11institutetext: Institute for Nuclear Physics, Technische Universität
Darmstadt, Darmstadt, Germany

# Ion Acceleration—Target Normal Sheath Acceleration††thanks: Re-published,
with permission, from P. McKenna et al (eds.), "Laser-Plasma Interactions and
Applications", Scottish Graduate Series (Springer International Publishing,
2013).

M. Roth and M. Schollmeier

###### Abstract

Energetic ions have been observed since the very first laser-plasma
experiments. Their origin was found to be the charge separation of electrons
heated by the laser, which transfers energy to the ions accelerated in the
field. The advent of ultra-intense lasers with pulse lengths in the
femtosecond regime resulted in the discovery of very energetic ions with
characteristics quite different from those driven by long-pulse lasers.
Discovered in the late 1990s, these ion beams have become the focus of intense
research worldwide, because of their unique properties and high particle
numbers. Based on their non-isotropic, beam-like behaviour, which is always
perpendicular to the emitting surface, the acceleration mechanism is called
target normal sheath acceleration (TNSA). We address the physics of the
mechanism and its dependence on laser and target parameters. Techniques to
explore and diagnose the beams, to make them useful for applications, are also
addressed.

Keywords
Laser; ion acceleration; ultraintense lasers; plasma; accelerator.

##  0.1 Introduction

Since the first irradiation of a target by a laser, the generation of
energetic ions has been well known. The origin of those ions is the electric
field generated by the charge separation as a result of the energy transferred
from a long-pulse laser to the electrons, and their respective temperature
[1]. The ions are then accelerated in the double-layer potential and can reach
significant particle energies, expanding isotropically in all directions from
the target front surface. Since the advent of ultra-short-pulse lasers with
pulse lengths of less than picoseconds, one of the most exciting results
obtained in experiments using solid targets is the discovery of very
energetic, very intense bursts of ions coming off the rear, non-irradiated
surface in a very high quality, beam-like fashion. At the turn of the century,
a number of experiments have resulted in proton beams with energies of up to
several tens of megaelectronvolts generated behind thin foils irradiated by
lasers exceeding hundreds of terawatts [2, 3, 4]. Since the first
observations, an extraordinary amount of experimental and theoretical work has
been devoted to the study of these beams’ characteristics and production
mechanisms. Particular attention has been devoted to the exceptional
accelerator-like spatial quality of the beams, and current research focuses on
their optimization for use in a number of groundbreaking applications,
addressed in Section 0.4. But first we will focus on the best understood of
all the possible acceleration mechanisms, so-called target normal sheath
acceleration (TNSA).

The greater part of this chapter is drawn from Ref. [5]. Review articles about
TNSA, the diagnostics of short-pulse laser plasmas, and applications in fast
ignition can also be found in Refs. [6, 7, 8, 9, 10].

##  0.2 Target normal sheath acceleration: the mechanism

###  0.2.1 Initial conditions

The primary interaction of a high-intensity short laser pulse with a solid
target strongly depends on the contrast of the laser pulse, that is, the ratio
of unwanted, preceding laser light to the main pulse. At peak intensities
exceeding $10^{20}\UW/\UcmZ^{2}$ even a contrast of $10^{6}$ is sufficient to
excite a plasma that is expanding towards the incoming main pulse. As a common
source of this unwanted laser light, amplified spontaneous emission or
prepulses, caused by a limited polarization separation in regenerative
amplifiers, have been identified. This ablative plasma sets the stage for a
wealth of uncontrolled phenomena at the interaction of the main pulse with the
target. The laser beam can undergo self-focusing due to ponderomotive force or
relativistic effects, thereby strongly increasing the resulting intensity, or
it can break up into multiple filaments, or, finally, it can excite
instabilities that ultimately lead to the production of energetic electrons.
Moreover, the ablative pressure of blow-off plasma caused by the incident
laser energy prior to the main pulse launches a shockwave into the target,
which can ultimately destroy the target before the arrival of the main pulse.
We address this issue, even though it is not directly related to the TNSA
mechanism, because of its influence on the electron spectrum and the thickness
of targets that can be used in practice.

###  0.2.2 General description

Before going into detail, it is worth taking a step back and qualitatively
looking at the general picture of the TNSA ion acceleration mechanism. Let us
interpret the process of generating a proton beam by TNSA as a new variation
on a familiar theme—acceleration by a sheath electrostatic field generated by
the hot-electron component. We assume the interaction of an intense laser
pulse well exceeding $10^{18}\UW/\UcmZ^{2}$ with a solid thin foil target as
the standard case for TNSA. The interaction of the intense laser pulse with
the preformed plasma and the underlying solid target constitutes a source of
hot electrons with an energy spectrum related to the laser intensity. This
cloud of hot electrons penetrates the foil at, as we shall see, an opening
angle of about $30^{\circ}$ and escapes into the vacuum behind the target. The
target’s capacitance, however, allows only a small fraction of the electrons
to escape before the target is sufficiently charged that escape is impossible
for even megaelectronvolt electrons. Those electrons are then
electrostatically confined to the target and circulate back and forth through
the target, laterally expanding and forming a charge-separation field on both
sides over a Debye length. At the rear surface there is no screening plasma,
owing to the short time-scales involved, and the induced electric fields are
of the order of several teravolts per metre. Such fields can ionize atoms and
rapidly accelerate ions normal to the initially unperturbed surface. The
resulting ion trajectories thus depend on the local orientation of the rear
surface and the electric field lines driven by the time-dependent electron
density distribution. As the ions start from a cold solid surface just driven
by quasi-static electric fields, the resulting beam quality is extremely high,
as we shall see. This process is illustrated in tnsa-scheme.

Figure 1: Target normal sheath acceleration. A thin target foil with thickness
$d=5$–$50\Uum$ is irradiated by an intense laser pulse. The laser prepulse
creates a preplasma on the target’s front side. The main pulse interacts with
the plasma and accelerates megaelectronvolt electrons, mainly in the forward
direction. The electrons propagate through the target, where collisions with
the background material can increase the divergence of the electron current.
The electrons leave the rear side, resulting in a dense sheath. An electric
field due to charge separation is created. The field is of the order of the
laser electric field ($\sim\UVZ[T]/\UmZ$), and ionizes atoms at the surface.
The ions are then accelerated in this sheath field, pointing in the target
normal direction.

###  0.2.3 Electron driver

Current laser systems are not yet capable of accelerating ions directly.
Therefore, all existing laser ion acceleration mechanisms rely on the driving
electron component and the resulting strong electric field caused by charge
separation. Thus the electron driver is extremely important, and will be
discussed here in detail. As a rule of thumb, particle-in-cell calculations
[11, 12, 13] have indicated that the so-called hot-electron component has a
logarithmic-slope temperature that is roughly equal to the ponderomotive
potential of the laser beam. This is represented by the cycle-averaged kinetic
energy of an electron oscillating in the laser electromagnetic field,
$T_{\textrm{hot}}\approx U_{\textrm{pond}}\approx
1\UMeV\times(I\lambda^{2}/10^{19}\UW\cdot\Uum^{2}/\UcmZ^{2})^{1/2}$ in the
relativistic regime [14]. The relativistic electrons are directed mainly in
the forward direction [15]; hence, the particle distribution function can be
simplified by a one-dimensional Maxwell–Jüttner distribution, which is close
to an ordinary Boltzmann distribution.

The conversion efficiency from laser energy to hot electrons is not perfect,
and only a fraction $\eta$ is converted. The total number of electrons is

$n_{0}=\frac{\eta E_{\textrm{L}}}{c\tau_{\textrm{L}}\pi
r_{0}^{2}k_{\textrm{B}}T_{\textrm{hot}}}\leavevmode\nobreak\ ,$ (1)

following a scaling with intensity as

$\eta=1.2\times 10^{-15}I^{0.74}\leavevmode\nobreak\ ,$ (2)

with the intensity in watts per square centimetre reaching up to 50% [16]. For
ultra-high intensities, $\eta$ can reach up to 60% for near-normal incidence
and up to 90% for irradiation under $45^{\circ}$ [17]. A discussion on which
distribution function best fits the experimental data is given in Ref. [18]
and, in more detail, in Ref. [19]. However, neither theoretical nor
experimental data give a clear answer to the question about the shape of the
distribution function.

Given the intensities of modern short-pulse lasers, therefore, copious amounts
of energetic electrons are generated and, in contrast with thermal electrons
in long-pulse laser plasmas, are pushed into the target. It is fair to
estimate that a fraction of $N=\eta
E_{\textrm{L}}/k_{\textrm{B}}T_{\textrm{hot}}$ electrons in the
megaelectronvolt range are created, where $E_{\textrm{L}}$ is the laser
energy. These electrons have typical energies such that their mean free path
is much longer than the thickness of the targets typically used in
experiments. While the electrons propagate through the target, they constitute
a current that exceeds the Alfvén limit by several orders of magnitude. Alfvén
found that the main limiting factor on the propagation of an electron beam in
a conductor is the self-generated magnetic field, which bends the electrons
back towards the source [20]. For parameters of relevance for inertial
confinement fusion, a good review is given in Ref. [21]. So as not to exceed
the limit of
$j_{\textrm{A}}=m_{\textrm{e}}c^{3}\beta\gamma/e=17\beta\gamma\leavevmode\nobreak\
(\UkAZ)$, the net current must be largely compensated for by return currents,
to minimize the resulting magnetic field. The return currents will be driven
by the charge separation in the laser–plasma interaction region and strongly
depends on the electrical conductivity of the target, as those currents are
lower in energy and thereby affected by the material properties. The large
counter-streaming currents also give rise to instabilities, which affect the
forward motion of the electrons. The influence of limited electrical
conductivity on the inhibition of fast-electron propagation has been addressed
in Ref.[22], also with respect to space charge separation. Without the return
currents, the electric field would stop the electrons in a distance of less
than 1[23]. The electric field driving the return current, in turn, can be
strong enough to stop the fast electrons. This effect, known as transport
inhibition, is prominent in insulators, but almost negligible in conductors
[24].

The propagation of electrons through the target is still an active field of
research. As depicted in etransport, the laser pushes the critical surface
$n_{\textrm{c}}$, leading to a steepening of the electron density profile. The
motion of the ablated plasma causes a shockwave to be launched into the
target, leading to ionization and therefore a modification of the initial
electrical conductivity. As soon as the electrons penetrate the cold solid
region, binary collisions (multiple small-angle scattering) with the
background material are no longer negligible. These tend to broaden the
electron distribution, counteracting the magnetic field effect [25]. For long
propagation distances ($z\geq 15\Uum)$, the current density is low enough that
broadening due to small-angle scattering becomes the dominating mechanism
[26].

Figure 2: Schematic of laser-generated fast-electron transport. The laser
(shown in red) impinges on a preplasma with exponential density profile from
the left side. The light pressure leads to profile steepening, depicted in the
graph at the top of the figure. An ablation plasma creates an inward-
travelling shockwave that heats, ionizes, and compresses the target. Fast
electrons are created by the laser, propagating into the dense plasma towards
the target’s rear side. The high electron current $j_{\textrm{fast}}$ can lead
to filamentation and magnetic field generation (shown by the light red- and
blue-coloured areas), as well as driving a return current $j_{\textrm{ret}}$.
The global magnetic field tends to pinch the fast-electron current. Electrons
propagating in the dense solid matter interact with the background material by
binary collisions. This leads to a spatial broadening of the electron
distribution, which becomes the major effect for longer distances. At the rear
side, the electrons form a sheath and build up an electrostatic field $E_{z}$
(grey line in graph). This can lead to refluxing (recirculation) of the
electrons, heating the target even further.

The majority of data show a divergent electron transport. The transport full-
cone angle of the electron distribution was determined to be dependent on
laser energy and intensity, as well as target thickness. For rather thick
targets $(d>40\Uum)$ this value is around $30^{\circ}$ for full width at half
maximum (FWHM), whereas for thin targets $(d\leq 10\Uum)$ published values are
of the order of $16^{\circ}$ (this figure was indirectly obtained by a fit to
proton energy measurements) and are ${\approx}150^{\circ}$ at most [27]. It
has been shown that different diagnostics lead to different electron transport
cone angles [28], so the nature of the ‘true’ cone angle dependence with laser
and target parameters remains unclear.

When the electrons reach the rear side of the target, they form a dense
charge-separation sheath. The outflowing electrons lead to a toroidal magnetic
field $B_{\theta}$, which can spread the electrons over large transverse
distances by a purely kinematic $E\times B_{\theta}$ force [29], sometimes
called the fountain effect [30]. The electric field created by the electron
sheath is sufficiently strong to deflect electrons back into the target, which
then recirculate. Experimental evidence for recirculating electrons is
presented in Refs. [17, 31, 32]. The relevance to proton acceleration was
first demonstrated by MacKinnon et al. [33], who measured a strong enhancement
of the maximum proton energy for foils thinner than $10\Uum$, compared with
thicker foils. With the help of computer simulations, this energy enhancement
was attributed to an enhanced sheath density caused by refluxing electrons.
Further evidence of refluxing electrons was also found in an experiment
discussed in Ref. [34].

