1
u = arc sinh
t+2
3
σc2
112
(3.3.80)
3 Ion beam generation by laser-matter interaction
in the reasonable case of initial momentum equal to zero. We can calculate the kinetic
energy of the particle recalling that:
ε 2 = m i c2
2
+ (pc)2
(3.3.81)
and:
ε = εk + mi c2
(3.3.82)
So we have:
s
ε k = m i c2  1 +
note that in this equation
p
mi c
2

− 1
(3.3.83)
p is the proper momentum, but pareal /σ = p/mi and thus
using (3.3.79):
εk = mi c sinh (u) +
2
1
−1
4 sinh (u)
(3.3.84)
1 6I
t
4 σc2
(3.3.85)
The asymptotic time tendency is:
r
εk ≈ mi c2
3
To nd an upper limit of the ion energy acquired due to the interaction with a laser
pulse of nite duration, [Esirkepov 2004] includes the dependence of the laser EM eld
on space and time. Further discussion on those topics are in [Macchi 2008, Macchi 2010,
Robinson 2008, Marx 1966, Forward 1984].
Simulations by []
113
3 Ion beam generation by laser-matter interaction
3.4 Characteristics of the ion source
We have shown theoretically that the laser/target interaction can act as a proton or ion
source when certain specic conditions are met.
Here we present a list of interesting
parameters that can fully characterize the proton beam.
They can be seen as gure of
merit of the laser/target interaction and in fact are used to summarize if the beam is close
or not to the specic needs of an application. We will talk about:
ˆ spectrum
ˆ laminarity or emittance
ˆ source and beam size
ˆ angle of divergence
ˆ virtual source point
Spectrum
The most important characteristic of a proton/ ion beam is its spectrum dN/dE , that
gives the measure of the number of the particle in the beam at given energy value.
It
can be expressed in unit of number of particle per unit energy [number/MeV] (i.e.: how
many particle of a xed energy are present in a xed energy spread interval) or number
of proton per unit energy per unit of solid angle of emission [number/MeV/sr].
The ion/proton spectra can be measured with several instruments:
ˆ Radio Chromic Films:
using a multilayer stack of calibrated RCF lms we can
retrieve information about the absorbed dose and on the energy of protons. Using
several RFC stacks positioned in dierent directions we can have information about
the spatial distribution of the beam allowing us to measure the divergence angle.
ˆ Faraday cups: is a cup-shape detector that catches particles emitted from the target;
it gives a current response equal to the number of impinging particle.
ˆ nuclear activation measurements:
ˆ magnetic spectrometer: it analyzes protons; an applied magnetic eld bends the
protons trajectories resulting in a space distribution of the proton beam due to
dierent energies. The electron magnetic spectrometer uses the same principle and
is described in appendix.
ˆ Thomson parabola: it can analyze ions bending their trajectories by applying an
electromagnetostatic eld; the traces on the detector plate are dierent parabolas
depending on charge to mass ratio; the higher the velocity the smallest the deection.
One of the most important features is the spread of the spectrum: in many applications a
very small ∆E/E is required, and if ∆E/E → 0 the proton beam is said to be monoenergetic around the energy value E .
Another interesting characteristic is the maximum
proton energy, that can be considered as the cut o of the spectrum. We will describe in
114
3 Ion beam generation by laser-matter interaction
the fourth chapter the experimental conditions that can be changed to maximize the maximum energy. Optimization of the maximum energy is one of the main research topics of
the last decade since the beginning of the studies of laser plasma interaction for ion/proton
acceleration. Eorts were focused on the possible scaling laws of proton energy, and the
most investigated direction is simply to improve the laser characteristics like beam energy,
focal spot size, pulse duration, ASE pedestal and prepulses suppression, high contrast.
Other parameters that aect the proton energy are related to the target.
During our
experiments we have tried to see the behavior of the laser-target interaction throughout
the analysis of maximum energy and proton spectrum. We present our results in chapter
4.
We have already seen in equation (3.3.38) that the plasma expansion model gives us the
spectrum of the proton with an exponential decay governed somehow by the hot electron
temperature. Another model explained in [Antici PhD 2007] arise from sperimental data
and takes into account that the observed proton spectrum has two dierent decays at low
and high energy; therefore we can write:
h q
h q
i
i
2E
2E
exp
−
exp
−
kB T1
kB T2
dN
∝ √
+ √
dE
2EkB T1
2EkB T2
(3.4.1)
Laminarity and emittance
The laminarity of a proton beam is the property that ensures the possibility to obtain a
point to point relationship between the position of a collected proton on the detector and
position of the emitted proton on the source: this menas that we can trace the trajectories
of the protons and that those trajectories are not intersecting and overlapping each other.
One experimental method to retrieve information about the degree of laminarity of a proton beam is to implement on the rear side of the target a microstructured micrometric
pattern that, in the presence of a laminar beam, must be reconstructed for example analyzing the proton deposited dose on RCF lm. The more detailed is this reconstruction
(i.e.
if the proton image on RCF shows micrometric details of the structure) the more
laminar is the beam.
The emittance of a ion/proton beam is a more quantitative characteristic than the
laminarity and it's connected to the dynamic of the particle itself: the emittance is the
measure of the area in the phase-space occupied by the particles of the beam, hence it
gives information on the correlations between position and momentum of the particles in
the beam. Assuming the cilindrical simmetry of the proton beam emitted from the rear
r is the distance from
z. Hence the transverse emittance (i.e. the emittance of the beam
side of the target, we can use a two coordinate system (r, z) where
the axis of simmetry,
0
in the trasverse plane) is dened as proportional to the area of the phase space (r, r ).
115
3 Ion beam generation by laser-matter interaction
Figure 3.4.1: Comparison of dierent proton beam transverse emittance (taken from
[Antici PhD 2007]).
A perfect laminar beam has zero transverse emittance because its phase space is composed by a straight line. A more turbulent beam has a phase space surface that is spreaded,
hence is not perfectly laminar. For a better understanding of the denition see the gure
(3.4.1). For calculate the emittance we can approximate the surface of the phase space
with the ellipsoid surface that bounds for example 90% or 95% or 99% of the total number
of particle.
We can also understand if a beam is divergent or collinear by noting that
in the former case, at a xed instant of time, its phase space ellipsoid is tilted from the
z-axis, whilst in the latter case is not tilted (see gure (3.4.1)).
Regarding the longitudinal emittance, a reference particle must be xed, and from that,
calculating the phase space is easy to obtain the longitudinal emittance with the same
denition as for the transverse emittance.
116
3 Ion beam generation by laser-matter interaction
Source and beam size, angle of divergence and virtual source point
Using a RCF stack to analyze the proton beam, we can measure
ˆ the size of the beam: measuring for each RCF layer the surface where protons are
absorbed.
We can see that low energy protons are absorbed in the rst layers of
the RCF stack, and they have a broader surface comparing to the more energetic
protons of the subsequent layers.
ˆ the size of the source: counting the number of grooves that are visible in each RCF
layer and multiplying it by the spacing between two grooves on the rear side, we can
roughly obtain informations on the size of the source for protons that have energy
in the range for that specic layer.
ˆ divergence angle: for each layer we can measure the size of the beam, and knowing
the distance between the RCF and the target we can calculate the divergence angle
for that energy.
ˆ the virtual source point: is a virtual point, outside the target in the half space of
the front surface, where all the trajectories of the ions come from.
117
3 Ion beam generation by laser-matter interaction
3.5 Applications for laser generated proton/ion beams
3.5.1 Isochoric heating
[Mancic 2010]
High energy lasers are an unique tool to recreate inside a laboratory exotic states of
matter that are usually not existing on earth. Hence, the possibility of producing, in a
controlled way, matter at solid density (for most common element such as metals being
between 1-10 g/cm3) while maintaining it at a high temperature (0.1-100 eV, 1 eV being
11600 K), known as isochoric heating (isochore = same density), has become a desired
goal. Studying matter in these conditions would provide an array of information useful
for a variety of problems, not only in the eld of astrophysics (matter in the planetary
cores), but also for fundamental physics (dense and relatively cold matter at the frontier of
plasma physics) and inertial connement fusion (stopping power of particles in warm dense
matter). Figure 1.24 shows the dierent Equation of state (EOS) regimes that exist in our
universe (Sun, Supernova), As can be seen from the graph, current existing laser facilities
(NIF, short pulse laser plasmas) are able to reproduce such extreme regimes. To achieve
isochoric heating, energy has to be deposited in a short time scale before heating-induced
expansion decreases the density of the heated matter. The heated material is then inertially
conned. Up to now, attempts to achieve isochoric heating have been pursued either using
short-pulses lasers or short-duration laser-produced X-rays but optical frequencies are very
limited in their penetration of solid matter, tipycally 5 nm of skin depth in Al for 1 µm
light (see Figure 1.26). X-rays have also a limited penetration compared to protons (see
Figure 1.16).
3.5.2 Inertial Connement Fusion
The rst application scientists were thinking of in 1992 for laser accelerated ions [Tabak
2006] (knowing the results of the CO2 laser) was fusion and to use protons as energy
transfer medium to heat the ignition region. In this introduction we will only discuss the
application of protons to ignition, in particular fast ignition.
Recent publications have
proposed the use of protons for the fast ignition [Roth 2001], in better energy deposition
and can be focused to the hotspot [Patel 2003, Antici ECA 2006].
Due to their Bragg
peak, they are able to deposit their energy very locally and are therefore very suitable for
fast ignition.
3.5.3 Hadrontherapy
One of the leading for proton acceleration in general is in the medical eld, with focus on
proton therapy and isotope production for Positron Emission Tomography (PET).
There are two possible types of therapy. By passive attack and active attack. In the
passive attack, the part of the human body is irradiated constantly by a large proton
beam (30 x 30 cm).
is obtained.
With the preformed mask, the localization of the dose deposition
In the active attack, the proton beam is much more intense (3 - 6 mm
diameter) and moves to irradiate the part of the human body. The localization of the dose
118
3 Ion beam generation by laser-matter interaction
deposition can in that case also be obtained by increasing or decreasing the energy of the
proton beam.
Present-day proton therapy centers throughout the world use as particle
source accelerator mainly cyclotrons and synchrotrons. Due to the size of the accelerator
and mainly of the beam transport lines, an entire building (the size of a football eld) is
needed to house this equipment, which can weigh up to 900 tons. Using compact lasers
could therefore allow to greatly reduce the size of the overall accelerator and transport
system. The beam requirements for a proton therapy center are generally about 250 MeV,
as monoenergetic as possible, since a non-monoenergetic beam would destroy human cells
adjacent to those targeted: for a precision of ~1 mm the energy spread needs to be < 0.3
MeV. Note that an energy spread of ~1% is generated when the particles pass into the
human tissues, due to straggling. The beam intensity in the order of nA. Note that for
the human body it is not relevant if protons arrive continuously or pulsed, since human
cells do not react dierently when a source is continuously or pulsed but with the pulse
duration < 100 ms (i.e. repetition rate higher than 10 Hz). For a full treatment, usually
about 60-71 Gy are needed that are deposited in doses of 2 Gy per session for 5 days a
week. The dose is given within ~1 minute (in some cases dierent ways of treatment can
be applied). The estimated cost (all inclusive) per treatment session is ~1000 ¿.
3.5.4 Isotope production for positron emission tomography
Positron Emission Tomography (PET) A second application of protons in medicine is
the Positron Emission Tomography (PET), a nuclear-medicine technique used for medical
imaging. This kind of tomography uses positron emitters to characterize the biochemical
function of cells, organs, and body structures in vivo, producing a three-dimensional image
or map of functional processes.
The basic process of PET is described as follows:
In
the rst phase (1) Chemical elements are irradiated with protons to produce radioactive
isotopes (lifetime 20 to 110 minutes).
Those isotopes are attached to glucose, water of
ammonia producing what is called a radiotracer (2), now the liquid can be injected in
the part of the body that needs to be analyzed (3).
After a waiting period of about
30 minutes to 1 hour, while the metabolically active molecule becomes concentrated in
tissues of interest, the patient is placed inside the imaging scanner. The decay process of
the radiotracer (explained hereafter) is traced by a computerized system that allows data
reconstruction: In the decay process (the radioisotope undergoes positron emission decay)
(see Figure 1.21), the positron emitted by the isotope encounters and annihilates with an
electron.
During this process two photons (gamma rays) of 511 keV energy (part a) of
the gure) are created that travel in opposition direction.
These photons are detected
when they reach a scintillator material in the imaging scanner creating a burst of light
that can be detected. From the time of ight of the two photons it is possible to detect
where the annihilation event has taken place (if the recovery time of the detector has
a resolution that is high enough, i.e.
in the ps range).
Since cancerous cells use more
glucose than normal tissue they appear brighter on the camera images (part c) of the
gure). It is therefore possible to detect them earlier than with conventional tomography
where signicant masses of cancerous tissues need to develop before being able to be
detected. As already said, typical isotopes used for PET have a very short lifetime (a few
minutes till about two hours), which makes it therefore hard to prepare the radioactive
119
3 Ion beam generation by laser-matter interaction
liquid much in advance. Hence, the production of isotopes needs to be made close to the
tomography machine, but, once more, unfortunately, the accelerators needed to create the
isotopes are relatively big (a commercial accelerator for PET has as size 6.1 x 0.9 x 1.8
m, weight 3 tons) and expensive (1 M¿), which makes the PET a not easy and aordable
way of tomography.
3.5.5 Proton radiography
Another interesting application of laser-generated proton beams relies on their unique
spatial properties of high laminarity or how emittance (details will be given in chapter 2).
They can therefore be used in radiography, producing images with high spatial resolution.
altre: Sarri
3.5.6 Particle accelerator
120
121
4 Description and results of the experiments
4 Description and results of the
experiments
4.1 Outline of the interaction chamber and diagnostics
Figure 4.1.1: Schematic outline of the chamber
122
4 Description and results of the experiments
In (4.1.1) is presented the schematic outline of the interaction chamber. In the following
paragraog we will detail each part of the interaction chamber, and the diagnostics used
during the experiments. In fact, several measurements have been performed during each
shot. The diagnostics used during the experiments are the following:
ˆ imaging line
ˆ shadowgraphy
ˆ TASRI (Time And Space Resolved Interferometry)
ˆ Time Of Flight (TOF) measurements
ˆ Spectralon.
4.1.1 Laser beam path and imaging line
The laser beam path is depicted in gure(4.1.1) with a red line: it comes from the transmission line after the compressor, and its alignment is done throughout a system of two
mirrors inside the trasmission line, each one located in a turning box.
The alignment
of the principal beam line is carried out roughly rst with a HeNe red laser located at
the entrance of the rst turning box. The laser beam of approximately 7 cm of diameter
impinges on the 4 inches diameter mirror, and after reaches the o axis parabola that
focalizes it on the target-chamber center (TCC) which is at a distance of about 30 cm.
The beam before reaching TCC is reected on a plasma mirror that increases the contrast
ratio of the beam itself.
The gure (4.1.2) below shows the working principle of an o-axis parabola or, more
properly, an o axis
parabolic mirror.
Figure 4.1.2: Sketch of the o-axis parabolic mirror working principle.
After the parabolic mirror, on the main beam path there is the plasma mirror.
The
optical axis of the laser beam has 45 degrees of incidence with both plasma mirror and
target at TCC. The target chamber center of course is the point in the space of the
interaction chamber where the main beam is focused and that's the point where the target
front surface must be centered. In fact the targets are mounted on a target-holder wheel
that can hold 24 target, including thin tips, wires and silica glasses for plasma mirror and
123
4 Description and results of the experiments
diagnostics alignments. The selected target is positioned at the center with the help of a
motorized drive system (see gure (4.1.3) ):
ˆ 4 motors are controlled with a Labview interface and are x-y-z for the translational
movements, w for the wheel rotation.
ˆ 2 additional motors, roughly controlled with two switches, allow the vertical and
horizontal tilt of the wheel.
Figure 4.1.3: Sketch of target alignment motors system.
To ensure the proper alignment, a reference-marks system with three monitor cameras
is used in conjuction with the HeNe laser, (see gure (4.1.1) and (4.1.3) ):
ˆ shadowgraphy camera:
this camera is used also as a diagnostic device (see next
y -direction, ensures
in conjunction with the Labview interface the proper positioning in the z -direction
(thanks to the number of motor-steps /micrometer calibration), and allows to monitor the vertical tilt of the target. The camera helps the x -direction alignment, that
paragraph). This camera provides the correct alignment in the
124
4 Description and results of the experiments
is parallel with the shadowgraphy direction: in fact we can see that when the target
is well aligned in the
x -direction, the image of the shadowgraphy camera is perfectly
focused.
ˆ top view camera: this camera, similarly to the previous one, is also used as a diagnostic device (see paragraph). The camera helps to align the target in the
y -directions. It gives also information abou the horizontal tilt.
ˆ chamber camera:
x and
this camera is a monitor camera, and helps the alignment in
conjunction with the other two cameras.
The so called
imaging line is an optical system which allows to view on a camera the TCC
point, and in fact is used to collect images of the laser beam focal spot. As shown in gure
(4.1.1) the imaging system in the chamber consists in the series of a wedge, a
zero degree
mirror, and other mirrors. The laser beam focalised by the o axis parabola at TCC loses
the main part of its energy passing through the zero degree mirror because clearly it is
used
in trasmission (and not in reection): this, in addiction to some optical densities
(OD), allows the beam to reach the camera for recording the image without damaging it.
If the laser beam has a perfect circular with hat-top intensity prole, the focusing lens
will produce on the camera a perfect Airy disks pattern (that could come out also from a
Frahunofer diraction pattern of coherent light through a circular aperture).
The normalized light intensity of the Airy disks pattern is given by:
Iairy (r)
=
I0
2J1 (x)
x
2
where J1 is the Bessel's function of rst kind and its argument
x=
where
πr
fλ
(4.1.1)
x is:
(4.1.2)
f is the f-number of the o axis parabola, and r is the radial distance from peak
(center of the Airy disks). A plot of the Airy disk function is given in gure (4.1.4):
125
4 Description and results of the experiments
Figure 4.1.4: Airy disks normalized intensity in function of the Bessel argument x =
πr
f λ
The laser energy encircled in the rst three lobes of the perfect Airy disk are respectively
83.8%, 91.0%, and 93.8%. Because of the non perfect hat-top shape of the laser intensity,
the image collected by the camera can not be the Airy disk pattern.
Using a Matlab
routine given in appendix we can evaluate the encircled energy percentage of the rst lobe
or at full width half maximum (FWHM). In gure (4.1.5) we show an example of focal
spot image, and in gure (4.1.6) the corrispondent plots of the normalized intensity prole
and the percentage of encircled energy in function of the radial distance from the intensity
peak.
126
4 Description and results of the experiments
Figure 4.1.5: Cropped focal spot image. We used the free software ImageJ to apply to our
greyscale, 1280 pixel×960 pixel 16-bit image, a simple colored look-up table
re.
Figure 4.1.6: Example of a normalized intensity prole and encircled energy in function
of the radial distance from the peak of intensity. The encircled energy for
the rst lobe (radius of the rst lobe aproximately 6,5 [µm]) is 60%, far less
from the perfect Airy disk value of 83,8%. The focal spot FWHM is ≈6 [µm]
(twice the radius where the normalized intensity falls to its FWHM).
127
4 Description and results of the experiments
4.1.2 Shadowgraphy
The shadographs are images taken by a camera using an imaging system with the help of
a probe beam, delayed of several nanoseconds from the main beam, used to illuminate the
target in a direction parallel to the front/rear surface. The imaging system of shadowgraphy line (that consists only in a lens that images the TCC position where the interaction
between laser and target takes place), and the probe beam path, are sketched in gure
(4.1.1). The probe beam is a low energy beam derived from the main beam, inside the
compressor chamber, before the main pulse compression; thereafter it is delayed using a
delay line, compressed in a other compressor (not in vacuum), and sent in the chamber.
The probe beam doesn't travel in vacuum because of its low energy. The shadowgraphy
line in the interaction chamber at ALLS 200TW laboratory was aligned simply using a
HeNe laser: this laser is also used for target alignment as already described in (4.1.1).
The probe beam is rst synchronized forcing it to interfere at TCC with the main beam
by manipulating the delay line: imaging the intereference pattern with the shadow line
gives an exact information on the zero position of the mirrors in the delay line. The displacement of the delay line mirrors from the zero position is calculated accurately from
the value of the desired delay; moreover, with a measurement system of the beam peaks
based on fast photodiodes, we retrieve informations on the delay xing it to the desired
value.
For each shot one shadograph is collected: an example of shadograph is given in gure.
128
4 Description and results of the experiments
4.1.3 Time and space resolved interferometry (TASRI)
129
4 Description and results of the experiments
4.1.4 Time of ight measurements
This kind of diagnostic consists in two long straight tubes in vacuum connected directly
to the interaction chamber, in the perpendicular direction of the targe, one in the front
side, the other one in the back side.
At the end of each tube, the combined use of
a scintillator and a photo-multiplying tube PMT, will analyze the electron and proton
beam as follows.
The scintillator catch the electron and proton beams, giving a light
signal signal proportional to the number of incident particles. The light output is a strong
function of the type of incident particle or photon and of its energy. The PMT consists in
a coupling of photocathode and a system of amplication consisting in a series of dinodes.
The overall response electric signal is a function of the number of electrons and protons
incident.
Hence the PMT detects the exact instant of time when particles strike the
scintillator.
The typical signal of a time of ight measurement diagnostic is given in gure (4.1.7).
This image was captured by an oscilloscope which trigger signal comes from a sync signal
distributed from the early stages of the laser chain.
Figure 4.1.7: PMTs typical output signal after a laser shot on target
Channel 2 waveform refers to front side, whilst channel 3 to back side.
The current
signal shows clearly two dierent peaks, the rst comes from the fast electrons, the second
130
4 Description and results of the experiments
to the slower, more massive, protons. To analyze this signal and have an estimation of the
maximum proton energy, we refer to the following schematic gure (4.1.8):
Figure 4.1.8: Time of ight schematic diagram of measurements
The instant of time when electrons arrive is te , while for proton is tp .
What we can
measure is only the interval, T , between the two dierent bunches arrivals. We can infer
that the electron velocity is approximated
c and thus, knowing the length of the path
LT OF traveled by both electrons and protons, (i.e. the distance from the target chamber
center (TCC) to the PMT), we can calculate the instant te , by the simple:
te ≈
LT OF
c
(4.1.3)
With that information we can calculate also the velocity of the protons, and write with
a similar argument:
vmax ≈
LT OF
LT OF
c
≈ LT OF
=
te + T
1 + L cT
+T
c
(4.1.4)
T OF
v
After that, we refer this velocity, to the speed of light, calculating the β =
and
c
γ = √ 1 2 factors. The energy of protons is given by E = (γ − 1) mp c2 . The proton mass
1−β
2
is approximately 938 [MeV/c ].