Neglecting the complicated interaction for thicknesses below $d\approx
15\Uum$, a reasonable estimate for the electron beam divergence is the
assumption that the electrons are generated in a region of the size of the
laser focus and are purely collisionally transported to the rear side of the
target. This is in agreement with most published data. The broadening of the
distribution is then due to multiple Coulomb small-angle scattering, given
analytically, by Molière’s theory in Bethe’s description [35]. To lowest
order, the angular broadening, $f(\Theta)$, follows a Gaussian (see Ref. [35],
Eq. (27)),

$f(\theta)=\frac{2\textrm{e}^{-\vartheta^{2}}}{\chi_{\textrm{c}}^{2}B}\sqrt{\theta/\sin\theta},$
(3)

where the second term on the right-hand side is a correction for larger angles
(from Ref. [35], Eq. (58)). The angle $\vartheta$ can be related to $\theta$
by $\vartheta=\theta/\chi_{\textrm{c}}B^{1/2}$. The transcendental equation,
$B-\textrm{ln}B=\textrm{ln}\left(\chi_{\textrm{c}}^{2}/\chi_{a^{\prime}}^{2}\right)$,
determines $B$. The screening angle $\chi_{a^{\prime}}^{2}$ is given by
$\chi_{a^{\prime}}^{2}=1.167(1.13+3.76\alpha^{2})\lambda^{2}/a^{2}$, where
$\lambda=\hbar/p$ is the de Broglie wavelength of the electron and
$a=0.885a_{\textrm{B}}Z^{-1/3}$, with the Bohr radius $a_{\textrm{B}}$.
$\alpha$ is determined by $\alpha=Ze^{2}/(4\pi\epsilon_{0}\hbar\beta c)$ with
the nuclear charge $Z$, electron charge $e$, and $\beta=v/c$, where
$\epsilon_{0}$, $\hbar$, and $c$ denote the usual constants. The variable
$\chi_{\textrm{c}}$ is given by

$\chi_{\textrm{c}}^{2}=\frac{e^{4}}{4\pi\epsilon_{0}^{2}c^{2}}\frac{Z(Z+1)Nd}{\beta^{2}p^{2}}\leavevmode\nobreak\
,$ (4)

with the electron momentum $p$ and $N=N_{\textrm{A}}\rho/A$ being the number
of scattering atoms, determined by Avogadro’s number $N_{\textrm{A}}$,
material density $\rho$, and mass number $A$. $\chi_{\textrm{c}}$ is
proportional to the material thickness $d$ and density $\rho$ as
$\chi_{\textrm{c}}\propto(\rho d)^{1/2}$. Since $\chi_{\textrm{c}}$ determines
the width of $f(\theta)$, the angular spread of the electron distribution
propagating through matter is proportional to its thickness as well as its
density. The analytical formula allows us to estimate the broadening of the
laser-accelerated electron distribution during the transport through the cold
solid target. For a laser intensity $I_{\textrm{L}}=10^{19}\UW/\UcmZ^{2}$, the
mean energy (temperature) is $k_{\textrm{B}}T_{\textrm{hot}}\approx 1\UMeV$.
The increase in distribution radius $r$ with target thickness $d$ is shown in
broaden. The electrons were propagated in aluminium (red dashed lines) and
gold (blue dashed lines). Aluminium does not lead to a strong broadening,
owing to its low density and $Z$; compare this with the broadening observed in
gold. The graph shows that in each case the radius at the rear side scales as
$r\propto d^{2}$ (green lines).

Figure 3: Increase in radius $r$ of an electron distribution with target
thickness $d$. The calculation utilized eq:broaden, taking an energy of
$k_{\textrm{B}}T_{\textrm{hot}}\approx 1\UMeV$, corresponding to a laser
intensity $I=10^{19}\UW/\UcmZ^{2}$.

The estimate based on an electron distribution broadening determined by small-
angle scattering can be used to explain the measured proton beam profiles. It
should be noted that although the model seems to be able to calculate the
broadening of the forward-propagating fast-electron distribution generated by
intense laser–matter interaction, it could fail to determine the real number
of electrons arriving at the rear side of the target. According to Davies
[19], the generation of electromagnetic fields, as well as the recirculation
of the electrons, must be taken into account, making estimation and even
calculation very difficult. Recent experiments by Akli et al. [36] have shown
that this is true at least for targets thinner than $20\Uum$, but for thicker
foils the assumption of strong recirculation overestimates the number of
electrons. Therefore, the question of whether electromagnetic fields and
recirculation are essential to determine fast-electron transport from the
front to the rear side of the target can still not be satisfactorily answered,
although assuming simple collisional broadening gives a a relatively good
estimate.

###  0.2.4 Target normal sheath acceleration

The electrons are transported through the target to its rear side. The laser
creates about $10^{13}$ electrons, which potentially all propagate through the
target. The broadening results in transverse extension, which can be estimated
by

$r_{\textrm{sheath}}=r_{0}+d\tan(\theta/2),$ (5)

where $r_{0}$ denotes the laser spot radius, $d$ the target thickness, and
$\theta$ the broadening angle of the distribution, calculated using
eq:broaden. The electrons exhibit an exponential energy distribution

$n_{\textrm{hot}}(E)=n_{0}\exp\left(-\frac{E}{k_{\textrm{B}}T_{\textrm{hot}}}\right)$
(6)

with temperature $k_{\textrm{B}}T$ and overall density $n_{0}$, as given by
eq:total_el. The electron density at the rear side (neglecting recirculation),
therefore, can be estimated as

$\displaystyle n_{\textrm{e},0}$ $\displaystyle=\frac{\eta
E_{\textrm{L}}}{c\tau_{\textrm{L}}\pi(r_{0}+d\tan\theta/2)^{2}k_{\textrm{B}}T_{\textrm{hot}}}$
(7)

This last approximation was obtained by inserting
$E_{0}=\sqrt{2I_{0}/\epsilon_{0}c}\approx 2.7\times 10^{12}\UV/\UmZ$ and using
Eqs. (1) and (2), together with a practical notation for the electron
temperature based on ponderomotive scaling,

$k_{\textrm{B}}T_{\textrm{hot}}=m_{0}c^{2}\left(\sqrt{1+\frac{I_{0}\leavevmode\nobreak\
\left[\UWZ/\UcmZ^{2}\right]\leavevmode\nobreak\
\lambda_{\textrm{L}}^{2}\leavevmode\nobreak\
\left[\UumZ^{2}\right]}{1.37\times 10^{18}}}-1\right)\leavevmode\nobreak\ ,$
(8)

in eq:edens1. $I_{18}$ indicates that the intensity has to be taken in units
of $10^{18}\UW/\UcmZ^{2}$. The estimate shows that the electron density at the
rear side of the target strongly scales with the laser intensity and is
inversely proportional to the square of the target thickness. Taking the
standard example of a laser pulse with $I=10^{19}\UW/\UcmZ^{2}$, focused to a
spot of $r_{0}=10\Uum$ and assuming a target thickness $d=20\Uum$, the angular
broadening according to eq:broaden is $\theta=42^{\circ}$ (FWHM) for electrons
with mean energy $k_{\textrm{B}}T$, as determined by eq:pond. Hence, the
electron density at the target’s rear side is $n_{\textrm{e},0}=1.4\times
10^{20}\UcmZ^{-3}$. This is orders of magnitude below the density of the solid
and justifies the assumption of a shielded transport through the target.

The electrons arrive at the rear side of the target and escape into the
vacuum. The charge separation leads to an electric potential $\Phi$ in the
vacuum region, according to Poisson’s equation. In one dimension, it is given
as

$\epsilon_{0}\frac{\partial^{2}\Phi}{\partial
z^{2}}=en_{\textrm{e}}\leavevmode\nobreak\ .$ (9)

To solve poisson, it is assumed that the solid matter in one half-space
$(z\leq 0)$ perfectly compensates for the electric potential, whereas for
$z\rightarrow\infty$ the potential goes to infinity. The derivative
$\partial\Phi/\partial z$ vanishes for $z\rightarrow\pm\infty$. In the vacuum
region $(z>0)$, the field can be obtained analytically [37]. The electron
density is taken as

$n_{\textrm{e}}=n_{\textrm{e},0}\exp\left(\frac{e\Phi}{k_{\textrm{B}}T_{\textrm{hot}}}\right)\leavevmode\nobreak\
,$ (10)

where the electron kinetic energy is replaced by the potential energy
$-e\Phi$. The initial electron density $n_{\textrm{e},0}$ is taken from
eq:edens. The solution of the Poisson equation is found by using the ansatz
$e\Phi/k_{\textrm{B}}T_{\textrm{hot}}=-2\ln(\lambda z+1)$, where $\lambda$ is
a constant defined by the solution and the ‘$+1$’ is necessary to fulfil a
continuous solution with the condition $\Phi(0)=0$ at the boundary to the
solid matter. The resulting potential is

$\Phi(z)=-\frac{2k_{\textrm{B}}T_{\textrm{hot}}}{e}\ln\left(1+\frac{z}{\sqrt{2}\lambda_{\textrm{D}}}\right)$
(11)

and the corresponding electric field reads

$E(z)=\frac{2k_{\textrm{B}}T_{\textrm{hot}}}{e}\frac{1}{z+\sqrt{2}\lambda_{\textrm{D}}}\leavevmode\nobreak\
.$ (12)

In this solution, the electron Debye length

$\lambda_{\textrm{D}}=\left(\frac{\epsilon_{0}k_{\textrm{B}}T_{\textrm{hot}}}{e^{2}n_{\textrm{e},0}}\right)^{1/2}$
(13)

appears; this is defined as the distance over which significant charge
separation occurs [38]. Replacing $k_{\textrm{B}}T_{\textrm{hot}}$ with
eq:pond and $n_{\textrm{e},0}$ with eq:edens leads to

$\lambda_{\textrm{D}}\approx
1.37\Uum\frac{r_{0}+d\tan\theta/2}{r_{0}}\frac{\sqrt{1+0.73I_{18}\lambda^{2}}-1}{I_{18}^{7/8}}\leavevmode\nobreak\
.$ (14)

The Debye length, or longitudinal sheath extension, on the rear side is of the
order of a micrometre. It scales quadratically with target thickness (since
$d\tan(\theta/2)\propto d^{2}$) and is inversely proportional to the laser
intensity. Thus, a higher laser intensity on the front side of the target
leads to a shorter Debye length at the rear side and results in a stronger
electric field. The standard example of Debyelength leads to
$\lambda_{\textrm{D}}=0.6\Uum$.

The maximum electric field is obtained at $z=0$:

$\displaystyle E_{\textrm{max}}(z=0)$
$\displaystyle=\frac{\sqrt{2}k_{\textrm{B}}T_{\textrm{hot}}}{e\lambda_{\textrm{D}}}$
(15) $\displaystyle\approx 5.2\times 10^{11}\UV/\UmZ\leavevmode\nobreak\
\frac{r_{0}}{r_{0}+d\tan\theta/2}I_{18}^{7/8}\leavevmode\nobreak\ ,$ (16)
$\displaystyle=9\times 10^{10}\UV/\UmZ\leavevmode\nobreak\
\frac{r_{0}}{r_{0}+d\tan\theta/2}E_{12}E_{12}^{3/4}.$ (17)

Hence, the initial field at $z=0$ is proportional to the laser intensity and
depends almost quadratically on the laser’s electric field strength. In
eq:maxelectricfield, the laser’s electric field strength is inserted in
normalized units of $10^{12}\UV/\UmZ$. By inserting the dependence of the
broadening with target thickness from broaden, the scaling with the target
thickness is obtained as $E_{\textrm{max}}(z=0)\propto d^{-2}$. The standard
example leads to a maximum field strength of $E_{\textrm{max}}\approx 2\times
10^{12}\UV/\UmZ$ just at the surface, that is it is of the order of
$\UV[T]/\UmZ$ or $\UV[M]/\UumZ$. It is only slightly smaller than the laser
electric field strength of $E_{0}=8.7\times 10^{12}\UV/\UmZ$. However, at
times later than $t=0$, the field strength is dictated by the dynamics at the
rear side of the target, ionization and ion acceleration.

As just mentioned, the electric field strength instantly leads to ionization
of the atoms at the target’s rear surface, since it is orders of magnitude
above the ionization threshold of the atoms. A simple model to estimate the
electric field strength necessary for ionization is the field ionization by
barrier suppression (FIBS) model [39]. The external electric field of the
laser overlaps the Coulomb potential of the atom and deforms it. As soon as
the deformation is below the binding energy of the electron, the electron is
instantly freed, hence the atom is ionized. The threshold electric field
strength $E_{\textrm{ion}}$ can be obtained with the binding energy
$U_{\textrm{bind}}$ as

$E_{\textrm{ion}}=\frac{\pi\epsilon_{0}U_{\textrm{bind}}^{2}}{e^{3}Z}\leavevmode\nobreak\
.$ (18)

As the electron sheath at the rear side is relatively dense, the atoms could
also be ionized by collisional ionization. However, as discussed by Hegelich
[40], the cross-section for field ionization is much higher than the cross-
section for collisional ionization for the electron densities and electric
fields appearing at the target surface. Taking the ionization energy of a
hydrogen atom with $U_{\textrm{bind}}=13.6\UeV$, the field strength necessary
for FIBS is $E_{\textrm{ion}}=10^{10}\UV/\UmZ$. This is two orders of
magnitude less than the field strength developed by the electron sheath in
vacuum calculated earlier. Hence, nearly all atoms (protons, carbon, heavier
particles) at the rear side of the target are instantly ionized and, since
they are no longer neutral particles, they are then subject to the electric
field and are accelerated. The maximum charge state of ions found in an
experiment is an estimate of the maximum field strength that appeared. This
has been used to estimate the sheath peak electric field value [40], as well
as the transverse field extension [41, 42].

The strong field ionizes the target and accelerates ions to mega electronvolt
energies, if it is applied for a sufficiently long time. The time can easily
be calculated by the assumption of a test particle moving in a static field,
generated by the electrons. Free protons were chosen as test particles. The
non-linear equation of motion is obtained from eq:field. The solution was
obtained numerically with matlab [43]. It shows that for a proton to obtain a
kinetic energy of 5, the field has to stay for $500\Ufs$ in the shape given by
eq:field. During this time, the proton has travelled $11.3\Uum$. The electric
field will be created as soon as electrons leave the rear side of the target.

Some electrons can escape this field, whereas others with lower energy will be
stopped and will be re-accelerated back into the target. Since the electron
velocity is close to the speed of light and the distances are of the order of
a micrometre, this happens on a time-scale of a few femtoseconds, leading to a
situation where electrons are always present outside the rear side of the
target. The electric field being created does not oscillate but is quasi-
static on the order of the ion acceleration time. Therefore, ultra-short laser
pulses, although providing the highest intensities, are not the optimum laser
pulses for ion acceleration. The electric field is directed normal to the
target’s rear surface; hence, the direction of the ion acceleration is normal
to the target, giving the process its name, target normal sheath acceleration.

###  0.2.5 Expansion models

The laser acceleration of ions from solid targets is a complicated,
multidimensional mechanism including relativistic effects, non-linearities,
and collective and kinetic effects. Theoretical methods for the various
physical mechanisms involved in TNSA range from analytical approaches for
simplified scenarios over fluid models up to fully relativistic, collisional
three-dimensional computer simulations.

Most of the approaches that describe TNSA neglect the complex laser–matter
interaction at the front side of the target as well as the electron transport
through the foil. These plasma expansion models start with a hot-electron
distribution that drives the expansion of an initially given ion distribution
[16, 44, 45, 46, 47, 48, 49, 37, 50]. Crucial features such as the maximum ion
energy, as well as the particle spectrum, can be obtained analytically,
whereas the dynamics have to be obtained numerically. The plasma expansion
description dates back to 1954 [51]. Since then, various refinements of the
models were obtained, with an increasing activity after the first discovery of
TNSA. These calculations resemble the general features of TNSA. Nevertheless,
they rely on somewhat idealized initial conditions from simple estimates. In
addition to that, the plasma expansion models are one-dimensional, whereas the
experiments have clearly shown that TNSA is at least two-dimensional. Hence,
these models can only reproduce one-dimensional features, the particle
spectrum of the TNSA process.