1

E = mp c  s
−
1

2


1
1 − 1+ cT
2
LT OF
131
(4.1.5)
4 Description and results of the experiments
Since the instant of time tp can't be exactly evaluated, this will aect the uncertainty
on the determination of the maximum energy. It is interesting to repeat these calculation
for each collected point after the extimated tp on the waveform. The correct evaluation of
the tp is also aected by noise in the oscilloscope traces. We developed a Matlab routine
(reported in appendix) to analyze the samples collected by the oscilloscope, and doing
that what we found the proton spectrum. An absolute calibration of the scintillator and
PMT system was not performed, hence the unit of the measures is arbitrary.
Our experimental setup provides two time of ight lines: one corresponding to the front
side of the target and one on the rear side of the target.
The length of the two paths
are 2.18 [m] and 2.51 [m] for TOF front and TOF rear side respectively, leading to two
f ront
≈ 8.367 [ns]. An example
≈ 7.267 [ns] trear
dierent electron-arrival time-references te
e
of the rear and front side proton spectra is given in gure (4.1.13):
Figure 4.1.9: Rear side and front side proton spectra for a
To extimate the horizontal error bar, we can calculate:
∆E =
dE
dE
∆T +
∆LT OF
dT
dLT OF
(4.1.6)
where ∆T and ∆LT OF are the uncertainty of the values of time interval between electrons
and protons arrival, and the uncertainty of the measurement of the TOF length. We can
1
infer that ∆T ≃ 2 ×
where fs is the oscilloscope sampling frequency (i.e. 1/fs is
fs
the time between two consecutive samples, the factor two is due to uncertainty in the
132
4 Description and results of the experiments
correct determination of both te and tp ), and, not critically for the relativistic velocities,
∆LT OF ≈ 1 [cm]. In our case we have set fs = 1.25 [GHz] that gives ∆T = 1.6 [ns]. For
the sake of simplicity, we set x =
cT
and calculating the (4.1.6)
LT OF
∆E =
dE
∆x
dx
(4.1.7)
with:
∆x = c
dx
dx
T
∆T
+ 2 ∆LT OF =
∆T +
∆LT OF = c
dT
dLT OF
LT OF
LT OF
∆T
∆LT OF
cT
+
=
LT OF T
LT OF
(4.1.8)
and:
"
"
2 #−3/2
2 #
1 dE
1
1
1
d
= − 1−
1−
=
mp c2 dx
2
1+x
dx
1+x
"
= 1−
1
1+x
2 #−3/2
1
d
1 + x dx
1
1+x
=
1
1
3/2 = 2
[x + 2x]3/2
(1 + x) − 1
2
(4.1.9)
Hence:
m p c2
cT
∆E = 3/2
2
LT OF
cT
+ 2 LTcTOF
LT OF
∆T
∆LT OF
+
T
LT OF
(4.1.10)
Substituting the several known quantities, we plot in gure (4.1.10) the function ∆E (T )
for the TOF front side and rear side.
133
4 Description and results of the experiments
Figure 4.1.10: Energy uncertainty in function of the time interval between electron and
proton arrival.
In gure is presented the energy uncertainty in function of the proton energy. We will
see that the region of interest of this graph is between 1 and 10 MeV, thus obtaining
an uncertainty approximately between 0.03 and 0.8 MeV. Of course the precision of the
measurement can be improved by increasing the sampling frequency.
134
4 Description and results of the experiments
Figure 4.1.11: Energy uncertainty in function of the measured proton energy.
135
4 Description and results of the experiments
4.1.5 Spectralon
Spectralon is a translucent material made of compressed and sintered polytetrauoroethylene (PTFE) powders, which has the highest diuse reectance of any known material
or coating over the ultraviolet, visible, and near-infrared regions of the spectrum.
light penetrates the spectralon before leaving it.
The
The radiative properties of a Spec-
tralon plate are nearly Lambertian, and the scattered light is eciently depolarized.
[Courreges-Lacoste 2003]. One of the most important things to know of an optical diuser
is the bidirectional reectance distribution function (BRDF) that is a four-dimensional
function that denes how light is reected at an opaque surface. The BRDF of spectralon
is characterized by a limited number of parameters describing the optical properties of the
material, such as the albedo, optical thickness, and anisotropy of the scattering.
The spectralon is used in the experiment as a perfectly diuser of the light reected
by the target. This measurement is very important in order to determine the absorption
of light by the target and thus the eective energy of the laser beam transferred to the
plasma during the laser-matter interaction.
The spectralon device used in the experiment is a 2 × 2 square inches, mounted 20
centimeters away from TCC on the plane perpendicular to the reected light direction
(see gure (4.1.1)). Further on during the experiment the spectralon was moved closer to
the target at a distance of 11 centimeters from TCC to have a better illumination from
the reected beam.
Spectralon material, as stated in [Courreges-Lacoste 2003], shows a very good Lambertian behavior (further insights shown in [Haner 1999] demonstrate that is not a perfect
Lambertian diuser) with a reectance very close to unity (up to 99% in a wide range
from UV, passing through visible, to near infrared light between 400 nm and 1800 nm) as
can be seen in gure (4.1.12):
136
4 Description and results of the experiments
Figure 4.1.12: Spectralon 8° hemispherical reectance taken from Labsphere spectralon
datasheet.
The radiant intensity [W/sr] reected by spectralon follows the Lambert's cosine law:
I (ϑ) = In cos ϑ
(4.1.11)
where In is the radiant intensity in the normal direction. The total irradiated power is
simply the integral over 2π of the radiant intensity:
ˆ
Pirr =
ˆ 2π
I (ϑ) dΩ =
2π
ˆ π/2
dϕ
0
ˆ π/2
I (ϑ) sin ϑdϑ = 2πIn
0
= 2πIn
cos ϑ sin ϑdϑ
0
π/2
1 2
sin ϑ
= πIn
2
0
(4.1.12)
If we consider that the spectralon has reectance R = 1, the total power incident over
its surface is equal to the total re-irradiated power, thus:
Pirr = Pinc = I[W/cm2 ] × S
where
(4.1.13)
2
S is the portion of the spectralon surface illuminated by the beam (S = πrbeam
).
For the calculation of rbeam , it must be taken into account the distance dT CC of spectralon
diuser from TCC and the relationship between focus
f and diameter of the beam D on
the parabola, hence we have the proportion:
f : dT CC = D : 2rbeam
and:
137
(4.1.14)
4 Description and results of the experiments
rbeam =
D
dT CC
2f
(4.1.15)
Finally we calculate:
In =
Pinc
2
= I[W/cm2 ] × rbeam
π
(4.1.16)
The amount of power collected by the lens of the spectralon imaging system depends
on the distance dlens from spectralon, on the ϑ angle between normal and observer line
of sight, area of the lens Slens .
We can approximate the collected power as the simple
product between the radiant intensity and the solid angle of sight of the imaging system:
Pcoll ≈
Pinc
Slens
Slens
2
cos ϑ × 2 = I[W/cm2 ] × rbeam
× 2 × cos ϑ
π
dlens
dlens
(4.1.17)
The power integrated over the pulse duration gives an information of the total energy,
hence the previous considerations are valid also for the energies:
Ecoll ≈ Einc
cos ϑ Slens
π d2lens
= C × Einc
(4.1.18)
The previous statements allow to extimate the order of magnitude of total energy collected usefull for designing the diagnostic set up. A typical collected image is shown in
gure (4.1.13):
Figure 4.1.13: Example of a collected spectralon image (16-bit unsigned 1280×960 grey
scale image).
The brighter area is the illuminated surface of spectralon
diuser. The target of this shot was a thin stack of two layer 30nm SiN +
80nm Al, pulse duration 50 fs with ≈1,8 J of energy.
138
4 Description and results of the experiments
A complete calibration of the spectralon diagnostic can be made considering the total
pixels count of the region of interest in the images collected with the camera. A Matlab
routine was developed to perform this task. From calibration measurements we want to
retrieve informations about:
count ↔ Einc
(4.1.19)
During calibration we send a known Einc , and we obtain the calibration function:
Einc = f (count)
(4.1.20)
Figure shows the calibration done during our experiments, with spectralon diuser at a
distance from TCC of about 20 cm.
Figure 4.1.14: Spectralon calibration.
It's important to extimate which is the error introduced by the background noise and
the mirror mount. We have repetitively performed analysis on several
sample shots done
in dierent days throughout all the experiment, to obtain a quantitative extimation of
the error. All the images were processed
by hand using the Matlab function roipoly that
allows us to crop the region of interest in the images.
To extimate the error we perform the following calculation. First we select the three
dierent areas of the spectralon image to analyze (see also gure (4.1.15) for a better
understanding of those regions):
ˆ mirror mount area Amount : this area contribute to the error only with a small portion
because sometimes the image has not a good contrast, hence the spectralon area and
mirror mount area can be superposed for a very small portion Aoverlapping of the region
of interest. If we know exactly this surface we can calculate the % error.
139
4 Description and results of the experiments
ˆ mirror area Amirror : the dark region inside the mirror mount allow us to calculate
the average background noise of the spectralon.
ˆ spectralon area Aspectralon : is the region of interest.
Hence with those three regions we calculate the counts Cmount , Cmirror , Cspectralon ; considering that the overlapping area is included in the spectralon count, we can express the
three counts with the following:
Cmirror = hnoisei Amirror
(4.1.21)
∗
+ hnoisei Amount
Cmount = Cmount
(4.1.22)
(mount)
∗
Cspectralon = Cspectralon
+ hnoisei Aspectralon + Coverlap
(4.1.23)
(mount)
where with Coverlap we note the part of the count in the overlapping area due to the
mirror mount. From the rst we calculate the average noise, and we substitute it in the
∗
second to calculate the values Cmount . Now we can extimate with:
(mount)
∗
Coverlap ≈ Cmount
Aoverlapping
Amount
(4.1.24)
∗
therefore we have all the information to calculate Cspectralon . The % error of using the
∗
count Cspectralon instead of the real value Cspectralon can be evaluate with:
∗
Cspectralon − Cspectralon
ε% =
∗
Cspectralon
(4.1.25)
After several analysis we can conclude that if the contrast of the imag is not poor the
error ε% is limited to values around 4% − 5%. For example for the shot in gure (4.1.13),
neglecting the irrelevant Aoverlapping , we obtain ε% ≃ 3, 51%.
In gure (4.1.15) is shown a typical spectralon image in gray scale and color scale. We
plot also the grayscale value of the pixels along the yellow line in the top left image.
140
4 Description and results of the experiments
Figure 4.1.15: Plot of grayscale pixel value along the indicated yellow line. In this specic
case the peak of the light signal due to the mirror mount is approximately
1/16 of the peak of spectralon diused light.
In the color-scale image is
evident the mirror mount but also the square shape of the spectralon diuser
(2 × 2 square inches). This image and the image in (4.1.13) are the same.
4.1.6 Other diagnostics
Other usefull diagnostics, in laser-solid target interaction experiments, are Thomson parabolas and electron spectrometer. We describe them in the appendix because the feasibility
of both diagnostics was extensively studied before the experiments but at the end they
were not used.
141
4 Description and results of the experiments
4.2 Results and discussion
142
List of symbols
We present a list of symbols used in this thesis.
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mass number
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . magnetic eld, or magnetic induction eld
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . speed of light in vacuum
cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ion acoustic velocity
D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electric induction eld
E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electric eld
e or q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . elementary charge
eN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler's number, the base of natural logarithm
H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . magnetic eld
J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electric density current
kB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boltzmann constant
me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electron rest mass
mi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ion rest mass
mp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . proton rest mass
ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electron density
ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ion density
q or e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . elementary charge
rL or ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Larmor radius or gyroradius
T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . absolute temperature
Te . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electron temperature
vt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thermal velocity of ions or electrons
Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . atomic number
Γee and Γii . . . . . . . . . . . . . . . . . . . . . . . . . . . . plasma coupling parameter for electrons or ions
γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorentz factor
0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vacuum permittivity
r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . relative permittivity, or relative dielectric constant
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . absolute permettivity, or dielectric constant
Φ or φ or V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electric potential
λD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Debye's length
µ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vacuum permeability
ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . charge density
σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface charge density
σi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . areal mass or surface mass density
τp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plasma oscillation period
ωL or Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Larmor frequency or gyrofrequency
ωp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plasma oscillation frequency of electrons or ions
∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nabla dierential operator
143
List of physical constant and
conversion units
Here we present a list of physical constant used in the thesis extracted from [CODATA 2006]
9
−1
c = 0, 299792458 × 10 [m s ]
e = 1, 602176487 × 10−19 [C]
kB = 1, 3806505 × 10−23 [J K−1 ] = 8, 617343 × 10−5 [eV K−1 ]
me = 0, 510998918 [MeV/c2 ]
mp = 938, 272029 [MeV/c2 ]
−1
0 = (µ0 c2 )
Here is a usefull list of conversion factors:
18
1[J] = 6, 24150975 × 10 [eV]
1[eV] = 1, 602176487 × 10−19 [J]
1[T] = 104 [G]
144
5 Appendix
5.1 Thomson's parabolas
Thomson's parabolas is a particular ion spectrometer that allows to determine energy and
charge to mass ratio of dierent ion species that are present in an ions beam. [reference
needed] The most simple desing of this spectrometer is sketched in gure:
Figure 5.1.1: Sketch of Thomson's parabolas ion spectrometer
It consist in the superposition of magnets and electrodes which is
l
All the ions that passes throughout the pin-hole experience a Lorentz force, thus we can
calculate their trajectories as follows.
d
qi
v=
[E + v × B]
dt
mi
(5.1.1)
v 0 = v0 x̂
(5.1.2)
with:
145
5 Appendix
E = E ŷ
(5.1.3)
B = B ŷ
(5.1.4)
as in gure; velocity and position of an ion are expressed with time dependent functions
by:
r (t) = x (t) x̂ + y (t) ŷ + z (t) ẑ
v (t) = vx (t) x̂ + vy (t) ŷ + vz (t) ẑ =
(5.1.5)
d
r
dt
(5.1.6)
Solving the problem:
qi
qi
d
[vx x̂ + vy ŷ + vz ẑ] =
[E ŷ + (vx x̂ + vy ŷ + vz ẑ) × B ŷ] =
[E ŷ + vx B ẑ − vz B x̂]
dt
mi
mi
(5.1.7)
qi
d
vx = − vz B
dt
mi
(5.1.8)
qi
d
vy =
E
dt
mi
(5.1.9)
qi
d
vz =
vx B
dt
mi
(5.1.10)
vx (t) = v0 cos ωt
(5.1.11)
Integrating:
qi
E t
mi
(5.1.12)
vz (t) = v0 sin ωt
(5.1.13)
vy (t) =
where:
ω=
The position of the particle at an istant
qi
B
mi
t is given by the three scalar components:
v0
sin ωt
ω
1 qi
y (t) =
E t2
2 mi
x (t) =
z (t) =
(5.1.14)
v0
(cos ωt − 1)
ω
146
(5.1.15)
(5.1.16)
(5.1.17)
5 Appendix
where we suppose that the initial position is at the origin of the axes. As in gure, the
∗
eld length (eld interaction region) is . Hence the instant of time t when the particle
l
goes outside the interaction region, starting its straight trajectory, is given by:
x (t∗ ) = l
(5.1.18)
lω
lω
≃
v0
v0
2
1 qi
l
∗ def ∗
y (t ) = y ≃
E
2 mi
v0
s

2
lω
v0 
def
1−
z (t∗ ) = z ∗ =
− 1
ω
v0
ωt∗ = arcsin
(5.1.19)
(5.1.20)
(5.1.21)
and the relative velocity's components are:
s
2
lω
1−
v0
qi
l
def
vy (t∗ ) = vy∗ ≃
E
mi
v0
∗
def
vx (t ) = vx∗ = v0
def
vz (t∗ ) = vz∗ = lω
(5.1.22)
(5.1.23)
(5.1.24)
The trajectories outside the interaction area with the B and E elds are straigth lines:
x (t) = l + vx∗ t
(5.1.25)
y (t) = y ∗ + vy∗ t
(5.1.26)
z (t) = z ∗ + vz∗ t
(5.1.27)
∗∗
∗∗
The particle reaches the detector in the instant of time t
when x (t ) = l + L where
L is the distance from the magnets/electrodes and the detector.
Hence we can calculate:
x (t∗∗ ) = l + L = l + vx∗ t∗∗
(5.1.28)
It follows that:
t∗∗ =
L
vx∗
(5.1.29)
The position of the particle on the detector plane is:
L
def
y (t∗∗ ) = Ydef l. = y ∗ + vy∗ ∗
vx
147
(5.1.30)
5 Appendix
L
def
z (t∗∗ ) = Zdef l. = z ∗ + vz∗ ∗
vx
(5.1.31)
Substituting:
1
Ydef l. ≃
2
qi
E
mi
s
v0 
Zdef l. =
ω
1−
lω
v0
l
v0
2
+
qi
E
mi
l
v0
L
r
1−
v0
qi
2 ≃ mi v 2 lE
0
lω
L
− 1+lω
r
1−
v0
l
+L
2
(5.1.32)
v0

2
v0 1
2 ≃ ω 2
lω
lω
v0
2
lω
qi
+ L=
lB
v0
mi v0
l
+L
2
v0
(5.1.33)
Note that:
qi
Z e
=
mi
A mp
ZE
Z
, v0 ≃
C
Ydef l.
A
A v02
Z
ZB
Zdef l.
, v0 ≃
C
A
A v0
where
(5.1.34)
(5.1.35)
(5.1.36)
C is a characteristic constant of the spectrometer's geometry:
e
l
C=
mp
l
+L
2
(5.1.37)
We can calculate the locus described by the two parametric functions of the deection
by eliminating the parameter v0 ; squaring the Zdef l. we obtain:
1
=
v02
A
ZBC
2
2
Zdef
l.
(5.1.38)
Substituting in the expression of Ydef l. :
Z
Ydef l. = EC
A
A
ZBC
2
"
#
A
m
E
p
2
2
Zdef
Zdef
l. =
l.
Z el 2l + L B 2
(5.1.39)
So it is clearly seen that the locus of the trajectories on the detector plane is a parabola.
Dierent ion species have dierent mass to charge ratios and therefore their loci are parabolas with dierent aperture. It is also clearly seen that the iso-velocity curves are straight
lines, in fact if we divide Zdef l. by Ydef l. we obtain:
Zdef l.
B
= v0
Ydef l.
E
148
(5.1.40)
5 Appendix
that is the equation of a straight line passing throughout the origin of the axes in the
frame Zdef l. /Ydef l. .
Another possible conguration is the one sketched in gure (5.1.2), where rst of all,
the particle experiences a magnetic eld, and after an electric eld. [Sakabe 1980]
Figure 5.1.2: Alternative setup of Thomson's parabolas ion spectrometer
The solution for the trajectories is derived similarly to the previous case:
qi
d
v=
[E + v × B]
dt
mi
(5.1.41)
v 0 = v0 x̂
(5.1.42)
B ŷ x ∈ [0; l1 ]
0 elsewhere
(5.1.43)
E ŷ x ∈ [l1 ; l1 + l2 ]
0
elsewhere
(5.1.44)
with:
B=
E=
So for the rst part of the path:
d
qi
qi
[vx x̂ + vy ŷ + vz ẑ] =
(vx x̂ + vy ŷ + vz ẑ) × B ŷ =
[vx B ẑ − vz B x̂]
dt
mi
mi
149
(5.1.45)
5 Appendix
d
qi
vx = − vz B
dt
mi
(5.1.46)
d
vy = 0
dt
(5.1.47)
d
qi
vz =
vx B
dt
mi
(5.1.48)
Those equations guide us to the solution:
x (t) =
z (t) =
v0
sin ωt
ω
(5.1.49)
y (t) = 0
(5.1.50)
v0
(cos ωt − 1)
ω
(5.1.51)
The instant of time t1 when the particle reaches the interface between the dierent
interaction regions is given by:
ωl1
ωl1
≃
v0
v0

s
2
ωl1
v0 
def
− 1
1−
z (t1 ) = z1 =
ω
v0
ωt1 = arcsin
(5.1.52)
(5.1.53)
and the velocity components are:
s
def
vx (t1 ) = vx1 = v0
1−
ωl1
v0
2
vy (t1 ) = 0
def
vz (t1 ) = vz1 = ωl1
(5.1.54)
(5.1.55)
(5.1.56)
The second part of the path is under the inuence of electric eld:
d
qi
[vx x̂ + vy ŷ + vz ẑ] =
E ŷ
dt
mi
(5.1.57)
d
vx = 0
dt
(5.1.58)
d
qi
vy =
E
dt
mi
(5.1.59)
150
5 Appendix
d
vz = 0
dt
(5.1.60)
vx (t) = vx1
(5.1.61)
Integrating:
vy (t) =
qi
E t
mi
(5.1.62)
vz (t) = vz1
(5.1.63)
Further integrating to obtain the position:
x (t) = l1 + vx1 t
1
y (t) =
2
(5.1.64)
qi
E t2
mi
(5.1.65)
z (t) = z1 + vz1 t
(5.1.66)
The instant of time t2 when the particle comes out from the region of the electrodes is:
l1 + vx1 t2 = l1 + l2
(5.1.67)
hence:
l2
vx1
2
1 qi
l2
def
y (t2 ) = y2 =
E
2 mi
vx1
t2 =
def
z (t2 ) = z2 = z1 + l2
vz1
vx1
(5.1.68)
(5.1.69)
(5.1.70)
the components of the velocity vector at t = t2 are constant except for:
def
vy (t2 ) = vy2 =
qi
E
mi
l2
vx1
(5.1.71)
After the particle continues its path along a straight line:
x (t) = l1 + l2 + vx1 t
(5.1.72)
y (t) = y2 + vy2 t
(5.1.73)
z (t) = z2 + vz1 t
(5.1.74)
151
5 Appendix
An the particle reaches the detector plane at the instant t3 :
L
vx1
t3 =
(5.1.75)
The other coordinates, that give the deection of the particle in the detector plane, are:
def
vy2
vx1
(5.1.76)
def
vz1
vx1
(5.1.77)
y (t3 ) = Ydef l. = y2 + L
z (t3 ) = Zdef l. = z2 + L
Substituting the previous expressions:
1
Ydef l. =
2
qi
E
mi
l2
vx1
2
L
+
vx1
qi
E
mi
s
Zdef l. = z1 + l2
vz1
v0 
vz1
+L
=
vx1
vx1
ω
v0 1
≃
ω2
ωl1
v0
2
1−
l2
vx1
ωl1
v0
qi
≃
E
mi
1 L
+
2 l2
l2
v0
2
(5.1.78)

2
ωl1
=
+ (l2 + L)
v0
− 1 + (l2 + L)
r
v0
1
l1 + l2 + L
2
ωl1
v0
ωl1
2 ≃
1
1 − ωl
v0
(5.1.79)
The approximated results are:
Ydef l. =
qi
E
Cy 2
mi v0
(5.1.80)
Zdef l. =
qi
B
Cz
mi v0
(5.1.81)
with the denitions of the geometric coecients:
Cy = l2
Cz = l1
1
l2 + L
2
1
l1 + l2 + L
2
(5.1.82)
(5.1.83)
2
Squaring the Zdef l. and calculating 1/v0 :
2
Zdef
1
l.
=
2
2
v0
qi
C
B
mi z
(5.1.84)
substituting:
Amp Cy E
2
Ydef l. =
Zdef
l.
Ze (Cz B)2
152
(5.1.85)
5 Appendix
quite similar to the expression of the rst geometric setup. The dierence in the two
setup is related to dynamic range of the instruments in the two dierent conguration.
In both cases we have:
v0 = C
Dening the velocity resolution
Zdef l.
Ydef l.
(5.1.86)
R as in [Sakabe 1980], we have:
Z
def l.
C Ydef
v0
Ydef l. Zdef l.
Ydef l. Zdef l.
l.
R=
= ∆Zdef l.
=
≥
∆Ydef l.
∆v0
Ydef l. ∆Zdef l. − Zdef l. ∆Ydef l.
(Ydef l. + Zdef l. ) ∆w
− CZdef l. 2
C
Ydef l.
Ydef l.
(5.1.87)
where ∆w is the aperture of the entrance pinhole. The minimum detectable ion velocity
is given by the velocity of the particle which unintentionally hits the electrodes in the exact
instant of time when the particle should exit from them:
v0 > vmin =
(5.1.88)
Another modied setup is described in [Carrol 2010] and depicted in FIGURE. This is
the real set up proposed for the experiments in the 200TW ALLS facility but unfortunately not completed on time for the experimental campaing during June/July 2010: it is
expected to be completed for the next experimental campaign during october 2010.
The modied Thomson parabola spectrometer is designed to produce high resolution
measurements of multi MeV ions beams. The electro-magnetostatic eld of the device is
optimized to give a greater dispersion for a given voltage compared to the other
designs.
153
traditional
5 Appendix
Electron spectrometer
Is usefull to have also informations about the electrons that come out from the target
after the laser interaction. An electron spectrometer can give us information about the
number of electrons present in beam in a particular energy interval.
The spectrometer
action consists only in the deection of the electrons by a magnetic eld. The electrons
trajectories are bent down and each electron reach the surface of an imaging plate. The
motion of an electron in this device is quite simple to obtain, and from the equation of
motion we have:
F =m
d
v = −ev × B
dt
(5.1.89)
We can set those initial conditions:
x(t = 0) = 0, y(t = 0) = y0 , z(t = 0) = z0
(5.1.90)
vx (t = 0) = ve , vy (t = 0) = 0, vz (t = 0) = 0
(5.1.91)
B = B ŷ = cost
(5.1.92)
F = −e (vx x̂ + vy ŷ + vz ẑ) × B ŷ = −eBvx ẑ + eBvz x̂
(5.1.93)
The magnetic eld
thus:
So along the three directions we have:
.
vx =
eB
vz
me
(5.1.94)
.
vy = 0
.
vz = −
(5.1.95)
eB
vx
me
(5.1.96)
Taking the derivative of the rst equation and substituting the third in it, we obtain:
eB .
vx =
vz = −
me
..
eB
me
2
vx = −ωL2 vx
(5.1.97)
with the denition of the gyrofrequency:
ωL =
eB
me
(5.1.98)
Solving the second order dierential equation:
vx (t) = A sin ωL t + B cos ωL t
154
(5.1.99)
5 Appendix
vz (t) =
1
1 .
vx =
(ωL A cos ωL t − ωL B sin ωL t)
ωL
ωL
(5.1.100)
and thus rearranging:
vx (t) = A sin ωL t + B cos ωL t
(5.1.101)
vy = vy (t = 0) = 0
(5.1.102)
vz (t) = A cos ωL t − B sin ωL t
(5.1.103)
B = ve , A = 0
(5.1.104)

 vx (t) = ve cos ωL t
vy (t) = 0

vz (t) = −ve sin ωL t
(5.1.105)
The initial conditions give:
Hence:
Integrating:

´t
 x(t) = 0 ve cos (ωL t0 ) dt0
y(t) = y0
´t

z(t) = − 0 ve sin (ωL t0 ) dt0
(5.1.106)
We nally obtain the parametric (in time) expression of the particle trajectory inside
the interaction region:
x(t) = ωvLe sin (ωL t)
y(t) = y0