Sophisticated three-dimensional computer simulation techniques have been
developed for a better understanding of the whole process of short-pulse high-
intensity laser–matter interaction, electron transport and subsequent ion
acceleration. The simulation methods can be classified as (i) particle-in-
cell, (ii) Vlasov, (iii) Vlasov–Fokker–Planck, (iv) hybrid fluid or particle,
and (v) gridless particle codes; see the short review in Ref. [23] for a
description of each method.

The particle-in-cell method is the most widely used simulation technique. In
this method, Maxwell’s equations are solved, together with a description of
the particle distribution functions. The method more or less resembles a
‘numerical experiment’ with only a few approximations; hence, a detailed
insight into the dynamics can be obtained. The disadvantage is that no
specific theory serves as an input parameter and the results must be analyzed
like experimental results, i.e., they need to be interpreted and compared with
analytical estimates.

#### Plasma expansion model

Plasma expansion is often described as an isothermal rarefaction wave into
free space. There is quite a large similarity with the expansion models used
to describe TNSA. The isothermal expansion model assumes quasi-neutrality,
$n_{\textrm{e}}=Zn_{\textrm{i}}$, and a constant temperature $T_{\textrm{e}}$.
Using the two-fluid hydrodynamic model for electrons and ions, the continuity,
momentum, and energy conservation equations are used, usually with the
assumption of an isothermal expansion (no temperature change in time), no
further source term (no laser), no heat conduction, collisions or external
forces, and a pure electrostatic acceleration (no magnetic fields). One can
find a self-similar solution [52]:

$\displaystyle v(z,t)$
$\displaystyle=c_{\textrm{s}}+\frac{z}{t}\leavevmode\nobreak\ ,$ (19)
$\displaystyle n_{\textrm{e}}(z,t)$
$\displaystyle=Zn_{\textrm{i}}(z,t)=n_{\textrm{e},0}\exp\left(-\frac{z}{c_{\textrm{s}}t}-1\right)\leavevmode\nobreak\
,$ (20)

where $v$ denotes the bulk velocity and $n_{\textrm{i}}(n_{\textrm{e}})$ the
evolution of the ion (electron) density. The rarefaction wave expands with the
sound velocity
$c_{\textrm{s}}^{2}=Zk_{\textrm{B}}T_{\textrm{e}}/m_{\textrm{i}}$. By
combining these two equations, replacing the velocity with the kinetic energy
$v^{2}=2E_{\textrm{kin}}/m$ and taking the derivative with respect to
$E_{\textrm{kin}}$, the ion energy spectrum
$\textrm{d}N/\textrm{d}E_{\textrm{kin}}$ from the quasi-neutral solution per
unit surface and per unit energy in dependence of the expansion time $t$ is
obtained [46]:

$\frac{\textrm{d}N}{\textrm{d}E_{\textrm{kin}}}=\frac{n_{\textrm{e},0}c_{\textrm{s}}t}{\sqrt{2Zk_{\textrm{B}}T_{\textrm{hot}}E_{\textrm{kin}}}}\exp\left(-\sqrt{\frac{2E_{\textrm{kin}}}{Zk_{\textrm{B}}T_{\textrm{hot}}}}\right)\leavevmode\nobreak\
.$ (21)

The ion number $N$ is obtained from the ion density as
$N=n_{\textrm{e},0}c_{\textrm{s}}t$. Moreover, the electric field in the
plasma is obtained from the electron momentum equation,
$n_{\textrm{e}}eE=-k_{\textrm{B}}T_{e}\nabla n_{e}$, as

$E=\frac{k_{\textrm{B}}T_{\textrm{e}}}{ec_{\textrm{s}}t}=\frac{E_{0}}{\omega_{\textrm{pi}}t},$
(22)

with
$E_{0}=(n_{\textrm{e},0}k_{\textrm{B}}T_{\textrm{e}}/\epsilon_{0})^{1/2}$,
where
$\omega_{\textrm{pi}}=(n_{\textrm{e},0}Ze^{2}/m_{\textrm{i}}\epsilon_{0})^{1/2}$
denotes the ion plasma frequency. The electric field is uniform in space
(constant) and decays with time as $t^{-1}$. The temporal scaling of the
velocity is obtained by solving the equation of motion, $\dot{v}=Zq/mE$, with
the electric field as before. This yields

$\displaystyle v(t)$
$\displaystyle=c_{\textrm{s}}\ln(\omega_{\textrm{pi}}t)+c_{\textrm{s}}$ (23)
$\displaystyle z(t)$
$\displaystyle=c_{\textrm{s}}t\left(\ln(\omega_{\textrm{pi}}t)-1\right)+c_{\textrm{s
}}t\leavevmode\nobreak\ .$ (24)

Both equations satisfy the self-similar solution. The scaling of the ion
density is found as $n(t)=n_{0}/\omega_{\textrm{pi}}t$.

However, at $t=0$, the self-similar solution is not defined and has a
singularity. Hence, the model of a self-similar expansion is not valid for a
description of TNSA at early times and must be modified. Additionally, in TNSA
there are more differences. Firstly, the expansion is not driven by an
electron distribution, being in equilibrium with the ion distribution, but by
the relativistic hot electrons that are able to extend in the vacuum region in
front of the ions. There, quasi-neutrality is strongly violated and a strong
electric field will build up, modifying the self-similar expansion solution.

Secondly, the initial condition of equal ion and electron densities must be
questioned, since the hot-electron density with $n_{\textrm{e}}\approx
10^{20}\Ucm^{-3}$ is about three orders of magnitude less than the solid
density of the rear-side contamination layers. This argument can only be
overcome by the assumption of a global quasi-neutrality condition
$Zn_{\textrm{i}}=n_{\textrm{e}}$.

Thirdly, it might not be reasonable to assume a model of an isothermal plasma
expansion. It can be assumed, however, that the expansion is isothermal, since
the laser pulse provides ‘fresh’ electrons from the front side of the target,
the assumption is valid for the laser pulse duration $\tau_{\textrm{L}}$. As
will be shown, the main acceleration time period is of the order of the laser
pulse duration. This justifies the assumption of an isothermal expansion.

The plasma expansion, including charge separation, was quantitatively
described by Mora [46, 47, 48] with high accuracy. The main point of this
model is a plasma expansion with charge separation at the ion front, in
contrast with a conventional, self-similar plasma expansion. The plasma
consists of electrons and protons, with a step-like initial ion distribution
and an electron ensemble that is in thermal equilibrium with its potential.
The megaelectronvolt electron temperature results in a charge separation being
present for long times. It leads to enhanced ion acceleration at the front,
compared with the case of a normal plasma expansion. This difference is
sometimes named the TNSA effect.

Although only one-dimensional, the model has been successfully applied to
experimental data at more than 10 high-intensity short-pulse laser systems
worldwide in a recent study [16]. It was separately used to explain
measurements taken at the ATLAS-10 at the Max-Planck Institute in Garching,
Germany [53] as well as to explain results obtained at the VULCAN PW [54]
(with a few modifications). Therefore, it is seen as a reference model,
currently used worldwide for an explanation of TNSA. Because of its success in
describing TNSA, it will be explained in more detail now.

After the laser acceleration at the foil’s front side, the electrons arrive at
its rear side and escape into the vacuum. The atoms are assumed to be
instantly field ionized, leading to $n_{\textrm{i}}=n_{\textrm{e}}/Z$. Charge
separation occurs and leads to an electric potential $\phi$, according to
Poisson’s equation:

$\epsilon_{0}\frac{\partial^{2}\phi}{\partial
z^{2}}=e\left(n_{\textrm{e}}(z)-n_{\textrm{i}}(z)\right).$ (26)

The electron density distribution is always assumed to be in local thermal
equilibrium with its potential,

$n_{\textrm{e}}=n_{\textrm{e},0}\exp\left(\frac{e\phi}{k_{\textrm{B}}T_{\textrm{hot}}}\right)\leavevmode\nobreak\
,$ (27)

where the electron kinetic energy is replaced by the potential energy $e\phi$.
The initial electron density $n_{\textrm{e},0}$ is taken from eq:edens. The
ions are assumed be of initial constant density
$n_{\textrm{i}}=n_{\textrm{e},0}$, with a sudden drop to zero at the vacuum
interface. The boundary conditions are chosen such that the solid matter in
one half-space $(z\leq 0)$ perfectly compensates the electric potential for
$z\rightarrow-\infty$, whereas for $z\rightarrow\infty$ the potential goes to
infinity. Its derivative $E=-\partial\phi/\partial z$ vanishes for
$z\rightarrow\pm\infty$. In the vacuum region (initially $z>0$), the field can
be obtained analytically [37]. The resulting potential is

$\phi(z)=-\frac{2k_{\textrm{B}}T_{\textrm{hot}}}{e}\ln\left(1+\frac{z}{\sqrt{2\exp(1)}\lambda_{\textrm{D},0}}\right)-\frac{k_{\textrm{B}}T_{\textrm{hot}}}{e}\leavevmode\nobreak\
,$ (28)

and the corresponding electric field reads

$E(z)=\frac{2k_{\textrm{B}}T_{\textrm{hot}}}{e}\frac{1}{z+\sqrt{2\exp(1)}\lambda_{\textrm{D},0}}\leavevmode\nobreak\
.$ (29)

The initial electron Debye length is
$\lambda_{\textrm{D},0}^{2}=\epsilon_{0}k_{\textrm{B}}T_{\textrm{hot}}/e^{2}n_{\textrm{e},0}$.
The full boundary value problem, including the ion distribution, can only be
solved numerically. The result obtained with matlab [43] is shown in
potential. The potential $\phi$ (red curve) is a smooth function and is in
perfect agreement with the analytical solution eq:analyt (grey curve) in the
vacuum region. Both are given in units of $k_{\textrm{B}}T_{\textrm{hot}}/e$.
The electron density $n_{\textrm{e}}$ (green curve), normalized to
$n_{\textrm{e},0}$, follows from eq:dens2. The normalized ion density
$n_{\textrm{i}}$ (in black) is a step function with
$n_{\textrm{i}}(z<0)/n_{\textrm{e},0}=1$ and zero for $z>0$. The electric
field $E$ (blue curve) has a strong peak at the ion front, with
$E_{\textrm{max}}=\sqrt{2/\exp(1)}E_{0}=0.86E_{0}$ . The normalization field
$E_{0}$ is given by
$E_{0}=k_{\textrm{B}}T_{\textrm{hot}}/e\lambda_{\textrm{D},0}$. The $z$
coordinate was normalized with $\lambda_{\textrm{D},0}$. The subsequent plasma
expansion into vacuum is described in the framework of a fluid model, governed
by the equation of continuity (left) and the momentum balance (right):

$\frac{\partial n_{\textrm{i}}}{\partial
t}+\frac{\partial(v_{\textrm{i}}n_{\textrm{i}})}{\partial z})=0\hskip
56.9055pt\frac{\partial v_{\textrm{i}}}{\partial
t}+v_{\textrm{i}}\frac{\partial v_{\textrm{i}}}{\partial
t}=-\frac{e}{m_{\textrm{p}}}\frac{\partial\phi}{\partial
z}\leavevmode\nobreak\ .$ (30)

Figure 4: Solution of eq:potential. The potential $\phi$ (red) was obtained
numerically. The analytical solution eq:analyt (grey) is in perfect agreement.
Both are given in units of $k_{\textrm{B}}T_{\textrm{hot}}/e$. The electron
density $n_{\textrm{e}}$ (green), normalized to $n_{\textrm{e},0}$, follows
from eq:dens2. The normalized ion density $n_{\textrm{i}}$ (black) is a step
function with $n_{\textrm{i}}(z<0)/n_{\textrm{e},0}=1$ and zero for $z>0$. The
electric field E (blue) is given in units of
$k_{\textrm{B}}T_{\textrm{hot}}/e\lambda_{\textrm{D},0}$. The $z$ coordinate
is given in units of $\lambda_{\textrm{D},0}$.

The full expansion dynamics can only be obtained numerically. Of particular
interest is the temporal evolution of the ion distribution and the evolution
of the electric field driving the expansion of the bulk. In [5] a Lagrangian
code in matlab was developed, which solves Eqs. (27), (28), and (30), similar
to Ref. [46]. The numerical method is similar to the method described in Ref.
[53]; however, the developed code uses matlab’s built-in bvp4c function for a
numerical solution of the boundary value problem in the ion fluid. The
initially constant ion distribution is divided into a grid, choosing the left
boundary to be $L\gg c_{\textrm{s}}t$. The boundary value for the potential is
$\phi(-L)=0$. At the right boundary (initially at $z=0$), the electric field
$-\partial\phi_{\textrm{front}}/\partial
z=\sqrt{2/e}k_{\textrm{B}}T_{\textrm{hot}}/e\lambda_{\textrm{D,front}}$ must
coincide with the analytical solution of eq:analytfield, where the local Debye
length must be determined by the potential at the front:

$\lambda_{\textrm{D,front}}=\lambda_{\textrm{D},0}\exp\left(\frac{e\phi_{\textrm{front}}}{k_{\textrm{B}}T_{\textrm{hot}}}\right)^{-1/2}\leavevmode\nobreak\
.$ (31)

Initially, the Debye length at the ion front is obtained by inserting
eq:analyt in eq:dens2 to give
$\lambda_{\textrm{D,0,front}}=e^{-1}\lambda_{\textrm{D},0}$. The code divides
the fluid region into a regular grid. Each grid element (cell) has position
$z_{j}$, ion density $n_{j}$, and velocity $v_{j}$. For each time-step $\Delta
t$, the individual grid elements are moved according to the following scheme
[53]:

$\displaystyle z_{j^{\prime}}$ $\displaystyle=z_{j}+v_{j}\Delta
t+\frac{e}{2m_{\textrm{p}}}E\Delta t^{2}\leavevmode\nobreak\ ,$ (32)
$\displaystyle v_{j^{\prime}}$
$\displaystyle=v_{j}+\frac{e}{m_{\textrm{p}}}E\Delta t\leavevmode\nobreak\ .$
(33)

After that, the density of the cell is changed according to the broadening of
the cell due to the movement:

$n_{j^{\prime}}=n_{j}\frac{\Delta x_{j}}{\Delta
x_{j^{\prime}}}\leavevmode\nobreak\ .$ (34)

At the front, the individual cells quickly move forward, resulting in a ‘blow-
up’ of the cells, which dramatically diminishes the resolution. Thus, after
each time-step, the calculation grid is mapped onto a new grid ranging from
$z_{\textrm{min}}$ to the ion front position $z_{\textrm{front}}$ with an
adapted cell spacing. This method is called rezoning. The new values of
$v_{j}$ and $n_{j}$ are obtained by third-order spline interpolation,
providing very good accuracy.

#### Temporal evolution and scaling

A crucial point in the ion expansion is the evolution of the electric field
strength $E_{\textrm{front}}$, the ion velocity $v_{\textrm{front}}$ and the
position $z_{\textrm{front}}$ of the ion front. Expressions given by Mora are
[46, 47, 48]

$\displaystyle E_{\textrm{front}}$
$\displaystyle\simeq\left(\frac{2n_{\textrm{e},0}k_{B}T_{\textrm{hot}}}{\textrm{e}\epsilon_{0}}\frac{1}{1+\tau^{2}}\right)^{1/2}\leavevmode\nobreak\
,$ (35) $\displaystyle v_{\textrm{front}}$ $\displaystyle\simeq
2c_{\textrm{s}}\ln\left(\tau+\sqrt{1+\tau^{2}}\right),$ (36) $\displaystyle
z_{\textrm{front}}$ $\displaystyle\simeq
2\sqrt{2\textrm{e}}\lambda_{\textrm{D},0}\left[\tau\ln\left(\tau+\sqrt{1+\tau^{2}}\right)-\sqrt{1+\tau^{2}}+1\right],$
(37)

where $\textrm{e}=\exp(1)$ and
$\tau=\omega_{\textrm{pi}}t/\sqrt{2\textrm{e}}$. The other variables in these
equations are the initial ion density $n_{\textrm{i},0}$, the ion-acoustic (or
sound) velocity
$c_{\textrm{s}}=(Zk_{\textrm{B}}T_{\textrm{hot}}/m_{\textrm{i}})^{1/2}$,
$T_{\textrm{hot}}$ is the hot-electron temperature and
$\omega_{\textrm{pi}}=(n_{\textrm{e},0}Ze^{2}/m_{\textrm{i}}\epsilon_{0})^{1/2}$
denotes the ion plasma frequency. Owing to the charge separation, the ion
front expands more than twice as quickly as the quasi-neutral solution in Eqs.
(24) and (LABEL:eq:charge). From eq:vfront, the maximum ion energy is given as

$E_{\textrm{max}}=2k_{\textrm{B}}T_{\textrm{hot}}\ln^{2}\left(\tau+\sqrt{1+\tau^{2}}\right).$
(38)

The particle spectrum from Mora’s model [46, 47, 48] cannot be given in an
analytic form, but it is very close to the spectrum of eq:espect, obtained by
the self-similar motion of a fully quasi-neutral plasma expanding into a
vacuum. The phrase ‘fully quasi-neutral’ should indicated that in this
solution there is no charge separation at the ion front, hence there is no
peak electric field.

A drawback of the model is the infinitely increasing energy and velocity of
the ions with time, which results from the assumption of an isothermal
expansion. Hence, a stopping condition must be defined. An obvious time
duration for the stopping condition is the laser pulse duration
$\tau_{\textrm{L}}$. However, as found by Fuchs et al. [16, 45], the model can
be successfully applied to measured maximum energies and spectra, as well as
to particle-in-cell simulations, if the calculation is stopped at
$\tau_{\textrm{acc}}=\alpha(\tau_{\textrm{L}}+t_{\textrm{min}})$. It was found
that, for very short pulse durations. the acceleration time
$\tau_{\textrm{acc}}$ tends towards a constant value
$t_{\textrm{min}}=60\Ufs$, which is the minimum time that the energy transfer
from the electrons to the ions needs. The variable $\alpha$ takes into account
that for lower laser intensities the expansion is slower and the acceleration
time must be increased. $\alpha$ varies linearly from 3 at an intensity of
$I_{\textrm{L}}=2\times 10^{18}\UW/\UcmZ^{2}$ to $1.3$ at
$I_{\textrm{\leavevmode\nobreak\ L}}=3\times 10^{19}\UW/\UcmZ^{2}$. For higher
intensities, $\alpha$ is constant, at 1.3. Hence, the acceleration time is

$\tau_{\textrm{acc}}=\left(-6.07\times 10^{-20}\times(I_{\textrm{L}}-2\times
10^{18})+3\right)\times(\tau_{\textrm{L}}+t_{\textrm{min}})$ (39)

for $I_{\textrm{L}}\in[2\times 10^{18}-3\times 10^{19}]\UW/\UcmZ^{2}$, and

$\tau_{\textrm{acc}}=1.3\times(\tau_{\textrm{L}}+t_{\textrm{min}})$ (40)

for $I_{\textrm{L}}\geq 3\times 10^{19}\UW/\UcmZ^{2}$.

[b] 5 shows the temporal evolution of the electric field and the ion velocity
at the ion front, respectively. The electric field was normalized to $E_{0}$,
the ion velocity is divided by the sound velocity. There is a very good
agreement between the simulated values (blue circles) and the expressions by
Mora [46, 47, 48] from Eqs. (35) and (36) (red curve). The maximum deviations
from the scaling expressions are 1.6% for the electric field and 0.4% for the
velocity. The electric field evolution and the development of the electron and
ion density profiles are shown in tempelfsim. The electric field (green curve)
peaks sharply at the ion front for all times. Initially, the ion density (blue
curve) is $n_{\textrm{i}}=n_{0}$ for $z\leq 0$ and zero for $z>0$. The
electron density (red curve) is infinite and decays proportionally to
$z^{-2}$. Note the different axis scalings for the electric field and the
densities; the latter are plotted on a logarithmic scale. For later times, at
$t=(500,1000,1500)\Ufs$, the ions are expanded, forming an exponentially
decaying profile.

Figure 5: Temporal evolution of the electric field and the ion velocity at the
ion front. There is very good agreement between simulated values (blue
circles) and Eqs. (35) and (36) (red curve). Figure 6: Temporal evolution of
the electric field and the ion and electron density, respectively. The
electric field (green curve) peaks sharply at the ion front. The ion density
(blue curve) is $n_{\textrm{i}}=n_{0}$ for $z\leq 0$ and zero for $z>0$ for
$t=0$. The electron density (red curve) decays proportionally to $z^{-2}$. For
later times, at $t=(500,1000,1500)\Ufs$, the ions are expanded, forming an
exponentially decaying profile.

A large part of the expanding plasma is quasi-neutral and can be identified by
the constant electric field, as derived in eq:eisotherm. At the ion front, the
charge separation is still present, leading to an enhanced electric field that
is a factor of two greater than the electric field in the bulk, in agreement
with Ref. [46]. This scaling is maintained for the whole expansion. The
scaling of the peak electric field value at the ion front at position $z$, as
given by the analytical expressions in Eqs. (35) and (37), is in perfect
agreement with the simulation.

The final proton spectrum is shown in specsim. The numerical solution (blue
curve) is close to the analytical solution from the quasi-neutral model of
eq:espect (red curve). The analytical spectrum is assumed to reach a maximum
energy, taken from eq:emax. The maximum energy in the simulation is
$E_{\textrm{max,num}}=19\UMeV$, which is in close agreement to the analytical
value of $E_{\textrm{max,analyt}}=18.5\UMeV$. As expected, there is excellent
agreement in the spectra for low energies, since in both cases the expansion
is quasi-neutral. For high energies, the numerical spectrum deviates from the
self-similar model. The numerical spectrum is lower than the self-similar one
even though the ion density of the numerical solution increases close to the
ion front, as can be seen in tempelfsim in the deviation of the electron and
ion densities close to the front. However, the velocity increase at the front
in the simulation is much faster than that in the self-similar solution, owing
to the enhanced electric field. Thus, the kinetic energy of the fluid elements
close to the ion front is greater than the kinetic energy of fluid elements in
a self-similar expansion. The spectrum is obtained by taking the derivative of
the ion density with respect to the kinetic energy. It turns out that the
kinetic energy increases more strongly than the ion density, hence
$\textrm{d}N/\textrm{d}E$ is a little less than the self-similar expansion.

Figure 7: Energy spectrum $\textrm{d}N/\textrm{d}E$ from the simulation (blue
curve) compared to the spectrum of a quasi-neutral plasma expansion (red
curve). norm, normalized.

In conclusion, the Lagrangian code and the model developed by Mora show that
TNSA-accelerated ions are mainly emitted in the form of a quasi-neutral
plasma, with a charge separation at the ion front that leads to an enhanced
acceleration compared with the expansion of a completely quasi-neutral plasma.
For later times, if $\omega_{\textrm{pi}}t\gg 1$, the analytical expression of
the maximum ion energy in eq:emax can be used to determine the cut-off energy
of TNSA-accelerated ions accurately. The spectral shape of the ions is close
to the spectrum of a quasi-neutral, self-similar expansion.

The equations show, that the maximum energy, as well as the spectral shape,
strongly scale with the hot-electron temperature. The expression for the
initial electric field scales as $E\propto
k_{\textrm{B}}T_{\textrm{hot}}n_{\textrm{e}}$; hence, a simplistic estimate
would assume that both are equally important for the maximum ion energy. In
contradiction, however, the investigation has shown that the maximum ion
energy only depends weakly on the hot-electron density and is directly
proportional to the hot-electron temperature. It is worth noting that this
finding is in agreement with results obtained earlier with an electrostatic
particle-in-cell code by Brambrink [55]. The hot-electron density—owing to the
quasi-neutrality boundary condition—determines the number of generated ions.
Both the number of ions and the energy increase with time, again showing that
not the shortest and most intense laser pulses are favourable for TNSA, but
somewhat longer pulses, of the order of a picosecond. This requires a high
laser energy to keep the intensity sufficiently high.

Nevertheless, the model is still very idealized, since it is one-dimensional
and isothermal, with the electrons ranging into infinity, and it neglects
laser interaction and electron transport. An approach with electrons in a
Maxwellian distribution always leads to the same asymptotic behaviour of the
ion density [56], hence two-temperature [57] or even tailored [58] electron
distributions will lead to different ion distributions. There are many
alternative approaches to the one described here, including, an adiabatic
expansion [48]; multitemperature effects [48, 57]; an approach in which an
upper integration range is introduced to satisfy the energy conservation for
the range of a test electron in the potential [50]; expansion of an initially
Gaussian-shaped plasma [47]; and expansion of a plasma with an initial density
gradient [59]. Most of these approaches assume an underlying fluid model,
where particle collisions are neglected and the fluid elements are not allowed
to overtake each other. Hence a possible wavebreaking or accumulation of
particles is not included in the models but requires a kinetic description
[60, 61]. Furthermore, the transverse distribution of the accelerated ions
cannot be determined from a one-dimensional model and requires further
modelling. This can be done in the framework of two-dimensional particle-in-
cell simulations. Particle-in-cell simulations allow a much more sophisticated
description, including relativistic laser–plasma interaction, a kinetic
treatment of the particles, and a fully three-dimensional approach.

##  0.3 Target normal sheath acceleration: ion beam characteristics

Part of the motivation of the extensive research on laser-accelerated ion
beams is based on their exceptional properties (high brightness and high
spectral cut-off, high directionality and laminarity, short pulse duration),
which distinguish them from those of the lower-energy ions accelerated in
earlier experiments at moderate laser intensities. In view of these
properties, laser-driven ion beams can be employed in a number of
groundbreaking applications in science, technology, and medicine. This section
reviews the main beam parameters; Section 0.4 focuses on established and
proposed applications using these unique beam properties.

###  0.3.1 Beam parameters

#### Particle numbers

One of the striking features of TNSA-accelerated ion beams is the fact that
the particle number in a forward-directed beam is very high. At present,
particle numbers of up to $6\times 10^{13}$ protons with energies above
$4\UMeV$ have been detected in experiments. This typically leads to a
conversion efficiency of laser to ion beam energy of up to 9%. At these high
particle numbers, drawn from a very limited source size, for high-energy
short-pulse laser systems, the depletion of the proton contamination layer at
the rear surface becomes an issue. This has been addressed by Allen et al.
[62], who determined that there are $2.24\times 10^{23}$ atoms$/\UcmZ^{3}$ at
the rear surface of a gold foil, in a layer 1.2 Å thick. Assuming an area of
about $200\Uum$ diameter, the accelerated volume is about $V=3.8\times
10^{-11}\Ucm^{-3}$. Hence, the total number of protons in this area is about
$N_{\textrm{total}}=8.4\times 10^{12}$, which is close to the integrated
number determined in the experiments. Experiments have shown [8] that a rear
surface coating of a metal target can provide enough protons up to a thickness
of $\approx$100, where the layer thickness causes the onset of instabilities
in the electron propagation, owing to its limited electrical conductivity.

#### Energy spectrum

Based on the acceleration mechanism and the expansion model described earlier,
the usual ion energy distribution is an exponential one with a cut-off energy
that is dependent on the driving electron temperature. Without special target
treatment, and independently of the target material, protons are always
accelerated first, as they have the highest charge-to-mass ratio. These
protons stem from water vapour and hydrocarbon contamination, which are always
present on the target surface, owing to the limited achievable vacuum
conditions. Protons from the top-most contamination layer on the target
surface are exposed to the highest field gradients and screen the electric
field for protons and ions coming from the successive layers. The acceleration
of particles from different target depths results in a broad energy
distribution, which becomes broader with increasing contamination layer
thickness. The inhomogeneous electron distribution in the sheath also leads to
an inhomogeneous accelerating field in the transverse direction. The resulting
exponential ion energy spectrum constitutes the main disadvantage in laser ion
acceleration.

Only three groups so far [63, 64, 65] have produced a quasi-monoenergetic ion
beam with lasers and an energy spread of 20% or less. Hegelich et al. [63]
have used 20$\Uum$ thick palladium foils that were resistively heated before
the acceleration. At temperatures of about $600\UK$, the targets were
completely dehydrogenized, but carbon atoms still remained on the surface. By
increasing the target temperature $(T>1100\UK)$, the carbon underwent a phase
transition and formed a monolayer or graphite (graphene) on the palladium
surface, from which $C^{5+}$ ions were accelerated to $3\UMeV/\textrm{u}$ with
an energy spread of 17%. An advantage of resistive heating is the complete
removal of all hydrogen at once but there are also several disadvantages. The
formation of graphene cannot be controlled and the set-up requires a very
precise temperature measurement.

Schwoerer et al. [64] used $5\Uum$ thick titanium foils coated with $0.5\Uum$
thick hydrogen-rich polymethylmethacrylate dots of $20\Uum\times 20\Uum$ on
the target’s rear surface. This configuration was designed to limit the
transverse extension of protons, so that the proton-rich dot would have a
smaller diameter than the scale of inhomogeneity of the electron sheath. In
this case, all protons experience the same potential. The parasitic proton
contamination layer could be reduced by nanosecond-laser ablation; the
accelerated protons showed a quasi-monoenergetic energy spectrum peaking at
$1.2\UMeV$.