z(t) = ωvLe (cos (ωL t) − 1) + z0


(5.1.107)
ve
is the Larmor radius or gyroradius. Note that rL is a function of velocity
ωL
and thus of the electron energy. The nal result is the particle trajectory: we can calculate
where rL =
it squaring and summing the rst and third equation:
x2 + [z − (z0 − rL )]2 = rL2
(5.1.108)
The relativistic correction can be made substituting the classical Larmor radius with its
relativistic counterpart [Tanaka 2005]. From a relativistic point of view the only dierence
is that we have to replace mass to velocity product with momentum but in fact if we
calculate the relativistic velocity as follows:
2
Etot
= (pc)2 + me c2
2
Etot = Ekin + me c2
155
(5.1.109)
(5.1.110)
5 Appendix
with:
p = γme ve
(5.1.111)
Solving to nd velocity:
s
ve = c
E + me c2
me c2
2
−1
(5.1.112)
and we can use it in the formula of gyroradius to nd the relativistic correction to
classical Larmor radius:
me c
rL (E) =
eB
s
E + me c2
me c2
2
1
−1=
eBc
√
q
E 2 + 2Eme c2
2
2
(E + me c2 ) − (me c2 ) =
eBc
(5.1.113)
In a practical case the previous expression takes the form:
p
rL [mm] =
2
EM
eV + 1, 022 × EM eV
0, 29979 × BT
(5.1.114)
The larmor radius is the radius of the electron's trajectory in a costant B eld. Let's
now analyse two dierent setup for the electron spectrometer.
Imaging plate inside the magnet's interaction area
The trajectories are:
x(t) = rL sin (ωL t)
(5.1.115)
z(t) = rL [cos (ωL t) − 1] + h
(5.1.116)
the minimum radius (that allows detection of an electron with the corresponding energy)
is:
h
2
(5.1.117)
h2 + l2
2h
(5.1.118)
rLmin =
the maximum is:
rLmax =
A rough sketch of the trajectories in the spectrometer chamber is shown in gure:
156
5 Appendix
The corresponding x-position on the IP is:
xIP (rL ) =
p
2rL h − h2
(5.1.119)
So we can calculate the projected surface of the pin hole area for two electrons with same
energy (and thus same larmor radius) but dierent z-position (i.e. the pin-hole aperture):
w
w
∆xIP (rL ) = xIP (rL , h+ )−xIP (rL , h− ) = h +
2
2
2
w
s
2rL
− 1− h −
h + w2
2
w
s
2rL
−1
h − w2
(5.1.120)
The corrisponding energy spread can be evaluate with the approximated formula:
∆EIP ≃ ∆xIP ×
dE
dE drL
= ∆xIP ×
dxIP
drL dxIP
(5.1.121)
with:
2
0.29979 × B[T] × rL
dE
[MeV/mm] = q
2
drL
0.5112 + rL2 × 0.29979 × B[T]
(5.1.122)
drL
xIP
=
dxIP
h
(5.1.123)
Imaging plate outside the magnet interaction area
A sketch of the geometry:
157
5 Appendix
The trajectories are:
x(t) = rL sin (ωL t)
(5.1.124)
z(t) = rL [cos (ωL t) − 1] + h
(5.1.125)
inside the interaction area with the magnetic eld
B.
The motion laws are linear in time in the region outside the interaction area with the
magnets, because in this region the particle do not experience any acceleration.
The
initial velocities of the motion can be calculated from the above equations taking the rst
0
derivatives and calculating them in the istant t where the particle reaches the x position
x(t0 ) = l (the length of the magnets). Hence we have:
0
ωL t = arcsin
l
rL
vx (t0 ) = rL ωL cos ωL t0 = ωL
(5.1.126)
q
rL2 − l2
(5.1.127)
vz (t0 ) = −rL ωL sin ωL t0 = −ωL l
(5.1.128)
q
x(t) = x0 + vx0 t = l + ωL rL2 − l2 t
(5.1.129)
q
z(t) = z0 + vz0 t = rL2 − l2 + h − rL − (lωL ) t
(5.1.130)
The trajectories are:
the minimum radius (that allows detection of an electron with the corresponding energy)
is:
158
5 Appendix
rLmin =
h20 + l2
2h0
(5.1.131)
that can be miminized taking h0 = l (wherever it's possible), giving:
rLmin = l
(5.1.132)
The corresponding x-position on the IP is:
q
1 2
2
2
2
r − l + (h0 + z0 − rL ) rL − l
xIP (rL ) =
l L
(5.1.133)
The projected surface of the pin hole area for two electrons with same energy (and thus
same larmor radius) but dierent z-position (i.e. the pin-hole aperture):
w
w
w
) − xIP (rL , h − ) =
2
2
l
∆xIP (rL ) = xIP (rL , h +
q
rL2 − l2
(5.1.134)
The corrisponding energy spread can be evaluate with the approximated formula:
∆EIP ≃ ∆xIP ×
dE
dE drL
= ∆xIP ×
dxIP
drL dxIP
2
0.29979 × B[T] × rL
dE
[MeV/mm] = q
2
drL
0.5112 + rL2 × 0.29979 × B[T]
(5.1.135)
(5.1.136)
calculated in the appropriate rL ,
−1 −1
drL
dxIP
drL
=
=
dxIP
dxIP
drL
(5.1.137)
The inversion can be made only in the domain of continuity of the rst derivative of
xIP :
"
#
q
dxIP
1
(h
+
z
−
r
)
r
0
L
L
p 0
=
2rL − rL2 − l2 +
2
2
drL
l
rL − l
(5.1.138)
Supposing we can approximate the main lobe of the focus spot on target as a gaussian
pulse in time and space, the intensity is:
2
r
t2
I (r, t) = Imax exp − 2 exp − 2
τ
w
(5.1.139)
where:
τ=
F W HM (time)
√
2 ln 2
(5.1.140)
F W HM (time) can be extimated with a second order correlation of the laser pulse, and
pulse duration τpulse ;
it is said to be the
159
5 Appendix
w=
F W HM (space)
√
2 ln 2
(5.1.141)
F W HM (space) depends on the beam waist and can be approximated with the rst
zero of the Airy function only if the laser near-eld has a top-hat shape:
F W HM (space) ≈ r0 ≈
where
1, 22λ
f
D
(5.1.142)
f is the focal lenght of the parabola that focuses the beam on target, D is the
beam diameter, and λ the laser beam wavelength.
The energy encircled by the main lobe can be found by integrating the laser intensity
over time and space, bu this is also a measurable fraction of the total laser pulse's energy.
´ +∞
Performing the two integrals, using the property of the gaussian integral
exp [−x2 ] =
−∞
√
π we obtain:
√
Ef ocal = Imax π πw2 τ = f% Etot
and thus rearranging to obtain the maximum intensity in units of
Imax
(5.1.143)
W
:
cm2
W
0, 829 × 1023 × f% Etot
=
2
cm2
τpulse[fs] × r0[µm]
(5.1.144)
where:
ˆ f% is the fraction of the incident laser energy Elaser contained in the rst lobe (perfect
Airy function, max value: 88%)
ˆ τpulse[fs] is the pulse duration full width at half maximum in femtoseconds
ˆ r0[µm] is approximately the diameter of the main lobe of laser pulse (focal spot diameter) in microns
ˆ Elaser is the laser energy (i.e.: if a plasma mirror is used to increase the contrast
ration, this energy must take into account the possible reduction caused by Reection<100%) in Joule.
Now if we can consider the approximation for the hot electron temperature we can write:
v

u
2
u
Imax[ W ] × λ[µm]

t
cm2
− 1
Th [MeV] ≈ Φponderomotive ≈ 0.511  1 +
1, 37 × 1018
160
(5.1.145)
5 Appendix
Hot electron temperature [MeV]
10
Th [MeV]
1
0,1
0,01
0,1
1
10
100
Intensity [x10^18 W/cm2]
The electron spectrum [reference needed] is:
E
dN
= N0 exp −
dE
Th
(5.1.146)
The total energy of electrons can be calculated as follows:
ˆ N0
electron
Etot
=
0
ˆ 0
dN
EdN =
EdE = −N0
+∞ dE
ˆ +∞
0
E
E exp −
dE = N0 Th2
Th
(5.1.147)
So we can calculate:
electron
Etot
x%Elaser
=
2
Th
Th2
(5.1.148)
electron
dN
Etot
E
=
exp −
dE
Th2
Th
(5.1.149)
N0 =
In fact now we have:
Now we have obtained the energy spread onto the IP, but in fact we must calculate
the total expected ux on that area of the IP to obtain an exstimation of the total PSL
produced. We have to calculate the total expected number of electrons in that interval of
energy around a xed value ε.
Electron spectrum:
161
5 Appendix
dN
ηtot Ef ocus
E
(E) [numb.elec./MeV] =
exp −
dE
Th2
Th
where
(5.1.150)
ηtot = ηtarget ηlaser must take into account that a not negligible percentage of
electrons are retained in the sheat at the rear surface of the target.
The total number of electrons in a interval of energies ε ± ∆E/2 is:
ˆ ε+∆E/2
Ntot ≃
ε−∆E/2
dN
dN
dE = Th
(ε) 2 sinh
dE
dE
∆E
2Th
(5.1.151)
Electron beam: approximately a cone with aperture α. Solid angle of the beam:
α
Ωbeam = 2π 1 − cos
2
(5.1.152)
Solid angle of visibility at the pinhole:
Ωpinhole ≃
Spinhole
π w2
=
d2
4 d2
where Spinhole is the surface of the pin hole,
target,
w is the diameter of the pinhole.
(5.1.153)
d is the distance of the pinhole from the
Number of electrons in a interval of energies ε ± ∆E/2 reaching the pin hole:
Ωpinhole
Ωbeam
(5.1.154)
Npinhole
× w × ∆xIP
4
(5.1.155)
Npinhole = Ntot
The total ux is:
F (εe , ∆E) = π
The total PSL value:
P SL =
P SL
electron
× Npinhole
The PSL per incident electron is termed sensitivity [Tanaka 2005] :
162
(5.1.156)
5 Appendix
Note that the incidence is not normal to the IP plane so this eect can also be taken
into account by calculating the incidence angle. The relative sensitivity decrease with the
angle with a cosine law. Let φ be the incident angle: it also depends on the energy of the
single electron. This incidence angle can be calculated as follows.
Inside setup:
Consider that the velocity along the vertical axis z is a component of the velocity vector
ve:
vz (z = 0) = −ve cos φ
(5.1.157)
0
We calculate the time t when the electron hits the IP:
t0 : z(t0 ) = 0
(5.1.158)
and thus:
h
ωL t = arccos 1 −
rL
0
(5.1.159)
vz (t0 ) = −ve sin ωL t0 = −ve cos φ
(5.1.160)
s
2
h
h
cos φ = sin arccos 1 −
= 1− 1−
rL
rL
(5.1.161)
Outside setup:
The z component of the velocity is:
vz = −ωL l = −ve cos φ
but ve = rL ωL ,
163
(5.1.162)
5 Appendix
cos φ =
164
l
rL
(5.1.163)
Bibliography
[Albright 2006] B. J. Albright, L. Yin, B. M. Hegelich, Kevin J. Bowers, T. J. T. Kwan,
and J. C. Fernandez; Theory of Laser Acceleration of Light-Ion Beams from
Interaction of Ultrahigh-Intensity Lasers with layered targets; PRL 97, 115002
(2006); DOI: 10.1103/PhysRevLett.97.115002.
[Alfvén 1939] Hannes Alfvén; On the motion of cosmic rays in interstellar space; The
Phys. Rev. vol. 55 no. 5 page 425 (1939).
[Allen 2004] Matthew Allen, Pravesh K. Patel, Andrew Mackinnon, Dwight Price, Scott
Wilks, and Edward Morse; Direct Experimental Evidence of Back-Surface Ion
Acceleration from Laser-Irradiated Gold Foils; PRL 93, 265004 (2004); DOI:
10.1103/PhysRevLett.93.265004.
[ALLS 2005] ALLS Annual Report 2005-2006.
[ALLS 2006] ALLS Annual Report 2006-2007.
[ALLS 2008] ALLS Annual Report 2008-2009.
[Amaldi 2001] U. Amaldi; The Italian hadrontherapy project CNAO; Physica Medica Vol. XVII, Supplement 1, 2001.
[Amemiya 1988] Y. Amemiya, J.Miyahara; Imaging plate illuminates many elds; Nature
vol. 336 page 89 3 november 1988.
[Amirano 1982] F Amirano, K Eidmann, R Sigelt, R Fedosejevs, A Maaswinkel, Yunglu Teng, J D Kilkenny, J D Hares, D K Bradley, B J MacGowan and T J Goldsack; The evolution of two-dimensional eects in fast-electrontransport from
high-intensity laser-plasma interactions; J. Phys. D: Appl. Phys., 15 (1982)
2463-2468.
[Antici PhD 2007] Patrizio Antici; Laser-acceleration of high energy short proton beams
and applications, PhD thesis 2007 Ecole polytechnique.
[Antici 2008] P. Antici, J. Fuchs1, M. Borghesi, T. Grismayer, C. A. Cecchetti, L. Gremillet, A. Mancic, P. Mora, A. C. Pipahl, T. Toncian, O. Willi, P. Audebert;
Space- and time-resolved dynamics of a solid target rear surface expansion induced by fast electrons and of the energy partition into bulk cold electrons;
Journal of Physics: Conference Series 112 (2008) 022099; DOI: 10.1088/17426596/112/2/022099.
165
Bibliography
[Antici 2009] P. Antici, J. Fuchs, M. Borghesi, T. Grismayer, S. Atzeni, C.A. Cecchetti,
L. Gremillet, A. Mancic, P. Mora, A.C. Pipahl, A. Schiavi, T. Toncian, O.
Willi, and P. Audebert; Time and space resolved interferometry for detecting
plasma expansion from solid targets; Eur. Phys. J. Special Topics 175, 139142
(2009); DOI: 10.1140/epjst/e2009-01131-6.
[Beg 1997] F. N. Beg, A. R. Bell, A. E. Dangor, C. N. Danson, A. P. Fews, M. E. Glinsky,
B. A. Hammel, P. Lee, P. A. Norreys, and M. Tatarakis; A study of picosecond
19
lasersolid interactions up to 10
W cm−2 ; Phys. Plasmas 4 (2), February
1997.
[Bellan 2004] Paul
M.
Bellan,
Fundamentals
of
Plasma
Physics,
2004
Cambridge
http://www.metu.edu.tr/
~rafatov/phys427/SUPPLEMENTARY/Cambridge%20University%20Press%
20Fundamentals%20Of%20Plasma%20Physics.pdf.
University
Press.
Free
book
available
at
[Betti 2005] S Betti, F Ceccherini, F Cornolti and F Pegoraro; Expansion of a nitesize plasma in vacuum; Plasma Phys. Control. Fusion 47 (2005) 521529;
DOI:10.1088/0741-3335/47/3/008
[Borghesi 1997] M. Borghesi, A. J. MacKinnon, L. Barringer, R. Gaillard, L. A. Gizzi, C.
Meyer, and O. Willi; Relativistic Channeling of a Picosecond Laser Pulse in a
Near-Critical Preformed Plasma; PRL volume 78, number 5 3 February 1997.
[Borghesi 2006] M. Borghesi, J. Fuchs, S. V. Bulanov, A. J. MacKinnon, P. K. Patel, and
M. Roth; Fast Ion Generation By High-Intensity Laser Irradiation of Solid
Target and Aapplications; Fusion Science and Technology Vol. 49 Apr. 2006.
[Brunel 1987] F.Brunel; Not-So-Resonant Resonant Absorption; PRL volume 59, number
1 6 July 1987.
[Buechoux 2010] S. Buechoux, J. Psikal, M. Nakatsutsumi, L. Romagnani, A. Andreev,
K. Zeil, M. Amin, P. Antici, T. Burris-Mog, A. Compant-La-Fontaine, E.
d'Humieres, S. Fourmaux, S. Gaillard, F. Gobet, F. Hannachi, S. Kraft, A.
Mancic, C. Plaisir, G. Sarri, M. Tarisien, T. Toncian, U. Schramm, M. Tampo,
P. Audebert, O. Willi, T. E. Cowan, H. Pepin, V. Tikhonchuk, M. Borghesi,
and J. Fuchs; Hot Electrons Transverse Reuxing in Ultraintense Laser-Solid
Interactions; PRL 105, 015005 (2010); DOI: 10.1103/PhysRevLett.105.015005.
[Bulanov 1994] S. V. Bulanov, N. M. Naumova, F. Pegoraro; Interaction of an ultrashort,
relativistically strong laser pulse with an overdense plasma; Phys. Plasmas 1
(3), March 1994.
[Carrol 2010] D.C. Carroll, P.Brummitt, D.Neely, F.Lindau, O.Lundh, C.-G.Wahlstrom,
P.McKenna; A modied Thomson parabola spectrometer for high resolution
multi-MeV ion measurements -Application to laser-driven ion acceleration;
Nucl. Instr. and Meth. A (2010); doi:10.1016/j.nima.2010.01.054.
166
Bibliography
[Cattani 2000] F. Cattani, A. Kim, D. Anderson, and M. Lisak; Threshold of induced
transparency in the relativistic interaction of an electromagnetic wave with
overdense plasmas; Phys. Rev. E vol. 62, no. 1 (2000).
[Ceccotti 2007] T. Ceccotti, A. Levy, H. Popescu, F. Reau, P. D'Oliveira, P. Monot, J. P.
Geindre, E. Lefebvre, and Ph. Martin; Proton Acceleration with High-Intensity
Ultrahigh-Contrast Laser Pulses; PRL 99, 185002 (2007); DOI: 10.1103/PhysRevLett.99.185002.
[Ceccotti 2008] T Ceccotti, A Levy, F Reau, H Popescu, P Monot, E Lefebvre and Ph
Martin; TNSA in the ultra-high contrast regime; Plasma Phys. Control. Fusion
50 (2008) 124006; DOI:10.1088/0741-3335/50/12/124006.
[Chen H. 2006] Hui Chen, Scott C. Wilks, Parvesh K. Patel, and Ronnie Shepherd;
Short pulse laser produced energetic electron and positron measurements;
Rev.Sci.Instr. 77, 10E703 2006; DOI:10.1063/1.2220141.
[Chen H. 2007] H. Chen, R. Shepherd, H. K. Chung, A. Kemp, S. B. Hansen, S. C. Wilks,
Y. Ping, K. Widmann, K. B. Fournier, G. Dyer, A. Faenov, T. Pikuz, and
P. Beiersdorfer; Fast-electron-relaxation measurement for laser-solid interaction at relativistic laser intensities; Phys. Rev. E 76, 056402 (2007); DOI:
10.1103/PhysRevE.76.056402.
[Chen H.]
Hui Chen, Norman L. Back, Teresa Bartal, F. N. Beg, David C. Eder, Anthony J. Link, Andrew G. MacPhee, Yuan Ping, Peter M. Song, Alan Throop,
and Linn Van Woerkom; Absolute calibration of image plates for electrons at
energy between 100 keV and 4 MeV; Rev. Sci. Instr. 79, 033301 2008; DOI:
10.1063/1.2885045.
[Chen M. 2007] Min Chen, Alexander Pukhov, Tong-Pu Yu, Zheng-Ming Sheng; Radiation reaction eects on ion acceleration in laser foil interaction;
[Chen S.N. 2007] S. N. Chen, G. Gregori, P. K. Patel, H.-K. Chung, R. G. Evans, R. R.
Freeman, E. Garcia Saiz, S. H. Glenzer, S. B. Hansen, F. Y. Khattak, J. A.
King, A. J. Mackinnon, M. M. Notley, J. R. Pasley, D. Riley, R. B. Stephens,
R. L. Weber, S. C. Wilks, and F. N. Beg; Creation of hot dense matter in shortpulse laser-plasma interaction with tamped titanium foils; Physics of Plasmas
14, 102701 (2007); DOI: 10.1063/1.2777118.
[CODATA 2006] Peter J. Mohr,
Recommended
Values
Barry N. Taylor,
of
the
and David B. Newell;
Fundamental
Physical
CODATA
Constants:
2006;
physics.nist.gov/constants.
[Courreges-Lacoste 2003] G. Bazalgette Courreges-Lacoste, J. Groote Schaarsberg, R.
Sprik, S. Delwart; Modeling of Spectralon diusers for radiometric calibration in remote sensing; Opt. Eng. 42(12) 36003607 (December 2003); DOI:
10.1117/1.1622961.
167
Bibliography
[Cowan 2004] T. E. Cowan, J. Fuchs, H. Ruhl, A. Kemp, P. Audebert, M. Roth, R.
Stephens, I. Barton, A. Blazevic, E. Brambrink, J. Cobble, J. Fernandez, J.C. Gauthier, M. Geissel, M. Hegelich, J. Kaae, S. Karsch, G. P. Le Sage,
S. Letzring, M. Manclossi, S. Meyroneinc, A. Newkirk, H. Pepin, and N.
Renard-LeGalloudec; Ultralow Emittance, Multi-MeV Proton Beams from a
Laser Virtual-Cathode Plasma Accelerator, PRL vol. 92, no. 20 (2004); DOI:
10.1103/PhysRevLett.92.204801.
[Crow 1975] J. E. Crow P. L. Auer J. E. Allen; The expansion of a plasma into a vacuum;
J. Plasma Physics vol. 14 part 1 pag. 65-76 (1975).
[Di Piazza 2000] A. Di Piazza; Exact Solution of the Landau-Lifshitz Equation in a Plane
Wave; Lett Math Phys (2008) 83:305313; DOI: 10.1007/s11005-008-0228-9.
[Dong 2001] Q. L. Dong, J. Zhang, and H. Teng; Absorption of femtosecond laser pulses
in interaction with solid targets; Phys. Rev. E, vol. 64, 026411 (2001); DOI:
10.1103/PhysRevE.64.026411.
[Dong 2003] Q. L. Dong,1 Z.-M. Sheng, M. Y. Yu, and J. Zhang; Optimization of ion acceleration in the interaction of intense femtosecond laser pulses with ultrathin
foils; Phys. Rev. E 68, 026408 (2003); DOI: 10.1103/PhysRevE.68.026408.
[Dorozhkina 1998] D. S. Dorozhkina and V. E. Semenov; Exact Solution of Vlasov Equations for Quasineutral Expansion of Plasma Bunch into Vacuum; PRL vol. 81
no. 13 semptember 1998.
[Doumy 2004] G. Doumy, F. Queré, O. Gobert, M. Perdrix, Ph. Martin, P. Audebert, J.
C. Gauthier, J.-P. Geindre, and T. Wittmann; Complete characterization of
a plasma mirror for the production of high-contrast ultraintense laser pulses;
Phys. Rev. E 69, 026402 (2004); DOI: 10.1103/PhysRevE.69.026402.
[Esirkepov 2004] T. Esirkepov, M. Borghesi, S.V. Bulanov, G. Mourou, and T. Tajima;
Highly Ecient Relativistic-Ion Generation in the Laser-Piston Regime; PRL
vol. 92, no. 17 April 2004; DOI: 10.1103/PhysRevLett.92.175003.
[Estabrook 1975] K. G. Estabrook, E. J. Valeo, and W. L. Kruer; Two-dimensional relativistic simulations of resonance absorption; The Physics of Fluids Vol. 18 No.
9 September 1975.
[Estabrook 1978] K.G. Estabrook and W.L. Kruer; Properties of Resonantly Heated Electron Distributions; PRL vol. 40 no. 1 January 1978.
[Feng 2001] Simin Feng and Herbert G. Winful; Physical origin of the Gouy phase shift;
Optics Letters Vol. 26 No. 8 April 2001.