Ter-Avetisyan et al. [65] produced quasi-monoenergetic deuteron bursts by the
interaction of a high-intensity, high-contrast (${>}10^{-8}$) laser with
limited-mass water droplets. The peak position in the spectrum, at 2, had an
energy spread of 20%. This experiment, however, suffered from a low
laser–droplet interaction probability.

#### Opening angle

[b] 8 shows the energy-resolved opening angles for data obtained at Trident
(blue circles), LULI-100[T] (green circles), and Z-Petawatt (red circles). The
plots have been normalized to the respective maximum energy of each beam.
These are $19\UMeV$ for Trident, $16.3\UMeV$ for LULI-100[T], and $20.3\UMeV$
for Z-Petawatt, respectively.

Figure 8: Energy dependence of the half opening angle. Data obtained at
Trident (blue circles), LULI-100[T] (green circles), and Z-Petawatt (red
circles). The plots have been normalized to the respective maximum energy of
each beam. The opening angle decreases with increasing energy. A parabolic
dependency could be fit to the LULI and Z-Petawatt results, the data for
Trident produce a linear slope.

Protons with the highest energy are emitted with the smallest opening angle
from the source, up to a $5^{\circ}$ half angle. Protons with less energy are
subsequently emitted in larger opening angles. Below about 30% maximum energy,
the opening angle reaches a maximum and stays constant for lower energies. In
most cases, the opening angles decrease parabolically with increasing energy,
as shown in opening_angle. In some shots, however, the decrease of opening
angle with increasing energy is close to linear, as in the example obtained at
Trident. The slope of the opening angle with energy is a result of the initial
hot-electron sheath shape at the target surface, as pointed out by Carroll et
al. [66]. According to Ref. [66], a sheath with Gaussian dependence in the
transverse direction results in a strongly curved opening angle–energy
distribution, whereas a parabolic hot-electron sheath results in a linear
dependency. However, only crude details of the exact modelling of the
acceleration process are given in Ref. [66]. In Ref. [5], a more detailed
expansion model is described, which is able to explain the experimental
results in more detail. It should be noted that the term ‘opening angle’ is
not equivalent to the beam ‘divergence’. The divergence of the protons
increases slightly with increasing energy, whereas the emitting area (source
size) decreases with proton energy [6, 67]. This results in a total decrease
of the opening angle measured experimentally.

#### Source size

[b] 9 shows energy-resolved source sizes for the three laser systems Trident
(blue circle), LULI-100[T] (green circles), and Z-Petawatt (red circles). As
in Section 0.3.1, the energy axis has been normalized to the individual
maximum energy of the shot, with the maximum energies as before. Source size
decreases with increasing energy. Protons with the highest energies are
emitted from sources of about $10\Uum$ diameter and less. For lower energies,
the source sizes progressively increase, up to about $200\Uum$ diameter for
the lowest energies measurable with radiochromic film imaging spectroscopy,
about $1.5\UMeV$. For even lower energies, the source sizes might be much
larger and could reach in excess of $0.5\Umm$ in diameter [42].

Figure 9: Energy-resolved source sizes for data from Trident (blue circles),
LULI-100[T] (green circles) and Z-Petawatt (red circles). The energy-source-
size distribution fits to a Lorentzian with full width at half maximum of
$54.8\Uum$ for Trident, $56.5\Uum$ for LULI-100[T], and $92.8\Uum$ for
Z-Petawatt.

The energy dependence of the source size well fits a Gaussian, indicated in
source_size. The data could be fit by

$E=\exp\left(\frac{-(4\ln(2){\mbox{source
size}})^{2}}{\sigma^{2}}\right)\leavevmode\nobreak\ ,$ (41)

where $2\sigma$ denotes the FWHM. This fit allows the complete energy-
dependent source size to be characterized using only one parameter. The FWHM
for Trident with a $10\Uum$ thin gold target is $\sigma=54.8\Uum$. For
LULI-100[T], the source size is $\sigma=56.5\Uum$ for a $15\Uum$ thin gold
foil. A larger source size has been measured at Z-Petawatt, with
$\sigma=92.8\Uum$ and a $25\Uum$ thick gold target.

The energy dependence of the source size is directly related to the electric
field strength distribution of the accelerating hot-electron sheath at the
source. Protons with high energies have been accelerated by a high electric
field. Cowan et al. [67] relate increasing source size with decreasing energy
to the shape of the hot-electron sheath, under the assumption of an isothermal
quasi-neutral plasma expansion where the electric field is
$E=-(k_{\textrm{B}}T_{\textrm{e}}/e)(\delta n_{\textrm{e}}/n_{\textrm{e}})$. A
transversely Gaussian electric field distribution would result in a non-
analytical expression for the electron density, $n_{\textrm{e}}$. Conversely,
a realistic assumption of a Gaussian hot-electron distribution would result in
a radially linearly increasing electric field, in contradiction to the
measured data. Hence, it is concluded that the quasi-neutral plasma expansion,
although it is the driving acceleration mechanism for later times, does not
explain the observed source sizes. In fact, the source size must develop
earlier in the acceleration process, at very early times, when the electric
field is governed by the Poisson equation (poisson), with $E(z)\propto
k_{\textrm{B}}T_{\textrm{e}}/\lambda_{\textrm{D}}\propto\sqrt{k_{\textrm{B}}T_{\textrm{e}}n_{\textrm{e}}}$.
With the data from source_size, it can be concluded that there must be a
radial dependency of $E(z)$, and hence a Gaussian decay of either the hot-
electron temperature or the density, or both.

#### Emittance

As we have seen, the acceleration of the ions by the TNSA mechanism basically
constitutes a quasi-electrostatic field acceleration of initially cold (room-
temperature) atoms at rest, which are field ionized and then pulled by the
charge-separation field. As the electrons are scattered while being pushed
through the target, at least for materials with enough electrical conductivity
to provide the return currents, the transport smooths the distribution at the
rear surface to result in a Gaussian-like field shape that expands laterally
in time as the electrons start to recirculate. Thus the initial random ion
movement in the beam, represented in an extended phase space and often
regarded as an effective beam temperature, is very low.

An important parameter in accelerator physics is the transverse emittance of
an ion beam. In view of the nature of the ion sources used in conventional
accelerators, there is always a spread in kinetic energy and velocity in a
particle beam. Each point on the surface of the source emits protons with
different initial magnitude and direction of the velocity vector. The
emittance $\epsilon$ provides a figure of merit for describing the quality of
the beam, its laminarity [68]. Assume that the beam propagates in the
$z$-direction. Each proton represents a point in the position-momentum space
$(x,p_{x}$ and $y,p_{y})$, the phase space. The transverse phase space (in the
$x$-direction) of the TNSA protons is obtained by mapping the source position
(indicated for example by imprinted surface grooves in the radiochromic film)
versus the angle of emission $p_{x}/p_{z}$, obtained by the position $x$ of
the imprinted line in the radiochromic film and the distance $d$ by
$p_{x}/p_{z}=x^{\prime}=\arctan(x/d)$.

In general, the quality of charged-particle beams is characterized by their
emittance, which is proportional to the volume of the bounding ellipsoid of
the distribution of particles in phase space. By Liouville’s theorem, the
phase space volume of a particle ensemble is conserved during non-dissipative
acceleration and focusing. For the transverse phase space dimensions, the area
of the bounding phase space ellipse equals $\pi\epsilon$, where the emittance
$\epsilon$, at a specific beam energy (or momentum $p$), is expressed in
normalized root-mean-square (r.m.s.) units as

$\epsilon_{\textrm{N,rms}}=(p/mc)\left[<x^{2}><x^{\prime
2}>-<xx^{\prime}>^{2}\right]^{1/2}\leavevmode\nobreak\ .$ (42)

In emittance, $m$ is the ion mass, $c$ is the velocity of light, $x$ is the
particle position within the beam envelope, and $x^{\prime}=p_{x}/p_{z}$ is
the particle’s divergence in the $x$-direction. At a beam waist, emittance
reduces to
$\epsilon_{\textrm{N,rms}}=\beta\gamma\sigma_{x}\sigma_{x^{\prime}}$, where
$\sigma_{x}$ and $\sigma_{x^{\prime}}$ are the r.m.s. values of the beam width
and divergence angle. Several effects contribute to the overall emittance of a
beam: its intrinsic transverse ‘thermal’ spread; intrabeam space charge
forces; and non-ideal accelerating fields, for example, at apertures in the
source or accelerator. For typical proton accelerators (the CERN SPS or FNAL-
Tevatron), the emittance at the proton source is
${\approx}0.5\Umm\cdot\UmradZ$ (normalized r.m.s.) and up to
20–80$\Umm\cdot\Umrad$ within the synchrotron. The longitudinal phase space
$(z$–$p_{z})$ is characterized by the equivalent energy–time product of the
beam envelope; for the CERN SpS, a typical value is ${\sim}0.1\UeV\cdot\UsZ$.

The highest quality ion beams have the lowest values of transverse and
longitudinal emittance, indicating a low effective ion temperature and a high
degree of angle-space and time-energy correlation. In typical TNSA
experiments, one may estimate an upper limit for the transverse emittance of
the proton beam from emittance, by assuming that initial beamlets were
initially focused to a size ${\ll}100\UnmZ$, and that the observed width is
entirely due to the initial width of the $x^{\prime}$ distribution. The upper
limit of the emittance for $12\UMeV$ protons is ${<}0.002\Umm\cdot\UmradZ$.
This is a factor of ${>}100$ smaller than typical proton beam sources, and we
can attribute this to the fact that during much of the acceleration the proton
space charge is neutralized by the co-moving hot electrons, and that the
sheath electric field self-consistently evolves with the ions to produce an
effectively ‘ideal’ accelerating structure. The remaining irreducible
‘thermal’ emittance would imply a proton source temperature of ${<}100\UeV$.
The energy spread of the laser-accelerated proton beam is large, ranging from
zero to tens of megaelectronvolts; however, owing to the extremely short
duration of the accelerating field (${<}10\Ups$), the longitudinal phase space
energy–time product must be less than $10^{-4}\UeV\cdot\UsZ$. More details can
be found in Ref. [67], from which part of this section was extracted.

#### Ion species

Since protons are the lightest ions, and have the highest charge-to-mass
ratio, they are favoured by the acceleration processes. The protons are
present in the target as surface contaminants or in compounds of the target
itself or of the target coating. The cloud of accelerated protons then screens
the electric field generated by the electrons for all the other ion species.
The key for the efficient acceleration of heavy ions is the removal of any
proton or light ion contaminants. In a few cases, heating of the target was
performed prior to the experiments, to eliminate the hydrogen contaminants as
much as possible, and to obtain a better, more controllable ion acceleration.
In particular, by removing the proton from the targets, or by choosing H-free
targets, the acceleration of heavier ions was favoured. For the latter case,
recent experiments have demonstrated heavy-ion acceleration of up to more than
$5\UMeV/$u, which corresponds to ion energies that are usually available at
the end of a conventional accelerator of hundreds of metres in length [40].

Figure 10: (a) Overlaid signals of heated and non-heated (blue)
$\EW/\ECa\EF_{2}$ targets: the proton signals vanish for heated targets, the
fluorine signals (especially $\EF^{7+}$) go up to much higher energies. (b)
Corresponding $\EF^{7+}$ spectra: more than $5\UMeV$ per nucleon are achieved
for $\EF^{7+}$ ions. From Ref. [40].

First attempts to remove the hydrogen contaminants used resistively heated Al
targets to temperatures of a few hundred degrees Celsius. The partial removal
of the hydrogenous contaminants already strongly enhanced the acceleration of
carbon ions [8]. A reduction by a factor of ten increased the energy of the
carbon ions by a factor of $2.5$ and their number by two orders of magnitude.
Using tungsten as a thermally stable target resist and coating the rear
surface with the material of interest, the target could be heated to more than
1200 degrees Celsius. Such targets show no accelerated protons at all, but
instead a strongly increased contribution of heavy ions. The maximum energy
could be enhanced by a factor of five compared with aluminium targets and the
conversion efficiency again by a factor of ten [40] (see heavy_ion). Where the
ohmic heating of target materials of interest is prohibited, owing to a low
evaporation point, laser heating has proved an appropriate option for removing
the contamination layers. In that case, the intensity of the laser heating the
rear surface and evaporating the proton layers and the timing, with respect to
the short pulse, have to be matched carefully. The energy spectrum of the
heavy ions, together with their charge-state distribution also provides
detailed information about the accelerating electric field at the rear
surface. It was shown that, in a typical experiment, collisional ionization
and recombination in flight is negligible, and so the detected charge states
directly image the electric field strength because of the field ionization
process. The field strengths that have been obtained match the estimated field
strength, also predicted by theory, very well, and range from
$10^{11}\UV/\UmZ$ to ${\sim}10^{12}\UV/\UmZ$.

The accelerating field deduced from ion acceleration is also consistent with
observed proton energies with non-heated targets. For example, in typical
experiments, fluorine ions were accelerated up to $100\UMeV$, more than
$5\UMeV$/nucleon at a maximum charge state of $7^{+}$. This corresponds to an
electric field of $2\UV[T]/\UmZ$, which would have accelerated protons, if
present, to energies of up to $25\UMeV$. These were exactly the maximum
energies found in experiments with non-heated targets under similar
experimental conditions. Furthermore, the conversion efficiency is very high.
Similar to the results obtained for proton beams, conversion efficiencies of
up to 4% from laser to ion beam energy have been measured. Because of the same
accelerating mechanism for protons and heavy ions, an excellent beam quality
was expected. This section was extracted from [6].

###  0.3.2 Target dependence

In the previous section, we have looked extensively at the influence of target
thickness and conductivity on the driving electron sheath distribution. To
summarize, a highly conducting ultra-thin target is the most favourable for
efficient ion acceleration. Moreover, as the electrons can distribute a part
of the energy provided by the laser into bremsstrahlung, a low-$Z$ material is
preferable. The ultimate thickness of the target is determined by the limited
laser contrast, as TNSA requires a sharp density gradient at the rear surface.
For effective ion acceleration, an undisturbed back surface of the target is
crucial to provide a sharp ion density gradient, since the accelerating field
strength is proportional to $T_{\textrm{hot}}/el_{0}$, where
$T_{\textrm{hot}}$ is the temperature of the hot electrons and $l_{0}$ is the
larger of either the hot-electron Debye length or the ion scale length of the
plasma on the rear surface. The limited contrast of the laser causes a
shockwave, launched by the prepulse, that penetrates the target and causes a
rarefaction wave that diminishes the density gradient at the back, thereby
drastically reducing the accelerating field. The inward moving shockwave also
alters the initial conditions of the target material due to shockwave heating
and therefore changes, the target density and conductivity. As a trade-off,
one has, however, to note that a certain preplasma at the front side is
beneficial to the production of hot electrons, somehow contradicting the need
for high-contrast lasers. Also, as the lateral expansion of the electron
sheath affects the evolution of the ion acceleration, it has been found that
to confine the electron by reducing the transverse dimensions of the targets
also enhances the ion particle energy. So the ideal target would resemble an
ultra-thin, low-$Z$, highly conducting target with small lateral dimensions
and a large preplasma at the front side.