[Fitzpatrick 1998] Richard Fitzpatrick;
The Physics of Plasmas;
Texas at Austin. Free book available at
teaching/plasma/380.pdf.
168
1998 University of
http://farside.ph.utexas.edu/
Bibliography
[Formaux 2008] S. Fourmaux, S. Payeur, A. Alexandrov, C. Serbanescu, F. Martin, T.
Ozaki, A. Kudryashov, and J. C. Kieer; Laser beam wavefront correction for
ultra high intensities with the 200 TW laser system at the Advanced Laser
Light Source; Optics Express 11987, Vol. 16, No. 16, 4 August 2008.
[Forslund 1975] D.W. Forslund, J. M. Kindel, K. Lee, E. L. Lindman and R. L. Morse;
Theory and simulation of resonant absorption in a hot plasma; Phys. Rev. A
vol. 11 no. 2 February 1975.
[Forward 1984] Robert L. Forward; Roundtrip Interstellar Travel Using Laser-Pushed
Lightsails; J. Spacecraft Vol. 21, No. 2, March-April 1984.
[Fuchs 2005] J. Fuchs, Y. Sentoku, S. Karsch, J. Cobble, P. Audebert, A. Kemp, A.
Nikroo, P. Antici, E. Brambrink, A. Blazevic, E. M. Campbell, J. C. Fernandez, J.-C. Gauthier, M. Geissel, M. Hegelich, H. Pepin, H. Popescu, N.
Renard-LeGalloudec, M. Roth, J. Schreiber, R. Stephens, and T. E. Cowan;
Comparison of Laser Ion Acceleration from the Front and Rear Surfaces of
Thin Foils; PRL 94, 045004 (2005); DOI: 10.1103/PhysRevLett.94.045004.
[Fuchs 2006] J. Fuchs, P. Antici, E. D'Humieres, E. Lefevbre, M. Borghesi, E. Brambrink,
C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora,
J. Schreiber, T. Toncian, H. Pepin and P. Audebert; Laser-driven proton scaling laws and new paths towards energy increase; Nature Physics vol. 2 January
2006; DOI:10.1038/nphys199.
[Fuchs 2007] J. Fuchs, C. A. Cecchetti, M. Borghesi, T. Grismayer, E. d'Humieres, P.
Antici, S. Atzeni, P. Mora, A. Pipahl, L. Romagnani, A. Schiavi, Y. Sentoku, T. Toncian, P. Audebert, and O. Willi; Laser-Foil Acceleration of HighEnergy Protons in Small-Scale Plasma Gradients; PRL 99, 015002 (2007);
DOI: 10.1103/PhysRevLett.99.015002.
[Fuchs 2009] Julien Fuchs, Patrick Audebert, Marco Borghesi, Henri Pépin, Oswald Willi;
Laser acceleration of low emittance, high energy ions and applications; C. R.
Physique 10 (2009) 176187; DOI: 10.1016/j.crhy.2009.03.011.
[Geindre 2001] J.-P. Geindre, P. Audebert, S. Rebibo, and J.-C. Gauthier; Single-shot
spectral interferometry with chirped pulses; Optics Letters Vol. 26, No. 20
October 15, 2001.
[Georgiev 2007] Georgi T. Georgiev, and James J. Butler; Long-term calibration monitoring of Spectralon diusers BRDF in the air-ultraviolet; Applied Optics vol. 46
no. 32, 10 November 2007.
[Gibbon 1992] P. Gibbon, A. R. Bell; Collisionless Absorption in Sharp-Edged Plasmas;
PRL vol. 68, no. 10 March 1992.
[Glinec 2006] Y. Glinec, J. Faure, A. Guemnie-Tafo, and V. Malka, H. Monard, J. P.
Larbre, V. De Waele, J. L. Marignier, and M. Mostafavi; Absolute calibration
169
Bibliography
for a broad range single shot electron spectrometer; Rev. Sci. Instr. 77, 103301
(2006); DOI: 10.1063/1.2360988.
[Green 2008] J. S. Green, V. M. Ovchinnikov, R. G. Evans, K. U. Akli, H. Azechi, F. N.
Beg, C. Bellei, R. R. Freeman, H. Habara, R. Heathcote, M. H. Key, J. A.
King, K. L. Lancaster, N. C. Lopes, T. Ma, A. J. MacKinnon, K. Markey, A.
McPhee, Z. Najmudin, P. Nilson, R. Onofrei, R. Stephens, K. Takeda, K. A.
Tanaka, W. Theobald, T. Tanimoto, J.Waugh, L. VanWoerkom, N. C.Woolsey,
M. Zepf, J. R. Davies, and P. A. Norreys; Eect of Laser Intensity on FastElectron-Beam Divergence in Solid-Density Plasmas; PRL 100, 015003 (2008);
DOI: 10.1103/PhysRevLett.100.015003.
[Grismayer 2006] T. Grismayer and P. Mora; Inuence of a nite initial ion density gradient on plasma expansion into a vacuum; Physics of Plasmas 13, 032103 (2006);
DOI: 10.1063/1.2178653.
[Grismayer 2008] T. Grismayer, P. Mora, J. C. Adam, and A. Héron; Electron kinetic
eects in plasma expansion and ion acceleration; Phys. Rev. E 77, 066407
(2008); DOI: 10.1103/PhysRevE.77.066407.
[Haines 2009] M. G. Haines, M. S. Wei, F. N. Beg, and R. B. Stephens; Hot-Electron
Temperature and Laser-Light Absorption in Fast Ignition; PRL 102, 045008
(2009); DOI: 10.1103/PhysRevLett.102.045008.
[Haner 1999] David A. Haner, Brendan T. McGuckin, and Carol J. Bruegge; Polarization
characteristics of Spectralon illuminated by coherent light; Applied Optics Vol.
38, No. 30, 20 October 1999.
[Hatchett 2000] Stephen P. Hatchett, Curtis G. Brown, Thomas E. Cowan, Eugene A.
Henry, Joy S. Johnson, Michael H. Key, Jerey A. Koch, A. Bruce Langdon, Barbara F. Lasinski, Richard W. Lee, Andrew J. Mackinnon, Deanna M.
Pennington, Michael D. Perry, Thomas W. Phillips, Markus Roth, T. Craig
Sangster, Mike S. Singh, Richard A. Snavely, Mark A. Stoyer, Scott C. Wilks,
and Kazuhito Yasuike; Electron, photon, and ion beams from the relativistic
interaction of Petawatt laser pulses with solid targets; Physiscs of Plasmas Vol.
7, No. 5, May 2000.
[Hegelich 2006] B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M.
Paett, H. Ruhl, J. Schreiber, R. K. Schulze and J. C. Ferna´ndez1; Laser acceleration of quasi-monoenergetic MeV ion beams; Nature Vol 439, 26 January
2006; doi:10.1038/nature04400.
[Henig 2009] A. Henig, S. Steinke, M. Schnurer, T. Sokollik, R. Horlein, 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; PRL 103, 245003 (2009);
DOI: 10.1103/PhysRevLett.103.245003.
170
Bibliography
[Huang 2007] Yong-sheng Huang, Xiao-fei Lan, Xiao-jiao Duan, Zhi-xin Tan, Nai-yan
Wang, Yi-jin Shi, and Xiu-zhang Tang, Ye-xi He; Hot-electron recirculation
in ultraintense laser pulse interactions with thin foils; Physics of Plasmas 14,
103106 (2007); DOI: 10.1063/1.2795128.
[Key 1998] M. H. Key, M. D. Cable, T. E. Cowan, K. G. Estabrook, B. A. Hammel, S. P.
Hatchett, E. A. Henry, D. E. Hinkel, J. D. Kilkenny, J. A. Koch, W. L. Kruer,
A. B. Langdon, B. F. Lasinski, R. W. Lee, B. J. MacGowan, A. MacKinnon,
J. D. Moody, M. J. Moran, A. A. Oenberger, D. M. Pennington, M. D.
Perry, T. J. Phillips, T. C. Sangster, M. S. Singh, M. A. Stoyer, M. Tabak,
G. L. Tietbohl, M. Tsukamoto, K. Wharton, and S. C. Wilks; Hot electron
production and heating by hot electrons in fast ignitor research; Physics of
Plasmas vol. 5, no. 5, May 1998.
[Khattak 2006] F. Y. Khattak, O. A. M. B. Percie du Sert, D. Riley, P. S. Foster, E. J.
Divall, C. J. Hooker, A. J. Langley, J. Smith, P. Gibbon; Comparison of experimental and simulated Kα yield for 400 nm ultrashort pulse laser irradiation;
Phys. Rev. E 74, 027401 (2006); DOI: 10.1103/PhysRevE.74.027401.
[Klimo 2008] O. Klimo, J. Psikal, J. Limpouch, V. T. Tikhonchuk; Monoenergetic ion
beams from ultrathin foils irradiated by ultrahigh-contrast circularly polarized
laser pulses; Phys. Rev. Special Topics - Accelerators and Beams 11, 031301
(2008); DOI: 10.1103/PhysRevSTAB.11.031301.
[Kovalev 2001] V. F. Kovalev, V. Yu. Bychenkov, and V. T. Tikhonchuk; Ion Acceleration
during Adiabatic Plasma Expansion: Renormalization Group Approach; JETP
Letters, Vol. 74, No. 1, 2001, pp. 1014.
[Kovalev 2003] V. F. Kovalev and V. Yu. Bychenkov; Analytic Solutions to theVlasov
Equations for Expanding Plasmas; PRL vol. 90 no. 18, May 2003; DOI:
10.1103/PhysRevLett.90.185004.
[Kruer 1985] W. L. Kruer and K. G. Estabrook; J×B heating by very intense laser light;
Phys. Fluids 28 (1), January 1985 page 430.
[Krushelnick 2000] K. Krushelnick, E. L. Clark, M. Zepf, J. R. Davies, F. N. Beg, A.
Machacek, M. I. K. Santala, M. Tatarakis, I. Watts, P. A. Norreys, A. E.
Dangor; Energetic proton production from relativistic laser interaction with
high density plasmas; Physics of Plasmas vol. 7, no. 5, May 2000.
[Landau 1946] L. D. Landau; Sov. Phys. JETP 16, 574 (1946).
[Lau 2003] Y. Y. Lau, Fei He, Donald P. Umstadter, and Richard Kowalczyk; Nonlinear
Thomson scattering: A tutorial; Physics of Plasmas vol. 10, no. 5, May 2003;
DOI: 10.1063/1.1565115.
[Handbook of chemistry and physics 2003-2004] David R. Lide; CRC Handbook of chemistry and physics 84th edition; 2003-2004 CRC press.
171
Bibliography
[Lefebvre 1997] Erik Lefebvre and Guy Bonnaud; Nonlinear electron heating in ultrahighintensity-laserplasma interaction; Phys. Rev. E vol. 55, no. 1 January 1997.
[Lefebvre 2010] Erik Lefebvre, Laurent Gremillet, Anna Lévy, Rachel Nuter, Patrizio Antici, Michaël Carrié, Tiberio Ceccotti, Mathieu Drouin, Julien Fuchs, Victor
Malka, and David Neely; Proton acceleration by moderately relativistic laser
pulses interacting with solid density targets; 09/01/2010 not yet published.
[Levy 2007] Anna Lévy, Tiberio Ceccotti, Pascal D'Oliveira, Fabrice Réau, Michel Perdrix, Fabien Quéré, Pascal Monot, Michel Bougeard, Hervé Lagadec, Philippe
Martin, Jean-Paul Geindre and Patrick Audebert; Double plasma mirror for
ultrahigh temporal contrast ultraintense laser pulses; Optics Letters Vol. 32
No. 3 February 1, 2007.
[Levy PhD 2008] Anna Levy; Acceleration d'ions par interaction laser-matiere en regime
de ultra haut contraste laser; Ecole polytechnique, PhD thesis.
[Levy 2009] A Lévy, R Nuter, T Ceccotti, P Combis, M Drouin, L Gremillet, P Monot, H
Popescu, F Réau, E Lefebvre and P Martin; Eect of a nanometer scale plasma
on laser-accelerated ion beams; New Journal of Physics 11 (2009) 093036;
doi:10.1088/1367-2630/11/9/093036.
[Li 2006]
Z. Li, H. Daido, A. Fukumi, A. Sagisaka, K. Ogura, M. Nishiuchi, S. Orimo,
Y. Hayashi, M. Mori, M. Kado, S. V. Bulanov, T. Zh. Esirkepov, Y. Oishi,
T. Nayuki, T. Fujii, K. Nemoto, S. Nakamura and A. Noda; Measurements of
energy and angular distribution of hot electrons and protons emitted from a pand s-polarized intense femtosecond laser pulse driven thin foil target; Physics
of Plasmas 13, 043104 (2006); DOI: 10.1063/1.2192758.
[Liseykina 2007] T. V. Liseykina, A. Macchi; Features of ion acceleration by circularly
polarized laser pulses; Physics of Plasmas (from arXiv:0705.4019v1).
[Liseykina 2008] Tatiana V Liseykina, Marco Borghesi, Andrea Macchi, and Sara Tuveri;
Radiation pressure acceleration by ultraintense laser pulses; Plasma Phys. Control. Fusion 50 (2008) 124033; DOI: 10.1088/0741-3335/50/12/124033.
[Liseykina 2009] Tatyana V. Liseykina, Domenico Prellino, Fulvio Cornolti, and Andrea
Macchi; Ponderomotive acceleration of ions:
circular vs linear polarization;
missing informations.
[Macchi 2005] Andrea Macchi,
Federica Cattani,
Tatiana V. Liseykina,
and Fulvio
Cornolti; Laser Acceleration of Ion Bunches at the Front Surface of Overdense
Plasmas; PRL 94, 165003 (2005); DOI: 10.1103/PhysRevLett.94.165003.
[Macchi 2008] Andrea Macchi, Tatiana V. Liseikina, Sara Tuveri, Silvia Veghini; Theory
and Simulation of Ion Acceleration with Circularly Polarized Laser Pulses;
(missing info), Preprint submitted to Elsevier Science November 30, 2008.
172
Bibliography
[Macchi 2010] Andrea Macchi, Silvia Veghini, Tatyana V Liseykina, and Francesco Pegoraro; Radiation pressure acceleration of ultrathin foils; New Journal of Physics
12 (2010) 045013; DOI: 10.1088/1367-2630/12/4/045013.
[Mackinnon 2001] A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H.
Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi; Eect of Plasma
Scale Length on Multi-MeV Proton Production by Intense Laser Pulses; PRL
vol 86 no 9 26 February 2001; DOI: 10.1103/PhysRevLett.86.1769.
[Malka 1996] G. Malka and J. L. Miquel; Experimental Conrmation of PonderomotiveForce Electrons Produced by an Ultrarelativistic Laser Pulse on a Solid Target;
PRL vol. 77 no. 1, 1 July 1996.
[Mancic 2010] A. Mancic, J. Robiche, P. Antici, P. Audebert, C. Blancard, P. Combis, F.
Dorchies, G. Faussurier, S. Fourmaux, M. Harmand, R. Kodama, L. Lancia, S.
Mazevet, M. Nakatsutsumi, O. Peyrusse, V. Recoules, P. Renaudin, R. Shepherd, J. Fuchs; Isochoric heating of solids by laser-accelerated protons: Experimental characterization and self-consistent hydrodynamic modeling; High
Energy Density Physics 6 (2010) 2128; DOI: 10.1016/j.hedp.2009.06.008.
[Marx 1966] G. Marx; Interstellar vehicle propelled by terrestrial laser beam; Nature vol
211 pag 22 July 2 1966.
[McKenna 2004] P. McKenna, K. W. D. Ledingham, J. M. Yang, L. Robson, T. McCanny, S. Shimizu, R. J. Clarke, D. Neely, K. Spohr, R. Chapman, R. P.
Singhal, K. Krushelnick, M. S. Wei, and P. A. Norreys; Characterization of
proton and heavier ion acceleration in ultrahigh-intensity laser interactions