Meanwhile, high-contrast laser systems are able to irradiate targets as thin
as only a few nanometres, and we begin to see the change in the accelerating
mechanism to radiation pressure acceleration [70] or laser breakout
afterburner type acceleration [71], which is beyond the scope of this paper.

Apart from maximizing the ion beam particle energy, targets can be used to
shape the beam for the applications listed next. Ballistic focusing [8, 72,
73, 74, 75] and defocusing [8] have been demonstrated by numerous groups,
tailoring on a nanoscale using microstructured targets [76] and layered
targets to modify the shape of the energy spectrum[64].

###  0.3.3 Beam control

Ballistic focusing of laser-accelerated proton beams has been known since 2003
[72] and has been investigated in detail because of the large importance for
proton-driven fast ignition [77] and the generation of warm dense matter [78].
Experiments have shown that the real source size of a few hundred micrometres
could be collimated down to $30\Uum$ using ballistic focusing from the rear
surface of hemispherical targets. However, one has to take into account the
real sheath geometry of the driving electrons to interpret the proton beam
profile accurately. The driving sheath consists of a superposition of the
sheath field normal to the inner surface of the hemisphere and the Gaussian
electron density distribution caused by the limited source size of the driving
laser focus. Therefore, the experiments in Ref. [72] have indicated that a
larger laser focal spot should minimize the second effect and thus result in a
better focus quality.

For almost all applications, a precise control of the beam parameter and the
possibility of tailoring the beams is of great importance. As we have seen,
using the guidance by the shape of the target material, we have large control
over the spatial ion beam distribution. However, it might also be instructive
and preferable to try to manipulate the ion beam just using optical methods.
Results reported so far were obtained with a round laser spot, focused as well
as possible to obtain the highest intensities. But, as found by Fuchs et al.
[79], the laser focal spot shape is eventually imprinted in the accelerated
proton beam. Fuchs et al. assumed that the bulk of the hot electrons follow
the laser focal spot topology and create a sheath with the same topology at
the rear side. The proton beam spatial profile, as detected by a film
detector, was simulated with a simple electrostatic model. Fuchs et al. took
the laser beam profile as an input parameter and assumed the electron
transport to be homogeneous, with a characteristic opening angle that needed
to be fit to match the measured data. The unknown source size of the protons
was fit to best match the experimental results. It was shown, that for their
specific target thickness and laser parameters, the fitted broadening angle of
the electron sheath at the rear side closely matched the broadening angle
expected by multiple Coulomb small-angle scattering. However, Fuchs et al.
could only fit the most intense part of the measured beam and neglected the
lower intense part that originates from rear-side accelerated protons.
Additionally, there is no information on the dependence of these findings on
target thickness.

Using microgrooved targets, a more detailed understanding has been achieved.
The asymmetric laser beam results in asymmetric proton beam profiles. The
energy-resolved source size of the protons was deduced by imaging the beam
perturbations from the microgrooved surface into a radiochromic film detector
stack. It was shown that the protons with the highest energies were emitted
from the smallest source. When the laser focus size was increased, the proton
source size increased as well. For symmetric as well as asymmetric laser beam
profiles, the source-size dependent energy distribution in both cases could be
fit to a Gaussian. This leads to the conclusion that the laser beam profile
has no significant contribution to the general expansion characteristics of
laser-accelerated protons, but can strongly modify the transverse beam profile
without changing the angle of the beam spread. For a more detailed analysis of
the experimental results, a code for sheath-accelerated beam ray-tracing for
ion analysis (SABRINA) was developed, which takes the laser beam parameters as
input and calculates the shape of the proton distribution in the detector. The
electron transport was modelled to follow the laser beam profile topology
closely and a broadening due to small-angle collisions was assumed. It was
shown that broadening due to small-angle collisions is the major effect that
describes the source size of protons for thick target foils $(50\Uum)$. By
contrast, thin target foils $(13\Uum)$ show much larger sources than expected
for small-angle collisions. The physical reason behind this observation
remains unclear; it is most likely the result of electron refluxing. Thus, the
shape of the sheath at the rear side of the thick targets can be estimated by
a simple model of broadening due to multiple small-angle scattering, but the
model fails for the description of sheath broadening in thin targets.

The imprint of the laser beam profile affects the intense part of the proton
beam profile. This effect must also be present in cases with a round focal
spot. Therefore, a focal spot with a sharp peaked laser beam profile will
result in a strongly divergent proton beam, as observed in experiments. The
findings also explain that in cases where a collimation of the proton beam is
required, proton fast ignition or the injection of the beam into a post-
accelerator, not only is a curved target surface necessary, but a large, flat-
top laser focal spot is indispensable, to produce a flat proton-accelerating
sheath.

##  0.4 Applications

Summarizing the beam parameters achievable using the TNSA mechanism one can
make a number of conclusions.

The measured particle energies so far extend up to tens of megaelectronvolts
($78\UMeV$ protons, $5\UMeV$/u palladium) and the particles showed complete
space charge and current neutralization due to accompanying electrons. In
experiments, particle numbers of more than $10^{13}$ ions per pulse and beam
currents in the mega-amp regime were observed. Another outstanding feature is
the excellent beam quality, with a transverse emittance of less than
$0.004\pi\Umm\cdot\Umrad$ and a longitudinal emittance of less than
$10^{-4}\UeV\cdot\UsZ$. Because of these unmatched beam characteristics, a
wealth of applications were foreseen immediately. Those applications range
from:

  * •

injection of high power ion beams for large-scale basic research facilities;

  * •

new diagnostic techniques for short-pulse phenomena, since the short pulse
duration allows for the imaging of transient phenomena;

  * •

modification of material parameters (from applications in material science to
warm dense matter research and laboratory astrophysics);

  * •

applications in energy research (‘fast igniter’ in the inertial fusion energy
context);

  * •

medical applications;

  * •

use as a new laser-driven pathway to compact, bright neutron sources.

###  0.4.1 Ion source

An important application for TNSA ion beams is for use as next-generation ion
sources or accelerators, where the excellent beam quality and strong field
gradients can replace more conventional systems. Several collaborations are
actively working on that task, spearheaded by the LIBRA collaboration in the
UK, the LIGHT collaboration in Germany and a group at JAEA in Japan.

We briefly address a few aspects of current research aimed at applying laser
ion beams as a new source.

#### Collimation and bunch rotation of accelerated protons

One of the main drawbacks of laser-accelerated ions and, in particular,
protons are the exponential energy spectrum and the large envelope divergence
of the beam. Different techniques have been developed to modify the energy
distribution, to produce a more monoenergetic beam. Therefore, special targets
were created with thin proton or carbon layers on the rear side, as well as
deuterated droplets. In addition, there have been attempts to reduce the
initial divergence of the beams by ballistic focusing, with the help of curved
hemispherical targets, resulting in a beam focus at a distance of the diameter
of the sphere from the laser focus. In a different experiment, a laser-
triggered microlens was used to select a small energy interval and to focus
the protons with these specific energies to a millimetre spot $70\Ucm$ from
the target [80].

The total proton yield of typically $10^{13}$ particles and the extremely high
observed phase space density immediately behind the source and prior to any
collimator are highly encouraging. As in all cases of sources of secondary
particles (antiprotons, muons, rare isotopes, ), transmission efficiency and
phase space degradation due to the first collimator need to be carefully
examined. In particular, higher than first-order focusing properties of the
collimator are a serious limitation to the realistically ‘usable’ fraction of
the production energy spectrum, as well as of the production cone divergence.
As these same limitations might cause a serious degradation of the transverse
emittance of the ‘usable’ protons, the very small production emittance becomes
a relatively irrelevant quantity. Instead, an ‘effective’ emittance, taking
into account transmission loss and blow-up caused by the collimator, should be
used. In this context, space charge (non-linear) effects are a further source
of emittance degradation—probably not the dominant one—to be carefully
examined. Recently, we have shown that the collimation of a laser-accelerated
proton beam by a pulsed high-field solenoid is possible and leads to good
results in terms of collimation efficiencies. More than $10^{12}$ particles
were caught and transported by the solenoid. The steadiness of the proton beam
after collimation could be proven up to a distance of $324\Umm$ from the
target position. Inside the solenoid, strong space charge effects occurred,
owing to the co-moving electrons, which are forced to circulate around the
solenoid’s axis at their gyroradius by the strong magnetic field, leading to a
proton beam aggregation around the axis (see solenoid). Details can be found
in Ref. [81].

Figure 11: Simulation of the propagation of target normal sheath acceleration
protons through a solenoid magnet up to a detector at six timepoints. For a
better view, the accompanying electrons are not shown. A clear aggregation of
protons at the axis due to space charge effects is visible. The proton energy
in this case ranges from 1 to 5. RCF, radiochromic film. Published in Ref.
[81].

More detailed calculations of the injection into ion optical structures have
been published by Ingo Hofmann [82, 83].

#### Chromatic error of solenoid collimation

In general, pulsed solenoids are a good match to the ‘round’ production cone
of laser-accelerated particles; a quadrupole-based focusing system appears to
be disadvantageous in the defocusing plane of the first lens, owing to the
relatively large production angles. As an example, we use the short-pulsed
solenoid currently under experimental study at the GSI Helmholtz Centre for
Heavy-Ion Research. It has a length of $72\Umm$ and a theoretical maximum
field strength of $16\UT$, which is sufficient to parallelize protons at
$10\UMeV$. The distance from the target spot to the solenoid edge is assumed
to be $17\Umm$.

The prevailing higher-order effect of a solenoid is the increase in focal
length with particle energy. Owing to the debunching process, different
sections along the bunch have different energies and thus focus at different
distances. This results in an effective increase in the bunch-averaged
emittance to the effect that the tiny initial production emittance should be
replaced by a chromatic emittance. To examine the expected behaviour in
detailed simulation, we employed the DYNAMION code [84], which includes
higher-order effects in amplitudes and energy dependence, as well as space
charge effects. The latter are based on particle–particle interaction, which
limits the space charge resolution.

Figure 12: Dependence of ‘chromatic’ emittances (here total emittances for 95%
of particles) on production cone angle, as obtained by DYNAMION simulations.

The solenoid three-dimensional magnetic field was obtained by direct
integration using the coil geometry of the experimental solenoid. To quantify
the chromatic effect, we consider an ensemble of protons with constant energy
spread $\Delta E/E=\pm 0.04$ around a reference energy of $10\UMeV$. Results
for final emittances (ignoring space charge) are shown in chromemittance to
depend exactly quadratically on the considered production cone opening angle
$\delta x^{\prime}$, which varied up to $\pm 172\Umrad\leavevmode\nobreak\
(\pm 10^{\circ})$. To test the influence of space charge, we also simulated a
case with the number of protons in the bunch, $N_{\textrm{b}}$, equal to
$3\times 10^{9}$, which is equivalent to a linac current of $I=50\UmA$ (using
$I=eN_{\textrm{b}}f_{\textrm{RF}}$ and assuming that each bucket of a
$f_{\textrm{RF}}=108\UMHz$ sequence is filled identically). For simplicity,
the bunch intensity was chosen to be independent of the opening angle. It is
noted that the quadratic law still approximately applies. Since for given
$\delta x^{\prime}$ the dependence on $\Delta E/E$ is found to be practically
linear, we can justify the following scaling of the chromatic emittances in
the absence of space charge:

$\epsilon_{x}=\alpha_{\textrm{c}}(\delta x^{\prime})^{2}\frac{\Delta E}{E},$
(43)

with $\alpha_{\textrm{c}}\approx 0.04\Um/\UradZ$ for the particular solenoid
described here. The law is still approximately true if space charge is
included for the assumed bunch intensity. Note that the chromatic emittance is
found to be practically independent of the initial spot radius
$r_{\textrm{spot}}$—contrary to the production emittance given by the product
$r_{\textrm{spot}}\delta x^{\prime}$.

#### Transmission through beam pipe

For planned experiments, it is important to note that the increase of
emittance with energy spread will inevitably lead to transmission loss in the
finite acceptance of the following beam pipe. To this end, we have assumed a
beam pipe of $3\Ucm$ radius up to $250\Ucm$ distance from the source. We have
also assumed a linac current $N_{\textrm{b}}\approx\Delta E/E$, with
$N_{\textrm{b}}=2\times 10^{9}$ for the lowest value $\Delta E/E=\pm 0.04$.
The increase in transmission loss with distance is mostly due to the large
spread of focusing angles as a function of the energy spread and is also due,
to a lesser extent, to space charge. The surviving energy distribution
evaluated at different distances from the source goes down to 35% for the
largest initial energy spread case in the previous example of $\Delta E/E=\pm
0.64$ and a correspondingly high current of $560\UmA$. Obviously, an extended
beam pipe serves as an energy filter.

#### RF bunch rotation

For most applications of laser-accelerated particles, in particular for ion
beam therapy, it is desirable to reduce the final energy spread on the target
to a fraction of one per cent, to enable focusing on a small target spot. This
is achieved by means of a ‘bunch rotation’ RF cavity applied to the beam after
debunching to a length suitable for the RF wavelength. The initial short bunch
length increases with debunching proportional to the distance from the source
and the considered energy spread. Capture into the RF bucket of a fraction of
a beam within a given transverse emittance defines the ultimate six-
dimensional extraction efficiency and the ‘usable’ part of the total
production of protons. As reference value we take an energy spread of $\Delta
E/E=\pm 0.04$, which can be reduced to a reasonably small value by using a
$500\UkV/108\UMHz$ bunch rotation RF cavity approximately $250\Ucm$ away from
the solenoid. This means that only the central part of the totally transmitted
energy distribution—about 20% for the $3\Ucm$ aperture limitation—can be
captured by the RF bucket. Diagnosing the intensity and six-dimensional
emittance of this ‘usable’ fraction in the presence of the background of the
fully transmitted spectrum is a challenge to the diagnostics.

Thus, at the current status, a careful study of the transfer efficiency of
these beams into conventional transport and focusing structures is crucial and
timely, and will be carried out within the next few years, given the unique
prerequisites present among all the international collaborations. The foremost
goal of the proposed effort is to determine the properties of the generated
proton or ion beams with the prospect of later applications and to examine the
possibilities of collimation, transport, debunching and, possibly, post-
acceleration in conventional accelerator structures, both theoretically and
experimentally.

###  0.4.2 Diagnostics

A highly energetic laminar beam of charge particles with a pulse duration of
only a few picoseconds constitutes an ideal tool to diagnose transient
phenomena. Like a short burst of X-rays, a pulse of laser-driven protons can
penetrate a target and reveal important information about its parameters.
Laser-driven protons are complementary to X-rays, as the interactions of
charged particles are fundamentally different from those of electromagnetic
radiation. In the past, ion beams produced by conventional accelerators have
already been used to radiograph static and transient samples [85], as well as
for the investigation of electric fields in laser-produced plasmas [86].
Several experiments with laser-accelerated beams as probes were performed, to
investigate the evolution of highly transient electric fields evolving from
charging of laser-irradiated targets [92, 88]. These fields change on a
picosecond time-scale. Proton beams from ultra-intense laser–matter
interactions are accelerated in a few picoseconds, depending on the laser
pulse length. Combined with the very low emittance [67], these beams allow for
two-dimensional mapping of the primary target with unprecedented spatial and
temporal resolution. Using this technique, remnants of relativistic solitons
that were generated in a laser plasma have been detected [89], and the
accelerating Debye sheath in a TNSA process as well as the ion expansion from
the rear side of the target foil could be pictured [90].

Because of the energy spectrum, and owing to the dispersion of the proton
pulse at the point of interaction with the target to be probed, different
proton energies probe the target at different times. Using the radiochromic
film stack technique, the ions can be separated in energy, where the high-
energy particles deposit their energy in the rearmost layer, while lower-
energy particles are stopped in the front layers. Thus, by unfolding the
layers, one can separate the ion energies and therefore the interaction time
down to picosecond temporal resolution.

#### Energy loss

The fundamental contrast mechanism for generating image information is energy
loss in the sample, and the consequent shift in the energy distribution of
transmitted protons. As one proceeds from the shallow layers to the deepest
radiochromic film layers, protons with progressively higher incident energies
are preferentially recorded. By examining a portion of the image where the
sample contained no intervening material, we can deduce the incident proton
energy distribution from the depth dependence of the deposited energy, based
on the respective response function.

If the incident protons pass through a thin sample of thickness $\delta x$,
they lose energy according to their energy loss rate, and the transmitted
proton energy, which is incident on the film detector, is

$E_{\textrm{f}}\approx E_{\textrm{i}}-(\textrm{d}E/\textrm{d}x)\cdot\delta
x\approx E_{\textrm{i}}-\Delta E\leavevmode\nobreak\ .$ (44)

If the sample is thick, so that $\textrm{d}E/\textrm{d}x$ is not constant over
the energy range $\Delta E$, then the energy loss is given by the integral
along the trajectory,

$\Delta E=\int_{0}^{\delta
x}\textrm{d}E(\textrm{d}x)\textrm{d}x\leavevmode\nobreak\ .$ (45)

A limitation of this technique is that energy loss, and therefore thickness
information, is encoded as a spectral shift of the proton energy distribution
due to energy loss. If the object has a large thickness range, and hence large
values of the energy loss, early-time (high incident proton energy)
information from thick portions of the sample is recorded in the same final
proton energy interval as late-time (low incident proton energy) information
from thinner portions of the sample. Deconvolution of the spatial, temporal,
and thickness information is complicated, and even self-consistent solutions
may not be unique.

In the ideal limit of no transverse scattering, the resolvable thickness
variation over a sample is related only to the energy loss and the exponential
fall-off of the proton spectrum. This is a strong function of initial proton
energy via the energy dependence of $\textrm{d}E/\textrm{d}x$. For example,
for our test case, in which we observe a $2\UMeV$ exponential distribution, a
resolvable intensity variation of 1% implies a resolvable energy loss of
$20\UkeV$. At a proton energy of $10\UMeV$, this would correspond to a plastic
(CH) thickness of ${<}5\Uum$, and, at $4\UMeV$, a thickness of $2\Uum$.

#### Transverse scattering

In addition to continuously slowing down, the protons also undergo multiple
small-angle Coulomb scattering from the nuclei in the sample. In the energy
range of interest, proton multiple scattering can be described by a Gaussian
fit to the Molière distribution, which is similar to the case of the electrons
we have seen earlier. That is, protons are scattered according to a near-
Gaussian polar angle distribution,

$\textrm{d}N/\textrm{d}\Omega\approx\frac{1}{\sqrt{2\pi}\Theta_{0}}\exp(-\Theta^{2}/2\Theta_{0}^{2})\leavevmode\nobreak\
,$ (46)

where $\Theta_{0}$ is given by

$\Theta_{0}=\frac{13.6\UMeV}{\beta
pc}\sqrt{x/X_{0}}[1+0.038\ln(x/X_{0})]\leavevmode\nobreak\ ,$ (47)

with $X_{0}$ defined as

$X_{0}=\frac{716.4\Ug\Ucm^{-2}\UA}{Z/(Z+1)\ln(287/\sqrt{Z})}\leavevmode\nobreak\
.$ (48)

Multiple scattering of the protons as they traverse the sample degrades the
spatial resolution possible from ideal energy loss imaging, but it can also
increase the contrast and hence the thickness resolution. This is because
those protons scattered away from the direct line of sight from the source to
the film detector are moved from the umbral to penumbral region on the film,
thus reducing the flux of protons in the direct shadow. The decrease in proton
flux associated with a given film plane being sensitive to higher initial
energy protons, owing to the energy loss in the sample, is augmented by a flux
reduction from scattering. The very small-angle scattered protons, however,
increase the net flux of protons in the penumbral region, which can lead to
‘limb brightening’ effects, which are usual for image techniques based on
scattering (rather than absorption).

#### Field deflection

Probably the most important applications to date of proton probing are related
to the unique capability of this technique to detect electrostatic fields in
plasmas [91, 92]. This has allowed researchers to obtain, for the first time,
direct information on electric fields arising through a number of laser–plasma
interaction processes [88, 93, 89]. The high temporal resolution is here
fundamental in allowing the detection of highly transient fields following
short-pulse interaction. When the protons cross a region with a non-zero
electric field, they are deflected by the transverse component $E_{\perp}$ of
the field. The proton transverse deflection at the proton detector plane is
equal to

$\Delta r_{\perp}\approx
eL\int_{0}^{b}(E_{\perp}/m_{\textrm{p}}v_{\textrm{p}}^{2})\textrm{d}l\leavevmode\nobreak\
,$ (49)

where $m_{\textrm{p}}v_{\textrm{p}}^{2}/2$ is the proton kinetic energy, $e$
its charge, $b$ the distance over which the field is present, and $L$ the
distance from the object to the detector. As a consequence of the deflections,
the proton beam cross-section profile undergoes variations showing local
modulations in the proton density. Assuming the proton density modulation to
be small $\delta n/n_{0}\ll 1$, where $n_{0}$ and $\delta n$ are,
respectively, the unperturbed proton density and proton density modulation at
the detector plane, we obtain $\delta n/n_{0}\approx-\textrm{div}(\Delta
r_{\perp})/M$, where $M$ is the geometrical magnification. The value of the
electric field amplitude and the spatial scale can then be determined if a
given functional dependence of $E$ can be inferred a priori, from theoretical
or geometrical considerations. More details can be found in the references
cited in this paper and in Ref. [6], from where a part of this section was
taken.

###  0.4.3 Warm dense matter

The creation of extreme states of matter is important for the understanding of
the physics covered in various research fields, such as high-pressure physics,
applied material studies, planetary science, inertial fusion energy, and all
forms of plasma generation from solids. The primary difficulties in the study
of these states of matter are that the time-scales or the changes are rapid
(${\approx}1\Ups$), while the matter is very dense and the temperatures are
relatively low, ${\sim}\UeVZ/k_{\textrm{B}}$. With these parameters, the
plasma exhibits long- and short-range orders, which are due to the correlating
effects of the ions and electrons. The state of matter is too dense or too
cold (or both) to admit standard solutions used in plasma physics.
Perturbative approaches using expansions in small parameters for the
description of the plasma are no longer valid, proving a tremendous challenge
for theoretical models. This region where condensed matter physics and plasma
physics converge is the so-called warm dense matter regime [94].

Warm dense matter conditions can be generated in a number of ways, such as by
laser-generated shocks [95] or laser-generated X-rays [96, 97], intense ion
beams from conventional accelerators [98], or laser-accelerated protons [72,
73, 74], to name just a few. Whereas lasers only interact with the surface of
a sample, ions can penetrate deeply into the material of interest, thereby
generating large samples of homogeneously heated matter. The short pulse
duration of intense ion beams furthermore allows for the investigation of the
equation of states close to the solid-state density, because of the material’s
inertia, which prevents the expansion of the sample within the interaction.
Moreover, the interaction of ions with matter is dominantly due to collisions
and does not include a high-temperature plasma corona as is the case in
laser–matter interaction.

The generation of large, homogeneous samples of warm dense matter is
accompanied by the challenging task of diagnosing this state of matter, as the
usual diagnostic techniques fail under these conditions. The material density
results in a huge opacity and the relatively low temperature does not allow
traditional spectroscopic methods to be applied. Moreover, the sample size,
deposited energy and lifetime of the matter state are strongly interrelated
and are dominated by the stagnation time of the atoms in the probe. Thus, high
spatial and temporal resolution is required to gain quantitative data in those
experiments. Owing to the high density of the sample, laser diagnostics cannot
be used. The properties of matter could be determined by measuring the
expansion after the heating [73] or by measuring the thermal radiation emitted
by the sample [72]. However, even more interesting are the plasma parameters
deep inside the sample, where the ion heating is most effective. An ideally
suited diagnostic technique recently developed is X-ray Thomson scattering
[96, 99, 100].

[b] 13 shows a typical experimental scheme to investigate the transformation
of solid, low-$Z$ material into the warm dense matter state. The experimental
scheme requires a high-energy short-pulse laser and one or more long-pulse
laser beams in the same experimental vacuum chamber. In recent years, more and
more laser facilities have upgraded their laser systems for such types of
pump-probe experiment. A chirped pulse amplification laser beam above
$100\UW[T]$ power generates an intense proton beam from a thin target foil.
The protons hit a solid-density sample and heat it isochorically, up to a
temperature of several $\UeV/k_{\textrm{B}}$. The long-pulse beams are used to
drive an intense X-ray source from a titanium or Saran (which contains
chlorine) foil. The sample is probed by narrow-band line-radiation from the
chlorine or titanium plasma. The scattered radiation is first spectrally
dispersed by a highly efficient, highly oriented, pyrolytic graphite crystal
spectrometer with von Hamos geometry before it is detected. Extensive gold
shielding (partly shown in wdmsetup) is required, to prevent parasitic signals
in the scatter spectrometer. From the measured Doppler-broadened, Compton-
downshifted signal, the temperature and density can be inferred.

Figure 13: Experimental scheme to investigate the properties of laser-
accelerated proton-heated matter by spectrally resolved X-ray Thomson
scattering. HOPG, highly ordered pyrolytic graphite.

Whereas lasers only interact with the surface of a sample, ions can penetrate
deeply into the material of interest, thereby generating large samples of
homogeneously heated matter. The short pulse duration of laser-produced ion
beams furthermore allows for the investigation of equation of states close to
the solid-state density, because the material’s inertia prevents expansion of
the sample during the interaction. In addition to these unique
characteristics, the interaction of ions with matter is dominantly due to
collisions and does not include a high-temperature plasma corona, as it is
present in laser–matter interaction. The absence of a large radiation
background is of importance to the experiment. Large conversion efficiencies
have been observed, and significant energy can be transferred from the ultra-
intense laser via the ion beam into the sample of interest. Because of the
high beam quality, ballistic focusing has been demonstrated, allowing for an
increase in local energy deposition, thus leading to higher temperatures. The
use of hemispherical targets, including cone-guided targets to enhance the
local proton flux on the material of interest, can even enhance the locally
deposited energy.

Using laser pulses in excess of $100\UJ$, the intense proton beams can heat
large targets up to several times the melting temperature. In a milestone
experiment at TAW last year, the molten fraction in carbon samples heated by
intense proton beams was measured [78]. [b] 14 shows some results compared
with a radiation hydrodynamics simulation, which uses SESAME as the equation-
of-state model. It can be seen that agreement is better at lower temperatures,
where ionization is not important, but at higher temperatures, the presence of
an ionic component may be important.

Figure 14: Molten fraction of a carbon sample versus deposited energy by the
proton beam, calculated from the experimental data and simulated with Multi2D
[78].

To summarize, laser-accelerated proton beams are very well suited to produce
macroscopic samples of warm dense matter. Their unique feature, having pulse
durations of only a few picoseconds while containing more than $10^{12}$
protons, cannot be matched by conventional ion accelerators.

###  0.4.4 Fast ignition

In conventional inertial fusion, ignition and propagating burn occurs when a
sufficient temperature (5–$10\UkeV$) is reached within a sufficient mass of
deuterium-tritium fuel characterized by a density-radius product greater than
the range of an alpha particle, $(\rho r)_{\alpha}>0.3\Ug\Ucm^{-2}$. The
necessary conditions for propagating deuterium-tritium burn are achieved by an
appropriate balance between the energy gain mechanisms and the energy loss
mechanisms. Mechanical work ($P\textrm{d}V$), alpha particle energy deposition
and, to a smaller extent, neutron energy deposition are the principal energy
gain mechanisms in deuterium-tritium fuel. Electron heat conduction and
radiation are the principal energy loss mechanisms. When the rate of energy
gain exceeds the rate of energy loss for a sufficient period of time, ignition
occurs. Because of the short burn time and the inertia of the fuel, the
contribution of expansion losses is negligible. Fast ignition [101, 102] was
proposed as a means to increase the gain, reduce the driver energy and relax
the symmetry requirements for compression, primarily in direct-drive inertial
confinement fusion. The concept is to precompress the cold fuel and
subsequently to ignite it with a separate short-pulse high-intensity laser or
particle (electron or ion) pulse. Fast ignition is being extensively studied
by many groups worldwide, using short-pulse lasers or temporally compressed
heavy-ion beams. There are several technical challenges for the success of
laser-driven fast ignition. Absorption of the igniter pulse generates copious
relativistic electrons, but it is not yet known whether these electrons will
propagate as a stable beam into the compressed fuel to deposit their energy in
a small volume. In principle, heavy-ion beams can have advantages for fast
ignition. A focused ion beam may maintain an almost straight trajectory while
traversing the coronal plasma and compressed target, and ions have an
excellent coupling efficiency to the fuel and deliver their energy in a well-
defined volume, owing to the higher energy deposition at the end of their
range (Bragg peak) [103]. With the discovery of TNSA ions with excellent beam
quality, the idea of using those beams for fast ignition was introduced [77,
104]. Protons do have several advantages compared with other ion species [105]
and electrons. First, because of their highest charge-to-mass ratio, they are
accelerated most efficiently up to the highest energies. They can penetrate a
target more deeply to reach the high-density region, where the hot spot is to
be formed, because of the quadratic dependence of the stopping power on the
charge state. And finally, they do, like all ions, exhibit a characteristic
maximum of the energy deposition at the end of their range, which is
desirable, to heat a localized volume efficiently.