with heated target foils; Phys. Rev. E 70, 036405 (2004); DOI: 10.1103/PhysRevE.70.036405.
[Mora 1979] Patrick Mora and R. Pellat; Self-similar expansion of a plasma into a vacuum;
Phys. Fluids 22 (12) page 2300, December 1979.
[Mora 1982] Patrick Mora; Theoretical model of absorption of laser light by a plasma;
Phys Fluids 25(6) page 1051, June 1982.
[Mora2003] P. Mora; Plasma Expansion into a Vacuum; PRL vol. 90, no. 18, 9 May 2003;
DOI: 10.1103/PhysRevLett.90.185002.
[Mora 2005] P. Mora; Thin-foil expansion into a vacuum; Phys. Rev. E 72, 056401 (2005);
DOI: 10.1103/PhysRevE.72.056401.
[Mora course 2008] Patrick Mora; Introduction aux plasmas créés par laser; lectures from
the course 2008 of Patrick Mora, Ecole Polytechnique.
[Nakamura 2007] Tatsufumi Nakamura, Kunioki Mima, Hitoshi Sakagami and Tomoyuki
Johzaki; Electron surface acceleration on a solid capillary target inner wall
irradiated with ultraintense laser pulses; Physics of Plasmas 14, 053112 (2007);
DOI: 10.1063/1.2731383.
173
Bibliography
[Nakanii 2008] N Nakanii, K Kondo, S Suzuki, T Kobayashi, T Asaka, K Yanagida, K
Tsuji, K Makino, T Yamane, T Yabuuchi, S Miyamoto, K Horikawa, T Aratani,
M Kashihara, Y Mori, H Hanaki, Y Kitagawa, K Mima, and K A Tanaka; Absolute calibration of Imaging Plate for electron spectrometer measuring GeVclass electrons; Journal of Physics: Conference Series 112 (2008) 032073; DOI:
10.1088/1742-6596/112/3/032073.
[Nakatsutsumi 2008] M Nakatsutsumi, J R Davies, R Kodama, J S Green, K L Lancaster,
K U Akli, F N Beg, S N Chen, D Clark, R R Freeman, C D Gregory, H Habara,
R Heathcote, D S Hey, K Highbarger, P Jaanimagi, M H Key, K Krushelnick,
T Ma, A MacPhee, A J MacKinnon, H Nakamura, R B Stephens, M Storm,
M Tampo, W Theobald, L Van Woerkom, R L Weber, M S Wei, N C Woolsey
and P A Norreys; Space and time resolved measurements of the heating of
solids to ten million kelvin by a petawatt laser; New Journal of Physics 10
(2008) 043046; DOI: 10.1088/1367-2630/10/4/043046.
[NRL Formulary 2009] J.D. Huba; NRL PLASMA FORMULARY 2009.
[Passoni 2004] M. Passoni, V. T. Tikhonchuk, M. Lontano, and V. Yu. Bychenkov; Charge
separation eects in solid targets and ion acceleration with a two-temperature
electron distribution; Phys. Rev. E 69, 026411 (2004); DOI: 10.1103/PhysRevE.69.026411.
[Passoni 2008] M. Passoni and M. Lontano; Theory of Light-Ion Acceleration Driven by
a Strong Charge Separation; PRL 101, 115001 (2008); DOI: 10.1103/PhysRevLett.101.115001.
[Pearlman 1978] J. S. Pearlman and R. L. Morse; Maximum Expansion Velocities of LaserProduced Plasmas; PRL vol. 40, no. 25 19June 1978.
[Ping 2008] Y. Ping, R. Shepherd, B. F. Lasinski, M. Tabak, H. Chen, H. K. Chung,
K. B. Fournier, S. B. Hansen, A. Kemp, D. A. Liedahl, K. Widmann, S. C.
Wilks, W. Rozmus, and M. Sherlock; Absorption of Short Laser Pulses on
Solid Targets in the Ultrarelativistic Regime; PRL 100, 085004 (2008); DOI:
10.1103/PhysRevLett.100.085004.
[Popescu 2005] H. Popescu, S. D. Baton, F. Amirano, C. Rousseaux, M. Rabec Le Gloahec, J. J. Santos, M. Koenig, E. Martinolli, T. Hall, J. C. Adam, A. Heron, D.
Batani; Subfemtosecond, coherent, relativistic, and ballistic electron bunches
generated at ω0 and 2ω0 in high intensity laser-matter interaction; Physics of
Plasmas 12, 063106 (2005); DOI: 10.1063/1.1927328.
[Pukhov 1996] A. Pukhov and J. Meyer-ter-Vehn; Relativistic Magnetic Self-Channeling
of Light in Near-Critical Plasma: Three-Dimensional Particle-in-Cell Simulation; PRL vol. 76, no. 21, 20 May 1996.
[Pukhov 1999] A. Pukhov, Z.-M. Sheng, and J. Meyer-ter-Vehn; Particle acceleration in
relativistic laser channels; Physics of Plasmas vol. 6, no. 7, July 1999.
174
Bibliography
[Pukhov 2001] A. Pukhov; Three-Dimensional Simulations of Ion Acceleration from a Foil
Irradiated by a Short-Pulse Laser; PRL vol. 86, no. 16, 16 April 2001 ; DOI:
10.1103/PhysRevLett.86.3562.
[Qiao 2009] B. Qiao, M. Zepf, M. Borghesi, and M. Geissler; Stable GeV Ion-Beam Acceleration from Thin Foils by Circularly Polarized Laser Pulses; PRL 102, 145002
(2009); DOI: 10.1103/PhysRevLett.102.145002.
[Robinson 2008] A P L Robinson, M Zepf, S Kar, R G Evans, and C Bellei; Radiation
pressure acceleration of thin foils with circularly polarized laser pulses; New
Journal of Physics 10 (2008) 013021; DOI: 10.1088/1367-2630/10/1/013021.
[Robson 2007] L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau,
O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlstrom, M. Zepf and
P. McKenna; Scaling of proton acceleration driven by petawatt-laserplasma
interactions; Nature Physics Vol. 3 January 2007; DOI: 10.1038/nphys476.
[Roth 2002] M. Roth, A. Blazevic, M. Geissel, T. Schlegel, T. E. Cowan, M. Allen, J.-C.
Gauthier, P. Audebert, J. Fuchs, J. Meyer-ter-Vehn, M. Hegelich, S. Karsch,
and A. Pukhov; Energetic ions generated by laser pulses: A detailed study on
target properties; Phys. Rev. Special Topics - Accelerators and Beams, vol. 5,
061301 (2002); DOI: 10.1103/PhysRevSTAB.5.061301.
[Roth 2005] M. Roth E. Brambrink, P. Audebert, A. Blazevic, R. Clarke, J. Cobble, T.E.
Cowan, J. Fernandez, J. Fuchs, M. Geissel, D. Habs, M. Hegelich, S. Karsch, K.
Ledingham, D. Neely H. Ruhl, T. Schlegel and J. Schreiber; Laser accelerated
ions and electron transport in ultra-intense laser matter interaction; Laser and
Particle Beams (2005), 23, 95100; DOI: 10.10170S0263034605050160.
[Sakabe 1980] S. Sakabe, T. Mochizuki, T. Yamanaka, and C. Yamanaka; Modied Thomson parabola ion spectrometer of wide dynamic range; Rev. Sci. Instr. 51 (10),
Oct. 1980.
[Santala 2000] M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K.
Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D.
Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke,
and P. A. Norreys; Eect of the Plasma Density Scale Length on the Direction
of Fast Electrons in Relativistic Laser-Solid Interactions; PRL vol. 84, no. 7,
14 February 2000.
[Sarri 2009] ; Application of proton radiography in experiments of relevance to inertial
connement fusion; G. Sarri, M. Borghesi, C.A. Cecchetti, L. Romagnani, R.
Jung, O. Willi, D.J. Hoarty, R.M. Stevenson, C.R.D. Brown, S.F. James, P.
Hobbs, J. Lockyear, S.V. Bulanov, and F. Pegoraro; Eur. Phys. J. D (2009);
DOI: 10.1140/epjd/e2009-00115-8.
[Schlenvoigt 2010] Hans-Peter Schlenvoigt, Oliver Jäckel, Sebastian M. Pfotenhauer, and
Malte C. Kaluza; Laser-based Particle Acceleration; Advances in Solid-State
175
Bibliography
Lasers:
Development and Applications ISBN 978-953-7619-80-0, pp. 630,
February 2010.
[Schreiber 2006a] J. Schreiber, F. Bell, F. Gruner, U. Schramm, M. Geissler, M. Schnurer,
S. Ter-Avetisyan, B. M. Hegelich, J. Cobble, E. Brambrink, J. Fuchs, P. Audebert, and D. Habs; Analytical Model for Ion Acceleration by High-Intensity
Laser Pulses; PRL 97, 045005 (2006); DOI: 10.1103/PhysRevLett.97.045005.
[Schreiber 2006b] J. Schreiber, S. Ter-Avetisyan, E. Risse, M. P. Kalachnikov, P. V. Nickles, W. Sandner, U. Schramm, D. Habs, J. Witte and M. Schnürer; Pointing
of laser-accelerated proton beams; Physics of Plasmas 13, 033111 2006; DOI:
10.1063/1.2181978.
[Schwoerer 2006] H. Schwoerer, S. Pfotenhauer, O. Jackel, K.-U. Amthor, B. Liesfeld, W.
Ziegler, R. Sauerbrey, K. W. D. Ledingham and T. Esirkepov; Laserplasma
acceleration of quasi-monoenergetic protons from microstructured targets; Nature Vol 439, 26 January 2006; DOI: 10.1038/nature04492.
[Sentoku 2003] Y. Sentoku, T. E. Cowan, A. Kemp, and H. Ruhl; High energy proton acceleration in interaction of short laser pulse with dense plasma target; Physics
of Plasmas Vol. 10, No. 5, May 2003; DOI: 10.1063/1.1556298.
[Sheng 2001] Z.-M. Sheng, K. Nishihara, T. Honda, Y. Sentoku, K. Mima, and S. V.
Bulanov; Anisotropic lamentation instability of intense laser beams in plasmas near the critical density; Phys. Rev. E, Vol. 64, 066409 (2001); DOI:
10.1103/PhysRevE.64.066409.
[Simmons 1993] J. F. L. Simmons and C. R. McInnes; Was Marx right? or How ecient
are laser driven interstellar spacecrafts?; Am. J. Phys. 61 (3), March 1993.
[Snavely 2000] R. A. Snavely, M. H. Key, S. P. Hatchett, T. E. Cowan, M. Roth, T.W.
Phillips, M. A. Stoyer, E. A. Henry, T. C. Sangster, M. S. Singh, S. C. Wilks,
A. MacKinnon, A. Oenberger, D.M. Pennington, K. Yasuike, A. B. Langdon,
B. F. Lasinski, J. Johnson, M. D. Perry, and E. M. Campbell; Intense HighEnergy Proton Beams from Petawatt-Laser Irradiation of Solids; PRL vol. 85
no. 14, 2 October 2000.
[Sprangle 1990] P. Sprangle, E. Esarey, and A. Ting; Nonlinear Theory of Intense LaserPlasma Interactions; PRL Vol. 64, No. 17 23 April 1990.
[Steinke 2009] S. Steinke, A. Henig, M. Schnurer, T. Sokollik, P.V. Nickles, D. Jung, D.
Kiefer, J. Schreiber, T. Tajima, X.Q. Yan, J. Meyer-ter-Vehn, M. Hegelich,
W. Sandner, and D. Habs;
Ecient ion acceleration by collective laser-
driven electron dynamics with ultra-thin foil targets; Physics of Plasma (from
arXiv:0909.2334v1).
[Strickland 1985] D. Strickland and G. Mourou; Compression of amplied chirped optical
pulses, Opt. Commun. 56, 219 1985.
176
Bibliography
[Tajima 2009] Toshiki Tajima, Dietrich Habs and Xueqing Yan; Laser Acceleration of Ions
for Radiation Therapy; Reviews of Accelerator Science and Technology Vol. 2
(2009) 201-228.
[Tanaka 2005] Kazuo
A.
Tanaka,
Yamada-oka,
2-6,
Suita,
Osaka
565-0871,
Japan
Toshinori Yabuuchi, Takashi Sato, Ryosuke Kodama, Yoneyoshi Kitagawa,
Teruyoshi Takahashi, Toshiji Ikeda, Yoshihide Honda, Shuuichi Okuda; Calibration of imaging plate for high energy electron spectrometer; Rev. Sci. Instr.
76, 013507 (2005); DOI: 10.1063/1.1824371.
[Toncian 2006] Toma Toncian, Marco Borghesi, Julien Fuchs, Emmanuel d'Humieres Patrizio Antici, Patrick Audebert, Erik Brambrink, Carlo Alberto Cecchetti, Ariane Pipahl, Lorenzo Romagnani, Oswald Willi1; Ultrafast LaserDriven Microlens to Focus and Energy-Select MegaElectron Volt Protons; Science Vol.
312 page 410, 21 April 2006.
[Wattelier 2004] B. Wattellier, J. Fuchs, J. P. Zou and K. Abdeli, C. Haefner, H. Pépin;
High-power short-pulse laser repetition rate improvement by adaptive wave
front correction;
Rev. Sci. Instr. Vol. 75, No. 12, December 2004;
DOI:
10.1063/1.1819379.
[WenXi 2008] Liang WenXi, Li YuTong, Xu MiaoHua, Yuan XiaoHui, Zheng ZhiYuan,
Zhang Yi, Liu Feng, Wang ZhaoHua, Li HanMing, Jin Zhan, Wei ZhiYi, Zhao
Wei, Li YingJun, and Zhang Jie; Study of hot electrons generated from intense
laser-plasma interaction employing Image Plate; Sci China Ser G-Phys Mech
Astron, Oct. 2008, vol. 51, no. 10,1455-1462; DOI: 10.1007/s11433-008-0146-y.
[Wilks 1992] S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon; Absorption of
Ultra Intense Laser Pulse; PRL Vol. 69, No. 9, 31 August 1992.
[Wilks 2001] S. C. Wilks, A. B. Langdon, T. E. Cowan, M. Roth, M. Singh, S. Hatchett,
M. H. Key, D. Pennington, A. MacKinnon, and R. A. Snavely; Energetic
proton generation in ultra-intense lasersolid interactions; Physics of Plasmas
Vol. 8, No. 2, February 2001; DOI: 10.1063/1.1333697.
[Yabuuchi 2007] T. Yabuuchi, K. Adumi, H. Habara, R. Kodama, K. Kondo, T. Tanimoto,
K. A. Tanaka, Y. Sentoku, T. Matsuoka, Z. L. Chen, M. Tampo, A. L. Lei, and
K. Mima; On the behavior of ultraintense laser produced hot electrons in selfexcited elds; Physics of Plasmas 14, 040706 (2007); DOI: 10.1063/1.2722303.
[Yan 2008] X. Q. Yan, C. Lin, Z. M. Sheng, Z. Y. Guo, B. C. Liu, Y. R. Lu, J. X. Fang;
and J. E. Chen; Generating High-Current Monoenergetic Proton Beams by
a Circularly Polarized Laser Pulse in the Phase-Stable Acceleration Regime;
PRL 100, 135003 (2008); DOI: 10.1103/PhysRevLett.100.135003.
[Yin 2007] L. Yin, B. J. Albright, B. M. Hegelich, K. J. Bowers, K. A. Flippo, T. J. T.
Kwan, and J. C. Fernández; Monoenergetic and GeV ion acceleration from
177
Bibliography
the laser breakout afterburner using ultrathin targets; Physics of Plasmas 14,
056706 (2007); DOI: 10.1063/1.2436857.
[Young 1996] P. E. Young and P. R. Bolton; Propagation of Subpicosecond Laser Pulses
through a Fully Ionized Plasma; PRL Vol. 77, No. 22, 25 November 1996.
[Zuo 2000] J. M. Zuo; Electron Detection Characteristics of a Slow-Scan CCD Camera,
Imaging Plates and Film, and Electron Image Restoration; Microsc. Res. Tech.
49:245268 (2000).
178