The basic idea is to use a number of short-pulse lasers to irradiate a thin
foil. The protons were accelerated from the rear surface of the foils and,
because of the parabolic geometry, are focused into the compressed fuel. One
of the requirements for proton fast ignition is the possibility of focusing
the proton beams into a small volume. It has recently been demonstrated that
proton beam focusing is indeed possible, and spot sizes of about $40\Uum$ have
been achieved. This is comparable to what is required by proton fast ignition.
Larger irradiated areas on the target front surface, as required for proton
fast ignition, would flatten the electron distribution at the rear surface.
Not only might this result in a single divergence angle for different
energies, but it would also form a much smaller initial divergence angle that
could be compensated, in order to reach the required focal spot diameters.

The pulse length at the source is of the right order of magnitude for proton
fast ignition, which has already been indicated in first experiments on ion
acceleration. The protons are not monochromatic but rather have an exponential
energy distribution. This seemed to be a concern at the beginning because of
dispersion and pulse lengthening. A close distance to the pellet, however,
raises concerns as to whether the thin metallic foil, which is to be the
source of the protons, can be kept cold enough that it does not develop a
density gradient at the rear surface, which would diminish the accelerating
field. A second concern was related to the stopping power. Because of the
difference in initial velocity, the energy deposition of protons with
different kinetic energies is spread over a larger volume. Slower protons are
stopped at a shorter distance and do not contribute to the creation of the
igniter spark. Fortunately, further work has relieved those concerns.
Simulations by Basko et al. (presented at the Fast Ignition Workshop 2002,
Tampa, Florida) have shown that the protective shield placed in front of the
source can withstand the X-ray flux of the pellet compression and keep the
rear surface of the source foil cold enough for acceleration via TNSA. A
thickness of a few tens of micrometres, however, provides thermal shielding as
well as sufficient mechanical stability. The distance between the source and
the ignition spot can be a few millimetres. If this distance is too short for
the compression geometry (if a closely coupled hohlraum is not used), the
distance can be adjusted using a similar cone target, as for conventional fast
ignition. As for the concerns on the hydrodynamic stability of the proton
source foil, the proposed usage of a cone target similar to the one proposed
for electron fast ignition [106, 107] has solved most of the problems, by
shielding the foil from primary soft X-rays generated in the compression of
the capsule (see pfi). Furthermore it was demonstrated that small-scale plasma
density gradients at the rear side of the proton source target caused by
target preheating have no significant (less than 10%) impact on the TNSA
mechanism [45].

A big surprise was the fact that a monochromatic proton beam is actually not
the optimum to heat a hot spot in a fusion target. Numerical simulations have
shown that one has to take into account the decrease of the stopping power of
the nuclear fuel with increasing plasma temperature. Thus, an exponential
energy spectrum, like the one generated by this mechanism, is the most
favourable. The first protons with the highest energies penetrate the fuel
deeply. By the time the proton number increases and the target temperature
rises, the stopping power is reduced, thereby compensating for the lower
initial energy of the incoming protons. Thus the majority of the protons
deposit their energy within the same volume.

Existing short-pulse lasers have already demonstrated intensities that are
sufficient for generating the proton energy spectra required for proton fast
ignition. Regardless of the nature of the igniter beam, calculations show a
minimum deposited energy required for fast ignition of the order of a few tens
of kilojoules. There have been many experiments with different laser systems
accelerating proton beams. Interestingly, the laser to ion beam conversion
efficiency seems to increase strongly with total laser energy from a fraction
of a per cent up to more than 10%. Carefully extrapolating the conversion
efficiencies to multikilojoule laser systems, conversion efficiencies of more
than 10% can be expected, which would result in the need for a few hundred
kilojoules of short-pulse laser energy for proton fast ignition.

The most detailed theoretical analysis so far has been published by Temporal
et al. [102] for a proton beam with an exponential energy spectrum. Following
their assumptions, a total proton energy of about $26\UkJ$ at an effective
temperature of $3\UMeV$ is required. This moderate temperature was found to be
an optimum between the need for high temperatures to minimize the pulse
lengthening caused by the velocity spread and the stopping range for the major
part of the spectrum. It is interesting to note that the protons, which
effectively heat the hot spot, contain only $10\UkJ$ of the total energy and
range from 19 to $10\UMeV$. If it could be possible to shape the energy
spectrum of the laser-accelerated protons, this would strongly influence the
required laser beam energy. The total number of protons needed for ignition is
close to $10^{16}$. Is it conceivable to achieve a consistent scenario for
those requirements? A typical proton beam temperature of $3\UMeV$ is commonly
obtained in experiments at $5\times 10^{19}\UW\Ucm^{-2}$. Assuming a pulse
length of $4\Ups$ (which would increase the damage threshold of modern
dielectric compressor gratings) and a conversion efficiency of 10%, a total
laser energy of $260\UkJ$ would be needed.

The use of a cone-guided geometry, as in conventional fast ignition, has been
considered to be of great advantage. The source foil can be shielded from the
radiation caused by the primary drivers, the source-to-hot spot distance can
be tailored precisely, and the pellet can be protected from heat during the
injection into the target chamber. A recent experimental campaign to study the
influence of the cone walls on the propagation and the transport of TNSA
protons has shown that, despite the influence of self-generated electric
fields in the cone walls by the recirculating electrons, good focusing may be
achievable.

Figure 15: Proposed concept of using cone-guided proton fast ignition.
Courtesy of M. Key (LLNL)

After the initial introduction of laser-accelerated proton beams for fast
ignition, theoretical studies have not only quantified the required beam
parameters [102, 103], but also recently introduced sophisticated scenarios
that have greatly relaxed those parameters. A recent study proposed a
combination of two spatially shaped proton beam pulses with a total beam
energy that would match laser systems that might be available in the not too
distant future [108, 109]. The most recent scenario looks for a ring-shaped
proton beam, to impact into the dense fuel and further compress the hot spot,
with a subsequent pulse of protons in the centre, to ignite the double-
compressed core. This would cause an energy that is further reduced by a
factor of two compared with the model described previously. Such laser proton
beams have been generated using advanced cone geometries [110].

###  0.4.5 Medical applications

Soon after the discovery of TNSA ion beams, the prospects for medical
applications have been the focus of research. Besides the possibility of
transmuting short-lived isotopes for positron emission tomography, the main
interest was in the use as a driver for hadron therapy. Hadron therapy [111,
112, 113, 114, 115] is a radiotherapy technique that uses protons, neutrons,
or carbon ions to irradiate cancer tumours. The use of protons and carbon ions
in radiotherapy has several advantages over the more widely used
X-radiotherapy. First, the proton beam scattering on the atomic electrons is
weak and thus there is less irradiation of healthy tissues in the vicinity of
the tumour. Second, the slowing down length for a proton of given energy is
fixed, which avoids undesirable irradiation of healthy tissues at the rear
side of the tumour. Third, the well localized maximum of proton energy losses
in matter (the Bragg peak) leads to a substantial increase of the irradiation
dose in the vicinity of the proton stopping point.

The proton energy window of therapeutical interest ranges between 60 and
$250\UMeV$, depending on the location of the tumour. Proton beams with the
required parameters are currently obtained using conventional charged-particle
accelerators, such as synchrotrons, cyclotrons, and linacs [116]. The use of
laser-based accelerators has been proposed as an alternative [117, 118, 119,
120, 121], which could lead to advantages in terms of device compactness and
cost.

A laser accelerator could be used simply as a high-efficiency ion injector for
the proton accelerator, or could replace a conventional proton accelerator
altogether. Because of the broad energy and angular spectra of the protons, an
energy selection and beam collimation system will be needed [122, 123].
Typically, $\Delta E/E\approx 10^{-2}$ is required for optimal dose delivery
over the tumour region. All-optical systems have also been proposed, in which
the ion beam acceleration takes place in the treatment room itself and ion
beam transport and delivery issues are minimized. In this case, the beam
energy spread and divergence would need to be minimized by controlling the
beam and target parameters. The required energies for deep-seated tumours
$({>}200\UMeV)$ are still in the future, but appear to be within reach,
considering ongoing developments in the field. A demanding requirement to be
satisfied is also the system duty factor, the fraction of time during which
the proton beam is available for use, which must not be less than $0.3$.

With the recent experimental results of ion beams in the range up to
$80\UMeV$, the lower threshold for medical applications has been achieved.
However, for deep-seated tumours it is uncertain whether the TNSA mechanism is
still the best option or whether new mechanisms should be explored that not
only lead to higher particle energies, but also offer a much smaller energy
dispersion to begin with.

###  0.4.6 Neutron source

Since the first experiments with ultra-intense lasers, nuclear reactions have
been observed and also used to diagnose the hot core part of the laser–plasma
interaction [124]. In addition to the generation and detection of radio-
isotopes and transmuted nuclei, neutrons are released either as a result of
intense bremsstrahlung or by electron or ion impact. Because of the large
conversion efficiency of laser to ion beam energy and the large cross-section
for subsequent proton neutron reactions, laser-driven neutron sources based on
the TNSA mechanism have become a focus of modern nuclear research.

One has to distinguish between the different neutron generation mechanisms. At
proton beam particle energies in the megaelectronvolt range, the interaction
and neutron generation relies on the excitation of giant resonances that
result in single (p, n) or multiple (p, $\times$n) neutron emission. The
cross-section can be quite large and is energy dependent, peaking at
characteristic proton impact energies. In the case of two particles in the
exit channel, the neutron spectrum is monoenergetic for a given projectile
energy and neutron emission angle. However, since the angle and energy spread
of laser-emitted particles is large, only strongly exothermal reactions ($Q\gg
E_{\textrm{proj}}$) will yield roughly monoenergetic neutrons. Which process
takes place in a particular instance depends on the combination of target,
projectile, and momentum transfer. The cross-sections for these processes are
in the range of $100\Umb$ up to $1\Ub$ and are therefore quite large.

As the driving ion beam is ultra-short and the release mechanism is prompt,
the neutron pulses are very short and originate from a very small region
maximizing the net flux on secondary samples. Such a probe exists in the form
of fusion neutrons. They are generated by the d(d,n)3He fusion reaction in
deuterated targets, and their use as a laser-plasma diagnostic is not
fundamentally new. When neutrons are produced from laser-accelerated ions in
the bulk of an irradiated $(\EC\ED_{2})_{n}$ target, they are emitted within a
few picoseconds from a volume of the order of $(10\Uum)^{3}$. During the
neutron pulse, at a distance of several millimetres from the target, fast
neutron fluxes of $10^{19}/(\UcmZ^{2}\cdot\UsZ)$ can be achieved, which is
four orders of magnitude higher than current continuous research reactors can
deliver.

In the past, the neutron emission caused by ($\gamma$, n) and (p, n) reactions
from the target have been measured at moderate laser intensities. A typical
detector set-up is a silver activation detector attached to a photomultiplier
tube. On typical shots, the neutrons are generated by ($\gamma$, n) reactions
within the target (caused by the bremsstrahlung photons from the relativistic
electrons) and by (p, n) reactions of our proton beam impacting on the
radiochromic film screen or a dedicated secondary production target. This can
be, a target of deuterated plastic ($\EC\ED_{2}$), which was irradiated by a
beam of TNSA-accelerated deuterons. One can observe the yield of neutrons from
(d, d) fusion reactions.

To optimize laser-driven neutron sources, one can perform simulation studies
using consolidated findings on particle beam characteristics obtained from
laser experiments [125]. The optimization will be according to the absolute
neutron yield and angle, as well as the spectral distribution. For neutron
generation, we attempt a two-stage target design in which the TNSA ion beam
irradiates a secondary sample. The advantage of this design is that we can
optimize the proton or deuteron generation using different targets in the
first stage. According to the beam properties obtained from the first stage,
it will be possible to optimize the neutron production via proton- and
deuteron-induced neutron disintegration reactions, respectively, in the second
stage. The neutron production target design (second stage) allows adaptation
to the desired application.

Figure 16: Simulated neutron spectra for different converter materials

In earlier experimental campaigns at the PHELIX laser facility at the GSI
Helmholtz Centre for Heavy-Ion Research, ${>}10^{9}$ neutrons per shot from
proton-induced reactions in copper have been produced. The integrated number
of protons was $10^{12}$ to $10^{13}$. Each neutron yield in these experiments
already exceeded that from the accelerator-driven neutron source FRANZ [126]
in Frankfurt (Germany) by five orders of magnitude. With the help of the
GEANT4 code [127], we can simulate the expected neutron yield for TNSA protons
using experimental input spectra. We have simulated the neutron spectra and
the production efficiencies using several isotopes within the second-stage
target. The thickness of these different targets was $10\Umm$. The highest
production efficiencies were obtained from proton-induced reactions at
lithium, beryllium, boron, and vanadium in their natural abundance. As a
benchmark, one can compare the simulation results for copper with experimental
data, where one finds a good agreement.

The neutron spectra from proton-induced reactions in beryllium and lithium
show high particle numbers in the lower energy range and in the range around
several megaelectronvolts, respectively. This is of interest in transmutation
studies and nuclear material science. [b] 16 demonstrates the difference in
simulated neutron spectra using lithium in the natural abundance and lithium
fluoride. The properties of the initial proton spectrum that was used in the
simulation were obtained from experimental results at the PHELIX facility. The
initial particle number was $10^{13}$ and the maximum proton energy was
$25\UMeV$.

In addition, the neutron yield using lithium fluoride is much higher than the
neutron yield from proton-induced reactions in lithium. The explanation is
that the mass density of lithium fluoride is higher, owing to the interatomic
compounds. Lithium fluoride has a mass density of $2.64\Ug/\UcmZ^{3}$ and the
mass density of lithium fluoride is only $0.53\Ug/\UcmZ^{3}$. This
demonstrates the attraction in using composite targets in future studies of
laser-driven neutron sources. In future developments of the optimization of
laser-driven neutron sources, we will use more sophisticated composite target
designs for adaptation to the desired applications.

## References